Proceedings of the Thirteenth International Workshop of
Logic and Engineering of
Natural Language Semantics 13 (LENLS 13)

hosted by the 8th JSAI International Symposia on AI (isAI2016)

Workshop Chair
Alastair Butler (National Institute for Japanese Language and Linguistics)(Nov. 13th) National Institute for Japanese Language and Linguistics,(Nov. 14th, 15th) Raiousha Building, Keio University,
November 13-15, 2016

ISBN 978-4-915905-78-0 C3004(JSAI)Preface
This proceedings volume contains selected and invited papers on topics in formal semantics,
formal pragmatics, and related ﬁelds, including the following: Formal syntax, semantics and pragmatics of natural language
 Model-theoretic and/or proof-theoretic semantics of natural language
 Computational Semantics
 Game-theoretic/Bayesian approaches to pragmatics

 Nonclassical Logic and its relation to natural language (especially Substructural/Fuzzy/Categorical/T
logics) Formal Philosophy of language
 Scientiﬁc methodology and/or experimental design for linguistics
LENLS is being organized by an alliance of ”Establishment of Knowledge-Intensive Structural
Natural Language Processing and Construction of Knowledge Infrastructure” project, fundedby JST CREST Programs ”Advanced Core Technologies for Big Data Integration”.

Workship Organizers/Program Committee:
Alastair Butler (National Institute for Japanese Language and Linguistics)
Eric McCready (Aoyama Gakuin University)
Daisuke Bekki (Ochanomizu University/JST CREST/AIST AIRC/NII)
Koji Mineshima (Ochanomizu University/JST CREST)
Richard Dietz (University of Tokyo)
Yoshiki Mori (University of Tokyo)
Yasuo Nakayama (Osaka University)
David Y. Oshima (Nagoya University)
Katsuhiko Sano (Hokkaido University)
Osamu Sawada (Mie University)
Wataru Uegaki (Leiden University)
Katsuhiko Yabushita (Naruto University of Education)
Tomoyuki Yamada (Hokkaido University)
Shunsuke Yatabe (West Japan Railway Company)
Kei Yoshimoto (Tohoku University)iProgram and Table of Contents
1st Day: November 13 (Sun), 2016. Unshared Task Workshop@NINJAL
9:30-10:00 Reception and coﬀee
10:00-10:15 Opening Remarks
Unshared Task at LENLS13 – Theory and System analysis with FraCaS, MultiFraCaS andJSeM Test Suites –Alastair Butler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1–2
10:15-11:15 Invited Talk 1
Testing the FraCaS test suite
Robin Cooper (joint work with Stergios Chatzikyriakidis and Simon Dobnik) . . . . . . . . . . 3
11:15-11:30 Coﬀee Break
11:30-12:30 Unshared Task 1
FraCaS, JSeM and the ’Inferences as Tests’ paradigm
Daisuke Bekki
Treebank annotation of FraCaS and JSeM
Alastair Butler, Ai Kubota, Shota Hiyama and Kei Yoshimoto . . . . . . . . . . . . . . . . . . . . . . . . 4
Transformational Semantics on a tree bankOleg Kiselyov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
12:30-14:00 Lunch
14:00-15:00 Invited Talk 2
Improving AMR performance on FraCaS
Tim O’Gorman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
15:00-15:15 Coﬀee Break
15:15-16:45 Unshared Task 2iiUnshared Task of Natural Language Inference
Ran Tian and Kentaro Inui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7–12
How ccg2lambda solves FraCaS/JSeM
Koji Mineshima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
FraCaS meets transformational grammar
Yusuke Kubota . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
16:45-17:00 Coﬀee Break
17:00-18:00 Invited Talk 3
Can semantics contribute to neural machine translation?
Masaaki Nagata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2nd Day: November 14 (Mon), 2016. LENLS13@Keio University
10:00-10:30 Registration
10:30-11:30 Dynamic Semantics
Solving the Proportion Problem
Hsiang-Yun Chen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16–26
Coordination and Anaphora in Attitude Contexts
Naoya Fujikawa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27–40
11:30-13:30 Lunch
13:30-14:30 Expressive meanings
Truth-conditionals and Use-conditionals: and expressive modal analysis
Lukas Rieser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41–52
Negotiating Epistemic Authority
Eric Mccready and Grégoire Winterstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53–65
14:30-15:00 Coﬀee Breakiii

15:00-16:30 Type Theoretic Semantics
Lessons from a Prototype Implementation of Montagovian Lexical Semantics
Bruno Mery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66–78
On the interpretation of dependent plural anaphora in a dependently-typed setting
Ribeka Tanaka, Koji Mineshima and Daisuke Bekki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79–92
An Analysis of Selectional Restrictions with Dependent Type Semantics
Eriko Kinoshita, Koji Mineshima and Daisuke Bekki . . . . . . . . . . . . . . . . . . . . . . . . . . . 93–106
16:30-17:00 Coﬀee Break
17:00-18:00 Invited Talk 4
Reference and Pattern Recognition: A Metasemantic Study
Youichi Matsusaka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
19:00- Banquet@Ebisu

3rd Day: November 15 (Tue), 2016. LENLS13@Keio University
10:30-11:30 Japanese Linguistics
Interpretation of Wh-words in Aza-Irabu Miyakoan
Tomohide Kinuhata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108–119
Interpretations of embedded expressives: A view from the Japanese comparative expressive
mottoOsamu Sawada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120–133
11:30-13:30 Lunch
13:00-14:30 Syntax
Non-canonical Coordination in the Transformational Approach
Oleg Kiselyov and Leo Tingchen Hsu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134–146
From meaning representations to syntactic trees
Alastair Butler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147–160

iv

Lambek Categorial Grammars as Abstract Categorial Grammars
Philippe de Groote . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161–172
14:30-15:00 Coﬀee Break
15:00-16:30 Philosophy of Language
Rich situated propositions: the ’right’ objects for the content of propositional attitudes
Kristina Liefke and Mark Bowker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173–186
A Nonmonotonic Sequent Calculus with Downward Conditional
Shuhei Shimamura . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187–199
Game Semantics and Vagueness in Natural Language
Matthias Hofer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200–211
16:30-17:00 Coﬀee Break
17:00-18:00 Invited Talk 5
Proper names in interaction
Robin Cooper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
Alternates
Conversation as a Game
Yasuo Nakayama . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213–225v

Unshared Task at LENLS 13
Theory and System analysis with FraCaS, MultiFraCaS
and JSeM Test Suites
Introduction
This gives details for the Unshared Task at LENLS 13, which is focused onundertaking theory and system analysis with FraCaS and FraCaS inspired test
suites.
The FraCaS test suite was created by the FraCaS Consortium as a benchmark for measuring and comparing the competence of semantic theories andsemantic processing systems. It contains inference problems that collectively
demonstrate basic linguistic phenomena that a semantic theory has to accountfor; including quantification, plurality, anaphora, ellipsis, tense, comparatives,and propositional attitudes. Each problem has the form: there is some natural
language input T, then there is a natural language claim H, giving the task todetermine whether H follows from T. Problems are designed to include exactly
one target phenomenon, to exclude other phenomena, and to be independent ofbackground knowledge.
Following FraCaS, overlapping test suites are now available for a numberof languages (notably in addition to the original English: Farsi, German, Greek,
Japanese, and Mandarin), which together cover both universal semantic phenomena as well as language-specific phenomena. With the problem sets categorised
according to the semantic phenomena they involve, it is possible to focus on obtaining results for specific phenomena (within a language or cross-linguistically),
as well as strive for wide coverage.

The data
We have invited papers to apply either theoretical or computational analyses
or other ideas to any of the following datasets, or subsets thereof, and describe
findings:
FraCaS textual inference test suite (English)
Download machine readable version:
http://www-nlp.stanford.edu/~wcmac/downloads
For the original:
ftp://ftp.cogsci.ed.ac.uk/pub/FRACAS/del16.ps.gz
MultiFraCaS (Farsi, German, Greek, Mandarin)
Download: http://www.ling.gu.se/~cooper/multifracas
Japanese Semantics Test Suite (JSeM)
Download JSeM beta.zip from:http://researchmap.jp/community-inf/JSeM-1-

GoalsShared tasks typically provide “gold” analysed data with clear evaluation criteria
for competing systems and have become popular within NLP fields. The conceptof a so-called “unshared task” is an alternative to shared tasks. In an unsharedtask, there are neither quantitative performance measures nor set problems thathave to be solved. Instead, participants are given a common ground (e.g., data)
and an open-ended prompt.
With the availability of FraCaS, MultiFraCaS and JSeM Test Suites, the aimof this unshared task is for participants to put these resources to work as thebasis for inspiring analysis, e.g., for showcasing a semantic theory or semantic
processing system, or syntactic annotation model for the data.
We are also interested to hear about the creation of complementary data forother languages not yet represented by the existing test suites, or with work concerning properties of the existing test suites, or with cross-linguistic comparisonsusing the test suites, etc.
Being an unshared task, use made of the datasets is up to the authors. Anyof the data sets might serve as a benchmark for testing the approach taken (oreven a computational model, for participants who go that far) and reporting
success levels on the problems (if applicable).

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Testing the FraCaS test suite
Robin Cooper (joint work with Stergios Chatzikyriakidis andSimon Dobnik)
University of Gothenburg
In this talk I will present some of the background to the project FraCaS which
led to the FraCaS test suite. I will also talk about the MultiFraCaS project andsome of our ideas for extending this in the future. The examples in the original test suite were created by semanticists. I will discuss a number of ways inwhich one could go about verifying these examples. In particular, I will present
some preliminary work we have been doing using web-based forms to collectjudgements via crowd-sourcing. We will discuss the implications of some preliminary results that we have obtained, in particular the possibility of developing aprobabilistic semantics.
Crowd-sourcing allows us to extend the notion of inference from strict logicalinference to inference that is gradient and is prevalent in lexical meaning. The
probabilities obtained through crowd-sourcing tell us the likelihood of a native
speaker to make a particular conclusion. Eventually, we hope to extend the entire
FraCas suite this way.
The original aim of the test suite was to provide a way of evaluating computational systems that perform natural language inference. I will talk about somework applying type theory to this task, with as yet partial coverage of the test
suite.
The crowd-sourcing methods we have been using to evaluate the test suite
can also be used to give an empirical basis to predictions made by semantic
theories that are difficult to ascertain by only relying on intuitions of a single
linguist. I will present some work we have been doing on the semantics of verbal
restructuring, a phenomenon whose semantics has been debated and disagreed
upon in the literature, and discuss the preliminary results we have.

-3-Treebank annotation of FraCaS and JSeM
Alastair Butler, Ai Kubota, Shota Hiyama and Kei Yoshimoto
National Institute for Japanese Language and Linguistics and Tohoku
University
This talk will describe treebank annotation of the FraCaS (English data) andJSeM (Japanese data) test suites, with a shared annotation scheme in the style of
the Penn Historical family of corpora (https://www.ling.upenn.edu/hist-cor
pora/other-corpora.html). Syntactic analysis is often taken to be a necessary
prerequisite for building semantic analysis, and we will argue that it is helpful to cash out what are likely to be shared syntactic assumptions with gold
standard trees as transformable references of analysis. We will also detail workof transforming syntactic trees into meaning representations following Treebank
Semantics (Butler 2015), and explore overlaps that arise when corresponding
English and Japanese data are considered together.

-4-

Transformational Semantics on a tree bankOleg Kiselyov
Tohoku University
Recently introduced Transformational Semantics TS formalizes, restraints andmakes rigorous the transformational approach epitomized by QR and Transformational Grammars: deriving a meaning (in the form of a logical formula or alogical form) by a series of transformations from a suitably abstract (tecto-) formof a sentence. Unlike QR, each transformation in TS is rigorously and precisely
defined, typed, and deterministic. The restraints of TS and the sparsity of the
choice points (in the order of applying the deterministic transformation steps)
make it easier to derive negative predictions and control over-generation.
The rigorous nature of TS makes it easier to carry analyses mechanically,
by a computer. The current implementation takes a form of a domain-specificlanguage embedded in Haskell. It is intended as a ‘semantic calculator’, to interactively try various transformations, observe their results or failures. We reporton the first experiments for using the calculator in the ‘batch mode’, to process
tree bank data.

-5-

Improving AMR performance on FraCaS
Tim O’Gorman
University of Colorado Boulder
Abstract Meaning Representation (Banarescu et al. 2014) is a useful formalism,in part, because of its amenability to large-scale manual semantics annotation.
AMR annotates the meaning of a sentence directly (rather than over a representation of the syntax), and represents the intended meaning of a sentence incontext, rather than building underspecified representations of what it meansout of context. Such an approach enables quick and useful annotation of eventhe most ungrammatical sentences, and provides a clear representation that canbe easily understood and easily parsed. However, this flexibility comes at theprice of relatively weak treatments of classic semantic phenomena such as quantification, tense, and monotonicity.
This talk will illustrate how the AMR-style annotation of meaning would
handle the complex issues in the FraCaS test suite. The problematic portions of
the test suite will then be used to discuss how AMR semantic coverage might be
improved. I will first discuss the ongoing efforts to improve the AMR treatment of
quantification. Secondly, I will discuss the use of event annotation methodologies
to express tense, aspect and modality. These exemplify the trade-offs and harddecisions that are faced when designing a semantic representation designed forlarge-scale annotation.

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Unshared Task of Natural Language
Inference
Ran Tian and Kentaro Inui{tianran,inui}@ecei.tohoku.ac.jp
Tohoku University, Japan

Abstract. We propose a natural language inference engine which, at its
lowest level, relies on algebraic calculations of word vectors to compose
meanings of phrases and exploit semantic similarities. Upon that, it canhandle negation and universal quantifiers logically, with an expressivity
equivalent to first-order logic and a complete inference mechanism. We
plan to use the FraCaS dataset to test the inference ability of our system.

1

Introduction

Being able to infer the logical relations of entailment and contradiction
is considered a central capacity for understanding natural language. It
is not much like inference in symbolic logic, due to ambiguity of word
meanings and great diversity in saying a same thing. Natural language
inference has to deal with these difficulties besides logical inference. To
counter the problem, distributional semantics can project meanings
into a continuous vector space, such that words and phrases withsimilar meanings have similar vectors. However, it remains open how
to use the vectors in logical inference.
Recently, Tian et al. [8] show that some logical operators canbe realized by algebraic calculations of word vectors, open up new
possibilities in combining distributional semantics with logic. By
adopting the Dependency-based Compositional Semantics (DCS)
framework [4], they convert dependency parses into database queries
to form sentence meaning. Then, they show that database operations
such as intersection and projection can be realized as addition andlinear mapping of word vectors respectively, and provide a way to
train the vectors and linear mappings from unlabeled corpora. These
already suffice to build compositions in the basic version of DCS.
However, the basic version of DCS as a logical representation
only permits conjunctions and existential quantifiers, and most of

-7-

(a)

(b)

drug
ARG

banned drugs

AspirinThalidomide

drug

…

ARG

πCOMP

SUBJ

A man sells banned drugs.

sell

COMP

government alcoholJapan
Thalidomide
…

COMP
ARG

drug

ARG
COMP

ban

= Thalidomide…

projection

ban

ban

man

∩…

intersection

COMP

SUBJ
ARG

alcoholThalidomide

denotation of“banned drugs”

…

man

sell

ARG

SUBJ

John
Mike

John Aspirin
Mary perfume

…

…

COMP

…

Fig. 1. (a) DCS trees and (b) databaseits previous extensions [4, 7, 2] are rather ad hoc. In this paper,
we extend the vector model of DCS [8] into a full-fledged natural
language inference engine, using Peirce’s Existential Graph (EG) [5]as an inference mechanism. EG is a logical representation implicitly
assuming conjunctions and existential quantifiers, but by elegant useof negation, it acquires an expressivity equivalent to first-order logic
and is equipped with a simple but complete inference algorithm. Our
proposed inference engine combines the negation mechanism of EG
with the vector model of DCS. At the lowest level, it relies on vector
calculations to compose meanings of phrases and exploit semantic
similarities; upon that, negations and universal quantifiers can behandled logically. We plan to use the FraCaS dataset [6] to test theinference ability of our system.

2

Vector Model of DCS

DCS is a framework relating dependency-like trees to database queries.
A DCS tree (Figure 1a) is similar to a dependency tree, where
each node is a content word (e.g. ban) and each edge (e.g. ban–drug) is labeled two syntactic-semantic roles (e.g. COMP–ARG) at
its two ends respectively. Semantically, a DCS tree is a query abouta (hypothetical) database (Figure 1b), where each node corresponds-8-to a database table called its denotation, and labels surrounding thenode correspond to columns of the table.
Formally, a denotation is a set of things, and a “thing” is represented by a tuple of features of the form Label=Value, with a fixedinventory of labels. For example, the denotation
ban = {(SUBJ=Japan, COMP=Thalidomide), . . .}is a set of tuples, where each tuple records participants of a banning
event (e.g. Japan banning Thalidomide). Operations applied to sets ofthings generate new denotations, modeling semantic composition. One
needs three operations in the basic version of DCS1 : intersection “∩”,e.g. pet ∩ fish is denotation of pet fish; projection “πN ” which maps atuple into its value of label N, e.g. πCOMP (ban) = {Thalidomide, . . .}consists of banned objects; and inverse image of projection πN−1 (V ) :={x | πN (x) ∈ V }, e.g.,−1
D1 := πSUBJ(πARG (man))consists of tuples of the form (SUBJ=x, . . .), where x is a man’s name
(we use the label ARG to represent names of things). Thus, sell ∩ D1
will denote men’s selling events. Similarly, denotation of banned drugs
as in Figure 1 is formally written as−1
D2 := drug ∩ πARG
(πCOMP (ban)),and the following denotation−1
D3 := sell ∩ D1 ∩ πCOMP(πARG (D2 ))consists of selling events whose SUBJ is a man and whose COMP is abanned drug.
The above calculation proceeds in a recursive manner controlled by
the DCS tree of “a man sells banned drugs” (Figure 1a). Namely, if a
DCS tree node x has children y1 , . . . , yn , and the edges (x, y1 ), . . . , (x, yn )are labeled by (P1 , L1 ), . . . , (Pn , Ln ) respectively, then the denotation[[x]] of the subtree rooted at x is recursively calculated as[[x]] := x ∩n\πP−1i (πLi ([[yi ]])).

(1)

i=1
1

It is noteworthy that the same operations are used in [1] to formulate databasesemantics in description logic.

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(b)

(c)

cry

(d)

SUBJ

beat

donkey(a)

SUBJ COMP

farmer
ARG

ARG

beat

SUBJ

it

own

own

COMP

COMP

SUBJ

own

(2e)

donkeyCOMP

beat

donkey

cry

ARG

SUBJ

COMP

ARG

ARG

ARG

cry

(e)

it

ARG

SUBJ

donkeyARG

farmer

it

ARG

COMP

SUBJ COMP

ARG

farmer(2i), (1i)

SUBJ

beat

SUBJ COMP

ARG

beat

ARG

donkey(3e), (1e)

cry

SUBJ

COMP

own

SUBJ
ARG

farmerdonkeyFig. 2. Logical inference by Existential Graphs. Applied rules written under arrows.

This will result in D3 as the denotation of the DCS tree in Figure 1a.
To obtain a vector model of DCS, one replaces denotation of aword w with word vector vw , and approximate the three set operations by calculations on vectors. More precisely, one uses vector
addition to approximate intersection (for its rationale we refer to [8]);then projection πN should naturally be realized as a linear mapping
MN because logical interaction between projection and intersection
resembles a linear relation:
πN (X1 ∩ X2 ) ⊆ πN (X1 ) ∩ πN (X2 )holds for any X1 , X2 and is similar to(v1 + v2 )MN = v1 MN + v2 MN .
Finally, inverse image πN−1 is realized by the inverse matrix MN−1
because one has πN (πN−1 (V )) = V for any set V . Thus, as parallel to(1), one can use the following vector to model denotation of a DCS
tree:
n
1X
v[[x]] := vx +v[[yi ]] MLi MP−1
.
i
n i=1

3

Existential Graphs

When written in a logical formula, DCS tree nodes are predicates andedges are variables. A denotation is a set of possible values of somevariables when other variables are bound to existential quantifiers.
For example, denotation of banned drugs is the set{y | ∃x : ban(x, y) ∧ drug(y)}.

- 10 -

It is easy to see that the basic version of DCS only allows conjunctions
and existential quantifiers as logical operators, which is a limitation
the same as many graph-based semantic representations proposed in
1960s. However, the Existential Graph (EG) invented by Peirce in1897 provides a brilliant solution: while graphs implicitly represent
conjunctions and existential quantifiers, an oval enclosure marks
negation and its scope; ovals can be nested but not overlapped. With
negation, EG acquires the same expressivity as first-order logic. For
example, Figure 2a shows an EG representation for the sentence“every farmer who owns a donkey beats it”. The nested ovals allow
quantifiers in the antecedent of an implication to include consequent
within their scope. It is noteworthy that this is exactly the same
assumption that Kamp [3] made about the scope of quantifiers.
In EG, a graph node is a predicate, and equal variables arejoined by lines. Lines can be prolonged into nested ovals (i.e. thevariables can be used there). Oval enclosures separate a blank sheet
into disjoint areas, and an area nested inside an odd (resp. even)number of ovals is called negative (resp. positive). By convention,negative areas are shaded. Graphs inside a negative area are negated,so by De Morgan’s Law, existential quantifiers become universal andconjunctions become disjunctions.
The inference rules for EG are as follows. One can (1i ) add graphs
in negative area and (1e) erase subgraphs in positive area; (2i ) copy
subgraph into nested area and (2e) erase duplicated graphs fromnested area; (3i) add or (3e) erase two nested ovals with nothing in
between (i.e. double negation). For the soundness and completenessof these rules we refer to [5].
In the Figure 2 example, EG (b) (“every beaten donkey cries”) is
copied into the inner positive area of (a), and a line joining the twodonkey variables is added, resulting (c); then, the duplicate nodes
donkey- and beat- are removed, revealing (d); finally, one erases a
double negation surrounding cry and part of a positive area to get(e), “every donkey owned by a farmer cries”.

4

Inference with Vectors

An essential step in the previous example is to find graphs inside anegative area that are duplicates of some subgraphs outside, then

- 11 -

erase them using rule (2e). This is a basic pattern in most nontrivial inference. Logically, a “duplicate” can be characterized as thedenotation of any variable in the negative area being a super setof some denotation outside. In Section 2, denotations are assigned
vectors, so one can calculate their similarities; our proposed inferenceengine will then regard similar denotations as duplicates, besides
exact super sets. Hopefully, this will integrate knowledge about phrase
similarity into the inference engine.

References
1. D. Calvanese, G. D. Giacomo, and M. Lenzerini. On the decidability of querycontainment under constraints. In Proceedings of PODS, 1998.
2. Y. Dong, R. Tian, and Y. Miyao. Encoding generalized quantifiers in dependencybased compositional semantics. In Proceedings of PACLIC, 2014.
3. H. Kamp and U. Reyle. From Discourse to Logic. Springer Netherlands, 1993.
4. P. Liang, M. I. Jordan, and D. Klein. Learning dependency-based compositionalsemantics. Computational Linguistics, 39(2), 2013.
5. J. F. Sowa. Peirce’s tutorial on existential graphs. Semiotica, 186, 2011.
6. The Fracas Consortium. Using the framework. FraCaS: A Framework for Computational Semantics, 1996.
7. R. Tian, Y. Miyao, and T. Matsuzaki. Logical inference on dependency-based
compositional semantics. In Proceedings of ACL, 2014.
8. R. Tian, N. Okazaki, and K. Inui. Learning semantically and additively compositional
distributional representations. In Proceedings of ACL, 2016.

- 12 -

How ccg2lambda solves FraCaS/JSeM
Koji Mineshima (joint work with Pascual Martı́nez-Gómez, Ribeka
Tanaka, Yusuke Miyao and Daisuke Bekki)
Ochanomizu University/JST CREST
I present an on-going work on developing formal compositional semantics and inference system for English and Japanese wide-coverage statistical CCG parsers.
The focus of the talk is on how our system can solve linguistically challenging
inference problems compiled in the FraCaS and JSeM datasets. I will introduce apipeline (“ccg2lambda”) presented in Mineshima et al. (EMNLP2015, 2016) andMartı́nez-Gómez et al. (ACL2016) and evaluate the current system on FraCaS
and JSeM. I will also discuss how to extend the system with semantic underspecification using the idea of Dependent Type Semantics (DTS; see ESSLLI2016
lecture course: http://esslli2016.unibz.it/?page id=216), and thereby illustrate one way in which a logic-based system solves some of the most challenging problems in FraCaS, in particular, those in the “nominal anaphora” sectionof the dataset.

- 13 -

FraCaS meets transformational grammar
Yusuke Kubota
University of Tsukuba
I aim to do two things in this talk: (i) attempt (at least some beginnings of) ameta-evaluation of the linguistic significance of the work reported in Mineshima
et al. (EMNLP 2015, 2016) and (ii) try to motivate the use of a more powerful
linguistic formalism than the one employed in Mineshima et al.’s work for moreor less the the same task. For the latter component, I discuss possible advantages(and disadvantages) of using Hybrid Type-Logical Categorial Grammar (Hybrid
TLCG) as a replacement for the CCG syntax of Mineshima et al.’s system. I
will argue that the use of Hybrid TLCG is especially promising in dealing withcomplex linguistic phenomena (such as ellipsis) with which Mineshima et al.’ssystem struggles due to the inflexibility of the CCG syntax it adopts. In particular, Hybrid TLCG enables incorporating various analytic techniques introducedin the ‘mainstream’ transformational generative syntax and semantics over thelast several decades much more straightforwardly than is practically possible
with CCG. This has the further potential advantage of bringing computational
linguistics and mainstream ‘pencil-and-paper’ theoretical linguistics much closer
to each other than they currently are.

- 14 -

Can semantics contribute to neuralmachine translation?
Masaaki Nagata
NTT Communication Science Laboratories
Neural machine translation (NMT) is a recently developed translation technology which outperformed “conventional” statistical machine translation (SMT).
Unlike SMT which requires syntax to overcome word order difference between
distant languages such as Japanese and English, NMT is good at word reorderingand it seems no linguistic theory is required. In this talk, I will illustrate by examples that it is semantics that can contribute to solve the remaining problems
in NMT such as zero pronoun resolution and article generation.

- 15 -

Solving the Proportion Problem:
A Plea for Selectivity
Hsiang-Yun Chen
Institute of European and American Studies, Academia Sinica

Abstract. [12] argues that quantificational adverbs are unselective binders
over individuals. The Lewisian analysis, however, fails to recognize theambiguity in some quantificationally modified conditionals. That the
Lewisian approach cannot predict some attested reading is known as
the “proportion problem.” I propose a solution based on the following
ideas: (a) quantificational adverbs bind selectively; (b) a singular indefinite and its anaphoric pronoun may introduce a plural discourse referent,and (c) plural predication is elusive.
Keywords: Proportion Problem, Donkey Anaphora, Dynamic Semantics

1

The Proportion Problem[12] argues that quantificational adverbs (hereafter Q-adverbs) such as ‘always’
and ‘’usually’ are unselective binders. The Lewisian analysis consists of the following theses:(1)

a. Conditionals are analyzed as having a tripartite structure ( i.e. Qadverb: restrictor: nuclear scope)1
b. Indefinites introduce free variables in the logical form.
c. Q-adverbs range primarily over individuals.
d. Q-adverbs are unselective binders that bind all variables in theirscope.

The analysis, however, suffers from the so-called “proportion problem.” ([5])
To illustrate, consider (2). Is it true given (3)?2(2)

If a farmer owns a donkey, he usually beats it.

1

Lewis’ analysis of ‘if’ is in fact three-way ambiguous: (a) ‘if’-clauses in quantificationally modified conditionals are analyzed as restrictors ([12]); (b) indicative conditionals are analyzed as material implication ([13]); (c) counterfactuals are analyzed
as variably strict conditionals ([11]).
2
Underline indicates that a donkey is beaten by its owner.

- 16 -

2

Solving the Proportion Problem: A Plea for Selectivity(3)
Farmer DonkeyA
d1 , d2 , d3 , d4 , d5
B
d6
C
d7
D
d8
According to the Lewisian analysis, (2) is true iff the majority of the admissible value assignments that satisfy the restrictor satisfy also the nuclear scope.
That is, most of the farmer-donkey pairs in which the farmer owns the donkey,
the former beats the latter. (2) is true given (3). Yet there is a very natural
reading that it is false. The Lewisian analysis is problematic because, if we take(2) to be ambiguous, it under-generates; if (2) is not ambiguous but simply false,
the analysis is plainly wrong.
Now consider a different scenario:(4)

Farmer DonkeyA
d1 ,...,d10Bd11 , d12 ,d13C
d14 , d15 ,d16Here again, the traditional Lewisian analysis predicts that the Q-adverb modified donkey sentence (2) is false, since merely 4 out of the 16 farmer-donkey pairs
satisfy the condition specified in the nuclear scope. But this answer is lacking
and shows insufficient sensitivity to the complexity of the model. To be sure, forthose who think that a donkey-owing farmer is a donkey-beater just in case hebeats all his donkeys, (2) is false. On the other hand, (2) is true for those thatthink mistreating just one donkey is enough for the bad name. For example,
those who fights for animal rights or work for animal welfare and protection
would not hesitate to call out a farmer who beats at least one of his donkeys.
The question is, ultimately, what qualifies a donkey-owning for a donkey-beating
farmer.

2

Diagnosis

Quantifying over farmer-donkey pairs, as the Lewisian analysis predicts, yields
just the symmetric reading. The proportion problem shows that we need also
the asymmetric reading where the farmers somehow carry more weight. However,there is no straightforward quantification over the (donkey-beating) farmers, andwe do not want to quantify over just the donkeys that are mis-treated. Drawing
on Heims idea that the occurrence of pronouns inserts indirect pressure on how
the restrictor should be interpreted, we argue that all variables in the nuclear
scope must be quantificationally bound; however, the quantification cannot be
the standard unselective binding.

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Solving the Proportion Problem: A Plea for Selectivity

3

Meanwhile, it is unfortunate that previous studies on the proportion problem
focus on consistent scenarios3 only. Due to the oversight of non-consistent scenarios, the elusiveness of plural predication has yet to received its due attention.
The puzzle about how the Q-adverb modified donkey sentence is to be evaluatedwith respect to (4) is essentially how relational predication works for plurals. If
one thinks, as we do, that donkey sentences such as S1 is not just about farmers
that own one donkey only, we need to consider not only if a relational predicate,such as ‘beat, holds between a farmer and each of his donkey, but also if thepredicate holds between a farmer and the collection of his donkeys as a whole.
That native speakers do oscillate their judgements regarding (4) is suggestive.4
It indicates a sentence like ‘a farmer beats his donkeys can be made true by
various types of scenarios. That there is some grey area in the truth conditions
is evidence that there is a fundamental indeterminacy in plural predication. It is
unfortunate that pervious studies of the proportion problem focus on scenarios
where the relation described by the matrix predicate holds consistently, thatis, each farmer beats either all or none of his donkeys. Due to the oversightof non-consistent scenarios (where a farmer does not beat all of his donkeys),the elusiveness of plural predication does not receive its due attention.What is
puzzling about (2) when evaluated with respect to (4) is essentially how relational predication works for plurals. Once we abandon the assumption of relative
uniqueness, we need to seriously consider the possibility of non-consistent relationship between a ‘boss’ and its many ‘dependents.’ If we deem it is possiblethat a farmer owns more than one donkey, we need to consider not only if a
relational predicate (e.g. ‘beat’) holds between a farmer and each of his donkey,
but also if it holds between a farmer and the whole collection of his donkeys.
To summarize, the proportion problem exemplifies an ambiguity in what theQ-adverbs should bind, which is connected to how the restrictor should be understood. The standard Lewisian analysis predicts only the symmetric reading,where the Q-adverbs binds unselectively; the preferred asymmetric reading, on
the other hand, rests on an alternative interpretation of the restrictor according
to which it is not about the farmer-donkey pairs. Therefore, to obtain the asymmetric reading, the Q-adverbs needs to bind more selectively. We maintain thatthe proper construal must take into account two points. First, all the variables in
the consequent need to be quantificationally bound. Second, once we admit the3

By consistent scenarios I mean where the relation described by the matrix predicate
holds consistently; e.g. a farmer beats all of his donkeys.
4
Here is a relevant quote from [16]: “[regarding “Most farmers who own a donkey
beat it.”] does it mean that most farmers who own a donkey beat all of the donkeys
they own, that most farmers who own a donkey beat most of the donkeys theyown, or that most farmers who own a donkey beat some of the donkeys they own?
I am simply not sure, and informants I have consulted have not expressed strongor consistent opinions. This does not obviate the need for an analysis, since people
do have intuitions about certain situations. My own rationalization of the data isthat people have firm intuitions about situations where farmers are consistent [myemphasis] about their donkey-beating.” p.256.

- 18 -

4

Solving the Proportion Problem: A Plea for Selectivity

need to collapse some farmer-donkey pairs, we must address plural predication.
As a synthesis of these two points, the binding of the ‘donkey variable will haveto be special.

3

Solution

We propose a new solution based on the following ideas:(5)a. (a) Q-adverbs bind selectively;
b. (b) singular indefinites and their anaphoric pronouns may introduce
plural discourse referents;c. (c) plural predication is elusive.

Our formal analysis is couched in DRT. In standard DRT, a singular indefinite
such as ‘a farmer’ introduces a discourse referent/variable, which is matched
to a single individual via the embedding function. We keep the indefinite-asvariable thesis intact but argue that a singular indefinite may introduce into
DRS a plural or sum discourse referent. An indefinite description can receivea ‘collective’ reading where it introduces a set-indicating variable. Such a set
is maximal in the sense that its members are all the individuals satisfying therelevant conditions.5
Here is the construction rule: a singular indefinite such as ‘a donkey’ introduces invariably into K a sub-DRS K1 , and then there are three options:(i) the sub-DRS remains unchanged, resulting in relative uniqueness à la Kadmon(ii) the sub-DRS converges to the main DRS K(iii) the sub-DRS triggers a “plural introduction” such that a sum discourse
referent Z is added to the universe of the main DRS
Z
(i)

y

y
donkey(y)K1

K (ii)

donkey(y)K (iii)P
Z= yydonkey(y)K1KSolving the proportion problem calls for the asymmetric reading, which the third
option facilitates. The verification conditions for various DRS conditions follows
the interpretation of plurals in ([7]).
For our purpose, the only verification condition that needs to be noted is:6
P(6) f verifies Y= y K iff f(Y)= σa∃g[f ⊆ g&Dom(g) = Dom(f ) ∪ U (K)&g(d) = a&g verifies K in M]
5

This way, we can attribute the ambiguity between the symmetric reading and theasymmetric reading to the ambiguity in interpreting indefinites.
6
We adopt the following from chapter 4 in [7] and [18].

- 19 -

Solving the Proportion Problem: A Plea for Selectivity

5

Following our proposal, (2) is analyzed as:

x, Z
(7)

farmer(x)yP
Z= y donkey(y)own(x,y)

u, V
most
x

u=x
V=Z
beat(u,V)An embedding function f verifies (7) iff:most extensions g of f, where Dom(g)=Dom(f)∪{x, Z} that verify the condiyPtions in K1 , farmer (x) and Z= y donkey(y)own(x,y)can be extended to h, where in this case h=g, such that h verifies K2 , i.e.,
u, V
u=x
V=Z
beat(u,V)yP
A function f verifies Z= y donkey(y) in a model M=< D, I, t > iff:own(x,y)f(Z)=σa∃g[f ⊆ g&Dom(g) = Dom(f ) ∪ {y}&g(y) = a&g(y) ∈ I(donkey)&< g(x), g(y) >∈ I(own)
Given the verification conditions so sketched, it should be clear that the primary quantification involved in (7) is the quantification over x, or the farmers.
However, the verification condition of DRS K2 is tricky. This is due to the problem of plural predication. We may address the issue using two approaches. The
first is to be minimalistic, the second explicit. Suppose a function maps u to anindividual A, and his three donkeys d1 , d2 and d3 . We then state:(8)beat(u,V) is true iff A beats d1 , d2 and d3 .

(8) acknowledge the underlying looseness in plural predication and declarethat it is not the job of the semanticists to decide when a beating relation
holds between a man and his donkeys. That is the job for people who study themetaphysics of beating. This is the minimalistic approach.
Alternatively, we may expand (7) like this:- 20 -

6

Solving the Proportion Problem: A Plea for Selectivity

x, Z
(9)

u, V

farmer(x)yK1P
Z= y donkey(y)own(x,y)

most
x

u=x
V=Z
v
v∈V

Q
v

K2

beat(u,v)

The crucial difference between (7) and (9) is the transition from ‘beat(u,V)’to the duplex condition
v
(10)

v∈V

Q
v

beat(u,v)More generally, we postulate a construction rule for “plural elimination”:(11)

Let x and y stand for singular discoure referents and α and β stand forplural discourse referents:ya. A condition R(x,β) is transformed toy∈ βxb. A condition R(α,y) is transformed to

x∈ α

Q
y

R(x,y)Q
x

R(x,y)The representation delineated in (9) has the advantage of making the second
quantification explicit, confirming the idea that all pronouns occurring in the
nuclear scope must be quantified over. The pronoun ‘he’, represented by u, isquantified by ‘most’; the pronoun ‘it’, represented by V, is quantified indirectly
via the secondary quantification Q over v. The second quantifier Q is left unspecified in (9). This underspecification is intended to reflect the flexibility withrespect to the secondary quantification over the donkeys. While we prefer thissecond, more explicit solution, we remain neutral whether the first approach isfeasible.
Below we illustrate how our proposal would deal with a number of different
scenarios. Take (12):(12)

Farmer DonkeyA
d1 ,...,d10Bd11 , d12 ,d13C
d14 , d15 ,d16

If someone judges the donkey sentence ‘If a farmer owns a donkey, he usuallybeats it’ to be true with respect to (12), we can infer that for her, the secondary- 21 -

Solving the Proportion Problem: A Plea for Selectivity

7

quantifier Q is equivalent to the existential quantification ‘∃.’ In contrast, if
someone judges the sentence to be false, then we know that for her, Q has aquantificational force stronger than ‘∃.’Consider again (2), which is repeated here as (13):(13)

Farmer DonkeyA
d1 ,...,d10Bd11 , d12 ,d13C
d14 , d15 ,d16With respect to (13), if one judges the adverbially modified donkey conditional to be false, it means that for her, Q is not existential. On the other hand,
if she judges the sentence to be true, then Q might be ‘∃’ or something stronger,
such as ‘most.’ Note that reelative to the same scenario, different speakers may
have divergent criterion regarding Q, and one speaker can have varying specifications of Q relative to different scenarios. Furthermore, depending on therelational predicate in question (‘beat’ in the current example), one may have aparticular preference for some Q.

4

Final Remarks

The analysis we advance here have four central theses: (i) every pronoun that appears in the nuclear scope of a conditional must be quantified over; (ii) a singularindefinite and its anaphoric pronoun may introduce a plural discourse referent;(iii) when a conditional contains two pronouns, the corresponding discourse referents may receive distinct quantifications, and finally (iv) plural predication iselusive.
Recall Heim’s observation that the presence of a pronoun in the nuclear scopeexerts pressure on how the restrictor should be constructed. Our first point that
all the pronouns in the nuclear scope must receive some quantification is much
like an extended argument stemming from that idea.
Our second point addresses the challenge presented by conditionals with two
pronouns. We argue that singular indefinites and their anaphoric pronouns may
receive a collective reading and introduce into DRS a sum discourse referent.
This effectively echoes Neale’s (1990) claim that singular donkey pronouns aresemantically numberless.7 What we argue is that for the sake of obtaining theproper asymmetric reading, we adopt the “plural introduction” strategy so thata singular indefinite and its anaphoric pronoun can be understood as possibly
plural.
In the foregoing discussion, we only interpret ‘a donkey’ collectively, but
it is possible that ‘a farmer’ receives a pluralization treatment. If we consider
7

See [14], Chapter 6, especially section 6.3. However, due to the difficulties [9] raises
to salience, I remain neutral about applying the number neutrality account acrossthe board.

- 22 -

8

Solving the Proportion Problem: A Plea for Selectivity

scenarios where the co-ownership of donkeys are relevant, this is what we willneed. In principle, we can have either ‘a farmer,’ ‘a donkey,’ or both to be readas introducing a plural individual. However, if we confine ourselves to scenarios
where each farmer owns at most one donkey and any donkey is owned by at mostone farmer, it does not matter which indefinite description receives the collective
reading or if they both do.
Regarding the third point, we believe the split of quantification should be
welcomed. Given a uniform treatment of conditionals as triggering duplex conditions on a par with sentences with quantified noun phrases, we now have ageneral scheme for their DRS representation:(14)

For any conditional ‘Q-adverb, if φ, ψ,’ let K1 represent the restrictor
and K2 represent the nuclear scope. The DRS for the conditional is:

K1

Q1
x

Q2
y

K2

where Q1 corresponds to the quantificational force of the Q-adverb inuse.
The secondary quantification Q2 is optional in two senses. First, when there is
only one donkey pronoun in the nuclear scope, nothing will trigger the secondary
quantification. Secondly, even if there are two donkey pronouns, we may choose
to be parsimonious with respect to the representation and leave the secondary
quantification to interpretation (and metaphysics).
Complex DRSs triggered by quantified phrases also undergo minor changes.
K1 (Q1 x)K2 now becomes:

(15)

K1

Q1
x

Q2
y

K2

where the secondary quantification Q2 is needed only when there is a donkey
pronoun. When there is no donkey pronoun in the quantified sentence, e.g.,‘Most farmers who own a donkey are rich,” nothing will trigger the secondary
quantification.

- 23 -

Solving the Proportion Problem: A Plea for Selectivity

9

Besides contributing to the desired asymmetric reading to handle the proportion problem, the separation of quantification naturally lends itself as an
explanation of the difference between the weak and strong readings.8(16)

a. If a man has a quarter, he puts it in the meter.
b. Every man who has a quarter puts it in the meter.

The respective representation for (16a) and (16b) are:

u,V

x,Z
(17)

a.

man(x)y

K1
P
Z= y quarter(y)has(x,y)

∀
x

u=x
V=Z
v
v∈V

K2

Q2
v

x,Z
b.

man(x)y

K1
P
Z= y quarter(y)has(x,y)put in the meter(u,v)

V
∀
x

V=Z
v
v∈V

Q2
v

K2

put in the meter(x,v)

The strong reading results from taking the secondary quantification Q2 tobe universal, and the weak reading results from taking Q2 to be existential.
In short, while the proportion problem demonstrates the need to separate thequantification so that the (non-universal) Q-adverb binds more selectively, thedemand for splitting the quantification is already present when we are charged
with accounting for the weak reading. It is desirable to have a unified machinerythat provides the required division.
Finally, regarding the fourth point, that plural predication is loose is clearly
exemplified in the following:(18)

a. At the end of the press conference, the reporters asked the president
questions.9b. At the end of the press conference, the reporters asked the presidents
questions.
c. At the end of the press conference, the reporters asked the president
a question about gun control regulations.

8

For more discussions on the weak and strong readings, see [15], [1], [8], [10] and [3],among others.
9This is from [17], which he attributes to [2].

- 24 -

10

Solving the Proportion Problem: A Plea for Selectivity

The truth of (18a) does not depend on every single reporters at the press
conference asked the president a question. One or more reporters might haveasked one or more questions, but there might be one or more reporters who did
not asked any. Similarly, the truth of (18b) does not require that every reporters
asked a question and/or that every president was asked a question. Perhaps
bearing more directly on my proposal is (18c). It is not transparent the singularindefinite ‘a question’ entails that there was only one question asked about the
new stimulus package. Our intuition is that several reporters could have each
asked such a question, but other speakers might have a different judgement.
Returning to the initial Lewisian account of adverbs of quantification andconditionals, we have come to realize that in order to handle the proportion
problem, besides the assumption of unselective binding, other modifications arenecessary:(19)a. Conditionals are analyzed as having a tripartite structure.
b. Indefinites introduce free variables in the logical form. ⇒ singularindefinites may introduce plural variables.
c. Q-adverbs range primarily over individuals.
d. Q-adverbs are unselective binders that bind all variables in theirscope. ⇒ Q-adverbs bind selectively one of the variables in theirscope.10

The separation of quantification together with the elusiveness of plural predication suggests that there may be an inherent indeterminacy of donkey sentences.
The indeterminacy is subject to various constraints such as world knowledge, thepredicate in question, the Q-adverb in use, and the logical properties (monotonic
features) of the determiner, to name but a few. After all, “it may be sometimes
be futile if not wrong to suppose that donkey sentences must have a definite
reading.”11 Nevertheless, the representation helps to elucidate, given a particular reading (i.e., a determinate truth or falsity with respect to a scenario), what
the discourse content, or structure of information, must and might be.

10

A different formulation is this: Q-adverbs may bind multiple variables, but need not
do so; when the primary and secondary quantification coincide in their quantificational force, we have what appears as unselective binding from one single quantifier.
11[4], p.151.

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Solving the Proportion Problem: A Plea for Selectivity

11

References
1. G. Chierchia. Anaphora and dynamic binding. Linguistics and Philosophy, 15:111–
183, 1992.
2. D. Dowty. Collective predicates, distributive predicates and all. In Proceedings ofthe Eastern Proceedings of the Eastern States Conference on Linguistics. ESCOL3,
Cascadilla Press., 1986.
3. P. Elbourne. Situations and Individuals. Cambridge: MIT Press, 2005.
4. B. Geurts. Donkey business. Linguistics and Philosophy, 25:129–156, 2002.
5. N. Kadmon. On Unqiue and Non-Unique Reference and Asymmetric Quantification. PhD thesis, University of Massachusetts at Amherst, 1987.
6. N. Kadmon. Uniqueness. Linguistics and Philosophy, 13(3):273 – 324, 1990.
7. H. Kamp and U. Reyle. From Discourse to Logic. Kluwer, Dordrecht., 1993.
8. M. Kanazawa. Weak vs. strong readings of donkey sentences and monotonicity
inference in a dynamic setting. Linguistics and Philosophy, 17(2):109–158, 1994.
9. M. Kanazawa. Singular donkey pronouns are semantically singular. Linguistics
and Philosophy, 24(3):383–403, 2001.
10. M. Krifka. At least some determiners aren’t determiners. In K. Turner, editor, The
semantics/pragmatics interface from different points of view. (= Current Researchin the Semantics/Pragmatics Interface Vol. 1)., pages 257–291. Elsevier Science
B.V.,, 1999.
11. D. Lewis. Counterfactual. Harvard University Press, 1973.
12. D. Lewis. Adverbs of quantification. In E. Keenan, editor, Formal Semantics of
Natural Language. CUP. Cambridge University Press, 1975.
13. D. Lewis. Probabilities of conditional and conditional probabilities. Philosophical
Review, 8:297–315, 1976.
14. S. Neale. Dsecriptions. Cambridge: MIT Press, 1990.
15. F. J. Pelletier and L. K. Schubert. Mass expressions. In F. Guenthner and D. Gabbay, editors, Handbook of philosophical logic,, volume 4, pages 327–407. Kluwer:
Dordrecht, 1989.
16. M. Rooth. Noun phrase interpretation in montague grammar, file change semantics, and situation semantics. In P. Gardenfors, editor, Generalized Quantifiers.
Dordrecht: Reidel, 1987.
17. Y. Winter. Atoms and sets: a characterization of semantic number. Linguisitic
Inquiry, 33:493–505, 2002.
18. Y.-E. Yoon. Weak and Strong Interpretations of Quantifiers and Definite NPs in
English and Korean. PhD thesis, The University of Texas at Austin, 1994.

- 26 -

Coordination and Anaphora in Attitude
Contexts
Naoya Fujikawa
Department of Language Sciences, Graduate School of Humanities, Tokyo
Metropolitan University,
fjnaoya@gmail.com

Abstract. In this paper, I oﬀer a DRT analysis of cross-attitudinal
anaphora (intentional identity) in which the coordination relation between mental files plays a central role.
Keywords: intentional identity, cross-attitudinal anaphora, propositional
attitude reports, discourse representation theory, coordination, mental
files

1

Introduction

As Geach (1967) pointed out, cases of cross-attitudinal anaphora like (1), where‘she’ in the second sentence is anaphoric to ‘a witch’ in the first sentence, havereadings on which both attitude reports in them are true.
(1) Hob believes that a witch blighted Bob’s mare. Nob believes that she killed
Cob’s sow.
For example, (1) is true in the following situation (cf. Edelberg, 1986, p. 2).
The Hob-Nob case: In a village, because of an epidemic, many livestock are blighted and dead. A newspaper of the village spreads a rumor
that a witch (exactly one witch) is threatening the village and trying
to kill the livestock there. Both Hob and Nob read the newspaper andbelieve the rumor. Hob finds Bob’s mare getting sick and believes that awitch blighted Bob’s mare. Nob finds Cob’s sow dead and then believes
that a witch killed Cob’s sow.
It is often claimed that the anaphoric relation between ‘a witch’ and ‘she’in the reading in question is not properly explained by analyzing ‘she’ as anindividual variable bound by an existential quantifier corresponding to its antecedent ‘a witch’. First consider (2) (For simplicity, I ignore the complexity in
VPs in the embedded sentences and treat them as simple predicates).
(2) a. ∃x(witch′ (x) ∧ BEL(h, blight-bm′ (x)) ∧ BEL(n, kill-cs′ (x)))
b. ∃x(BEL(h, witch′ (x) ∧ blight-bm′ (x)) ∧ BEL(n, witch′ (x) ∧ kill-cs′ (x)))- 27 -

Since there exists no witch, for the reading in question to be true, it is not necessary that Hob’s belief and Nob’s belief are about the same particular existentobject. However, (2a) entails this, and (2b) does too.1
The following analysis for the reading of (1) avoids this problem, since theexistential quantifier appears within the scope of the belief operator.
(3) BEL[h, ∃x(witch′ (x) ∧ blight-bm′ (x) ∧ BEL(n, kill-cs′ (x)))]This is not a correct analysis, since the reading of (1) in question can be true
even when Hob believes nothing about Nob, Cob and his sow (Edelberg, 1986,
pp. 3-4). In (3), ‘BEL(n, kill-cs′ (x))’ appears within the scope of ‘Hob believes
that’, and thus it implies that Hob believes something about Nob’s belief andCob’ sow. But (1) doesn’t entail this. For a similar reason, the E-type analysis
of pronouns, which is illustrated by (4), does not give appropriate treatment of
the reading of (1) in question (cf. Geach, 1967, p. 630, Edelberg, 1986, pp. 3-4):(1) can be true even when Nob believes nothing about Hob, Bob, and his mare,but (4) can not.
(4) a. BEL[h, ∃x(witch′ (x) ∧ blight-bm′ (x))] ∧ BEL[n, kill-cs′ (ιx[witch′ (x) ∧blight-bm′ (x)])]b. BEL[h, ∃x(witch′ (x)∧blight-bm′ (x))]∧BEL[n, kill-cs′ (ιx[BEL(h, witch′ (x)∧blight-bm′ (x))])]In this manner, it is diﬃcult to analyze the pronoun ‘she’ in (1) as an individual
variable bound by existential quantifier or as an E-type pronoun.
In the Hob-Nob case, Hob’s belief and Nob’s belief contain the concept witch.
Regarding this as a common feature of the situations which make (1) true, one
may propose the following second-order analysis of the reading of (1) in question.
(5) ∃F (F = witch′ ∧ BEL[h, ∃x(F (x) ∧ blight-bm′ (x))] ∧ BEL[n, ∃x(F (x) ∧ killcs′ (x))])However, this is not correct, since for the reading of (1) in question to be true,
it is not suﬃcient that the descriptive content of Hob’s belief overlaps the oneof Cob’s (cf. Edelberg, 1986, pp. 8-9). For instance, suppose that no rumor
concerning witch-crisis has spread in the village. When Hob finds Bob’s mareto get sick, the belief that a witch blighted it happens to come to his mind.
Independently from this, Nob finds Cob’s sow dead and happens to believe thata witch killed Cob’s sow. In this case, even though their beliefs contain the samedescriptive content, say, witch(x), (1) is not true—at least it sounds quite weird.2
This is so, if we understand ∃ as ‘existential’ quantifier in the standard manner.
Note that, according to Meinongianism, quantifiers need not to be existentiallyloaded. Does this anti-Quinean understanding of quantifier solve the problem inquestion? Unfortunately, as Edelberg (1986) shows, anti-Quinean quantification is
not suﬃcient to give a correct analysis of cross-attitudinal anaphora.
2This doesn’t mean that such overlap is not required. For example, suppose Nob
believes that a wolf man, not a witch, killed Cob’s sow. In such a case, I tend to saythat (1) is not true. But, in this paper I leave this matter open.
1- 28 -

In this way, it is not suﬃcient for the truth of (1) that Hob’s belief andNob’s belief share the same descriptive content. Then, what else is required? It
is widely accepted that for the reading of (1) in question to be true, it must
be the case that Hob’s belief and Nob’s belief can be said to be ‘about thesame thing’ in some sense, in other words, their beliefs have ‘a common focus,
whether or not here actually is something at that focus.’ (Geach, 1967, p. 627).
Asher says, anaphoric links exemplified by (1) ‘only make sense if the agent’s
attitudes are coordinated together, whether by means of communication or someother mechanism, in such a way that the two agents can be said to have the“same” individual in mind’ (Asher, 1987, p. 127. My emphasis). For example,
in the Hob-Nob case, Hob’s belief and Nob’s belief are coordinated together
because their beliefs are based on the common source of information—the local
newspaper. As Asher claimed, such coordination will be established by meansof communication between the agents in question (see also EXAMPLE 1 in
Edelberg, 1986, p. 1). There can be other ways of establishing coordination
among mental states. For some reasons, a mental state is coordinated with someother mental state without their being about some particular existent object.
And such coordination between mental states is crucial for the truth of the casesof problematic cross-attitudinal anaphora like (1).
This paper explores a way to develop a semantic analysis of cross-attitudinal
anaphora in which coordination among mental states plays a central role. The
aim of this paper is not to elucidate the nature of coordination. Rather, putting
aside the question of what the nature of coordination is, in this paper, I show thatby incorporating the coordination relation between mental states into semantics
as its primitive notion, we can give a relatively simple and adequate semantics
for cross-attitudinal anaphora of the kind in question. Extending DRT semantics
for propositional attitude reports presented in Kamp, van Genabith and Reyle(2011), I propose the following: Firstly, indefinites and pronouns appearing in attitude contexts introduce what I call meta discourse referents, that is, discourse
referents which represent mental files; and secondly, assuming that coordination
is a relation between mental files, anaphoric links between indefinites and pronouns in attitude contexts are recorded by using meta discourse referents andthe coordination predicate, which represents coordination between mental files.
In section 2, we quickly review DRT semantics for cross-attitudinal anaphora
based on Kamp et al. (2011), a useful survey of recent development of DRT.
Section 3 gives a brief exposition of the notion of coordination. Section 4 provides
a semantics for cross-attitudinal anaphora by using DRT extended with meta
discourse referents and the coordination predicates.

2

Kamp et al. (2011) on cross-attitudinal anaphora

At the beginning of the history of DRT, Kamp stated the view that DRSs aremental representations a hearer of a discourse constructs as her interpretationof the discourse (cf. Kamp, 1981). In particular, discourse referents are taken
as mental representations of entities (cf. Kamp, et al. 2011, pp. 326-327). Even

- 29 -

though the idea of DRS as mental representation is not a core doctrine of DRT, it
has motivated DRT semantics for propositional attitude reports which describes
mental states of reportees of attitude reports by using DRSs.
A mental state is described as a pair of its attitudinal mode like belief, desire,or doubt and a DSR which specifies the content of the state. For example, a belief
that a delegate arrived is described as follows, where BEL is the indicator of
the attitudinal mode of believe.

⟨

(6)

BEL,

x

⟩

delegate′ (x)arrive′ (x)An attitude ascription ascribes one or more mental states to a subject For
example, we can ascribe to Mary a belief that a delegate arrived and a doubtthat the same delegate registered. The following DRS condition corresponds tothis attitude ascription (I use ‘DOU ’ as the mode indicator for doubt).3

(7)





⟨


BEL,
Att(m,


xdelegate′ (x)arrive′ (x)

⟩ ⟨
,

DOU,

registered′ (x)




⟩


)






Att(s, {⟨Φ, K⟩}) is an attitude ascription condition which means that a subjects has a mental state whose mode—belief, desire, intention, and so on—is Φ andwhose content is specified by the DRS K.
Kamp et al. (2011) claims that in cases of interpersonal cross-attitudinal
anaphora like (1), what supports the anaphoric link in question is that two
subjects has some mental state with the same content (cf. Kamp, et al, 2011,
p. 383). In the Hob-Nob case, both Hob and Nob believe that there is a witch,and this is supposed to support the anaphoric link between ‘a witch’ and ‘she’in (1). Based on these considerations, Kamp et al. (2011) proposes an analysis
of interpersonal cross-attitudinal anaphora illustrated by (8) for (1) (again, forsimplicity, I ignore the complexity in VPs in the embedded sentences and treat
them as simple predicates).

3

The following exposition omits the complexity concerning the notion of externalanchor (Kamp, et al. 2011, in particular, pp. 332-343), which is required to treat
cases of de re ascriptions.

- 30 -

h, nHob′ (h)N ob′ (n)




⟨


Att(h,
BEL,






(8)



⟨


Att(n,
BEL,







⟨


Att(n,
BEL,









⟩



x
witch′ (x)blighted-Bob’s-mare ′ (x)x′witch′ (x′ )







)


⟩






)

u
u = x′
killed-Cob’s-sow ′ (u)




⟩








)

However, this analysis fails to explain the anaphoric link between ‘a witch’ inthe first sentence and ‘she’ in the second sentence of (1). As we have seen, Kamp
et al. (2011) claims that in cases of interpersonal cross-attitudinal anaphora like(1), anaphoric links of the kind in question are supported by subjects’ havingsome mental state with the same content. In (8), the content that there is a witch
is shared by Hob and Nob, and this shared content is assumed to be support theanaphoric link in question. However, as we have seen in Section 1, sharing thesame descriptive content in this sense is not suﬃcient for interpersonal crossattitudinal anaphora.

3

Coordination and cross-attitudinal anaphora

In the passage quoted in Section 1, Asher claims that the anaphoric links between NPs in attitude contexts can be supported by what he called coordination
between mental states, which can be inter personal. Kamp et al. (2011) fails to
capture this feature of cross-attitudinal anaphora. But, what is coordination between mental states? Even though it is hard to give a precise characterization of
coordination, the following consideration will give us an intuitive understandingof it.
As Geach pointed out, it sometimes makes sense to regard a mental state ofa subject as being directed toward a ‘common focus’ as a diﬀerent mental stateof a diﬀerent subject is (Geach, 1967, p. 627). For example, suppose that John
told to Mary about a man who he met yesterday and Mary believed what John
said. In this cases, even if Mary had no idea about who the man John mentioned,
Mary’s belief is reasonably regarded as being directed towards a common focus
as John’s mental states. As Asher says, the coordination relation holds ‘by meansof communication or some other mechanism’: Communication is a good reason
for coordination, but it is not the only reason. As we have seen, in the Hob-Nob
case, Hob’s belief and Nob’s belief are coordinated together because their beliefs
are based on the common source of information—the local newspaper. In another- 31 -

example, let us suppose that John and Mary together see a mirage of an oasis
in a desert. In this case, it is reasonably said that John’s perceptual experience
is directed toward a common focus as Mary’s, and vice versa. This relation canbe intrapersonal as well. For example, when you see an apple in front of you
and think that it looks delicious, your thought is reasonably regarded as being
directed toward a common focus as your perceptual experience is. The samething holds even when you have a hallucination of an apple. Roughly speaking,two mental states are coordinated when they have the common focus in the sense
illustrated here. Let us say that a mental state x is coordinated with a mental
state y if x is reasonably regarded as being directed toward a common focus asy is directed toward.
The mirage case and the hallucination case above show that the coordination
relation between mental states is referent-independent, in the sense that in orderthat a mental state of a subject is coordinated with other mental state of thesame/other subject, it is not necessary for these mental states to be about aparticular external existent object.
I proposed to say that a mental state x is coordinated with a mental state
y when x is reasonably regarded as being directed toward a common focus as yis directed toward. I don’t present this as the definition of coordination. This,
together with examples, gives at most intuitive understanding of it. Many things
are left unexplained in the clause ‘being reasonably regarded as being directed
toward a common focus as’: Who does regard so?; what is a common focus?; and,what may other mechanism than communication establish coordination between
mental states? In this paper I can not answer to these questions. Rather, I propose to take coordination as a primitive notion in semantics of cross-attitudinal
anaphora, and use it to record cross-attitudinal anaphoric linkage.
To do this, we need to represent such coordination between mental statesin a DRS. But how? In (8) the discourse referent x can be regarded as a mental representation of a witch for Hob and u as the one for Nob. Given this, it
seems natural to represent coordination between Hob’s belief and Nob’s belief
as a relation between x and u (cf. Asher, 1986, pp. 151-159). However, withina standard framework of DRT, it is not clear how to record information about
coordination between attitudes. Let us call a DRS containing some attitude ascription condition as its DRS condition a matrix DRS, and a DRS which appears
in attitude ascription condition and represents the content of some mental statea content DRS.4 Within the standard framework of DRT, information about
linguistic anaphora between two NPs is recorded by the identity between the
discourse referents introduced by these NPs. Given this framework, there are
only two possible ways to record anaphoric links. The first one is to put theidentity in question in a matrix DRS, and the second one is to put it in a content DRS. Unfortunately, both ways cannot properly treat cases of interpersonalcross-attitudinal anaphora. If we put ‘x = u’ in a matrix DRS, then x and u
represent some individual objects, not mental representations, and the result
4

For simplicity, I ignore cases where the complement of an attitude verb is itself an
attitude report.

- 32 -

entails that there is a particular object of which Hob’s mental state and Nob’s
mental state in question are. On the other hand, it is also problematic to put
the identity in a content DRS. According to this option, the condition x = u
appears in the content DRS which specifies the content of Hob’s belief or one
which specifies the content of Nob’s belief. If the former was the case, then Nob’s
mental representation would appear in Hob’s mental representation, but this is
impossible. What can appear in Hob’s mental representation is, at most, a representation of Nob’s mental representation (and this is not useful in the present
context, since the truth of the reading of (1) in question doesn’t require Hob tobelieve anything about Nob). The same problem would arise if the latter was
the case.
In the next section, we explore a way to write down the coordination relation
between mental states in matrix DRSs (that is, outside the scope of an attitude),not in content DRSs, so as to record anaphoric links by this relation. To do this,we need something diﬀerent from the discourse referents x and u appearing incontent DRSs. A key notion to do this is the notion of meta discourse referent.
In DRS, the coordination is represented by using meta discourse referents.

4

DRT with coordination

In 4.1 I extend DRT language so that anaphoric links are recorded by meta
discourse referents and the coordination predicate in matrix DRSs, and give an
informal exposition of what extended DRSs means. In 4.2 we have a brief lookat a model theoretic semantics for DRSs containing meta discourse referents andthe coordination predicate. In section 4.3, we compare the account presented inthis paper with Edelberg (1992)’s seminal work on cross-attitudinal anaphora.
Note that the following theory is meant to be for cases where attitude verbs do
not appear in the embedded sentences of attitude reports. Also the following
theory says nothing about de re ascriptions. To extend the theory to cover suchcases is a topic for further investigation.
4.1

Syntax (and an intuitive exposition of how to interpret DRSs
with meta discourse referents and the coordination predicate)

Suppose that r is a discourse referent. Then, [r] is a meta discourse referent ofr. Meta discourse referents are used in two purposes. First, they are used toregister which discourse referents appear in content DRSs: whenever a discourse
referent r appears in a content DRS in a matrix DRS, its meta discourse referent[r] is in the universe of the matrix DRS. Secondly, they are used to describe
coordination among mental states and to record anaphoric link between NPs in
attitude contexts. Let me explain these points by example. Consider again the
Hob-Nob sentence.
(1) Hob believes that a witch blighted Bob’s mare. Nob believes that she killed
Cob’s sow.

- 33 -

I propose that whenever a discourse referent appears in the universe of a content
DRS, its appearance is registered by introducing its meta discourse referent into
the universe of the matrix DRS whose condition contains the content DRS. Thus,
when the first sentence of (1) updates the empty DRS, we have the following
DRS.
h, [x](9)

Hob′ (h)




⟨


Att(h,
BEL,






x
witch′ (x)blighted-Bob’s-mare ′ (x)




⟩








)

x in the content DRS is introduced by the indefinte NP ‘a witch’ in the embedded
sentence of it. [x] in the universe of the matrix DRS of (9) is the meta discourse
referent of x, which registers an appearance of the discourse referent x in theuniverse of the content DRS of (9). Then, what does this DRS represent? Hereit is useful to mention the notion of mental file. Informally speaking, in (9), [x]represents a mental file whose possessor and contents are specified in the attitude
ascription condition of (9). More specifically, (9) is true iﬀ there is a mental fileof Hob which contains the information being a witch and blighted Bob’s marewith the attitudinal mode of belief (the model theoretic interpretation of DRSs
containing meta discourse referents is defined in the next section). Let us develop
this picture in more detail.
It is common to assume mental files to explain object-oriented information
processing by cognitive systems (cf. Recanati, 2012). If two or more pieces of
information are taken by a subject as being about the same object, then, it issaid that they are stored in the same mental file for some object. For example, suppose that by seeing an apple in front of her, Mary thinks that it is red
and round. In this case, she has a perceptual mental file for the apple whichcontains information of being red and information of being round. If, as Kamp
has suggested, DRSs are mental representations hearers construct during interpreting discourses, then, for a DRS ⟨U, Con⟩, any pair of a discourse referent in
U and the conditions of it in Con can be seen as a temporal mental file constructed and used to interpret the discourse in question by its hearer. Thesefiles are temporal ones in the sense that they are sustained only during perceiving something or interpreting of discourses. But some files are sustained for along period, which would be in subjects’ long-term memories. For example, I
takes many pieces of information as being about the Empire State Building—being located in Manhattan, having elevators, and so on—and this means that
I have a mental file for it containing these pieces of information. Many mental
files are properly connected to a particular object, but this is not necessarily
the case. Some files fail to have any particular referent (for example, the Vulcan file LeVerrier had); some files contain pieces of information whose sources
are diﬀerent objects (for example, the Madagascar file Marco Polo had). It is
a common and implicit assumption that mental files are connected to understanding of referential terms like proper names, indexicals, demonstratives, and- 34 -

definite descriptions. However, this needs not be the case, given that the DRS⟨{x}, {delegate′ (x), arrive′ (x)}⟩ is a mental file which a hearer constructs when
she interprets the sentence ‘a delegate arrived’.
Let us take a content DRS within the attitude ascription condition Att(a, ADS)
as representing a mental file(s) of the agent a. More precisely, each discourse referent r in the universe of a content DRS represents a mental file whose contents
are partially described by the conditions of r specified in the content DRS. For
example, ⟨{x}, {delegate′ (x), arrive′ (x)}⟩ appearing inside Att(a, ADS) represents a mental file of the subject a containing information of being a delegate
and arrived (and probably more). Meta discourse referents are the devices torepresent such mental files outside attitude ascription conditions. If a meta discourse referent [x] is in the universe of a matrix DRS, this means that there isa mental file whose contents are specified by a content DRS with the discourse
referent x, and whose subject is specified by the attitude ascription condition
containing the content DRS.
Another element in an attitude ascription condition unexplained yet is themodes of mental states. For them, let us assume that each piece of information ina mental file is accompanied with some attitudinal mode. In (9), the conditionsof the content DRS is associated with the mode of belief. This means that in
the mental file represented by the DRS, information of being a delegate andinformation of arrived are associated with the mode of belief. It may be the case—indeed, it is often the case—that a mental file contains pieces of information with
diﬀerent attitudinal modes. For example, Mary believes that a delegate arrived
and doubt that she registered. In this case, she has a mental file containing
information associated with the mode of belief and information associated withthe mode of doubt, which is represented by the attitude ascription condition (7).
Let us move on to the second purpose of using meta discourse representations:to describe coordination relation between mental states and to register anaphoric
link between NPs in attitude contexts. As are usual discourse referents, meta
discourse referents in the universe of a DRS as a context of interpretation are
possible antecedents for anaphoric pronouns in subsequent sentences. In addition
to this we also assume that: (i) meta discourse referents are available only formeta discourse referents introduced by anaphoric pronouns appearing in theembedded sentences of attitude reports; and (ii) cross-attitudinal anaphora isregistered not by using the identity symbol and discourse referents, but by using
the coordination predicate and meta discourse referents. Let us see how theseideas work.
(10) is the preliminary DRS corresponding to the second sentence of (1).5
u in the context DRSs of (10) is a discourse referent introduced by ‘she’ inthe embedded sentence of the second sentence of (1). Again, [u] registers theappearance of the discourse referent u in the universe of the content DRS of(10), and represents a mental file the agent of the attitude in question—in thiscase, Nob—has.
5

For the notion of preliminary DRS, see Kamp et al. (2011) sec. 2.3.

- 35 -

n, [u]N ob′ (n)







⟨
⟨




Att(n,
BEL,











(10)








u
pers(u)f emale(u)










⟩⟩



u
,

′







killed-Cob’s-sow (u)6)

Anaphora resolution proceeds as follows. Let us assume that coordination relation which makes cross-attitudinal anaphoric linkage between NPs sense is arelation between mental files. To represent this relation, we introduce ‘>’, thecoordination predicate for meta discourse referents. [r] > [r′ ] represents that themental file represented by [r] is coordinated with the mental file represented by[r′ ]. If [r] takes [r′ ] as its antecedent, this information is recorded as [r] > [r′ ] inthe condition of the matrix DRS. For example, the anaphoric linkage between‘a witch’ in the first sentence of (1) and ‘she’ in the second sentence of (1) isrecorded by using two meta discourse referents [x], introduced by ‘a witch’, and[u], introduced by ‘she’, and the coordination predicate >. In this case, [x] isthe antecedent of [u], and thus this information is registered as [u] > [x] in thecondition of (10). So, taking (9), whose universe contains [x], as its context, we
obtain (12) as the result of anaphora resolution of (10).
n, [u]
N ob′ (n)


⟨


BEL,
Att(n,




(11)


⟩



u
killed-Cob’s-sow ′ (u)





)

[u] > [x]Finally, merging7 this with (9) results in the following DRS, which is the
DRS translation of the discourse (1).86

For simplicity, here I treat proper names as not presuppositional.
K is the result of the merge of K1 and K2 (K1 ⊎ K2 ) iﬀ K = ⟨UK1 ∪ UK2 , ConK1 ∪ConK2 ⟩ (Kamp et al. 2011, sec. 2.3).
8
We can easily construct DRSs for cases where content DRSs contain two or morediscourse referents. Here is an example (I owe this example to an anonymous referee).
7(12) a. Hob believes that a witch found a unicorn. Nob believes that she killed it.
h, [x], [y], n, [u], [v]b.

′Hob′ (h), N ob (n)

x, y




⟨


witch′ (x)Att(h,BEL,
unicorn′ (y)
f ound′ (x, y)













⟩
⟨




BEL,
), Att(n,













[u] > [x], [v] > [y]- 36 -

u, v
kill′ (u, v)



⟩


)





h, [x], n, [u](13)Hob′ (h)N ob′ (n)




⟨


Att(h,
BEL,







⟨

Att(n,
BEL,




x
witch′ (x)blighted-Bob’s-mare ′ (x)




⟩








)


⟩



u
killed-Cob’s-sow ′ (u)





)

[u] > [x]This DRS, and thus the reading of (1) in question, is true iﬀ (i) there are
two subjects, Nob and Hob; (ii) there are two mental files X (represented by [x])and U (represented by [u]) such that X is Hob’s file containing information that. . . is a witch and that . . . blighted Bob’s mare associated with the mode of beliefand U is Nob’s file containing information that . . . killed Cob’s sow associatedwith the mode of belief; and (iii) Nob’s mental file U is coordinated with Hob’s
mental file X.
4.2

Semantics

In this section, I define intensional semantics for DRS involving meta-discourse
referents and >.
A model M is any seven-tuple ⟨D, A, W, M F, C, , V ⟩ which satisfies thefollowing conditions.
D is a set of individual objects. A is a set of agents (bearers of propositional
attitudes), which is a subset of D. W is a set of worlds. For simplicity, we assume
that the domain is constant across worlds.
M F is a non-empty set taken as the set of mental files. For each w, M Fw
is the set of mental files agents have in w. M F must be disjoint with the set of
discourse referents constituting the basic vocabulary of DRSs.
C is a function which specifies the contents of ones’ mental files. To explain
this, first we need to introduce the notion of information state (Kamp et al.,2011, pp. 157-158). Standardly, an intentional semantics for a language L assigns
to each formula of L a set of possible world, that is, the set of all possible
worlds where the formula is true. Instead of a set of possible worlds, intentionalsemantics for DRSs assigns to each (proper) DRS an information state. In Kamp
et al. (2011, p. 157), an information state assigned to K relative to an intensionalmodel M = ⟨D, W, V ⟩, in symbols [[K]]sM , is a set of pairs of a possible world w
and an embedding function f : UK → D such that every condition in ConK isverified by f in w with respect to M .
(14) [[K]]sM := {⟨w, f ⟩ | ⟨Λ, f ⟩ |=M,w K}- 37 -

All embedding functions appearing in [[K]]sM have the same domain, that is, UK .
So the information state assigned to K registers information about the universeof K and thus about candidates of antecedents of anaphoric pronouns.
Turning back to C, the range of C is the set of information states, but here
we understand an information state I as a set of pairs of a world and a function
which maps a mental file to a member of the domain D. Partly following Kamp
et al. (2011), C assigns an information state in this sense to each ordered tripleof an agent a ∈ A, a world w, and an attitudinal mode Φ. For example, supposethat m is one of a’s mental files and that m contains the condition P with themode BEL. In this situation, if ⟨w, f ⟩ ∈ C(⟨a, w, BEL⟩), then f (m) satisfies
P in w. We require that all functions in the members of the information stateC(⟨a, w, Φ⟩) share the same domain, which is a subset of M Fw . Let us call thisdomain the base of C(⟨a, w, Φ⟩). We also require that for any attitudinal modes
Φ and Ψ , the base of C(⟨a, w, Φ⟩) is identical to the base of C(⟨a, w, Ψ ⟩). This
shared base is the set of all mental files the agent a has in w. Let us call it M Fwa .
The last requirement is that for any agents a and b, M Fwa is disjoint from M Fwb .
This reflects the intuition that no two agents can share the same mental file.
 : W → P(M F × M F ) determines coordination among mental files in eachworld. For each world w, w is a binary relation on M Fw . Note that w is both
intrapersonal and interpersonal, that is, coordination is both intrapersonal andinterpersonal. Semantic/inferential behaviors of cross-attitudinal anaphora force
us to put some constraints on . Due to a space constraint we can not discuss
details, but let me very quickly mention some structural features of coordination. First, to treat intrapersonal cases properly, we require w to be reflexive.
Second, given Edelberg (1986)’s well-known discussion about non-symmetricity
of intentional identity, w needs not be symmetric. Putting further constraints
on w , of course, will lead to diﬀerent ‘logics’ of cross-attitudinal anaphora.
V is an interpretation function assigning a model theoretic meaning to each
non-logical vocabulary, which is defined as usual.
An embedding function is a function from the universe of some DRS K to thedomain of the model. Now the universe of DRS contains not only discourse referents but also meta discourse referents. For any discourse referent x we requirethat any embedding function must assign x to [x].
In addition to this, we need functions which map each discourse referent
appearing the universe of a content DRS to a member of M F . Let us call such afunction an m-embedding function, since it maps a discourse referent to a mental
file.
Verification depends not only on embedding functions but also m-embeddinghfunctions. First let us define gm, the m-counterpart of an embedding function gwith respect to an m-embedding function h, where for all x and y if h(x) = h(y)hthen g(x) = g(y). gm
is the function which satisfies the following conditions.
hh(15) a. D(gm) (the domain of gm) is h[D(g)] (the image of the domain of gunder h), which is a subset of M F .
hb. For all x ∈ h[D(g)], gm(x) = g(x′ ), where x′ ∈ h−1 [x]- 38 -

Verification conditions are defined as follow. Suppose that g is an embedding
function and h is an m-embedding function, then(16) a. g, h |=M,w [x] > [y] iﬀ h(g([x])) w h(g([y]))b. g, h |=M,w Att(x, ADS) iﬀ– g(x) ∈ A, and– For each ⟨Φ, K⟩ in ADS, ∀w∀f (⟨w, f ⟩ ∈ [[K]]sM → ∃f ′ (⟨w, f ′ ⟩ ∈hC(⟨g(x), w, Φ⟩) ∧ fm
⊆ f ′ )) 9
c. For other types of conditions, g, h |=M,w is defined as g |=M,w
Finally, truth condition. A DRS K is true with respect to the model M in
w iﬀ there are an embedding function g and an m-embedding function h which
verify all conditions in ConK with respect to M in w.
4.3

Edelberg (1992)

The account presented in this paper has a great similarity with Edelberg (1992)’s
seminal work on intentional identity, according to which (1) is semantically analyzed as (17).
(17) ∃α∃β(BEL(Hob, [[α blighted Bob’s mare]] ∧ BEL(Nob, [[β killed Cob’ssow]] ∧ α ≈ β)),where ‘α’ and ‘β’ are variables ranging over ‘person-bound’ menta images and‘≈’ represents the counterpart relation between ideas. Both our account and
Edelberg’s account appeal to intrapersonal mental representations and somerelation between them.
So, the presented analysis of cross-attitudinal anaphora might be taken as a
DRT formulation of Edelberg’s semantics of cross-attitude anaphora. But even
if this were true,10 this would make diﬀerence. Since our account is formulated
within the framework of DRT, it inherits advantages of DRT, which is not available in Edelberg’s account. For example, it gives a bottom-up account of how
to get the truth conditional analysis of cases of cross-attitudinal anaphora, byshowing the corresponding DRSs construction procedures. Moreover, we can appeal to machineries in DRT, in particular, the notion of accessibility, to predict
distribution of possible anaphoric links. For example, in (18), ‘he’ in the second
sentence can not take ‘someone’ in the first sentence as its antecedent.
(18) *Mary does not believe that someonei murdered Jones. Bob believes thathei murdered Smith.
The first sentence of (18) is translated as (19a) and the second one is as (19b).
The relation between [[K]]sM and C(⟨g(x), w, Φ⟩) specified here is a modified versionof ⪯ in Kamp, et al. (2011, p. 160)
10Actually, this is not true, at least since the coordination relation needs not be symmetric in our account, but the counterpart relation is symmetric in Edelberg’s theory.
(Edelberg, 1992, sec. 9 and 10)9- 39 -

m
M ary ′ (m)[x](19) a.
¬



⟨


Att(m,
BEL,




x
murdered-Jones ′ (x)


⟩


)




b, [u]

b.

Bob′ (b)






⟨
⟨




Att(n,
BEL,











u
pers(u)male(u)













u
,

murdered-Smith ′ (u)




⟩⟩


)






Since [x] appears within the scope of negation and is not in the universe of thematrix DRS (19a), [x] is not accessible for [u]. This immediately explains why(18) is not acceptable.

5

Conclusion

In this paper, we oﬀered a DRT account of cross-attitudinal anaphora, basedon the idea that coordination between subjects’ mental states plays a crucialrole for cross-attitudinal anaphora. We extended DRT language by adding meta
discourse referents and the coordination predicate—in the model theoretic semantics, they correspond to mental files and the coordination relation between
them, which are taken as primitive in the semantics—, and used them to record
anaphoric linkage between NPs in attitudinal contexts.

References[1987] Asher, N. (1987). ‘A Typology for Attitude Verbs and Their Anaphoric Properties’, Linguistics and Philosophy, 10, 125-197.
[1986] Edelberg, W. (1986). ‘A New Puzzle about Intentional Identity’, Journal of
Philosophical Logic, 15(1), 1-25.
[1992] Edelberg, W. (1992). ‘Intentional Identity and The Attitudes’, Linguistics andPhilosophy, 15(6), 561-596.
[1967] Geach, P. T. (1967). ‘Intentional Identity’, Journal of Philosophy, 64: 20, 627632.
[1981] Kamp, H. (1981). ‘A Theory of Truth and Semantic Representation’, in J. Groenendijk et al. (eds), Formal Methods in the Study of Language. Amsterdam: Mathematisch Centrum[2011] Kamp, H., J. van Genabith, and U. Reyle (2011). ‘Discourse Representation
Theory’, in D. Gabbay and F. Guenthner (eds.) Handbook of Philosophical Logic,
Volume 15, Springer, pp. 125-394.
[2012] Recanati, F. (2012). Mental Files, Oxford: Oxford University Press.

- 40 -

Truth-conditionals and Use-conditionals
an expressive modal analysis
Lukas Rieser
Kyoto University Graduate School of Letters

Abstract. I propose that on some non-standard interpretations of conditionals, the antecedent influences not the truth-conditional, but theuse-conditional evaluation of the consequent by restricting the modalbase of a necessity operator introduced by the conditional form. On
this view, conditionals can be grouped into truth-conditionals and useconditionals, depending on interpretation. I argue that such an analysis
allows to predict properties of hypothetical conditionals, biscuit conditionals, and conditional hedges within a unified account.

1

Interpretations of conditionals

The three examples of conditionals given in (1) through (3) below each have distinct salient interpretations, for which I seek to account for in a unified analysis.
(1) If John remembered to go shopping, there’s beer in the fridge.
(2) If you are thirsty, there’s beer in the fridge.
(3) If I’m not mistaken, there’s beer in the fridge.
The salient interpretation of (1) is what I take to be the standard interpretation, also labeled the hypothetical conditional interpretation, on which thetruth of the consequent (=proposition of the main clause) is evaluated under theassumption that the antecedent (=proposition of the if-clause) holds (variantsof analyses on these lines have been prominent in the formal literature, startingwith Stalnaker 1968 [17]). The salient interpretation of (2) is that of a biscuit
conditional (term due to Austin’s [1] original example “There are biscuits on
the sideboard if you want some”), on which the information the consequent provides is relevant for the addressee only in case the antecedent holds. Finally, (3)is most plausibly interpreted as a conditional hedge, a kind of disclaimer, on
which interpretation the conditional form indicates possible error on part of the
speaker, that is the possibility of the consequent not holding, in order to avoid
the consequences of providing potentially false information.
There are differences in how the conditional antecedent and consequent relate
to each other on the three interpretations, as summarized below. The analysis
aims at accounting for these differences while deriving the interpretations thethree examples intuitively receive.

- 41 -

Conditional (in)dependence, conditional perfection
The hypothetical conditional in (1) conveys (truth-)conditional dependence between the antecedent and the consequent in the sense that the truth of the consequent is to be evaluated only under the assumption that the antecedent holds.
The intuition underlying truth-conditional analyses of such conditionals is that(1) says that the world is such that there’s beer only if John went shopping. In
other words, antecedent and consequent are truth-conditionally dependent.
In the biscuit conditional in (2), on the other hand, there is no such dependence, as the addressee’s thirst is irrelevant for the truth of whether or notthere is beer in the fridge. Franke (2007) [9] defines this as conditional independence, formally implemented as the property of a conditional that it is not
sufficient to learn the truth value of the antecedent in order to find out that of
the consequent. The intuition I build on is that what biscuit conditionals like(2) assert is the consequent only, which is only relevant (essentially in a Gricean
sense, i.e. relevant to the conversational goals of the participants) in case theantecedent holds. Following this intuition, previous analyses in the philosophicalliterature have assumed that what is conditioned on is assertion itself, i.e. theconsequent is only asserted when the antecedent holds (DeRose and Grandy 1999[5]), or that biscuit conditionals indicate the existence of a “potential literal act”
(such as the assertion that there is beer) in case the antecedent holds (Siegel 2006[16]). The analysis I propose is also one where assertion is conditioned on, but
in a sense closer to extant analysis of conditionals within formal semantics.
The intuitions regarding what the conditional hedge in (3) conditions on are
similar to that on the biscuit conditional in that the antecedent targets felicity
rather than truth conditions, but different in that the conditional hedge targets
Quality (of the speaker’s belief that the consequent holds), rater than Relevance(both in a Gricean sense). However, in contrast to both (1) and (2), conditional
perfection arises from (3): the truth of consequent and antecedent is in a materialbiconditional (or perfect conditional), relation ↔ (see Van der Auwera 1997
[2] for discussion): “If there’s beer in the fridge, I am not mistaken”, while asomewhat odd thing to say, intuitively has the same truth conditions as (3).
Below, I develop a formal analysis which is capable of predicting this property
of conditional hedges while also being applicable to biscuit and hypotheticalconditionals.

2

Conditionals as modals

I take conditionals to be modal constructions, in which the antecedent restricts
the modal base of the consequent, following a possible-world semantics analysis
of modality in conditionals (cf. Kratzer 2012 [13] for an overview), thus accounting for the intuitions on hypothetical conditionals as described in regard to (1)above. I further assume that when there is no overt modal in the consequent,
the conditional introduces a covert “human necessity” (Kaufmann and Schwager
2009 [12]) modal, the conversational backgrounds being a modal base reflecting- 42 -

the relevant circumstances and a stereotypical ordering source to control for
issues like those raised by strengthening of the antecedent.
I take the relevant circumstances to be those relevant to the connection between antecedent and consequent conveyed by the conditional, which in the caseof a hypothetical conditional is truth-conditional independence, in the case ofnon-standard interpretations a connection mediated by Gricean maxims. I further assume the modal base to be the propositions compatible with the speaker’s
beliefs regarding the relevant circumstances, rather than the relevant circumstances as such, making the modal base doxastic as well as circumstantial.1
To illustrate the standard interpretation of (truth-)conditional on this view, aparaphrase for the meaning of a hypothetical conditional is given in (4).
(4) A truth-conditional “If Ψ then Φ” is true iff in all worlds (stereotypical,
compatible with the speaker’s beliefs) in which Ψ holds, Φ also holds.
It is worth noting that the details of what kind of modal base and ordering
source are chosen are not central to my analysis of truth- and use-conditionals,
as long as the modal force is necessity, and the proposition in the conditional
antecedent is used to restrict the modal base, thus excluding worlds in which theantecedent does not hold. This sets the stage on which the conditional consequentis evaluated. Also, my analysis is fully compatible with a double-modal analysis
of conditionals (Frank 1996 [8]), on which the modal the consequent introduces
a modal base and ordering source potentially different from that introduced by
the conditional form.
2.1

Use conditions

The view of use conditions I base my analysis on connects them to truth conditions in the following way. Taking assertion as an example, the meaning of anutterance is split into the descriptive, or truth-conditional, and the expressive,or use conditional, dimension. Truth conditions of a proposition are determinedby valuation against worlds. When the proposition (usually one, if possibly complex) in the descriptive, truth-conditional dimension of utterance meaning is truein this sense, this is to say that something true is asserted. When the propositions (typically many) in the expressive, use-conditional meaning dimension are
true, this means that the assertion is felicitous. Thus, when the source of the
proposition in the expressive meaning dimensions are lexical conventional implicatures (dog vs. cur, see Gutzmann 2015 [11]) and parentheticals, the utterance’s
use conditions are straightforwardly determined by the truth or falsity of these
propositions, i.e. their truth conditions have become use conditions.
In addition to such use-conditional propositions derived from the lexical content, which are valuated against worlds in the same way as truth conditions
are, I propose that there are types of propositions in the expressive dimension, which are not valuated against worlds, but evaluated in regard to Gricean
1

In the paraphrases for the modal base, I will use “the speaker’s beliefs” to meand“the speaker’s beliefs regarding the relevant circumstances” for brevity.

- 43 -

conversational maxims (Grice 1975 [10]). Concretely, I propose that the entire
propositional content in the descriptive dimension is evaluated in such a way inthe expressive dimension, and that it is this part of the expressive dimension
where conditioning on relevance, quality, etc., in non standard-interpretations ofconditionals occurs. This essentially amounts to propositions based on Gricean
maxims being valuated in the usual way, and the utterance being felicitous if theyhold. According paraphrases for the truth- and use conditions of an assertion
with the prejacent proposition Φ are shown in (5) and (6).
(5) Truth conditions: Assert(Φ) is true w.r.t w iff Φ holds in w.
(6) Use conditions: Assert(Φ) is felicitous w.r.t w iff Φ is relevant to theparticipants’ goals, as informative as required, backed by adequate
evidence,. . . in w.
Notice that (6) makes no mention of expressive content originating in the lexical
content of the utterance. This is for ease of exposition — for the same reason, I
will only consider examples where no expressive content arises from the lexicalmaterial. Also, no mention is made of the truth of Φ in the use conditions, as I
assume that asserting a false proposition is not necessarily infelicitous, provided
that the speaker is not aware of its falsity. I will return to this latter point insection 4.1. In this section, I implement the view of use conditions sketched above
in a model of indicative conditionals, starting with the standard interpretation.
2.2

Truth-conditionals

The descriptive, truth-conditional meaning of a hypothetical conditional “if Ψ ,(then) Φ” in the formalization I assume is shown in (7), where kA(Φ)kt stands
for the truth-conditional denotation of an utterance where a speech act A basedon the prejacent proposition Φ. As for the notation representing the conditional,
H stands for a human necessity modal as outlined above, w for the actual
world, f (w) for the (doxastic / circumstantial) modal base (worlds compatible
with the speaker’s beliefs regarding the relevant circumstances in w), g for the(stereotypical) ordering source.
The subscript [Ψ ] on the modal H with the conditional consequent Φ inits nuclear scope indicates restriction of the modal base f (w) by the conditional
antecedent Ψ , yielding a restricted modal base f + in which non-Ψ worlds havebeen discarded.
tHt+(7) kAssert(H[Ψ ] [Φ])k = kAssert( [Φ])k w.r.t w, f , g(where f + = λw.f (w) ∪ kΨ kt )The truth-conditional meaning of an assertion of (8) thus comes out as (9)(8) If John remembered to go shopping, there’s beer in the fridge.
(9) (8) is true iff in all worlds (stereotypical, compatible with the speaker’s
beliefs) in which John remembered to go shopping, there’s beer in thefridge.

- 44 -

Next, in order to derive the use conditions of truth-conditionals, I adopt thegeneral framework of Potts (2005) [15], distinguishing between descriptive (=atissue) or truth-conditional (types marked with superscript a for “at-issue”), andexpressive or use-conditional (types marked with superscript c for “conventional
implicature”) levels of utterance meaning. I use the types ta and tc for propositions within the descriptive and expressive meaning dimensions, ta being thetype for truth-conditional content, tc for lexical CIs and parentheticals.2
In addition to these types, I introduce the use-conditional meaning type uc(utterance) following McCready (2015) [14]. This is the type which I propose isevaluated in terms of Gricean maxims, rather than valuated against worlds inthe usual manner for propositions. Also following McCready, elements of type uc
arise from the type-shifting operation utterance lifting (ul), by which propositional content, once asserted, is moved into the expressive domain, i.e. undergoes
type shift from type ta to type uc . I use a simplified version of ul as a function
from (at-issue) propositions to (expressive) utterances in the present proposal.3(10) ulA = λΦ.A(Φ) : < ta , uc >(where A(Φ) is a speech act based on proposition Φ)
With this rule in place, performing a speech act based on a proposition containing no lexical expressives or parentheticals can be represented as in (11),where (a.) shows the two dimensions of meaning before assertion, containing
propositional (descriptive, truth-conditional) content only, and (b.) shows utterance meaning4 after assertion. The representation follows the convention hτ a , τ c iwhere all truth-conditional elements τ a which are part of the utterance’s meaning are shown on the left, all use-conditional elements τ c on the right. A parallel
representation for a speech act based on a conditional proposition (i.e. a truthconditional) is given in (12).
(11) a. hΦ, ∅i(12) a. hH[Ψ ] [Φ], ∅iHb. hH[Ψ ] [Φ], A([Ψ ] [Φ])ib. hΦ, A(Φ)i

In order to derive the use conditions which A(Φ) and A(H[Ψ ] [Φ]) respectively
contribute in the expressive dimension, A needs to be resolved to a specificspeech act. I discuss the case of assertions of conditionals below, aiming to arrive
at a formalization of use-conditions as paraphrased in (6) above, and to accountfor the differences between indicative use- and truth-conditionals.
2

As mentioned, however, the examples will not contain any of the latter, and I remain
agnostic in regard to the question of whether or not content of type tc gets evaluatedfor felicity in terms of Gricean maxims.
3
McCready’s assumption that u is of a resource-sensitive shunting type us , making an additional operation of assertion-to-content necessary to reintroduce at-issue
meaning, is ignored here for ease of exposition.
4Note that I use the label “utterance meaning” to refer to the meaning of proposition
used in a speech act, rather than just for the parts of its meaning of type uc .

- 45 -

2.3

Use-conditionals

In the case of truth-conditionals, the modal base of the covert modal introduced by the conditional form on truth-conditional level is restricted by theconditional antecedent, reflecting (truth-)conditional dependence. In the case ofuse-conditionals, not the truth, but the felicity of the consequent depends on
the truth of the antecedent. I explain this as restriction of the modal base on
the expressive rather than the descriptive level, as paraphrased in (14) for a useconditional, parallel to the paraphrase for a truth-conditional in (13), repeatedfrom (4).
(13) A truth conditional “If Ψ then Φ” is true iff in all worlds (stereotypical,
compatible with the speaker’s beliefs) in which Ψ holds, Φ also holds.
(14) A use conditional “If Ψ then Φ” is felicitous iff in all worlds (stereotypical,
compatible with the speaker’s beliefs) in which Ψ holds, A(Φ) is felicitous.
The next question to be addressed is how to evaluate the felicity of A(Φ) (whichin the examples considered so far is assertion), and where in the derivation ofutterance meaning to introduce conditional restriction in order for it to operateon the expressive level of meaning in use-conditionals. I assume that what has
to happen compositionally to get the interpretations we are after is what (15)schematically shows for a truth-conditional, (16) for a use-conditional. (17) shows
the assumption I make for what happens on the use-conditional level in the caseof the speech-act A being an assertion, namely restriction of the modal base ofa non-asserted human necessity modal on the use-conditional level.
(15)

A

(16)
if Ψ H Φ

if Ψ m

(17)
AΦ

if Ψ

H A Φ

Now, the crucial question can be put as follows: what does Ψ restrict whenmodifying A(Φ)? In (16), the placeholder is labeled m for modal, anticipating
the analysis I propose, but there are other options — one could follow Siegel’s
proposal and have the conditional antecedent somehow quantify over potential
speech acts, or DeRose and Grandy’s proposal, making the assertion of ϕ dependon the truth of the antecedent. I propose an approach I consider more straightforward in light of extant formal theories of conditionals, in which speech acts
are neither quantified over nor suspended in this sense. Rather, the modal operator is introduced not on the descriptive, but on the expressive level. In thecase of A being an assertion, only a human necessity modal, as the one familiarfrom propositions of truth-conditionals, is introduced (other modals could be
introduced in the case of speech-acts like imperatives).
The denotation of the two dimensions of meaning of a truth-conditional understandard, hypothetical, interpretation, corresponding to the structure in (15), isshown in (18), that of a use-conditional, corresponding to the structure in (17),is shown in (19) below, alongside the denotation of a plain, non-conditional
assertion of Φ in (20). Each representation is shown before (a.) and after (b.)
assertion with utterance lifting.

- 46 -

(18) a. hH[Ψ ] [Φ], ∅iHb. hH[Ψ ] [Φ], [Ψ ] [Φ]i(19) a. hΦ, ∅ib. hΦ, H[Ψ ] [Φ]i(20) a. hΦ, ∅ib. hΦ, Φi

The denotation of the use-conditional in (19) is derived as follows. The prejacentΦ of type ta is asserted and undergoes utterance lifting to uc . At this point,
the human necessity modal introduced by the conditional, with the modal base
restricted by the antecedent Ψ , enters the derivation, taking Φ in its nuclear
scope. The resulting representation of utterance meaning in (19)b. contains thecproposition Φ on the descriptive side, the modal expression H[Ψ ] [Φ] of type u on
the expressive side. On the view of use-conditions sketched above, elements oftype uc are evaluated according to Gricean maxims to determine the utterance’s
use-value, or felicity, here against the modal base restricted by Ψ .5

3

Explaining interpretatrions

Note that the felicity conditions of the standard, truth-conditional interpretation shown in (18)b. are the same as those of the non-standard, use-conditional
interpretation as shown in (19)b., but the truth-conditions differ for each interpretation. This reflects that basis of evaluation regarding relevance, quality,
etc. should be the entire conditional, not just the consequent, in the case of atruth-conditional. In the case of the use-conditional, conditioning only occurs
on the expressive, but not on the descriptive level, as the modal introduced byconditional form enters the derivation only after assertion, i.e. after utterance
lifting has occurred.
In the case of the biscuit conditional in (2), repeated here as (21), this meansthat the truth conditions of the entire constructions are the same as those of theconditional consequent, and the felicity conditions, where modal base restriction
by the antecedent occurs, are as paraphrased in (22).
(21) If you are thirsty, there’s beer in the fridge.
(22) Asserting “there is beer in the fridge” is felicitous (w.r.t to w,f + ,g) iff inall stereotypical worlds compatible with the speaker’s beliefs where theaddressee is thirsty, it is relevant to the participants’ goals, as informative
as required, [. . . ] in w that there is beer in the fridge.
Next, the felicity conditions of the conditional hedge (23), repeated from (3),as predicted in the current proposal are paraphrased in (24).
(23) If I’m not mistaken, there’s beer in the fridge.
(24) Asserting “there’s beer in the fridge” is felicitous iff in all stereotypicalworlds compatible with the speaker’s beliefs where the speaker is not
mistaken,. . .
5

See section 4.1 for discussion of the alternative idea that propositions in elements oftype uc are valuated like those of type tc , but with respect to speaker belief.

- 47 -

This is a special case in that, given a belief of the speaker that there is beer in thefridge, eliminating worlds in which the speaker is mistaken (i.e. only considering
worlds in which the speaker’s beliefs are true) only leaves worlds in which theconsequent holds.6 Thus, when (23) receives a truth-conditional interpretation,
its truth depends solely on the existence of a speaker belief that Φ. From this,the intuitive conditional perfection of this example can be predicted: given a
speaker belief that Φ, modal base restriction to worlds where Φ holds leaves only
worlds in which the speaker is not mistaken. Conversely, given a speaker belief
that Φ, restriction to worlds where the speaker is not mistaken leaves only Φworlds, as mentioned. Thus, the conditional in (23) is a material biconditional↔ on the truth-conditional level. As this case shows, the truth-conditional andthe use-conditional interpretations can happily coexist (even though only theuse-conditional interpretation seems to be informative).
Choosing an interpretation
As for the question of how it is decided which interpretation a conditional receives, I suggest that the use-conditional interpretation is be preferred whenrestricting the modal base is uninformative on the at-issue level due to conditional independence. It should be noted here that I assume that both a useand a truth-conditional reading are in principle available for all conditionals,
but that one will usually be more salient. For illustration, consider the following
ambiguous example.
(25) If you are interested in art, we will go to the museum.
This conditional has a reading on which the information that the speaker andsome third party will go to the museum is relevant if the addressee is interested in art, and a reading on which the speaker intends to take the addressee
to the museum only if the addressee is interested in art. The former is a useconditional, the latter a truth-conditional reading. Different contexts will bringout the salient interpretation. Once we have settled on a use-conditional interpretation, we need to decide which part of felicity, or which Gricean maxim, istargeted by modal base restriction with the antecedent. In the case of a useconditional interpretation of (25), for instance, relevance is a salient option, if
the consequent is intended as an invitation to join.
Examples for ambiguity arising from the possibility of different parts offelicity being targeted include the contrast between discourse-structuring andproblem-solving conditionals Csipak (2015) [4] observes. A case where it is not
immediately clear which part of felicity is targeted are appropriateness hedges
like “If I may be frank. . . ”, which target the same part of felicity as parenthetical disclaimers (cf. McCready 2015 [14]) which is not easy to grasp with Gricean
maxims, but could be subsumed under Manner (a more thorough survey of whichaspects of felicity use-conditionals can target has to be left for further research).
6

On a side note, this is potentially an argument against making the doxastic / circumstantial modal base realistic.

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4

Conclusion & outlook

Summing up, I have proposed that the properties of conditionals with salient
non-standard interpretations such as biscuit conditionals and conditional hedges
can be accounted for in a unified analysis with hypothetical conditionals whenassuming that a covert modal, the modal base of which is restricted by theconditional antecedent, can not only be introduced on the descriptive (truthconditional), but also on the expressive (use-conditional) level. The latter readingis triggered by conditional independence, that is when modal base restriction on
the descriptive level does not result in truth conditions that differ from that of
the bare consequent, thus rendering restriction uninformative.
In the remainder of this section, I discuss two alternative approaches within
the current proposal, followed by possibilities for integration with other analyses
and expansion of data coverage.
4.1

An assertion modal

There is an additional possibility for the introduction of a modal which canbe modified by the conditional antecedent on the use-conditional level, namely
the assumption that assertion introduces a modal like human necessity, witha doxastic modal base. The argument for this goes as follows. Assuming thatelements of type uc , introduced by utterance-lifting, are evaluated by Gricean
maxims to determine felicity of the utterance, how does truth factor in? It would
be welcome to have a way of including elements of type uc in the expressive partof utterance meaning in propositional valuation to determine their use-values, inparallel to elements of type tc . The intuition this is based on is that a proposition
is felicitously asserted if the speaker believes it is true, whether or not this isactually the case. This can easily be reflected in utterance meaning by assumingthat utterance lifting introduces a covert modal with a doxastic modal base and astereotypical (or possibly empty) ordering source, similar to the human necessity
operator introduced by the conditional form. An according representation of aplain assertion before and after utterance meaning is shown in (26), modified
from (20).
(26) a. hΦ, ∅ib. hΦ, H [Φ]iOn this analysis of assertion, the paraphrase for use-conditions of assertions
given in (6) can be revised as in (27), for simplicity assuming that Gricean
quality requires the speaker to believe Φ, so that Φ is required to hold in all
worlds compatible with the speaker’s beliefs for felicity.
(27) Use conditions (revised): Assert(Φ) is felicitous w.r.t. w, f, g iff Φ is truein all stereotypical worlds compatible with the speaker’s beliefs, relevant to
the participants’ goals, as informative as required, backed by adequate
evidence,. . .

- 49 -

This modified view of use conditions has another possibly welcome effect on theanalysis presented so far: there is no need to assume that the conditional form
introduces a human necessity modal on the non-propositional part (assumingthat elements of type uc are no propositions) use-conditional level, just as it doeson the truth-conditional, propositional level. Rather, the conditional antecedent,
now not restricting a modal base on propositional level, restricts the modal baseof the conditional introduced by the speech act assertion.
4.2

Symmetric use- and truth-conditionals

There is a potentially more natural view of use-conditionals (as well as truth
conditionals) on which their meanings do not differ from truth-conditionals,
only their interpretations do. Above, I proposed that because of what amounts
to uninformativity of restriction of the conditional modal base on the truthconditional, the conditional is interpreted on the use-conditional level in the caseof biscuit conditionals and conditional hedges. There seems no harm, however,in assuming that the standard truth-conditional interpretation goes for cases ofconditional independence as well, it just happens to have the same truth-value as
the consequent. On such a view, truth- and use-conditionals to not only share thesame expressive meaning, but also the same descriptive meaning. Just as modalbase restriction on the expressive level does not change the use-conditions ofhypothetical conditionals, modal base restriction on the descriptive level does
not change the truth-conditions of biscuit conditionals and conditional hedges.
The differences in interpretation arise as modal base restriction only has an effecton one level of meaning.
There is one possible problem with this view, in form of a difference between
the use conditions of, for instance, a biscuit conditional, and those of a hypothetical conditional, in which conditioning occurs on the descriptive level. In thecase of the biscuit conditional, conditioning occurs on the expressive level, thatis the propositions based on the Gricean paraphrases are valuated against the restricted modal base, yielding the desired interpretation. When conditioning has
already happened on the descriptive level, as in hypothecial There is one possible
difference between the use conditions paraphrased here and those arising froma hypothetical conditional in which conditioning occurs on the descriptive level.
In the case of the biscuit conditional, conditioning occurs on the expressive level,
i.e. propositions based on the Gricean paraphrases are valuated against the restricted modal base, yielding the desired interpretation. When conditioning has
already happened on the descriptive level, as in hypothetical conditionals, therestriction of the modal base potentially happens within the proposition basedon Gricean maxims, thus not influencing the evaluation of felicity in the desired
way anymore.
If this issue can be resolved, a symmetric approach to use- and truth-conditionals
would be a direction to consider for further research as a variant of the present
proposal.

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4.3

Connecting other analyses, broadening data coverage

An obvious starting point for integration of the present proposal with otheranalyses of related phenomena, is McCready’s (2015) [14] analysis of such parenthetical hedges and disclaimers within the same framework. The proposal also
naturally integrates with analyses of conditionals within a framework of modal
semantics, and can be readily applied to double-modal analyses of conditionals. I have built on Kaufmann and Schwager’s (2009) analysis of conditionals,
so that expansion of the scope to the conditional imperatives they analyze isstraightforwardly possible. Such an expansion could also shed light on the potential of the alternative paths sketched in the two previous points. Analysesof other non-standard conditionals in a similar framework such as Condoravdi
and Lauer’s (2015) analysis of (near-)ananakastic conditionals are also possible
targets for expansion, especially considering that there are variants of anankastic conditionals which share properties with biscuit conditionals, as discussed in
Francez 2015 [7] under the label “chimerical conditionals”. Finally, in order totest the limits of which kinds of conditionals can be accounted for on the present
analyses, expansion to the non-standard conditionals with properties differingfrom better-studied ones, as well as sentences “expressing conditional thoughts”,
but without conditional form, as discussed in Elder and Jaszczolt (2016) [6], isan interesting perspective.

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References
1. Austin, J.: Ifs and Cans. In: Philosophical Papers. Oxford University Press (1979)
2. Van der Auwera, J.: Pragmatics in the last quarter century: The case of conditional
perfection. Journal of Pragmatics 27(3), 261–274 (1997)
3. Condoravdi, C., Lauer, S.: Anankastic conditionals are just conditionals. Semantics
and Pragmatics (2015)
4. Csipak, E.: Free factive subjunctives in German — Ich hätte da eine Analyse.
Ph.D. thesis, Georg-August-Universität Göttingen (2015)
5. DeRose, K., Grandy, R.E.: Conditional assertions and “biscuit” conditionals. Noûs33(3), 405–420 (1999)
6. Elder, C.H., Jaszczolt, K.M.: Towards a pragmatic category of conditionals. Journal of Pragmatics 98, 36–53 (jun 2016)
7. Francez, I.: Chimerical conditionals. Semantics and Pragmatics 8, 1–35 (2015)
8. Frank, A.: Context Dependence in Modal Constructions. Ph.D. thesis, Universität
Stuttgart (1996)
9. Franke, M.: The pragmatics of biscuit conditionals. 16th Amsterdam colloquium(2007)
10. Grice, H.P.: Logic and conversation. In: Syntax and Semantics, Vol. 3, SpeechActs, pp. 41–58. Academic Press, New York (1975)
11. Gutzmann, D.: Use-conditional meaning: Studies in multidimensional semantics.
Oxford Unviersity Press (2015)
12. Kaufmann, S., Schwager, M.: A unified analysis of conditional imperatives. In:
Semantics and Linguistic Theory. vol. 19, pp. 239–256 (2009)
13. Kratzer, A.: Modals and Conditionals. Oxford University Press (2012)
14. McCready, E.: Reliability in pragmatics. Oxford University Press (2015)15. Potts, C.: The Logic of Conversational Implicatures. Oxford University Press(2005)
16. Siegel, M.E.A.: Biscuit conditionals: Quantification over potential literal acts. Linguistics and Philosophy 29(2), 167–203 (apr 2006)
17. Stalnaker, R.C.: A theory of conditionals. In: Rescher, N. (ed.) Studies in Logical
Theory, pp. 41–55. Blackwell (1968)- 52 -

Negotiating Epistemic Authority?
Eric McCready and Grégoire Winterstein
1

Department of English
Aoyama Gakuin University
mccready@cl.aoyama.ac.jp
2
Department of Linguistics and Modern Language Studies
The Education University of Hong Kong
gregoire@eduhk.hk

Abstract. Why do we trust what other people say, and form beliefs on the basisof their speech? One answer: they are taken to have epistemic authority. Intuitively this means that the other person (or institution, or group) is taken to beauthoritative in what they say, at least with respect to a particular domain. Here,we want to claim that there are (at least) two varieties of epistemic authority, one
based on reliability and one on assuming (nonepistemic) authority. We claim thatboth are subject to linguistic negotiation. This paper begins by reviewing McCready’s (2015) theory of reliability, and then turns to strategies for attempting
to assume epistemic authority, focusing on those involving the use of not-at-issue
content. We then show the results of two experiments which test the interactionof stereotypes about gender with epistemic authority, and how this is mediated bylanguage use, focusing on the case of gendered pronouns. Finally, the results are
explored for Bayesian views of argumentation and analyzed within McCready’s
Reliability Dynamic Logic.

1

Introduction

Why do we trust what other people say, and form beliefs on the basis of their speech?
One answer: they are taken to have epistemic authority. Intuitively this means that theother person (or institution, or group) is taken to be authoritative in what they say, atleast with respect to a particular domain. Here, we want to claim that there are (atleast) two varieties of epistemic authority, one based on reliability and one on assuming (nonepistemic) authority. We claim that both are subject to linguistic negotiation.
This paper begins by reviewing McCready’s (2015) theory of reliability, and then turns
to strategies for attempting to assume epistemic authority, focusing on those involving
the use of not-at-issue content. We then show the results of two experiments which test
the interaction of stereotypes about gender with epistemic authority, and how this ismediated by language use, focusing on the case of gendered pronouns. The first experiment concerns English and the second Cantonese. Finally, the results are explored?

The authors would like to acknowledge the support of JSPS Kiban C Grant #16K02640, which
partially supported this research, and to thank Heather Burnett, Christopher Davis, Michael Erlewine, Regine Lai, Zoe Luk, Lukas Rieser, Henriette de Swart, and audiences at the Education
University of Hong Kong and Paris 7.

- 53 -

for Bayesian views of argumentation and an analysis is sketched within McCready’s
Reliability Dynamic Logic.

2

Passive assumption of authority

One way to be authoritative, in the sense of having one’s speech consistently believed,
is to be a speaker who is judged reliable with respect to speaking truth. If one is judged
reliable, one is likely to have a kind of epistemic authority, in the sense that the things
one says are likely to be believed. Here, reputation is key given that belief is a formof cooperation; it is known, for the game-theoretic case. that the use of reputation in
strategizing in repeated Prisoner’s Dilemma (18; 19) yields extremely good results, andis therefore likely to be evolutionarily stable.
One way to model reputation with respect to reliability is given by (13), which wewill briefly summarize. On this theory, reputations can be derived in part from histories,
defined as sequences of objects act ∈ A, A the set of possible actions for a given agent
in a given (repeated) game. These objects are records of an agent’s actions in past repetitions of the game. Game histories are n-tuples of sequences of records representing
the history of the agent’s actions at each decision point. For the case of communication,these are of course histories of speech acts. A player’s reputation in a game is derived
from his history in that game. A player’s reputation with respect to some choice as his
propensity, based on past performance, to make a particular move at that point in thegame. Such propensities are computed from frequencies of this or that move in the history. Specifically, the propensity of player a to play a move m in a game g at move i is:the proportion of the total number of game repetitions that the player chose the action
m at choice point i.
F Hag,n (move) =card({act ∈ Hag,n |act = move})card(Hag,n )Always, 0 ≤ F Hag,n (move) ≤ 1, so the above number can be viewed as a probability: in
effect, the information that the game participants have about a’s likelihood of choosing
move m.
An agent’s propensity to play a strategy is a real number in [0, 1]. This fact supports
a scalar view of propensities, and indeed of cooperation itself: an agent has a propensity
for using strategy σ iff i.e. the contextual standard for having that propensity (9). Thus,
Prop(a, σ) iff F Hag σ  s,where s is the contextual standard for propensity-having. These propensities can also
be used to decide whether to assign someone epistemic authority with respect to someclaim. In the context of the repeated PD, (18; 19) make use of reputations and find that
there are optimal strategies, given an index of reliability (here in [0,1], but for them in
the range 1–5), is to trust if, for example, a has a propensity for reliability (where this forthem amounts to setting some arbitrary number above which cooperation is dictated), orP
if Coop(σ) F Hag σ is above some threshold (not necessarily s) (for the sum of frequencies
of all a’s cooperative strategies), or if the other agent’s reliabilty index is higher than

- 54 -

yours. Since such strategies are public, the other agent has an incentive to maintain her
R-rating high: i.e. to genuinely be reliable. Any of the above seem reasonable bases tochoose to accept someone’s epistemic authority, or not.
The above must be combined with other information about reliability. This is so
because of the need to decide whether to give someone epistemic authority even in thefirst communication, before any kind of history is available. This decision corresponds
closely to the distinction between Humean and Reidian views on trust in testimony (13;15). One way to model the Reidian view, on which decisions about trust aren’t madeautomatically but rather on the basis of some metric, is that of (4), who takes speakers tomake judgements about people’s epistemic authority based on stereotypical information
about factors like their gender, race, occupation, and personal grooming. This seems
sensible: one might be more likely to believe a clean-shaven man in a suit about his
having had his wallet stolen and needing money for the train than the same statement
made by a homeless woman carrying a bottle in a brown paper bag (depending of course
on one’s other beliefs). This heuristic gives a first guess about reliability which can thenbe modified by interaction.
All this can be embedded in a more general model of information change; (13)proposes a new flavor of dynamic semantics for this purpose (6). The basic idea is
to virtually always update with content acquired from any source, but only ‘conditionally.’ To make this work, information states σ are complex and consist of possibly many
substates. Each IS is a set of worlds (simplification), ordered with a ‘plausibility ranking’ reflecting epistemic preferences on states. Each substate is indexed by an indexj ∈ Source ∪ A. Here Source is the set of evidence sources and A the set of agents,
which is constrained to only hold indices which the epistemic agent has had experience
with. This set is ordered by a total ordering a , where i ≺ j iff P(Rel(i)) < P(Rel( j)),when P(Rel(i)) is the probability that source i yields reliable information.
Updates are of the form Ei ϕ, for Ei an operator indicating source in i-type evidence.
A sentence Ei ϕ always induces update of state σi . Some cases are indeterminate cases,
such as the use of direct evidentials in some languages that have them, where it may not
be clear what the source is: visual, auditory, . . . In such cases, all possible substates are
updated. But in the testimonial case, states indexed with agentive sources a are updated.
So, at the level of substates, update with ϕ always takes place when ϕ is observed — but
this is not the same as coming to believe ϕ at a global level. Global beliefs are defined on
the global state σT resulting from unifying all substates σi . This unification is done via amerge operation (e): all substate content survives when non-contradictory, but in case of
conflict, information from higher-ranked sources trumps lower-ranked source-indexed
information. Thus the global state almost never exhibits conflicts; it only will if twosources are precisely equally ranked, which is unlikely given the range of real values,
but can be explicitly banned by enforcing a version of Lewis’s Limit Assumption, here
for sources rather than worlds (10).
More formally, global information states σ: consist of sets of elements (substates)of the form σi = hX, ≤a i where X ⊆ W (the set of states). The substates are plausibility frames in the sense of (1; 2): multi-agent Kripke frames hX, Ra ia∈A , where theaccessibility relations Ra are called ‘plausibility orders’, written ≤a , and assumed to belocally connected preorders. This simplifies a bit: sometimes the substates can be more

- 55 -

complex, in particular in the case of testimonial agents, as the substates associated withthem also have a similar structure. Total information states are written σT , and are of
the form hX, ≤a i for X ∈ ℘(W). They are derived by recursively merging all plausibility
relations found in σi ∈ σ via a lexicographic merge operation, which respects priority
ordering; so an agent’s beliefs thus are derived on the basis of the most reliable source,and so on down the source hierarchy. From this, we get resolution in cases of conflicting
sources.
Update in this system follows the [.]⇑ of (1; 2), defined as follows.
– σ[ϕ]⇑ = σ0 , where S 0 = S and s ≤0a t iff either (i) s < ϕ and t ∈ s(a) ∩ ϕ, or (ii)s ≤a t.
This definition thus leaves the set of states the same, but upgrades those states which satisfy ϕ above those which don’t, otherwise leaving the relative plausibilities untouched.
Using this operation ensures that substates will be comparable without recourse to revision.
Support and entailment are defined as follows. A total information state hX, ≤a i
is said to support a proposition ϕ, σ |= ϕ, iff {s ∈ X|s ∈ besta (s(a))} ⊆ φ, where
besta φ := {s ∈ φ|t ≤a s for all t ∈ φ}.3 The definition of entailment is the standard
fixed-point dynamic one modulo the use of [.]⇑ , as defined above (with ‘;’ dynamic
conjunction as usual):
φ1 , . . . , φn |=σ ψ iff σ[φ1 ]; . . . ; [φn ] = σ[φ1 ]; . . . ; [φn ]; [ψ].
Evidential update is defined via the following clause, which ensures that only thesubstate corresponding to the information source is updated, and all others are left alone.

σ0j = σ j [ϕ] if i = j
0
1. σ[Ei ϕ] = σ where, for all σ j ∈ σ, 
σ0 = σ jif i , jj
For an example, suppose agent a learns ϕ =‘It is raining’ from evidence source b (agentb). Then: σ0 = σ except that σ0b ∈ σ0 = σb [ϕ], by the definition of evidential update.
Thus: in all cases, the result of evidential update with ϕ is belief in ϕ. But this belief
may just be belief relative to the source, i.e. within σi for source i. ‘Genuine’ belief requires global belief wrt the global state. Essentially: Ba ϕ iff {s ∈ σT |s ∈ besta (s(a))} ⊆φ, where besta φ := {s ∈ φ|t ≤a s for all t ∈ φ}. The total belief state is derived
by lexicographic merge, so the content of our examples will be believed unless somehigher-ranked source disagrees. What happens when a conflict arises? Consider a caseof conflicting agents. Agent a claims φ and agent b claims ¬φ. a, let’s suppose, is pretty
trustworthy. b is unknown; let’s suppose that he looks somewhat untrustworthy. The
result is that a  b in the priority ordering for lexicographic merge. Thus the merge of
σa and σb verifies φ.
So far: update of substates, substates unified via merge, merge priority determinedby ordering. But what’s the source of the ordering? Without a substantive theory of how
the ordering is derived, the theory seems to have little empirical content. The claim of3

Note that this is essentially identical to the definition of belief in (??).

- 56 -

(13) is that the ordering is probability-based. The probabilities in question are probabilities of reliability. They indicate the (perceived) likelihood that information derived
from the source is correct.
These probabilities arise from two factors. The first factor is experience with reliability of the source, as derived from histories; the second is the initial probabilities
of reliability. These come in two types: prior beliefs about the reliability of different
evidence sources, and beliefs about the reliability of the providers of testimony basedon various aspects of their presentation. For an example of the first, one generally cantake direct evidence to be more reliable than hearsay: if I see that it’s raining outside, I
am likely to discount the fact that this morning’s weather report said it would be sunny.
For the second, as mentioned above, judgements about the reliability of individuals areoften made on the basis of stereotypical factors about their appearance and how they
are categorized (4). One might judge the kempt to be more reliable than the unkempt,
the professional to be more reliable than the amateur, or someone from the same social group as you to be more reliable than someone from an outgroup. As we’ll see in
the next section, these kinds of judgements can be manipulated, yielding effects on theattribution of epistemic authority.
The two factors are taken to interact as follows: given an initial probability and asequence of events of information acquisition, conditionalize on the initial probability
for each new acquisition event, with respect to truth-tracking. The idea is to modify theprobability that the source is reliable based on whether the new information is correct
or not:PI (R ∩ C)
PI (C)
The whole notion of authoritativeness analyzed here is (in a sense) a passive one.
One becomes authoritative by speaking the truth and by looking reasonably trustworthy.
This is a kind of authority acquired by being a good citizen in the testimonial sense,
essentially that of (5). But is there a more active way to acquire epistemic authority by
linguistic means? We think yes: by use of argumentative and other linguistic devices.
Some of these will be explored in the next section.

3

Using expressive content for authority negotiation

How can one actively try to acquire epistemic authority (or deny it to others), as opposed
to simply acquiring it by living a virtuous testimonial life? One way, of course, is justto assert one’s authority:(1) (You should believe me because) . . .
a. I know all about this topic.
b. I’m your teacher.
c. I’m your dad.
This strategy will be effective to precisely the degree that the speaker already has epistemic authority, because in the absence of epistemic authority, either the hearer won’t
accept what is said (1a), or the speaker’s external authority is already rejected (1b,c).
Consequently, a less direct strategy (or set of strategies) is needed. In the remainder- 57 -

of this section, we examine the use of expressive content (21) in the assumption of
epistemic authority, considering several cases.
We are choosing to focus on expressive content for two reasons. Expressive content is often talked about as ‘inflicted’ on the hearer (21; 16), which means (if correct)
that the content of the expressive cannot easily be contested. This is an important feature when it comes to manipulations of epistemic authority (and in argumentation in
general), as it removes the need to have epistemic authority already in order to have
one’s claims accepted, as with (1) above. This feature is not universally present in notat-issue content either; (26) notes that presuppositions for example can be challenged
in discourse, meaning that their content lacks the key feature of expressives we areinterested in here. The second reason is the close connection of many expressives tosocial meanings, which are obviously relevant for epistemic authority. This point will
be detailed as we proceed.
In this section, we will briefly consider the cases of particles, honorifics, and, finally,
our main concern, those expressives which serve to indicate membership in various
social groups.
First, particles like the Japanese yo (with falling intonation) work to try to ‘force’ thehearer to accept the content of the sentence (11; 3). Indeed, (17) presents an analysisof this particle in terms of epistemic authority. His idea is that yo indicates that the
speaker has at least as much epistemic authority as anyone else present with respect tothe content of the sentence. This implies that the particle can be used strategically to try
to claim such epistemic authority for the speaker; use of the particle (if unchallenged)
indicates that the speaker already has epistemic authority.
This view has some empirical effects. In the following example, the speaker requests
belief via the claim of teacherhood.
(2) watashi-wa
anata-nosensei desuyo
1P.Formal-Top 2P.Formal-Gen teacher Cop.Hon PT‘I am your teacher, don’t forget.’However, the use of strengthening yo implicates that the speaker doesn’t have authority
already, which further implies that the speaker takes his epistemic authority qua teacher
to be insufficient, resulting in a failed authority grab. Compare here the observation of(25) that falling yo infelicitous in e.g. instructions from a commanding officer in thearmy, because the attempt at claiming authority represented by yo (in the terms of thispaper) is not compatible with the presence of absolute authority.
The second case is honorifics, which, although they on a separate dimension from
epistemic claims (at least according to (8; 22; 12; 14), and others), to the extent thatone’s social status influences her epistemic authority the use of (anti-)honorifics shouldcount as a strategy for assuming it, or taking it from others. Notably: ‘raising’ the addressee could cede some epistemic authority to them. In terms of examples, while thefollowing are both grammatical and felicitous, there is a sad mismatch between content,
honorific tone and particle: it’s as if the speaker is desperately trying to assert himself.
This is unlikely to yield genuine epistemic authority.
(3) watashi-no
itteiru koto-o shinjite kudasai yo
1P.Formal-Gen saying thing believe please.Pol PT- 58 -

‘Believe what I’m saying, please.’vs the pure authority grab:(4) ore-no
itteru koto-o shinjiro
1P.Inf-Gen saying thing believe-Imp‘Believe what I’m saying!’
Finally, many expressives tag aspects of character which can be relevant to determinations of epistemic authority via social status; we can call these social expressives.
This strategy is less direct than the above in that it is entirely a side effect. The main
method here is to ascribe other individuals membership in groups which are or are notprivileged in a social sense, and use that (lack of) privilege to implicate something about
their epistemic authority. The same is true for slurs: by placing the addressee or otherindividual in a subordinate group, explicitly or implicitly (cf. (24)), it becomes possible
to emphasize one’s own epistemic authority over them. It is widely noted in the feminist philosophy literature (and elsewhere on the internet etc.) that the overt or covert
primary position of males in society, and their consequent authority, can lead to differences in epistemic authority as well. For instance, the claims of men are often believed
over the claims of women, all else being equal. If this is true, the use of e.g. gendered
2P pronouns in situations where other options are available (cf. (23)) could lead to thechanges in who is taken to have epistemic authority, meaning that the use of genderedlanguage can be a strategy for its assumption.
Here, we are interested in testimony: the main question in ceding epistemic authority involves how one should assign probabilities of likely reliability to individuals.
As mentioned above, (4) cites one technique, which is to make use of stereotypes
about groups, for example that ‘women are not logical’, ‘Asians are well educated’,and so on; she presents some compelling examples of such cases, though examples
which operate at the level of at-issue claims rather than expressive implications. However, many expressives tag aspects of character which can be relevant to determinationsof epistemic authority via social status. We can call these social expressives; they aremainly terms which categorize individuals into categories that — at least on a stereotypical or prejudicial level — are relevant to the (non)attribution of epistemic authority.
The basic method is to ascribe other individuals membership in groups which are associated with some stereotype, and then use that (lack of) privilege to implicate something
about their epistemic authority.
Two examples of social expressives are slurs and gendered language. By definition,slurs are negative and subordinating (cf. (24)), so can be used to emphasize one’s own
epistemic authority over categorized individual, given that other relevant individuals
share the prejudices the slurs express. With gendered language, the situation is more
subtle, because gender is not in any sense pejorative in the way of slurs. Still, the deployment of stereotypes about gender to acquire epistemic authority. It is a truism (anda common claim in feminist philosophy as well (4)) that the overt or covert primary
position of males in society, and their consequent authority, can lead to differences in
epistemic authority as well. For example, it is often said that the claims of men areoften believed over the claims of women, all else being equal. If this is true, the useof e.g. gendered 3P pronouns could easily lead to the changes in who is taken to have

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epistemic authority, meaning that the use of gendered language can be a strategy for its
assumption. In order to see whether this is correct, we conducted several experiments,
focusing on the use of gender stereotypes in argumentation.

4

Experiments: gender in argumentation

4.1

Experiment 1: English

We ran an experiment to test the relation between gendered speech and epistemic authority in argumentation. We tested two different types of argument which involve theauthority of a source: the direct, or abusive, form of the ad hominem argument and theargument from authority (or position to know). Schematically these arguments are as
follows (27):– Ad-hominem:• Source a is a person of bad character / has bad character for veracity• a argues that α• Conclusion: α should not be accepted– Argument from authority (position to know):• Source a is in a position to know about things in a certain subject domain S
containing proposition A• a asserts that A is true
• Conclusion: A is true
In each case, the source a is part of one of the premises of the argument.
The goal of the experiment was to test whether manipulating the gender of thesource induces a difference in the convincingness of the argument. To test this a protocol
similar to the one used by (7) to investigate the argument from authority was used.
First, a preliminary experiment was run to determine three distinct sets of topics
according to their gender bias. This was done as a categorization task on Amazon Mechanical Turk. Participants were presented with a topic and asked to choose which category most closely matched that topic: Men, Women or Both. 17 topics in total were
tested, out of which 15 were selected, 5 in each group. Each topic had an agreementof 80% or above, meaning that four participants agreed the topic was associated withthe relevant category. Participants could categorize multiple topics and were paid 0.05
USD for each categorized topic.
These topics were then used to produce 15 distinct arguments, in two major forms:
the ad hominem one, and the argument from authority one. Examples of each form
follow (using a male biased topic):– Authority argument• A and B are friends. A wants to buy a power drill and is thinking about which
one to buy. A wants a high performance drill to perform heavy duty work.
• A: I wonder if this one is a good choice.
• B: I have a friend who says he knows a lot about power tools, and he says thismodel is really powerful.
– Ad hominem- 60 -

• A and B are friends. A wants to buy a power drill and is thinking about which
one to buy. A wants a high performance drill to perform heavy duty work.
• A: I heard from Jamie that this model is really powerful.
• B: She doesn’t know anything about it.
Besides the argument scheme, two independent variables were tested: the gender of thesource (by using he, she or that friend/Jamie to refer to it) and the gender bias of the
topic, based on the results of the preliminary experiments. 450 US-based participants
were recruited on the Amazon Mechanical Turk and paid 0.2 USD for their participation. They judged the convincingness of 5 different arguments (4 fillers+1 target item)
presented in pseudo-random order. Convincingness was rated on a 5 point Likert scale.
Linear mixed effect models with maximal random effect structure were fitted to thedata using the lmer package in R. Effects of condition and group were confirmed bylikelihood-ratio tests.
The results show that generally, authority arguments are judged more convincing
than ad hominem (χ2 = 145.38, p < 0.01) and that the gender of the source and thegender bias of the topic have no main effect. Further analyses showed that these variables have no effect in the case of the ad hominem argument. However, the results of theargument from authority show that there is a significant interaction between the genderof the source and the gender bias of the topic (χ2 = 11.023, p = 0.026). It was observedthat men are generally more trusted for topics biased towards men (which is expected)
but that women are not more trusted than men for topics biased towards women, andthat there was no preference for neutral topics.
Discussion To explain why authority arguments are preferred to ad hominem ones,
we argue that when considering authority arguments the only question is how reliable
the source of the argument is. The reliability of the speaker is not directly relevant. Thiscan readily be integrated in an approach like that of (author?) (7, 20) who propose a
Bayesian treatment of argumentation. In that approach, the convincingness of an argument is proportional to how much the content of the argument affects the audience’s
prior belief in the conclusion targeted by the argument. The reliability of the source isfactored in the likelihood of using an argument a to target a conclusion C. There thespeaker’s reliability remains constant across possible sources and does not weigh in on
the evaluation of the argument.
However in the case of the ad hominem argument the speaker’s reliability is at odds
with that of the source, which might explain why those arguments are generally dispreffered since they put the speaker’s credibility against that of the source. As statedabove, gender biases can integrated into the Bayesian approach of argumentation. This
amounts to modifying the belief that source is reliable by conditionalizing on its gender.
However in the Bayesian approach the ad hominem and authority arguments are seen
as dual to each other: one lowers the reliability of the source while the other increases
it. As such, it should be expected that both forms would equally be affected by genderbiases, contra the results of our experiment. One way to model that difference would
thus be to explicitly distinguish between the reliability of the speaker and that of thesource of an information in the way an argument is evaluated, and the fact that they
may potentially be at odds. The approach lends itself to such a modification, but fur-

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ther experimentations are needed to validate whether this move is an effective way to
account for the data presented here.
4.2

Experiment 2: Chinese

A second experiment, similar to the one presented above, was run using material in
Chinese rather than English. Mandarin Chinese does not have gendered pronouns, but
in its written form uses characters which distinguish between male and female referents
for third person pronouns.Cantonese does not have gendered pronouns either, and thecharacter commonly used to transcribe the Cantonese third person pronoun is not gendered either. The goal of the experiment was to check whether the use of a genderedcharacter or a neutral one affected the way subjects evaluated the convincingness of anargument.
The overall protocol of the experiment was similar to the one described above, except that the participants were speakers of Cantonese and were recruited among thestudents of the Education University of Hong Kong and participated voluntarily. A firstcategorization task was run to identify the gender biases of different topics. Nine topics(3 male oriented, 3 female oriented and 3 neutral) were retained to be used in an experiment testing the convincingness of an argument. Speakers of Cantonese are able toread written Chinese fluently and feedback from some of the participants confirmed thatthey did not realize that some of the experimental material was written using Cantonese
specific characters or Mandarin Chinese specific ones.
88 participants judged the convincingness of 9 pseudo-randomized different arguments (6 fillers+3 target items) on a 5 point Likert scale. Since gendered effects were
only observed on authority arguments in the English experiments, only authority arguments were used in the Chinese experiment. An online questionnaire was hosted on the
IbexFarm platform and the link sent to participants. Linear mixed effect models withmaximal random effect structure were fitted to the data using the lmer package in R.
Effects of condition and group were confirmed by likelihood-ratio tests.
The analysis of the results shows no significant effect, either of the gender biasof the topics, the gender of the source or of their interaction. This suggests that, if
participants have gender biases, the use of gendered character did not trigger theses
biases when participants evaluated the arguments. Further investigations will be madeby changing the expression referring to the source by one that is explicitly gendered(e.g. my sister/daughter etc.) in order to see whether such expressions make biases
appear in a way that is comparable with the English experiments.

5

Conclusion

This paper has reviewed the analysis of reliability of (13) — a combination of stereotypebased probability ascriptions and examination of communicative histories — and proposed it as one means of acquiring epistemic authority. The other method is more proactive: to manipulate stereotypes and other aspects of the context via the deployment ofexpressive content. We looked at one such instance in detail via experimental methods:- 62 -

the use of gendered pronouns to influence judgements about reliability, both in English
and Cantonese. The results are intriguing, but still preliminary.
We propose to model these phenomena in the RDL logic proposed by McCready(2015). In that model, individuals are assigned credibilities on the basis of stereotypes(among other things). We propose to extend the model by allowing individuals to havecredibilities with regard to specific content, typically individuals of gender g will havea high credibility for g-relevant topics. But before fully formalizing the details, a better
understanding of the phenomena is however needed. The case of men being more credible for men-biased topics comes out naturally from the gender-based credibility model
proposed above. However, the fact that the converse is not true for women is more
problematic. One way to model this is to consider that, by default, men enjoy a higher
credibility than women, which then gets cancelled in the case of women-biased topics.
That possibility will be investigated in the near future via an additional experiment, andthe basic data will be verified using a larger population of subjects as well.

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Bibliography[1] Baltag, A., Smets, S.: A qualitative theory of dynamic belief revision. In: Bonanno, G., van der Hoek, W., Wooldridge, M. (eds.) Logic and the Foundationsof Game and Decision Theory, pp. 13–60. No. 3 in Texts in Logic and Games,
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Cossentino, M., Dastani, M., Dunin-Keplicz, B., Fortino, G., Gleizes, M.P., Leite,J., Mascardi, V., Padget, J.A., Pavón, J., Polleres, A., Fallah-Seghrouchni, A.E.,
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513–521. Chulalongkorn University (2014)[13] McCready, E.: Reliability in Pragmatics. Oxford University Press (2015)[14] McCready, E.: The semantics and pragmatics of honorification (2015),
manuscript, AGU[15] McCready, E.: Rational belief and evidence-based update. In: Hung, T.W., Lane,
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Lessons Learned from a Prototype Implementationof Montagovian Lexical Semantics
Bruno Mery
bruno.mery@u-bordeaux.fr
LaBRI & Université de Bordeaux
Bordeaux, France.

Abstract. We present a general-purpose implementation of the process of lexical semantics analysis theorised in the Montagovian Generative Lexicon Λ TYn(hereafter MGL). The prototype software itself serves as a proof of concept of
the MGL theory. The implementation process, the data structures and algorithms,
also provide valuable results as to the expressive power required by MGL. Whilethe implementation of terms and types for the purpose of meaning assembly assumed by MGL is in itself straightforward, some lexical phenomena imply additional mechanisms in order to process the logical representation using implicit
knowledge. We therefore also present a minimal architecture for knowledge representation, and how it can be applied to different phenomena.

Keywords: Lexical Semantics, Prototype Software, Montagovian Generative Lexicon,
Knowledge Representation for Natural Language Semantics.

1

Theories and Implementations of Lexical Semantics

Formal lexical semantics theories aim to integrate to the toolbox of compositional
analysis of natural language developed since Montague considerations of (logical) polysemy. Based on original studies such as [4,8], then on a theory thoroughly developedin [22], there have been many formulations that build upon powerful type-theoretic
foundations, with a generative, dynamic account of the lexicon at their heart. Such recent type-theoretic accounts of lexical meaning include Type Composition Logic (TCL)
presented in [1], Dynamic Type Semantics (DTS) presented in [3], Type Theory with
Records (TTR) presented in [7], Unified Type Theory (UTT) presented in [12], and theMontagovian Generative Lexicon (MGL) presented in [24].
Several partial or complete implementations of those theories have been provided
for demonstration purposes, using logical or functional programming, or theorem
provers such as Coq ([6] is an example among many). Concerning MGL, however, oneof the stated goals was (paraphrasing slightly [24]) to provide an integrated treatment
from syntax to semantics extending existing analysers based on Montagovian semantics such as [19] with mechanisms for lexical semantics that are easily implemented in a
typed functional programming language like Haskell. Our goal in this publication is topresent an actual prototype implementation (using functional and object programmingin Scala) of the lexical semantics of that framework.

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We detail some of the necessary data structures and algorithms used, what welearned from this implementation on the underlying logic properties of MGL, andsketch an architecture for simple knowledge representation that is necessary for the
representation of certain lexical phenomena. The demonstrably functioning prototype
illustrates both the validity of type-theoretic formulations of lexical meaning, and the
deep interaction of lexical meaning with at least some sort of knowledge representation
already evoked in [5].

2

A MGL Prototype

2.1

The Montagovian Generative Lexicon

MGL makes use of Λ TYn (an adaptation of the many-sorted logic TYn proposedin [21] in second-order λ -calculus, given in the syntax of System-F). The idea is toperform an usual Montague analysis (performing syntax analysis via proof-search andsubstituting semantic main λ -terms to syntactic categories). Lexical mechanisms arethen implemented in the meaning-assembly phase via a rich system of types based on
ontologically different sorts and optional λ -terms that model lexical adaptation. The
mechanisms, given in Fig. 1, can be summarised as follows:– First, the input utterance is super-tagged and analysed using categorial-grammar
mechanisms, which is the only step of proof-search of the process, yielding a syntactic term whose components are syntactic categories. The lexicon is then usedin standard Montagovian fashion to substitute λ -terms, yielding a main semantic
term, typed with many sorts and the type t for propositions.
– Second, as a many-sorted logic is used, some type mismatches might (and should)
occur, allowing mechanisms of lexical semantics to disambiguate between terms.
The lexicon provides optional λ -terms that are used as lexical transformations.
These optional terms are inserted depending on their typing and yield a λ -termwith no type mismatches.
– Finally, β -reduction yields a normal, η-long λ -term of type t (the type of propositions), i. e. a logical formula that can be used in any usual semantics, such asmodel-theoretic or game-theoretic semantics.
As the first step is already well-studied and implemented, the object of concern is thesecond step: given a term reflecting the syntactic structure of an utterance, to construct
a semantic λ -term in a many-sorted logic, making use of available transformations, andyielding a suitable formula. This is the object of our prototype implementation.
2.2

Modelling Types and Terms

The data structures and algorithms responsible for implementing the terms and typesof Λ TYn are the core mechanisms of the software. They are given as two Scala sealed
abstract classes, TermW and TypeW, with a flat hierarchy of case classes implementing the various possible terms and types; this simple categorisation allows us to easily
construct and detect patterns of objects.

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Fig. 1. MGL Process Summary

Terms and types are constructed as binary trees (abstractions and applications ofmore than one argument to a given term/type can be easily curryied).

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For terms, leaves are AtomicTerms (constants), TermVarIds (variables) with an
identifier and type, or specific Transformations and Slots, while inner nodes
are TermBindings (λ -abstracted terms), or TermApplications of a predicate
and argument. For types, leaves are constant Sorts, pre-defined objects such as
PropositionType for t, or second-order variable identifiers TypeVarIds, while
nodes are TypeFunctions between two types A and B modelling A → B, or
TypeApplications modelling A {B}.
A simplified UML class diagram presents this straightforward architecture in Fig. 2.

Fig. 2. Class diagram of the core package for terms and types.

Several algorithms are provided in order to work with types and terms; they aremostly simple recursive tree-walking algorithms, making the most of memoisation
when possible (e. g., lists of available ressources are incrementally built as terms andtypes are constructed in order to minimise computations). Algorithms include the typechecking of applications, comparison between types, automated α-conversion of variables in order to prevent issues of scope, replacement of term and type variables, β reduction, and the automated specialisation of types for polymorphic terms (i. e., apredicate with a type containing one or several type variables will be specialised to thecorrect types if applied to an argument with a compatible but specified type).

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Most are linear in complexity (with exceptions in the adaptative mechanisms below); all algorithms are at most polynomial in time.
2.3

Explicit Adaptation

The core of MGL is to provide transformations as optional terms, on top of themain λ -term associated to each lexeme. A canonical example is the book is heavy andinteresting. Supposing three basic sorts:– R for readable materials,– ϕ for physical objects,– I for informational contents;
the book can be modelled as the bookR , heavy as heavyϕ→t , interesting as
interestingI→t . The example utterance is a case of co-predication, as two predicates are simultaneously asserted on two different facets (with different, incompatible sorts) of a same object, and MGL will resolve this by having
the lexicon provide two optional terms associated with book in order to acR→ϕΠ =cess these two facets: f phys and finR→I
f o . A polymorphic conjunction &Λ αΛ β λ Pα→t λ Qβ →tΛ ξ λ xξ λ f ξ →α λ gξ →β .(andt→t→t (P ( f x))(Q (g x))) is needed
for the co-predication. This yields, after suitable substitutions, application, and reduction, the term (and (heavy( f phys book)) (interesting ( fin f o book))), which is normal andof type t.
In our implementation, there are several important differences with the theory outlined above. As we distinguish between type constants and variables, there is no needto explicitly abstract types. This is because the only second-order operation ever usedin Λ TYn is specialisation (i. e., the replacement – or instantiation – of type variables). Moreover, second-order (type) variables are all introduced by λ -bound firstorder (term) variables.
We also distinguish between term variables that are necessary for the definition ofan abstracted term (such as P, Q and x, the two predicates and the argument of theconjonction above) and adaptation slots, the places where optional λ -terms (such as
f and g above) can be inserted. This is because the optional terms can be provided
by various different mechanisms, and might not be provided at all if the term is wellformed (there is no lexical adaptation taking place in utterances such as heavy andblack rock; in that case, MGL provides an useful, if slightly redundant, id optional
polymorphic term that can be inserted in order to get the identity on any type).
We provide optional terms as Transforms, which are distinguished from otherterms. Each term has a list of available transformations, constructed recursively fromthe leaves (the transformations available to atomic terms should be given in the lexicon). We also distinguish Slots for explicit adaptations; the list of slots is maintained
during the construction of the terms. Our polymorphic and conjunction then becomes:
lambda p^(B->t).lambda q^(G->t).lambda x^A.((And^{(t->(t->t))}(p^(B->t) (f^{(A->B)} x^A))) (q^(G->t) (g^{(A->G)} x^A)))- 70 -

During the attempted resolution of the application of the conjunction to terms forheavy and interesting, the polymorphic and is specialised to sorts representing ϕ andI, and cannot be reduced further with the application of the argument book. A further
algorithm is provided in order to model the choice of transformations, trying to match
all available transformations to the adaptation slots. As all permutations are considered,this is potentially the most costly computation taking place. The result is a list of possible interpretations (given as term applications with slots filled by transformations):
there might be zero, one, or finitely many. A further check on the list of terms obtained
will filter those, if any, with a suitable typing, that will form the desired result(s). In thetests conduced with the input of the example, four interpretations where produced, with
only the correct one of a resolvable type (t):((And^{(t->(t->t))} (heavy^{(P->t)} (morph_R->Phy^{(R->P)}book^{R})))(interesting^{(I->t)} (morph_R->I^{(R->I)} book^{R})))
2.4

Implicit Adaptation

Polymorphic operators such as and, with explicit adaptation slots, are needed forco-predications. However, most lexical adaptations can take place implicitly, simply
by reacting to a type mismatch such as (pA→B aC ) and applying any suitable transformation to resolve the type mismatch. In order to do this automatically, there are
two possibilities to resolve such type mismatches: by adapting the predicate, yielding(( f (A→B)→(C→B) ) p) a), or the argument, resulting in (p ( f C→A a)). There is also a thirdsituation to consider, that of a partial application (λ xA .τ aC ), in which the argument canbe adapted as above, but the typing of the predicate might be as not be determined at
the moment of the adaptation.
A procedure analyses such applications with type mismatches and no explicit adaptation slots, and inserts suitable, automatically generated adaptation slots,
then proceeds as with explicit adaptations. For example, a simple term application such as (P^{(e->t)} a^{A}), with a transformation f_{A->e} available tothe atomic term a yields the straightforward (and only felicitous) interpretation(lambda x^A.(P^{(e->t)} (f_{A->e}^{(A->e)} x^A)) a^{A}), that reduces
to (P^{(e->t)} (f_{A->e}^{(A->e)} a^{A}).
Implicit adaptations are necessarily reduced to those simple cases. Trying to accountautomatically for co-predications would imply to try any possible permutation of types
and transformations at all nodes of a term, which would be exponential in complexity;
thus, the need for explicit operators such as the polymorphic and.
2.5

Lexicalisation

In addition to the core mechanisms, a tecto package provides support for a tectogrammatical/syntactic structure in the form of an unannotated binary tree of lexemes; this serves as a factory for the input of already analysed text, and as a more streamlined
form of output for adapted terms.

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A lexicon package enables the storage of lexical entries that associate lexemes (asstrings) to terms, complete with typing, transformations and ambiguities. Lexica canbe merged, in order to have combine the treatment of different phenomena, treated as
standalone modules, for complex sentences. Lexica also provide automated translations
from a syntactic structure (a tecto term) to a semantic one (a TermW term, initially not
adapted, reduced or even type-checked). Semantic terms can be presented either by astraightforward translation to syntactic terms, or printed to a string in the usual fullyparenthesed prefix notation with apparent typing (as in the examples of this article).
2.6

Phenomena Coverage

Many lexical phenomena discussed in [22,13] can be modelled using the simple
mechanisms of Λ TYn in their prototypal implementation given above; some others require additional mechanisms.
Lexical adaptations, including alternations, meaning transfers, grinding, qualiaexploitation and “Dot-type”-exploitation are all supported by the adaptation mechanisms, as given previously. Simple predications only require to have suitabletransformations available, and to use the implicit adaptation mechanisms ; copredications require explicit adaptation using polymorphic operators. Theoreticalgrounds have been laid in [2,24].
Constraints of application are required in order to perform co-predications correctly.
As explained in [17], the simultaneous reference to different facets of a same entity
can be infelicitous in some circumstances, such as the use of destructive transformations (grinding, packing) or metaphorical use of some words. Thus, the following
co-predications are infelicitous to some degree : *The salmon was lighting-fast anddelicious, ? Birmingham won the championship and was split in the Brexit vote. In
order to block such co-predications, we have proposed to place constraints on transformations in order to block their usage depending on the other transformations thathave been used on the same term. The first version of this system given in, e. g.,[2], distinguishes between flexible (allowing all other facets) and rigid (blocking allother facets) transformations. The latest version, given in [14], proposes a Λ ( TYn ,a system with terms of the linear intuitionistic logic as types, that (among otherthings) allow any arbitrary type-driven predicate to act as a constraint on the use oftransformations.
In this prototype implementation, all transformations come with a member functionthat can be defined as a constraint, and a compatibility check of all transformations
can be performed using every constraint, the default constraint being the boolean
constant true (that simply models flexible transformations). As the constraint caneffectively be any function, the precision is the same as in [14].
Ontological inclusion, called type accommodation in [22] and modelling the lexical
relation of hyponymy, can be supported by tweaking the system of sorts. The theoretical and empirical basis for doing so are discussed in [18], in which we argue thatcoercive sub-typing is an accurate and helpful mechanism for resolving ontologicalinclusion, but no other lexical phenomena.

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In order to support sub-typing, each sort can be defined with an optional parent
sort. A careful review of the typing comparison mechanism will then be enough,together with a rewriting of the equality method for sorts, in order to support subtyping. This is not implemented yet, but does not require (much) additional processing power.
Performative lexical adaptations, such as quantification, Hilbert operators for determiners, and the alternate readings of plurals and mass nouns, are supported as far
as the meaning assembly phase is concerned. However, in order to be useful, thiscategory of lexical phenomena (as well as hypostasis and several others) require
additional mechanisms in order to incorporate the knowledge gathered from theanalysis of the sentence into the logical representation. The basic architecture is
supported, but mechanisms of resolution remain preliminary and will be discussed
next, especially in Section 3.3.

3

Layers of Lexica and Knowledge Representation

3.1

The Additional Layers

Theories of semantics deriving from [22] generally encompass some degree of common sense world knowledge: it is considered known that a committee (and other suchgroup nouns) is made of several people and is a felicitous argument of predicates requiring a plural argument, and that engines are part of cars and thus that predicatessuch as powerful or fuel-guzzling can apply to cars via their constitutive quale. It has
been argued (e. g. in [9]) that such complex knowledge does not belong in a semantic lexicon; we will paraphrase Im and Lee from [10], defining semantics to be themeaning conveyed by an utterance to a competent speaker of the language in itself, excluding, for instance, the specific situation in which the utterance is made, but including
any previous discourse. Thus, the full contents of a given fairy tales should be able tobe described within semantics, while a political essay will probably require additional
knowledge about the position of the author and the specifics of the period of writing.
From our point of view, designing a complete tool for type-theoretic lexical semantics imply the careful definition of various lexica that can convey the necessary,elementary world-knowledge for each word. A lexicon for general use will associate to
all relevant lexemes their semantics (in the form of main and optional λ -terms) as canbe given in a dictionary of a language. However, there are two common cases in which
the general lexicon is not sufficient.
First are the specific lexica: vocabularies relevant only to a community (professional
jargons, local dialects, and other linguistic constructs specific to small groups of people), and/or to a specific literary universe (fairy tales, space opera, mythology, politic
speeches, etc.). Such lexica are activated on an as-needed basis, switched often, and are
more specific than the general-use lexicon.
Lexical semantics also requires a lexicon used for the current enunciation. A competent speaker of any language is able to use generative mechanisms in order to introduce new lexical concepts, either by hypostasis (the use of a new word, the meaning of
which can be inferred from context and morphology), or by creative use (giving a new,contextually evident meaning to an existing word).

- 73 -

In our view, the lexicon of the enunciation starts empty and can be augmented when
the analysis of the discourse encounters words that are not present in the current active
lexica. We think that such mechanisms can enable the learning of lexical semantic data.
In addition to these lexical layers of meaning, we tend to implement different lexical
phenomena using different lexica for simplicity’s sake, and create a merged lexicon
from every relevant one when processing text.
3.2

Individuals, Facts and Contexts

To summarise our argument in Section 3.1 above, in addition to mostly static lexical
data, some sort of knowledge representation is needed to process even simple lexical
phenomena such as collective and distributive readings for plurals. Namely, we need tokeep track of the individuals mentioned in a given discourse, and of the facts asserted ofthose individuals. To be complete, we would also need to keep track of agents, in orderto model dialogues or multiple points of view in which certain agents assert certain
facts. Our implementation prototype currently supports individuals, as atomic terms oftype A (for named entities: human agents, towns. . . ) or A → t (for common nouns, thatcan be resolved to a specific individual of type A by the means of an Hilbert-based
determiner) for any sort A. We also account for facts, as predicates (TermBindings
or atomic terms) of type α → t for any arbitrarily complex type α, that are used in aterm application, and apply to an individual. In the analysis of a term, individuals andtypes are extracted and added to the context of enunciation. The hierarchy of lexical
layers given above can be implemented as a hierarchy of contexts, some containing
initial individuals and facts relevant to each lexicon; in a such complete system, thecontext of the real world would, to resolve the paradox mentioned in [25], include thefact that there is no King of France (and therefore that The king of France is bald, while
grammatical, is not felicitous because there are no qualifying referents for the entities
described, and thus cannot be assigned a truth value). Such contexts are specific objects(aggregating individuals, facts and a related lexicon) in our implementation.
3.3

The Parsing-Knowledge Loop

We use a specific lexicon to list some common semantic terms for quantifiers, counting terms, logical and Hilbert operators (detailed in e. g. [23], more recently in [16]).
Other lexica can make use of these terms in order to construct, for instance, Link-based
semantics for plurals (originally given in [11]), using lexical transformations as suggested by [20] and detailed in [15]. Some functions associated to the logical lexicon
then resolve the operators, given a term and a context. This updated process of analysis
is given in Fig. 3.
To explain what the analysis of plural readings in MGL entail, consider the following example from [15] : Jimi and Dusty met is analysed as |λ ye .(y = j) ∨ (y = d)| >1 ∧ meet(λ ye .(y = j) ∨ (y = d)). One elementary issue is that the predicate met applies
to group individuals (such as a committee) and constructions made of more than one
individuals (such as Jimi and Dusty) but not to singular individuals (such as a student).
Thus, the lexical entry for the predicate is λ Pe→t .|P| > 1 ∧ meet(P) – a logical conjunction with a cardinality operator.

- 74 -

Fig. 3. Parsing-Knowledge Representation Feedback

Those two simple elements can be defined in System-F (the calculus in which Λ TYn ,the logic of MGL, is implemented). The issue is that, in order for our system to infercorrectly that Jimi and Dusty are two different individuals, and thus that the above term
resolves to meet(λ ye .(y = j) ∨ (y = d)), we must use processing power beyond the simple construction and reduction of terms: a minimal system of knowledge representationand logical inference. Within our architecture encompassing individuals and facts, andwith a functional lexicon for logical connectives (including the logical and operator ofthat example), as well as quantification and counting (including the cardinality operator), this example can be treated.
However, this requires a given term to be parsed at least twice: the first time, thesyntactic structure is converted into a semantic term and lexical transformations are applied, the second, facts that emerge from the transformations are added to the lexicon,
and the logical lexicon can be used in order to process the operators that have been introduced. Our prototype implementation does not incorporate such feedback yet, as thefirst step can result in several different interpretations; this remains a work in progress.
As a result, straightforward composition for plurals are tentatively supported (such as
in the previous example), but ambiguous covering readings for plurals are not yet available.
3.4

Hypostasis and Quantificational Puzzles

An enunciation-context lexicon that is filled with individuals and facts inferred fromthe primary semantic analysis can serve, in a limited way, to account for hypostasis.
Words absent from the lexicon, but syntactically placed in the position occupied byindividuals, will be added as primary entities to the lexicon, and their precise typing
inferred from the predicates they are applied to. An elementary mechanism should beenough to have a correct representation from Lewis Carroll’s Jabberwocky. Of course,
most competent human speakers also use morphosyntactic inference to attach at least
some degree of connotative meaning to the words being proposed (e. g., Star Wars’s
plasteel can be inferred as a fictive material somehow combining the characteristics ofplastics and steel by any English speaker).

- 75 -

This is completely beyond the power of our early software. Rather, we can havethe process of meaning assembly outline which lexemes are not in the lexicon, and use
human input for correcting the precise types and terms associated.
The process of counting, quantifying and selecting entities using Hilbert operators
can also shed some light on the quantificational puzzles mentioned in [1] and several
other related works. The issue with having universal quantification used together withco-predication on multi-faceted entities can be seen in examples such as There are fivecopies of War and Peace and a copy of an anthology of Tolstoı̈’s complete works on
the shelf (what is the answer to questions such as How many books. . . ?, and what
exactly is the type of book in such questions ?), or I read, then burnt, every book in
the attic (the entities being predicated form two different sets). In order to resolve such
quantificational puzzles satisfactorily, the methods for counting and quantifying must
be adapted to each predicate, and only apply to individuals of the appropriate type.
For our purpose, this implies a close monitoring of the entities introduced by lexicaltransformations and their context of appearance. This is also a work in progress.

4

Results

4.1

A Fragment of Second-Order

We have proven that MGL can actually be computationally implemented. This was
not really in doubt, but the way that the combination of types and terms are implemented illustrates that the time and space complexity of most of the process is limited:the algorithms used are mostly linear tree walks, with a few quadratic worst-case operations. The most complex step is the choice of optional terms for adaptation slots, ofcomplexity |t| × |s| × n at worst (the product of the number of optional terms available,
adaptation slots, and length of the term); the hypothesis behind MGL is that the numberof available optional terms at any point remains manageable. Thus, the step not actually
implemented in this prototype (but for which many implementations exist), syntactic
analysis, is the costliest of the process detailed in Fig. 1 and the complete process ofparsing is polynomial in time.
MGL accounts such as [24] point out that the whole expressive power of secondorder λ -calculus is not used, and that all could be implemented using first-order terms
if all possible adaptations were listed at each step (which is syntactically much longer
to write). Indeed, our implementation only supports the single second-order operationof type specialisation (by distinguishing type variables from other types and using pattern matching to recognise and rewrite types), which is required for having polymorphic terms. There are no features of Λ TYn that require additional power: sub-typing
can be implemented by an optional parent field in Sorts, arbitrary complex on copredications are supported by including a check on transformations that can be any
arbitrary function, quantification, counting and Hilbert operators can be included. . .
4.2

Minimal Processing Architecture

Our prototype implementation includes the skeleton of an architecture that represents the individuals, facts and agents appearing during the semantic analysis.

- 76 -

This goes beyond the straightforward process of producing a logical representation
for an utterance, as some of the terms of that logical representation might be analysed
differently depending on the context; we argue that that process is still part of a semantic analysis. The individuals, facts and agents are stored in objects called contexts,
organised in a hierarchy that includes the most specific context (modelling the analysisof the current discourse), universe-specific contexts (describing whether the discourse ispart of a fictional, historical or activity-specific setting), dialect- and language-specificcontexts, each associated to an appropriate lexicon. A complete analysis would minimally involve the construction of the logical representation of an utterance, the updateof the enunciation context with individuals and facts introduced by that utterance, and are-interpretation of the logical representation in the active contexts. This minimal processing architecture can be completed with no difficulties; our implementation includes
relevant data structures and algorithms, but requires significant work on examples of
performative lexica in order to be thoroughly tested.
4.3

Perspectives

This prototype implementation has already served its primary purpose: to illustratethat MGL can be computationally implemented, and that the examples usually given
with the theory actually work. As it is, however, this implementation is more of a proof
of concept than useful software.
To be actively used by the community, more work would be required to give it an
helpful interface, both for the user and for existing analysers; we also would like to convert from and to representations of the other most active type-theoretic accounts of lexical semantics. The knowledge-representation architecture remains a work in progress,
and requires solid efforts in order to correspond to our ambitions. However, what MGL
really requires in order to be useful is a large-cover library of types and terms; ourhope is that this prototype will help to build software that can learn those features from
corpora.

References
1. Nicholas Asher. Lexical Meaning in Context: A Web of Words. Cambridge University Press,
March 2011.
2. Christian Bassac, Bruno Mery, and Christian Retoré. Towards a Type-Theoretical Accountof Lexical Semantics. Journal of Language, Logic, and Information, 19(2), 2010.
3. Daisuke Bekki. Dependent Type Semantics: An Introduction. In Zoé Christoff, Paolo
Galeazzi, Nina Gierasimczuk, Alexandru Marcoci, and Sonja Smet, editors, Logic and Interactive RAtionality (LIRa) Yearbook 2012, volume I, pages 277–300. University of Amsterdam, 2014.
4. M. Bierwisch. Wördliche Bedeutung - eine pragmatische Gretchenfrag. In G. Grewendorf,
editor, Sprechakttheorie und Semantik, pages 119–148. Surkamp, Frankfurt, 1979.
5. Peter Bosch. The bermuda triangle: Natural language semantics between linguistics, knowledge representation, and knowledge processing. In Text Understanding in LILOG, Integrating Computational Linguistics and Artificial Intelligence, Final Report on the IBM Germany
LILOG-Project, pages 243–258, London, UK, UK, 1991. Springer-Verlag.

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6. Stergios Chatzikyriakidis and Zhaohui Luo. Natural language inference in coq. J. of Logic,
Lang. and Inf., 23(4):441–480, December 2014.
7. Robin Cooper. Copredication, dynamic generalized quantification and lexical innovation bycoercion. In Fourth International Workshop on Generative Approaches to the Lexicon, 2007.
8. D. A. Cruse. Lexical Semantics. Cambridge, New York, 1986.
9. J. A. Fodor and E. Lepore. The emptiness of the lexicon : Reflections on James Pustejovsky’s
The Generative Lexicon. Liguistic Inquiry, 29(2), 1998.
10. Seohyun Im and Chunngmin Lee. A developed analysis of type coercion based on type theory and conventionality. In Robin Cooper and Christian Retoré, editors, ESSLLI proceedingsof the TYTLES workshop on TYpe Theory and LExical Semantics, Barcelona, August 2015.
11. Godehard Link. The logical analysis of plurals and mass terms: A lattice-theoretic approach.
In P. Portner and B. H. Partee, editors, Formal Semantics - the Essential Readings, pages
127–147. Blackwell, 1983.
12. Zhaohui Luo. Contextual analysis of word meanings in type-theoretical semantics. In Pogodalla and Prost, pages 159–174. 2011.
13. Bruno Mery. Modélisation de la Sémantique Lexicale dans le cadre de la Théorie des Types.
PhD thesis, Université de Bordeaux, July 2011.
14. Bruno Mery. Lexical Semantics with Linear Types. In NLCS ’15, the Third Workshop on
Natural Language and Computer Science, Kyoto, Japan, July 2015.
15. Bruno Mery, Richard Moot, and Christian Retoré. Computing the Semantics of Plurals
and Massive Entities using Many-Sorted Types. In Tsuyoshi Murata, Koji Mineshima, andDaisuke Bekki, editors, New Frontiers in Artificial Intelligence JSAI-isAI 2014 Workshops,
LENLS, JURISIN, and GABA, Kanagawa, Japan, October 27-28, 2014, Revised Selected
Papers, volume 9067 of Lecture Notes in Artificial Intelligence, page 357. Springer-Verlag
Berlin Heidelberg, 2015.
16. Bruno Mery, Richard Moot, and Christian Retoré. Typed Hilbert Operators for the Lexical
Semantics of Singular and Plural Determiner Phrases. In Epsilon 2015 – Hilbert’s Epsilon
and Tau in Logic, Informatics and Linguistics, Montpellier, France, June 2015.
17. Bruno Mery and Christian Retoré. Recent advances in the logical representation of lexical
semantics. In NCLS – Workshop on Natural Language and Computer Science, LiCS 2013,
Tulane University, New Orleans, June 2013.
18. Bruno Mery and Christian Retoré. Are books events ? Ontological Inclusions as Coercive
Sub-Typing, Lexical Transfers as Entailment. In LENLS ’12, in jSAI 2015, Kanagawa, Japan,
November 2015.
19. Richard Moot. Wide-coverage French syntax and semantics using Grail. In Proceedings ofTraitement Automatique des Langues Naturelles (TALN), Montreal, 2010.
20. Richard Moot and Christian Retoré. Second order lambda calculus for meaning assembly:
on the logical syntax of plurals. In Coconat, Tilburg, Netherlands, 2011.
21. Reinhard Muskens. Meaning and Partiality. In Robin Cooper and Maarten de Rijke, editors,
Studies in Logic, Langage and Information. CSLI, 1996.
22. James Pustejovsky. The Generative Lexicon. MIT Press, 1995.
23. Christian Retoré. Sémantique des déterminants dans un cadre richement typé. CoRR,
abs/1302.1422, 2013.
24. Christian Retoré. The Montagovian Generative Lexicon Lambda Tyn : a Type Theoretical
Framework for Natural Language Semantics. In Ralph Matthes and Aleksy Schubert, editors,
19th International Conference on Types for Proofs and Programs (TYPES 2013), volume 26
of Leibniz International Proceedings in Informatics (LIPIcs), pages 202–229, Dagstuhl, Germany, 2014. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik.
25. Bertrand Russell. On denoting. Mind, 14(56):479–493, 1905.

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On the interpretation of
dependent plural anaphora
in a dependently-typed setting
Ribeka Tanaka1 , Koji Mineshima1,2 , and Daisuke Bekki1,2,3
1Ochanomizu University
CREST, Japan Science and Technology Agency
3
National Institute of Advanced Industrial Science and Technology (AIST),The Artificial Intelligence Research Centertanaka.ribeka@is.ocha.ac.jp, mineshima.koji@ocha.ac.jp,
daisuke@is.ocha.ac.jp
2

Abstract. Anaphora resolution may involve a reference to a dependency
relation between objects. One typical example is a dependent interpretation of pronoun it in the mini-discourse Every boy received a present.
They each opened it, which is a well-known phenomenon in the literature of plural anaphora. The standard way to account for dependentinterpretation is to record dependency relations by sets of assignment
functions (van den Berg, 1996a,b; Nouwen, 2003; Brasoveanu, 2008).
This approach, however, has to make substantial changes to the central notion of context in a way that is specialized for the treatment ofdependent interpretation. In this paper, we provide an alternative account from the perspective of dependent type theory. We handle dependency relations in terms of dependent function types (Π-types), which
are independently motivated objects provided in dependent type theory (Martin-Löf, 1984). We will adopt dependent type semantics (Bekki,2014) as a semantic framework and illustrate how dependent function
types encode dependency relations and naturally provide a resource fordependent interpretation.

1

Introduction

Interpretation of pronouns can be sensitive to linguistically introduced dependency relations between objects. Consider the following examples (van den Berg,
1996a,b; Krifka, 1996; Nouwen, 2003; Brasoveanu, 2008)4 :(1)

a. If every1 boy received a2 present, they1 opened it1 .
b. Every1 boy received a2 present. They1 opened it2 .

In (1a), given a reading where every boy receives a wide scope over a present(henceforth, ∀–∃ reading), the whole sentence can mean that if every boy received
4

An anaphor is subscripted by index, while its antecedent is superscripted by thesame index.

- 79 -

a present, each boy opened the present which he received. Similarly, the second
sentence in (1b) can be understood to mean that each boy opened the present hereceived. In both cases, ∀–∃ reading induces a dependency relation between boys
and presents. This quantificational dependency plays a crucial role in interpretingthe singular pronoun it in the consequent of (1a) and the second sentence of (1b).
More generally, the reference to dependency relations is possible when asemantic link between the restrictor of the universal quantifier and the subjectof a subsequent discourse can be established.
(2)
(3)
(4)

a. If every boy receives a1 present, some boy will open it1 .
b. Every boy will receive a1 present. Some boy will open it1 .
a. If every boy receives a1 present, every young boy will open it1 .
b. Every boy will receive a1 present. Every young boy will open it1 .
a. If every boy receives a1 present, John will open it1 .
b. Every boy will receive a1 present. John will open it1 .

In (2), the subject of the subsequent discourse, (some) boy, is the same nounphrase as the restrictor of the universal quantifier. Thus, it can mean that someboy will open a present where he receives (attributed to Lauri Karttunen in
Hintikka and Carlson, 1979; Ranta, 1994). In (3), young boy is a subset of boy.
Again, a similar interpretation is allowed (van den Berg, 1996b). There is no
explicit link in the case of (4), but if we have the background information thatJohn is a boy, i.e., the information that links John to boy, it can mean thatJohn will open the present which he received. Conversely, when it is diﬃcult toestablish such a link, the dependent interpretation of pronoun in question seems
to be inadequate. See the contrast between (5a) and (5b).
(5)

a. Every man will receive a1 present. Some wife will open it1 .
b. Every man will receive a1 present. ∗ Some woman will open it1 .

In (5a), since it is relatively easy to find a relation between man and wife, thesecond sentence can be understood to mean that some man’s wife will open thepresent he received. On the other hand, the similar reading is not possible in (5b)unless a strong relation between man and woman is provided in the context.
Another similar example is an instance of so-called quantificational subordination, which is originally discussed by Karttunen (1976).
(6)

a. Harvey courts a1 girl at every convention. She1 is very pretty.
b. Harvey courts a1 girl at every convention. She1 always comes to thebanquet with him. The1 girl is usually also very pretty.

Although this example is much more complicated than what we had so far, thesimilar structure seems to be involved here. (6a) can only mean that there is
one specific girl such that Harvey courts her at every convention and she isvery pretty. If we discard this reading and force ourselves to keep ∀–∃ reading,there is no way to establish an anaphoric link between a girl and the singular- 80 -

pronoun she. However, she can refer to a girl in each convention in (6b), since
the subsequent discourse contains quantificational adverbs such as always andusually, which provide links to every convention.
The observation above suggests that a dependency relation between objects
should be tracked through a discourse as an anaphoric resource, in order to bereferred to later on to interpret pronouns. Since the reference to dependency
relations is crucially involved in phenomena of plural anaphora in general, constructing a formal mechanism to account for dependencies is one of the central
issues in the dynamic semantics literature. The standard way is to model it assets of assignments (van den Berg, 1996a,b; Nouwen, 2003; Brasoveanu, 2008).
Another proposed approach is to model it by a more complex notion of assignment function called parametrized sum individuals (Krifka, 1996). However,since it is not straightforward to integrate functional relations directly into theunderlying semantics, both approaches have to make substantial changes to thecentral notion of context in a way that is specialized for the treatment of plural
anaphora.
In this paper, we propose an alternative account. We handle dependency relations in terms of dependent function types in dependent type theory. In contrast
to the mechanisms introduced in previous model-theoretic approaches, dependent function types are independently motivated objects that are provided in
dependent type theory from the beginning. We will adopt dependent type semantics (Bekki, 2014, henceforth DTS) as a semantic framework and illustrate
how dependent function types encode dependency relations in question and benaturally provided as anaphoric resources in discourse. In the following section,
we will first provide an overview of DTS. In section 3, we will provide a central
idea of handling the reference to dependency relations and show how it can beapplied to those examples mentioned above. Comparison of our approach withprevious works will be provided in section 4.

2

Dependent type semantics

2.1

Dependent types and natural language sentences

DTS (Bekki, 2014) is a proof-theoretic natural language semantics based on
dependent type theory. Dependent type theory (Martin-Löf, 1984) is a formal
system that extends simple type theory with a notion of types depending on
terms. There are two type constructors Π (dependent function type) and Σ (dependent product type) to construct dependent types and this rich type structure
provides a foundation for handling context dependence in natural language. Oneof the distinctive features of DTS, as compared to other frameworks based on
dependent type theory, is that it is augmented with underspecified terms called@-terms, so as to provide a unified analysis of entailment, anaphora, and presupposition from inferential and computational perspective. DTS also gives a fully
compositional account of inferences involving anaphora (Bekki, 2014).
The type constructor Π is a generalized form of the functional type. A termof type (Πx : A)B(x) is a function f which takes any element a of A and returns- 81 -

Π-typeΣ-typeMartin-Löf (1984) (Πx : A)B(x) (Σx : A)B(x)
[
]
x:A
DTS
(x : A) → B(x)B(x)
Fig. 1. Notation of Π-type and Σ-type.

a term f (a) of type B(a) dependent on the choice of the argument a. In otherwords, a dependent function is a function whose codomain is dependent on thegiven argument. The type constructor Σ is a generalized form of the producttype. A term of type (Σx : A)B(x) is a pair (m, n) which consist of a term mof type A and a term n of type B(m), where the type of the second element n
depends on the choice of the first element m.
According to Curry-Howard correspondence, a type can be regarded as aproposition and a term of the type can be regarded as a proof of the proposition.
From this point of view, A Π-type corresponds to a universal quantifier anda proof term of universal sentence is a function. If x ∈/ f v(B), i.e., there isno dependencies involved, (Πx : A)B corresponds to implication. A Σ-type
corresponds to an existential quantifier, and a proof term of existential sentence
is a pair. If x ∈/ f v(B), (Σx : A)B corresponds to conjunction. See, e.g., MartinLöf (1984) for more details and inference rules for Σ and Π constructors. Figure
1 shows the notation of Σ-type and Π-type adopted in DTS.
Since Π-types correspond to propositions with the universal quantifier, thesentence every boy entered can be represented as follows.
( [])x : entity(7)u:→ enter(π1 u)boy(x)
Here, entity is a basic type for all entities. Σ-types are associated with projection
functions [π1 and] π2 , which enables one to access to each part of the pair: fromx:A
, one can derive π1 t : A and π2 t : B(π1 t). Since u in (7) is a
B(x)[]x : entity, π1 u corresponds to an entity which is a boy. Thus,
pair of typeboy(x)a pair t :(7) corresponds to the proposition that for all entity that is a boy, that entity
entered.
A sentence with an existential quantifier such as a boy entered is represented
in terms of Σ-types. Again, π1 u corresponds to an entity which is a boy, andthus, (8) corresponds to the proposition that there exists an entity which is a
boy and entered.
 []x : entityu:boy(x)(8) enter(π1 u)
One advantage of having Σ-types is that it can capture an externally dynamic
property (Groenendijk and Stokhof, 1991) of existential quantifier and conjunc-- 82 -

tion. For instance, a discourse such as (9a) is known to be problematic in asense that its syntactically-corresponding formula in predicate logic, (9b), fails
to represent an anaphoric link between a boy and he.
(9)a. A boy entered. He whistled.
b. ∃x(boy(x) ∧ enter(x)) ∧ whistle(x)On the other hand, Σ-type can straightforwardly provide the semantic representation of this discourse as follows.
  [
]
x : entityu:v : 
boy(x)(10) 

enter(π1 u)
whistle(π1 π1 v)
Although a term u is no longer accessible from the argument position of whistle,
one can still pick up the term via a newly introduced term v, since v is a pair andeach of its parts is accessible by applying (a sequence of) projection function. In
this way, Σ-type can pass a variable binding relation to a subsequent discourse.
2.2

DTS and anaphora resolution

The remaining question is how one can obtain the term π1 π1 v in (10) for the
representation of the pronoun he. In DTS, anaphoric expressions are represented
in terms of underspecified terms called @-terms. Anaphora resolution in DTS isthen defined as a process that replaces the @-term with the specific term whichis constructed via type checking and proof construction (Bekki and Sato, 2015).
For instance, the pronoun he is assigned the semantic representation in (11),where the type annotation to the @-term represents the requirement that herefers to some entity being male.
( [])x : entity(11) π1 @imale(x)Dynamic conjunction between sentences is defined in terms of Σ-type. Thus, thesemantic representation of the whole discourse in (9a) is given as follows.
]
  [

x : entityu:v : 
boy(x)
enter(πu)1(12) 

(
(
[
]
))


x : entity
whistle π1 @imale(x)This underspecified representation is required to have sort type. This condition (felicity condition of a sentence) invokes type checking and leads to proof
construction associated with the @-term. In the current example, one needs tofind a proof term that satisfies the following inference.

- 83 -



[

]
[
]
x : entity
x : entityboy(x)(13) Γ, v :⊢?:male(x)enter(π1 (u))
u :Here, Γ is a global context that represents background knowledge, and v is a term
accessible from the position of the @-term, which corresponds to the information
provided up to this point of the mini-discourse.
]From these premises Γ and v,[
one needs to construct a proof term ofx : entity. Now, suppose that the global
male(x)context contains the proof term in (14), which corresponds to the knowledge thatevery boy is male.
( [])y : entity(14) kb : u :→ male(π1 (u))
boy(y)
By using this knowledge kb together with v, one can eventually construct a
proof term of the required type that can replace the underspecified term in (12),yielding the fully-specified representation in (10).
Note that this anaphora resolution procedure in DTS can account for thefollowing externally static property of universal quantifiers.
(15)

a. Every1 boy received a present. ∗ He1 looks happy.
b. Every boy received a1 present. ∗ It1 was a toy car.

In these cases, since the first sentences are universal sentences, proof terms provided to the subsequent discourse are functions. Thus, neither an entity beinga boy embedded in the domain of the function (i.e., an entity in the restrictor),
nor an entity being a present embedded in the codomain of the function (i.e.,
an entity in the nuclear scope) can be picked up by the same operation as theprevious example of existential sentences.
In this way, rich type structures of Σ-type and Π-type, together with theanaphora resolution process in DTS, provide a proof-theoretic account of dynamic properties of the existential and universal quantifiers.

3

Dependency relations and dependent interpretation

3.1

Basic example

As we mentioned above, the structure of Π-type corresponds to the externallystatic property of the universal quantifier, and it can block an anaphoric link
exemplified in (15b). However, since DTS is a proof-theoretic semantics and theanaphora resolution process is associated with inference, there is one possibility
to pick up an entity embedded in the codomain of the function. That is to apply
an appropriate argument to the function and obtain the return value. This is
one of the key ideas that underlies our analysis of the interpretation of pronouns
involving the reference to dependency relation.
We claim that that Π-type directly captures quantificational dependency
between objects and that its proof term can be used for inference during the- 84 -

anaphora resolution process in DTS. To illustrate this idea, let us consider thesimplest example in (2b), which is repeated below as (16).
(16) Every boy will receive a1 present. Some boy will open it1 .
Since a universal quantifier corresponds to a Π-type, the first sentence can be
represented as follows.
 [
]
( [
])
y : entity
x : entityv:
present(y) 
(17)
u:
→
boy(x)
receive(π1 u, π1 v)
The terms π1 u and π1 v pick up the entity being a boy and the entity being apresent, respectively. The type as a whole represents the proposition that, forevery boy, there exists a present such that the boy received it. This representation
corresponds to the distributive reading in question. Thus, a term of this type is afunction that receives a pair consists of an entity and a proof of that entity beinga boy, and then returns a tuple that consists of an entity, a proof of that entity
being a present, and a proof of the boy and the present being in the receiving
relation. This means that the representation of the first sentence introduces afunction that corresponds to the dependency relation between boys and presents.
The second sentence is represented by Σ-type and the pronoun it can be
defined as an underspecified term of type entity. Thus, by combining the semantic representation of the two sentences in terms of dynamic conjunction, (16)is represented as the following Σ-type.
 [
] 
 ( [
])
y : entity
x : entityv:
f : u :present(y)  →boy(x)

receive(π1 u, π1 v) ](18)   [
 z : x : entityboy(x)
open(π1 z, @1 entity)
In this way, the proof term f of the first sentence which corresponds to a dependency relation between boys and presents serves as anaphoric resources. In thecurrent case, anaphora resolution of the pronoun it yields the following inference.
(19)

 [
]
( [
])
[
]
y : entity
x : entity
x : entityv:
present(y)  , z :Γ, f : u :→⊢ ? : entity
boy(x)boy(x)
receive(π1 u, π1 v)

There are two proof terms which are accessible from the position of @-term:
the term f , which is a proof term of the first sentence, and z, which is a term
corresponding to the subject of the second sentence. The proof construction goesas follows. First, by applying z to the function f , one obtains the proof term f z
that is a pair corresponding to the present which is received by the boy, π1 z.
Second, by taking the first projection of the first projection of f z, one obtains
a term π1 π1 (f z) of type entity. Therefore, by replacing the @-term with the- 85 -

obtained term π1 π1 (f z), the second argument of open will be filled with anentity which depends on the term z, namely, an entity which depends on thesubject of the second sentence.
Some reader may think that proof terms have something in common withdiscourse referents in Discourse Representation Theory (Kamp and Reyle 1993;
Kamp et al. 2011, henceforth DRT) in that both are objects introduced by sentences and referred to afterward in order to resolve anaphora. We should say,however, that there are at least two crucial diﬀerences. Firstly, while discourse
referents are limited to individuals, proof terms can have any types, which allows richer inner structure (Ranta, 1994). Thus, DTS can directly handle theinformation of dependency relations as a proof term of dependent function type.
Secondly, together with the anaphora resolution mechanism provided in DTS,
proof terms can contribute to the logical inference, which yields a new proof
term serving as an antecedent.
Note that it is Ranta (1994) who first pointed out the possibility of using adependent function to interpret an anaphoric pronoun it in a sentence like “ifyou give every child a present, some child will open it” (Hintikka and Carlson,
1979). However, the brief discussion there was confined to this specific example
and did not generalize to other examples. It is also not clear how to obtain afully-specified representation from the given sentence in a compositional way.
The point we make here is that the idea that a proof term of a dependentfunction type can serve as a discourse referent can be applied to the well-known
phenomena of dependent interpretation involved in plural anaphora in general.
3.2

More examples

We have observed that a similar anaphoric link can be established even though
the subject of the subsequent discourse does not exactly match the restrictor of
the universal quantifier of the first sentence. Those examples are repeated below
as (20) and (21).
(20) Every boy will receive a1 present. Every young boy will open it1 .
(21) Every boy will receive a1 present. John will open it1 .
Both the first and the second sentences in (20) can be represented in terms of Πtypes. Thus, the whole sentence receives the following semantic representation.
 [
]
 ( [

])
y : entity
x : entityv:
f : u :present(y) →boy(x)

receive(π1 u, π1 v)
] (22)    [
x : entity t :  z : boy(x) → open(π1 π1 t, @1 entity) young(π1 z)
The premises of the[ inference]associated with the resolution of @1 here are terms
f and t. Since π1 t :x : entity
can be derived from the given t and can be appliedboy- 86 -

to f , one eventually obtains a term π1 π1 (f (π1 t)), which corresponds to a present
dependent on each young boy, π1 π1 t. Similarly, (21) is represented as follows.
 [
] 
 ( [
])
y : entity
x : entityv:
f : u :present(y)  →boy(x)(23) 

receive(π u, π v) 1

1

open(john, @1 entity)
To find a semantic link between John and boy, one needs the background knowledge that John is a boy. If the global context Γ supplies the background knowledge kj : boy(john) which corresponds
to the] proposition John is a boy, one[
can construct the term (john, kj ) :x : entity. Again, the function f can beboy(x)

supplied an argument.
When a relation between the restrictor of the universal quantifier and thesubject of the subsequent discourse is not clear, then it simply fails to searcha proof. For instance, in the case of (5b), repeated here as (24a), there exists
neither an explicit link nor an implicit link between men and women.
(24)

a. Every man will receive a1 present. ∗ Some woman will open it1 .
 [
] 
 ( [
])
y : entity
x : entityv:
f : u :present(y)  →man(x)

receive(π1 u, π1 v) ]
b.   [ z : x : entity
woman(x)open(π1 (z), @1 entity)

In this case, one needs an argument applying to the function f in order toconstruct a proof of the present received by some man. Thus, unless some relation
which bridges men and women is available in the global context, there is no way
to obtain the required proof term from z and f .
Let us now turn back to our first example (1) involving plural anaphora.
The example is repeated below as (25). (26) is a related example with adjectival
quantifiers (Krifka, 1996).
(25) Every1 boy received a2 present. They1 opened it2 .
(26) Three1 students each wrote an2 article. They1 each sent it2 to L&P.
Although providing a comprehensive analysis of plural anaphora including ananalysis of the so-called collective reading is not the main target of this paper,
let us briefly sketch the main idea of handling the reference to dependency relation involved in plural anaphora. In our analysis explained above, there are
two things that play an essential role to account for the reference to dependency
relation. Firstly, the initial sentence must have the ∀–∃ reading which induces adependency relation between objects in terms of a dependent function. Secondly,
a singular pronoun in the subsequent discourse can be interpreted anaphorically,by supplying an adequate argument to the dependent function introduced by- 87 -

the initial sentence. These two points are maintained in our analysis of plural
anaphora as in (25) and (26).
As for the first point, we follow an analysis of generalized quantifiers and adjectival quantifiers in DTS (Tanaka et al., 2014; Tanaka, 2014), which provides
semantic representation of those quantificational expressions by using dependentfunction. According to their analysis, generalized quantifiers such as most 5 andadjectival quantifiers such as three are uniformly represented as involving existential quantification over dependent functions whose domain is restricted bycardinality condition6 . Thus, this dependent function can be used for anaphoraresolution as the cases we have seen so far.
The essential role of the plural pronoun they is, then, to supply terms that are
adequate for the arguments of the dependent function. Semantic representationof they are also given in terms of the @-term. In contrast to the singular pronounsuch as it, the requirement of type annotation of the @-term associated with theyis to find a predicate and a proof term of the cardinality condition. This is because
the domain of dependent function provided by quantificational expression isrestricted by the predicate and the cardinality condition. Therefore, the term
replacing the @-term can supply adequate arguments to the function, whichenables the dependent interpretation of singular pronoun which comes after.

4

Previous approaches

In this section, we provide a brief overview of some of the existing solutions indynamic semantics to handle the reference to dependency relation.
In classical DRT (Kamp and Reyle, 1993), the reference to dependency relation is handled by using copy mechanism. First, the sentence every boy received
a present yields the following Discourse Representation Structure (DRS).

(27)

y
@
x
every@ present(y)boy(x) @ x
receive(x, y)@

5

In the case of every, we can provide its semantic representation in two ways: one
possibility is to treat it simply as Π-type as we have seen above, and another possibility is to represent it in the same way as other generalized quantifiers such as
most. Since these two formulas are mutually deducible, the story about generalized
quantifiers presented here can be applied to the case of every as well.
6
As dependent function types correspond to ∀–∃ reading (or distributive reading), thesemantic representation of three provided by Tanaka (2014) should correspond to thesemantic representation of three ... each. To obtain the semantic representation of
three ... each in a compositional way, we can integrate the existing analysis of pluralobjects into our framework. Since this is beyond the scope of this paper, we leave it
for another occasion.

- 88 -

The construction of this DRS triggers the operation called abstraction, whichconstructs a new plural discourse referent X ′ consisting of object that satisfies
the condition of x. The pronoun they refers to this X ′ and yields the DRS in
Figure 2.a, where universal quantification over X ′ takes place. In this DRS,
however, there is no discourse referent which can be associated with singular yin open(x, y). In such a case, there is an option to apply copy operation which
copies the conditions of x constituting X ′ to the restrictor part of the duplex
condition. The corresponding DRS is in Figure 2.b. In this way, the singularvariable y in open(x, y) can refer to each present associated with each boy.

X’
X’
y
@
every@ present(y)boy(x) @ x
receive(x, y)@xxy
X ′ = Σxxy
X ′ = Σxboy(x)
present(y)receive(x, y)

xy

@

x

y
@
every@ present(y)boy(x) @ x
receive(x, y)@xevery@x ∈ X’ @ xopen(x, y)boy(x)
present(y)receive(x, y)@boy(x)
every@
open(x, y)present(y) @ x
receive(x, y) @@a. DRS for (25) before applying copy operation. b. DRS for (25) after applying copy operation.

Fig. 2. DRS associated with (25).

Krifka (1996) criticizes this solution of performing representation-based copying operation and proposes the analysis based on the enriched assignment function called parametrized sum individuals. Parametrized sum individuals are setsof pairs of an individual and a variable assignment associated with the individual. A possible instance of parametrized individuals for every boy received apresent looks as follows.
⟨x, {⟨b1 , {⟨y, p1 ⟩}⟩, ⟨b2 , {⟨y, p2 ⟩}⟩, ⟨b3 , {⟨y, p3 ⟩}⟩, ...}⟩The individuals can be either singular or plural. Since individuals are followed byassignments associated with them, this structure captures dependency relations
between objects. In the case of the distributive interpretation, each parametrizedindividual is independently evaluated against predicates. Thus, singular pronouncan be interpreted along each parametrized individual, which produces an eﬀect
of interpretation sensitive to dependency relation.
The standard way to encode dependency relations is to adopt information
states for plurals proposed by van den Berg (1996a,b) in Dynamic Plural Logic.

- 89 -

In this approach, formulas are interpreted relative to information states, which
are sets of assignments, instead of to assignments. A possible information statefor every boy received a present looks as follows.
{{⟨x, b1 ⟩, ⟨y, p1 ⟩}, {⟨x, b2 ⟩, ⟨y, p2 ⟩}, {⟨x, b3 ⟩, ⟨y, p3 ⟩}, ...}
When distribution over x is involved, predicates are evaluated against each assignment of information states. Since the assignment of new values takes place
independently of each assignment function, the variables introduced may be dependent on x. This is the source of dependency.
Our intuition about the ∀–∃ reading of every boy received a present is that it
introduces a quantificational dependency, that is, a function f such that x is aboy receiving a present f (x). However, there is no natural place in the standard
dynamic semantics theory to store such a function for subsequent anaphora.
Therefore, those formalisms mentioned above need to capture dependency relations in an indirect way, which requires integrating a special mechanism or
structure into the underlying framework.
There are several other issues which can be raised from a more empiricalside. Firstly, the copy mechanism in DRT is triggered by the resolution of pluralpronoun they. However, we have seen that there are cases such as (16), (20), and(21), where a plural pronoun does not appear but still the reference to dependency relation takes place. It seems that the further stipulation or operation isneeded in DRT to handle more general cases including these examples. Secondly,
there exists no proof theory for both of the frameworks proposed by Krifka andvan den Berg. Van den Berg’s analysis can account for a case such as (20), wherea subset relation allows the reference to a dependent relation. In general, however, a semantic link between the restrictor of the universal quantifier and thesubject of the subsequent discourse is not limited to subset relation, as we canobserve in the example such as (5a): rather, it seems that resolving dependent
plural anaphora involves a more general kind of inference, of which a semantic
link in terms of subset relations is a special instance.
An advantage of proposed analysis in DTS is that dependent function typeis an independently motivated object provided in dependent type theory, andthus, we do not need to extend our framework to account for dependency relations. By following the standard operation of dynamic conjunction and anaphoraresolution procedure in DTS, it can naturally provide a function as discourse referent, which straightforwardly leads to the dependent interpretation of singularpronouns. In addition, since DTS is a proof-theoretic semantics and anaphoraresolution process involves proof search, it can provide more general and uniform
account of semantic links between the restrictor of the universal quantifier andthe subject of the subsequent discourse.

5

Conclusion

In this article, we have argued for a new account of dependency relations between
objects as dependent functions in dependent type theory. This contrasts with- 90 -

approaches in dynamic semantics tradition, where a function does not serve as
a discourse referent, and the enriched notion of assignment functions plays anessential role in handling dependencies. We have seen that the proposed accountis capable of explaining the dependent interpretation of pronouns by integratingwith anaphora resolution mechanism of DTS. This new account may oﬀer a basis
for proof-theoretic analysis of plural anaphora.
Acknowledgments. We would like to thank Alastair Butler, Yusuke Kubota,
Robert Levine, the audience of the workshop ‘New landscapes in theoretical computational linguistics,’ and two anonymous reviewers of LENLS13 for their valuable comments, suggestions, and discussions. The first author acknowledges thefinancial support from JSPS (Grant-in-Aid for JSPS Research Fellow; 15J11772).

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Barcelona, Spain, 2015.
Adrian Brasoveanu. Donkey pluralities: plural information states versus nonatomic individuals. Linguistics and Philosophy, 31(2):129–209, 2008.
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and Philosophy, 14(1):39–100, 1991.
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Semantics, volume 5 of Studies in Linguistics and Philosophy. Springer Netherlands, 1979.
Hans Kamp and Uwe Reyle. From Discourse to Logic: Introduction to Modeltheoretic Semantics of Natural Language, Formal Logic and Discourse Representation Theory. Studies in Linguistics and Philosophy. Kluwer Academic
Publisher, 1993.
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and Philosophy, 19(6):555–598, 1996.
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Ribeka Tanaka. A proof-theoretic approach to generalized quantifiers in dependent type semantics. In Ronald de Haan, editor, Proceedings of the ESSLLI2014 Student Session, pages 140–151, Tübingen, Germany, 2014.
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CSLI publications, 1996a.
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Dynamics of Nominal Anaphora. PhD thesis, University of Amsterdam, 1996b.

- 92 -An Analysis of Selectional Restrictions
with Dependent Type Semantics
Eriko Kinoshita∗1 , Koji Mineshima∗1∗3 , and Daisuke Bekki∗1∗2∗3∗1

Ochanomizu University ∗2 National Institute of Informatics∗3
CREST, Japan Science and Technology Agency

Abstract. Predicates in natural languages impose selectional restrictions on their arguments. In this paper, we analyze selectional restrictions of predicates within the framework of Dependent Type Semantics(DTS), a framework of natural language semantics based on dependent
type theory. We also analyze two phenomena, coercion and copredication for logical polysemous nouns, which present challenges to a simple
analysis of selectional restrictions, in terms of operators that shift themeanings of predicates.

1

Introduction

Predicates in natural languages impose selectional restrictions on their arguments. For example, the transitive verb marry expects its subject and object tobe expressions denoting human. Thus, from the utterance of (1), we can inferthat Bob and Ann are both human.
(1) Bob married Ann.
One potential way to explain this inference is to treat selectional restrictions ofpredicates as entailment. According to this analysis, the verb marry is assigned
the meaning in (2a) and the whole sentence in (1) has the interpretation in (2b).
(2)

a. λyλx.marry(x, y) ∧ human(x) ∧ human(y)b. marry(bob, ann) ∧ human(bob) ∧ human(ann)

A problem with this analysis is that it cannot handle the inference in (3).
(3) Bob didn’t marry Ann. ⇒ Bob and Ann are human.
From the negation of (1), one can also infer that Bob and Ann are human. If
selectional restrictions of predicates were part of entailment, we would assign
the interpretation (4) to the negative sentence in (3). This does not account for
the inference in (3).
(4) ¬(marry(bob, ann) ∧ human(bob) ∧ human(ann))
In general, the contents of selectional restrictions project out of the scope ofnegation, modals, and conditionals (Asher [2], Magidor [7]). This is a common- 93 -

feature of inferences known as presupposition projection (see e.g., Beaver [3] foran overview).
The goal of this paper is to propose an analysis that treats selectional restrictions as presupposition within the framework of Dependent Type Semantics (Bekki [4]). Using this framework, we also present a formal analysis of two
lexical phenomena related to selectional restriction, namely, coercion and copredication for logical polysemies.

2

Selectional restriction: types vs. predicates

Although the presuppositional analysis of selectional restriction goes back at
least to McCawley [9], it seems fair to say that its precise formulation has beenmostly neglected in the simply typed setting of standard formal semantics where
only e (entity) and t (truth-value) are taken as base types.
Recently, there are proposals in the literature that set out to handle selectional restrictions with extended type-theoretic frameworks (Asher [1], Luo[6], Retoré [11]). There are two possible approaches here. One is to represent
selectional restrictions as types; for instance, using animate and human asbase types, one can assign a type animate → prop to the predicate cry andhuman → human → prop to the predicate marry, and so on. According
to this approach, the violation of a selectional restriction is to be treated as a
type mismatch. One problem with this approach is the problem of subtyping.
Thus, to combine the predicate cry of type animate → prop with the termjohn of type human, one needs a subtyping relation human < animate andextra subtyping rules (cf. Luo [6], Retoré [11]). One drawback is that with additional subtyping rules, the resulting compositional semantics becomes fairly
complicated.
Alternatively, one can preserve the base type for entities and represent selectional restrictions as predicates over entities. This view seems to be underdeveloped; but it has an advantage in that it can dispense with subtyping and preserve
the clear, well-understood conception of syntax-semantics mapping. Our theory
is based on this second approach.

3

Dependent Type Semantics

The main challenge here is how to provide a presuppositional analysis of selectional restrictions combined with the selectional-restriction-as-predicate view.
We use Dependent Type Semantics (DTS) (Bekki [4]) as a theoretical framework,
which provides two crucial tools: dependent types (which are a generalization of
simple types) and underspecified terms. DTS is a proof-theoretic semantics ofnatural language based on dependent type theory (Martin-Löf [8]). It characterizes the meaning of a sentence from the perspective of inferences.
DTS uses two kinds of dependent types.

- 94 -

(i) Π-type (dependent function type), written as (x : A) → B, is a generalized
form of a function type A → B; a term of type (x : A) → B is a function fthat takes a term a of A and returns a term f (a) of type B(a).[]x:A(ii) Σ-type (dependent product type), written as (x : A) × B or
, is aB
generalized form of a product type A×B; a term of type (x : A)×B is a pair(t, u) such that t is of type A and u is of type B(t). The projection operators
π1 and π2 are defined in such a way that π1 (t, u) = t and π2 (t, u) = u.
Under the so-called propositions-as-types principle (Martin-Löf [8]), types
and propositions are identified; a term t having type A (i.e., t : A) serves as a
proof term for the proposition A.
In this dependently typed setting, Π-type and Σ-type correspond to universal
and existential quantifiers, respectively. For example, in DTS, the sentence in(5a) is given the semantic representation (SR) in (5b):(5)

a. Every man entered.
([])
x : entityb.
u:→ enter(π1 (u))
man(x)The term u here has a Σ-type: it consists of a term (let it be x) having type
entity and some proof term having type man(x) that depends on x. The term
π1 (u) in enter(π1 (u)) picks up the entity that is the first component of u. In
DTS, common nouns such as man are treated as a predicate rather than asa type. In other words, that a term x has a property man is represented asa proposition man(x) rather than as a judgement x : man. See Bekki and
Mineshima [5] for more discussions on the interpretation of common nouns in
our framework.
For Π-types and Σ-types, we use the following formation rules (ΠF , ΣF ),introduction rules (ΠI, ΣI), and elimination rules (ΠE, ΣE).
(i)x :. A..
.
A : type B : s(ΣF ), i
[
]
x:A
:s
B(x)

(i)

x :. A
..
.
A : s1 B : s2
(ΠF ), i(x : A) → B : s2(i)x :. A..
.
(x : A) → B : s M : B(ΠI), iλx.M : (x : A) → B

M :A

N : B[M/x][] (ΣI)
x:A(M, N ) :B(x)[M : (x : A) → B N : A(ΠE)
M N : B[N/x]

M:

x:A
B(x)

[

]

π1 (M ) : A

M:
(ΣE)

x:A
B(x)

]

π2 (M ) : B[π1 (M )/x](ΣE)

where s, s1 and s2 are kind or type (see Bekki and Mineshima [5] for more details).

- 95 -DTS has an underspecified term @ to handle anaphora and presupposition.
We use type annotation for underspecified terms; we write @ : A, where theunderspecified term @ is annotated with its type A. By using underspecified
terms, we can uniformly handle semantic phenomena that depend on the preceding contexts.
Presupposition and anaphora are resolved by constructing a proof term for@ : A with type checking and then replacing @ : A by the constructed term.
Type checking ensures that an SR is well-formed (i.e., having type type). For
underspecified terms, we use the ruleA : s A true (@)(@ : A) : A
where s ∈ {kind, type}. The judgement A true triggers a proof search to constructa term having the type A in a given context. The constructed term is to be
replaced with @ in the final representation. The annotated type A may containanother underspecified term, for which the type checking is triggered by thejudgement A : s (e.g., A : type) in the @-rule.
As an illustration, consider the sentence in (6a). For this sentence, one cancompositionally derive the SR in (6b).1(6)

a. He whistled.
( (
[
]))
x : entity
b. whistle π1 @ :man(x)The SR (6b) contains an underspecified term @ annotated with the Σ-type that
corresponds to the proposition that there is an entity x such that x is a man.
Fot the SR in (6b), the type checking runs as follows.
(1)(CON )
man : entity → typex : entity..
(ΠE)(CON )
entity : typeman(x) : type..
(ΣF ), 1[][]
x : entity
x : entity: typetrue
man(x)man(x)
(
[
]) [
]
(@)
x : entity
x : entity@::
man(x)man(x)([])(ΣE)(CON )
x : entity
whistle : entity → type
π1 @ :: entityman(x)
[
]))
( (
(ΠE)
x : entity: typewhistle π1 @ :man(x)

The application of the @-rule in this derivation triggers a proof search for thejudgement:[]x : entity
true.
man(x)
1

See Bekki [4] for details on the compositional derivations of SRs in DTS.

- 96 -

Assuming that we have john : entity and t : man(john) in the backgroundglobal context, we can construct a term (john, t) having the Σ-type in question,i.e., a type annotated for the underspecified term @. This term serves as anantecedent of the pronoun he. Replacing @ with the specific term (john, t),
the semantic representation in (6b) ends up with whistle(π1 (john, t)), which
reduces to whistle(john). In this way, we can derive the interpretation for thesentence containing a pronoun in (6a).

4

Selectional restriction in DTS

To handle selectional restrictions of predicates as presuppositions, we need to
calculate whether selectional restrictions are satisfied at the stage of type checking. We propose that selectional restrictions of predicates are specified in thelexicon. For instance, we can define lexical entries of intransitive and transitive
verbs as follows.

crysyntaxsemantic representation

S\N P

λx.cry(x, @ : animate(x))

marry (S\N P )/N P λy.λx.marry(y, @i : human(y))(x, @j : human(x))

To be concrete, we use CCG (Steedman [12]) as a syntactic framework. The
types of predicates cry and marry in the above SRs are defined as follows:[]x : entity
cry :→ typeanimate(x)[][]y : entity
x : entity
marry :→→ typehuman(y)human(x)For example, the predicate cry takes a pair consisting of an entity x and a
proof term for the proposition animate(x) as an argument and returns a type(as a proposition). In the lexical entry for the intransitive verb cry, the proof
term for the proposition animate(x) is underspecified; given that there is anunderspecified term @ : animate(x) in the SR, we have to prove animate(x)during the stage of type checking, in order to ensure that the subject of cry is
animate.
As an illustration, consider the sentence in (1). For this sentence, we canderive the following SR in a compositional way.
(7) marry(ann, @1 : human(ann))(bob, @2 : human(bob)).
The following is the compositional derivation of this SR.

- 97 -

married(S\N P )/N P: λy.λx.marry(x, @1 : human(x))(y, @2 : human(y))

Ann
NP
: ann
>

Bob
NP
: bob

S\N P
: λx.marry(ann, @1 : human(ann))(x, @2 : human(x))
marry(ann, @1 : human(ann))(bob, @2 : human(bob))<Now a type checking to ensure that the SR in (7) is well-formed runs asfollows:..
..
(@1 : h(a)) : h(a)
[
]
[
]
[
]
(ΣI)
(Con)
x:e
y:e
x:e
marry :→→ type(a, @1 : h(a)) :
h(x)
h(y)
h(x)
[
]
(ΠE)
y:e
marry(a, @1 : h(a)) :→ typeh(y)a : e (Con)marry(a, @1 : h(a))(b, @2 : h(b)) : type

b:e

..
..
(@2 : h(b)) : h(b)
[
]
(ΣI)
x:e
(b, @2 : h(b)) :
h(x)
(ΠE)
(Con)

Here we abbreviates entity as e, human as h, ann as a, and bob as b. There are
two open branches containing underspecified terms, @1 and @2 , which show that
we have to search the preceding context to construct a proof term of human(bob)and human(ann). That is to say, for the semantic representation to be wellformed, it is presupposed that x and y, which are respectively the subject andthe object of the verb married, are both human. In this way, the selectional
restriction of a predicate is derived as a presupposition.
Similarly, the SR of the negative sentence in (3) is given as follows.
(8) ¬marry(ann, @2 : human(ann))(bob, @1 : human(bob))According to the formation rule of negation, A and ¬A have the same wellformedness condition:
A : type(¬F )¬A : typeThat is, if we have A : type, then we have ¬A : type as well. Therefore, the typechecking for the negative SR in (8) ends up with the derivation that triggers a
proof search in the same way as the type checking for the SR in (6b) given in
Section 3. In this way, one can derive the inference pattern of presupposition
projection out of the scope of negation. A similar explanation applies to the caseof modals and conditionals.
Interestingly, a negative sentence like (9) has two readings (cf. McCawley [9]).
(9) The chair does not cry.
First, this sentence has a reading in which the selectional restriction projects
out of the scope of negation, hence resulting in a violation of selectional restriction. In our terms, after composing the meaning of (9), one obtains the SR¬cry(chair, @ : animate(chair); according to the formation rule of negation,
the content of selectional restriction, i.e., animate(chair), projects out of the- 98 -

scope of negation. Thus, for the SR to be well-formed, one needs to construct a
proof term of animate(chair), which is not available in the standard context.
Hence, it is predicted that under this reading, a violation of selectional restriction
occurs in the sense that the derived SR is not well-formed.
Secondly, and more interestingly, (9) can have a reading in which the selectional restriction does not project and hence is interpreted inside the scopeof negation. The presuppositional analysis correctly predicts this reading; withthe process of local accommodation, we can derive the SR ¬(animate(chair) ∧cry(chair)) for (9). In this case, one does not have to construct a proof ofanimate(chair); hence, it is correctly predicted that under this reading, theutterance of (9) is meaningful and can be true. We leave a detailed explanationof local accommodation in the framework of DTS for another occasion.

5

Coercion and copredication for logical polysemies

5.1

Coercion

There are two phenomena that are not explained by a simple analysis of selectional restrictions of predicates. The first one is coercion (Nunberg [10]). For
example, if we have a context in which there is a man who ate the omelet in acafe, we can understand the meaning of (10a) as (10b).
(10)

a. The omelet escaped.
b. The man who ate the omelet escaped.

To account for this phenomena, we define an operator called argument operator that transforms one predicate into another. The argument operator arg1 fora one-place predicate and arg2 for a two-places predicate are defined as follows:



 [
]

x:e


 z : @pr (x)




[2
] [
]
arg1 ≡ λP.λx.P π1 π1 @5 : 

x:e
x:e
pr
(@4 :
→
→ type)(x, (@3 (x)))(z)prpr@1 (x)@2 (x)
arg2 ≡




 [

]
x:e


 z : @pr (x)(x)[3] [] []λP.λy.λx.P π1 π1 @7 : 

x:e
x:e
x:e
pr
pr
(@6 :
→
→
→ type)(y, (@4 : @1 (y)))(x, (@5 : @2 (x)))(z)prprpr@1 (x)@2 (x)@3 (x)

Here an underspecified term @pr
i is an abbreviation for @i : e → type.
Let us first focus on the definition of the argument operator arg1 for one-placepredicates. In the definition of arg1 , the underspecified terms @1 and @2 in @pr
1
and @pr2 are annotated with type e → type; these are underspecified terms forproperties. Intuitively, given a one-place predicate P and its argument x of typee, the argument operator arg1 produces a new predicate P ′ that existentiallyintroduces a new entity z having some relation R to x.
When one underspecified term appears inside the type annotated with another underspecified term, the inside term has to be resolved first. More specifically, the underspecified terms contained in the argument operator arg1 areresolved in the following way.

- 99 -

1. First, given the entity x (e.g., the omelet in (10a)), one needs to find asuitable property F (e.g., edible) holding of x. This property F replaces @pr
1 .
2. Second, if there is a proof term for the proposition that x has the property
F (e.g., the omelet is edible), it replaces @pr
3 .
3. Also, one needs to find a property G that is substituted for @pr
2 . The property
G (e.g., animate) has to be chosen so that the newly introduced entity (thefirst element of the term z) satisfies G.
4. Next, one needs to find a relation R that is substituted for @pr
4 . In our
example, one has to find a relation (e.g., eat) that have selectional restrictions
specified by predicates edible(x) and animate(y). This relation R replaces@4 .
5. Finally, one needs to construct a term substituted for @5 , which is a tuple
consisting of an entity z whose first element satisfies the property G and a
proof term for the proposition that the relation R holds of x and z.
In this way, arg1 transforms the predicate escape into a predicate whose argument is an animate entity that has the eating-relation to the omelet.
Let us explain the derivation in more details. To begin with, we can derive
the SR of the sentence (10a) as follows.

escaped
S\N P: λx.escape(x, @5 : animate(x))ϵ(S\N P )\(S\N P ): arg1

S\N P: arg1 (λx.escape(x, @6 : animate(x)))

The omeletNP:oS: arg1 (λx.escape(x, @6 : animate(x)))(o)<<By unfolding the definition of arg1 , the sentence in (10a) is assigned the SR in(11).
(11) escape(Z1 , @6 : animate(Z1 ))Here Z1 abbreviates:




[
]

x:e
z
:
pr



@[2 (x) ].
[]π1 π1  @5 : 

x:e
x:e
pr
(@4 :
→
→ type)(o, (@3 : @1 (o)))(z)prpr@1 (x)@2 (x)- 100 -

Let us suppose that we have the following information in the global context K1 :K1 ≡ type : kind, e : type,j : e, o : e,animate : e → type, edible : e → type,
[
]
[
]
y:e
x:e
eat :
→
→ type,edible(y)animate(x)[]x:eescape :→ type,animate(x)p1 : animate(j), p2 : edible(o), p3 : eat(o, p2 )(j, p1 ).
Now type checking is triggered to determine whether the SR (10) is well-formed.
This is an example of nested presupposition, and underspecified terms are resolved from the most embedded one. Here we focus on the step to find a relation
R that is substituted for the following underspecified term:
[
]
[
]
x:e
x:e
@4 :
→
→ type.
@pr
@pr
1 (x)
2 (x)
The type checking tree for the relevant part looks as follows:
D1
D2
]
[
]
x:e
x:e
:
t
→t:k
@pr
@pr
1 (x)
2 (x)
[
]
[
]
(ΠF )
..
x:e
x:e
..
→
→
t
:
k
@pr
@pr
1 (x)
2 (x)
(
[
]
[
]
) [
]
[
]
(@)
x:e
x:e
x:e
x:e
@4 :
→
→
t
:
→
→
t
@pr
@pr
@pr
@pr
1 (x)
2 (x)
1 (x)
2 (x)
[

where we use abbreviations k for kind and t for type. The type checking for D1
runs as follows.
e:t

e:t

(CON )(CON )

e→t:k

t : k (ΠF )@pr
1 :e→t(CON )[..
..
e → t true (@)@pr
1 (x) : t

]
x:e
:t
pr
@1 (x)

x : e (1)(ΠE)(ΣF ), 1

Similarly, the type checking for D2 runs as follows.
e:t

e:t

(CON )

e→t:k(CON )

t : k (ΠF )@pr
2 :e→t..
..
e → t true (@)(CON )

[

]
x:e
:t
pr
@2 (x)@pr2 (x) : tx : e (1)(ΠE)(ΣF ), 1

t:k
[

]
x:e
→t:k
@pr
2 (x)

- 101 -

(CON )(ΠF )The judgements e → t true in D1 and D2 trigger a proof search; given a suitableglobal context, we can find the antecedents edible of type e → t for @sr
1 , and
animate of type e → t for @sr,
respectively.
Replacingeachunderspecified
2
term with its antecedent predicate, the above type checking tree is transformed
as follows.
D1
D2
]
[
]
x:e
x:e
..
:t
→t:k
edible(x)animate(x)
.
[
]
[
]
]
[ .
]
(ΠF ) [
x:e
x:e
x:e
x:e
→
→ t true→→t:kedible(x)animate(x)edible(x)animate(x)
(
[
]
[
]
) [
]
[
]
(@)
x:e
x:e
x:e
x:e
@4 :
→
→t :
→
→t
edible(x)animate(x)edible(x)animate(x)[Then we can find an antecedent eat for @4 that has a type
[
]
[
]
x:e
x:e
→
→t
edible(x)animate(x)in the context K1 . In a similar way, we can find a proof term for other @-terms:
p2 for @3 , ((j, p1 ), p3 ) for @5 , and p1 for @6 . By eliminating each @-term in(11) and reducing β-redexes, we obtain the SR espace(j, p1 ) as a fully specified
semantic representation for the sentence (10a).
5.2

Copredication for logical polysemies

The second phenomenon we consider is copredication of logically polysemous
nouns. There are nouns having multiple meanings in natural language; it can
be classified into accidental and logical polysemy (Asher [1]). For example, thenoun bank in (12a) is accidentally polysemous, and the noun book in (12b) islogically polysemous.
(12)a. # The bank is closed and is muddy.
b. Mary memorized and burned the book.

The sentence (12b) shows that the logically polysemous noun book allows copredication, despite the fact that memorized and burned require diﬀerent objects(i.e., informational objects and physical objects, respectively) as their object argument. To account for this fact, we can apply argument operators to the verbs
memorized and burned, thereby avoiding the violation of selection restrictions.
We introduce the logical polysemies of nouns as functions. For example, weassign the following functions to the noun book.
[]y:ebookinfoOf : (x : e) → (book(x) →)infoOf (x)(y)[bookphyObjOf]
y:e: (x : e) → (book(x) →)phyObjOf (x)(y)- 102 -

The function bookinfoOf (resp. bookphyObjOf ) takes an entity x and a proofof book(x) and returns an entity y that is the informational aspect (resp. thephysical aspect) of x.
Now we can derive the SR of the sentence (12b) as follows.
memorized
S\N P /N P
: M EMϵ(S\N P /N P )\(S\N P /N P ): arg2
S\N P /N P
: arg2 (M EM )burnedS\N P /N P
: BU RN<ϵ(S\N P /N P )\(S\N P /N P ): arg2
S\N P /N P
: arg2 (BU RN )

and
CON[J
]
p(y)(x): λp.λq.λy.λx.
q(y)(x)⟨Φ⟩[ S\N P /N P]
arg2 (M EM )(y)(x): λy.λx.
arg2 (BU RN )(y)(x)the book
NP
:b

S\N P
]
arg2 (M EM )(b)(x): λx.
arg2 (BU RN )(b)(x)
[

Mary
NP
:m

S
]
arg2 (M EM )(b)(m):arg2 (BU RN )(b)(m)

<

[

where
[
M EM ≡ λy.λx.memorize(y, @i :]w:e)(x, @j : animate(x)),
infoOf (w)(y)

and
[

]
w:e
BU RN ≡ λy.λx.burn(y, @i :)(x, @j : animate(x)).
phyObjOf (w)(y)Thus the sentence in (12b) is assigned the following SR.


[

]

x:e
 memorize(Z2 , @15 : infoOf (x)(Z2 ) )(m, @16 : animate(m)) [](13) 

x:eburn(Z3 , @17 :)(m, @18 : animate(m))
phyObjOf (x)(Z3 )where Z2 abbreviates


 [
]

x:e
 z : @pr (x)



,
[2
]
[
]
[
]
π1 π1  @7 : 

x:e
x:e
x:e
pr
pr
(@6 :
→
→
→
t)(b,
(@
:
@
(b)))(m,(@:@(m)))(z)
pr
pr
pr
4
5
1
2
@1 (x)@2 (x)@3 (x)and Z3 abbreviates


 [
]

x:e

 z : @pr (x)

9[

.
]
[
]
[
]
π1 π1 
@14 : 

x:e
x:e
x:e
pr
pr
(@13 :→→→ t)(b, (@11 : @8 (b)))(m, (@12 : @9 (m)))(z)prprpr@8 (x)@9 (x)@10 (x)- 103 -<>

Let us suppose that we have the following information in the global context K2 :K2 ≡ t : k, e : t,m : e, b : e, ib : e, pb : e,animate : e → t, book : e → t,
infoOf : e → e → t, phyObjOf : e → e → t,


[
]
y:e
[
]
x:e


w:e
memorize :→→ t,animate(x)infoOf (w)(y)[]y[ : e
]
x:e


w:e
burn :→→ t,animate(x)phyObjOf (w)(y)[]y:ebookinfoOf : (x : e) → (book(x) →),infoOf (x)(y)[]y:ebookphyObjOf : (x : e) → (book(x) →),phyObjOf (x)(y)p1 : animate(m), p2 : book(o),p3 : infoOf (b)(ib ), p4 : phyObjOd(b)(pb ).
Then we can find a proof term for each @-term in SR Z2 as follows. Here @i 7−→ Tmeans that the underspecified term @i is replaced with a term T .
@1 7−→ book,@2 7−→ animate,@3 7−→ infoOf (b),@4 7−→ p2 ,@5 7−→ p1 ,]u : book(y)@6 7−→ λy.λx.λz.
,
bookinfoOf (y)(u) =e z
@7 7−→[[]
u : book(y)((ib , p3 ), (λy.λx.λz.
)(b, p2 )(m, p1 )(ib , p3 )).
bookinfoOf (y)(u) =e z
And we can also find a proof term for each @-term in SR Z3 :@8 7−→ book,@9 7−→ animate,@10 7−→ phyObjOf (b),@11 7−→ p2 ,- 104 -

@12 7−→ p1 ,[]u : book(y)@13 7−→ λy.λx.λz.
,
bookphyObjOf (y)(u) =e z
@14 7−→[]
u : book(y)((pb , p3 ), (λy.λx.λz.
)(b, p2 )(m, p1 )(pb , p4 )).
bookphyObjOf (y)(u) =e z
The rest of the underspecified terms can also be replaced with specific terms asfollows.
@15 7−→ p3@16 7−→ p1@17 7−→ p4@18 7−→ p1
By eliminating each @-term in (13) and reducing β-redexes, we obtain the following as a fully specified semantic representation for the sentence (12b).
[]
memorize(ib , p3 )(m, p1 )(14)burn(pb , p4 )(m, p1 )6

Conclusion

In this paper, we proposed an analysis that treats the selectional restrictions ofpredicates as presuppositions. In addition, using argument operators, we gave
a unified analysis of lexical phenomena that are not accounted for by simple
analyses of selectional restrictions. Future work includes extending our analysis
to such phenomena as metaphors, which Asher [1] opened up a way to analyze
in type theoretical settings.

Acknowledgements
We thank the two anonymous reviewers for helpful comments and suggestions.
This work was supported by CREST, Japan Science and Technology Agency.

- 105 -

References[1] Asher, N.: Lexical Meaning in Context: A Web of Words. Cambridge University Press (2011)[2] Asher, N.: Selectional restrictions, types and categories. Journal of Applied
Logic 12(1), 75–87 (2014)[3] Beaver, D.I.: Presupposition and Assertion in Dynamic Semantics. Studies
in Logic, Language and Information, CSLI Publications & FoLLI (2001)[4] Bekki, D.: Representing anaphora with dependent types. In: Logical Aspectsof Computational Linguistics, pp. 14–29. Springer (2014)[5] Bekki, D., Mineshima, K.: Context-passing and underspecification in dependent type semantics. In: Chatzikyriakidis, S., Luo, Z. (eds.) Modern
Perspectives in Type Theoretical Semantics. Studies of Linguistics and Philosophy, Springer (2016)[6] Luo, Z.: Formal semantics in modern type theories with coercive subtyping.
Linguistics and Philosophy 35(6), 491–513 (2012)[7] Magidor, O.: Categiry Mistakes. Oxford University Press (2013)[8] Martin-Löf, P.: Intuitionistic type theory. Bibliopolis (1984)[9] McCawley, J.D.: Concerning the base component of a transformational
grammar. Foundations of Language 4(3), 243–269 (1968)[10] Nunberg, G.: Transfers of meaning. Journal of Semantics 12(2), 109–132(1995)[11] Retoré, C.: The montagovian generative lexicon lambda tyn: a type theoretical framework for natural language semantics. In: 19th international
conference on types for proofs and programs (TYPES 2013). pp. 202–229(2014)[12] Steedman, M.: Surface Structure and Interpretation. The MIT Press (1996)- 106 -

Reference and Pattern Recognition:
A Metasemantic Study
Youichi Matsusaka
Tokyo Metropolitan University
In this talk I wish to address the question of how the reference of a proper name isdetermined. I will propose what might be regarded as a descendant of the “cluster
theory” of descriptions: the reference of a proper name is determined on the basisof the properties each individual speaker associates to the name. Whereas the
traditional cluster theory holds that the referent of a name must satisfy mostof the properties a speaker ties to the name, I will argue that such properties
just play the role of the basis on which the speaker decides whether a piece of
information, whether it is given perceptually or verbally, is similar enough tothem so that it comes from the same object to which they are associated. This isa task of “pattern recognition”, widely studied in cognitive science and machinelearning. The aim of this talk is to reconsider the question of reference from thisperspective, and in doing this I will claim that a certain notion of stability of aconcept is required in order for there to be a referent of community-wide usesof a name. Finally I will draw out some implications of this view for Kripke’s
causal account.

- 107 -

Interpretation of Wh-words in Aza-Irabu
Miyakoan
Tomohide Kinuhata
Fukuoka University,
Nanakuma. 8-19-1, Fukuoka 814-0180, Japantkinuhata@cis.fukuoka-u.ac.jp

Abstract. The purpose of this paper is to discuss the semantic interpretation of wh-words of Aza-Irabu Miyakoan, a southern Ryukyuan
language. In Aza-Irabu Miyakoan, focus particles show concord with a
speciﬁc type of the sentence: Ga with wh-interrogatives, ru with yes/nointerrogatives and du with declaratives. In the complement of question
embedding verbs, however, du can appear with bare wh-words particularly when the attitude holder knows the answer of the embedded question. I argue that such wh-words metalinguistically quantify variables
over expressions.
Keywords: kakarimusubi, wh-words, focus particles, embedded questions, quotative marker, metalinguistic quantiﬁcation

1

Introduction

Aza-Irabu Miyakoan1 has kakarimusubi in the sense that focus particles appearing in the middle of the sentence show concords with sentence types:2 Du withdeclaratives, ru with yes/no interrogatives, ga with wh-interrogatives as shownby each example in (1).
(1)

a.

b.

Kincjaku=nu=du uti-taa.
wallet=NOM=FOC fall-PST‘A wallet fell.’Kincjaku=nu=ru uti-taa?
wallet=NOM=FOC fall-PST‘Did the wallet fall?’1

Miyakoan is a language spoken in Miyako Islands, located about 180 miles southwestof Okinawa Island. Aza-Irabu is a variety of Miyakoan, spoken in the Irabu community of the Irabu island. Aza-Irabu Miyakoan, as well as other Ryukyuan languages,
shares major typological features of morphosyntax with Japanese: Head ﬁnal, SOV
word order, dependent marking, accusative case alignment, agglutinative suﬃxal
morphology (See [13] for grammatical sketches of Ryukyuan languages). Due to the
combination of those typological similarities with cultural blending, the number of
traditional speakers of Aza-Irabu is severely declining.
2
See [9] for the deﬁnition of kakarimusubi adopted in this paper.

- 108 -

2

Tomohide Kinuhatac.

Noo=nu=ga
uti-taa?
what=NOM=FOC fall-PST‘What fell?’Replacing the focus particles in (1) changes the type of the sentence or leads
to ungrammaticality: While using du instead of ru makes the sentence in (1-b)
declaratives, the use of du in (1-c) makes it ungrammatical due to the incompatibility of du and wh-words as in (2).
(2)*Noo=nu=du
uti-taa?
what=NOM=FOC fall-PST‘What fell?’In embedded clauses, however, du as well as ga can be used despite theexistence of wh-words as in (3). I mark embedded clauses by brackets hereafter.
(3)

Obaa=ja[ic1={ga/du} fu1 kutu]=tii=jasa-n.
Grandma=TOP when=FOC come SFP=QUOT=TOP know-NEG‘Grandma does not know when he will come.’

This paper discusses the semantic interpretation of wh-words and particles insuch examples as in (3), clarifying the condition of the use of wh-words with adeclarative marker du.

2

Data

To clarify the condition of the use of du with bare wh-words, I prepared thefollowing set of embedding contexts, which is illustrated with the clause ‘whocame’ embedded.
(4)

A.
B.
C.
D.
E.

I don’t know who came.
He doesn’t know who came.
Do you know who came?
He knows who came.
I know who came.

The contexts from A to E are arranged according to the attitude holder’s certainty about the answer of the question embedded: While the attitude holders
are uncertain about the answers of the questions in A and B, they are not in D
and E; C is in-between. The diﬀerence between A and B, as well as D and E, isthe referent of the attitude holder: A and E have the ﬁrst person pronoun, i.e.
the speaker, as the matrix subject whereas B and D have the third person. As
the result, the embedded clause is similar to direct questions more in A than inB, provided that the speaker does not know the answer when he utters a question like ‘Who came?’ At the other end of the scale, E is the farthest from directquestions since it encodes that the speaker knows the answer. Thus, ga-markings
are most expected to appear for wh-phrases of A and least expected for those ofE, given the obligatory ga-marking in unembedded wh-interrogatives as in (1-c).

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Interpretation of Wh-words in Aza-Irabu

3

I conducted an experimental survey designed to conﬁrm whether this expectation is on the right track. The informants are two native speakers of Aza-Irabu:
TG born in 1935 and KK born in 19243 . Not informing the speakers of the intention of this research, I requested them to translate Japanese sentences into theirdialect. The Japanese sentences provided in the survey are varied by embedding
the following clauses into the contexts given in (4).
(5)

a.
b.
c.
d.
e.
f.

who cleaned the room
what Grandma is readingwhere this car is headingwhen K comes
which one the children ordered
why Grandma is angry

The result of the experiment is given in Table 1. The capital alphabets inthe top line correspond to those in (4) and the wh-words in the ﬁrst column
represent diﬀerent sentences embedded in the complement of know, i.e. (5).4
Table 1. Distribution of ga and duA.
B.
C. D. E.
who φ
φ
φ
du du
what ga
ga
du/ga du duwhere ga ga/dudu du duwhen ga/φ du/ga/φ ga du duwhich ga
ga
du/ga du duwhy gaga
du du duTG (male, 1935)

A. B.
who φ
φ
what ga ga
where ga ga
when ga ga/ duwhich ga ga
why ga ga

C. D.
du du
ga du
du du
du du
ga ga/ du
ga
du

E.
du
du
du
du
du
du

KK (male, 1924)

When the informant utters several sentences for one stimulus sentence and theuse of focus particles is not consistent, I write them with slash indicating thatthe left one is more often uttered than the right. The shading indicates that theclause is marked by a quotative marker tii and the rest is marked by gara.
I give in (6) examples of the what’s line of Speaker TG.
(6)

A. Abaa[mma=Qa
noo=ju=ga
jummi bu1]=gara
1SG.TOP Grandma=NOM what=ACC=FOC read CONT=Q
sa-n.
know-NEG‘I don’t know what Grandma is reading.’

3

I also conducted an experiment with a person born in 1941. But he always uses ga
even in embedded contexts. The detailed research on the age and the social status
which evoke this diﬀerence is left for another project.
4
φ in Table 1 means that no focus particle attaches to wh-phrases. In the who’s line,
the lack of focus particles is due to a morphophonological constraint of this dialect:/ga/ does not appear after the long vowel /aa/ or /aQa/. The reason for the drop ofparticles in the when’s line of TG has not yet been made clear.

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4

Tomohide Kinuhata

B.

Mma=a
[jarabi-taa noo=ju=ga
jummi bu1]=garaGranma=TOP children-PL what=ACC=FOC read CONT=Q
sa-nnoo-ham.
know-NEG EVI-ACOP‘Grandma doesn’t know what the children are reading.’
C. Ja=a[mma=Qa
noo=nuhun=nu={du/ga}
2SG=TOP Grandma=NOM what=GEN book=ACC=FOC
jummi bu1]=tii=jas1dzi=ru
bu1?
read CONT=QUOT=TOP know=FOC CONT
‘Do you know what Grandma is reading?’
D. Uja ja=tigaa[mma=Qa
noo=ju=dujummi
father COP=COND Grandma=NOM what=ACC=FOC readbu1]=tii=jas1dzi=dubu1=padz1.
CONT=QUOT=TOP know=FOC CONT=CONJ‘Our father knows what Grandma is reading.’
E. Abaa[mma=Qa
noo=ju=dujummi
1SG.TOP Grandma=NOM what=ACC=FOC readbu1]=tii=jas1dzi=dubu1=sugandz1
CONT=QUOT=TOP know=FOC CONT=CONC say
taf=fanii-n.
OPT=TOP no-NEG‘I know what Grandma is reading but I don’t want to say it.’
Table 1 exhibits the following: Ga and du are obligatorily used for A andE respectively; The use of ga is superior to that of du in B and vice versa in
D; both ga and du are evenly used in C. This conﬁrms our expectation thatga is most used in A and least in E. Moreover, I construe the above gradable
distribution from A to E as indicating that du is used when the embedded clausedenotes a proposition whereas ga is used when it denotes a set of proposition.
This hypothesis is considered as a natural consequence of the fact that du and ga
are used respectively for declaratives and interrogatives in matrix clauses. The
distribution of ga and du in Table 1 can be elucidated from this viewpoint in thefollowing way.
In context E, the attitude holder, i.e. the speaker, envisages a particular
proposition since he/she knows the answer of the question. Thus, the use of duis obligatory. In context D, du is preferred because the attitude holder, not the
speaker in this case, has a particular proposition in mind. But D diﬀers fromE in that the speaker might not know the answer of the question. Therefore,
ga is exceptionally used as in the which’s line of KK. The attitude holder’sstance toward the answer of the embedded question is neutral in context C.
Depending on the speaker’s presumption of the knowledge of the attitude holder,
he/she uses ga or du. B and A can be explained by the mirror image of D
and E respectively. Particularly, there is no possibility for du to appear in thecontext A. Under this context, the attitude holder=speaker does not associate
any particular proposition to the embedded clause, since he/she is uncertain- 111 -

Interpretation of Wh-words in Aza-Irabu

5

about the answer. Therefore, the semantic object of the embedded clause cannot
be a proposition, but is a set of propositions.
Another generalization drawn from Table 1 is that du is used only when thecomplement is marked by a quotative marker tii. While there are examples ofga appearing within the quotative marker, there is only one exception of duappearing in a clause marked by gara, i.e. the where’s line of B in Speaker TG.
It is not unreasonable to consider it as an exception, because this example isobtained after the speaker utters the sentences in the where’s line of C and,after the utterance, he corrected the relevant example replacing du with ga,
which suggests that the questionable sentence is generated by the inﬂuence ofthe preceding utterance. Granting that the use of du in the where’s line of B
is exceptional, the combination of focus particles and complementizers in AzaIrabu is schematized as in (7).
(7)

a. [....WH=ga....]=gara V.
b. [....WH=ga....]=tii V.
c. *[....WH=du....]=gara V.
d. [....WH=du....]=tii V.

Assuming that the status of gara as a question marker5 , the appropriateness of the pattern (7-a) naturally follows: Ga and gara cooperatively make thesentence questions. The acceptability of (7-b) is not surprising as well, because
quotative markers can embed questions as well as statements. What prevents
the pattern (7-c) from naturally occurring in the experiment? One reasonable
explanation is to attribute it to the mismatch between the meaning of du andthat of gara: Gara requires the clause to be a set of propositions as a question
marker; the sentence with du nonetheless denotes a proposition as a declarative
marker. This is reasonable because the hypothesis that the clause with du denotes a proposition also explained the distribution of focus particles in Table 1.
If this line of reasoning is on the right track, the clause marked by du in (7-d)
denotes a proposition though it includes a wh-word.
The above discussion allows us to rephrase the compatibility of du and whwords as a problem of how to interpret a wh-word used in a proposition. I
maintain in this article that wh-words in du-marked clauses are employed to hide
the value of expressions for some reasons. For example in E of (6), the speakerknows the proposition but, as the context indicates, he is reluctant to convey
the explicit information to the addressee, which leads him to use wh-words tohide the expression. In the case of D of (6), though the attitude holder knows
the answer, i.e. a proposition, the speaker might not know it, which forces him
to use wh-words to cover the value. In the next section, those uses of wh-words
in declaratives are analyzed to be made possible by the function of a quotative
marker tii, only by which the coexistence of du and wh-words is admitted.
5

Although I have not found an example of gara used in direct questions in AzaIrabu, the same form is adapted to mark self-addressed questions in many varietiesof Miyakoan.

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Tomohide Kinuhata

3

Account

Given the ungrammaticality of using du with wh-words in matrix questions, e.g.
(2), there would be two simple ways to account for the compatibility of wh-words
and du in some embedded clauses: One is to move du and the other to move whphrases, which result in interpreting them in diﬀerent places. It is untenable tomove du, however. Even if we move du to the matrix clause to circumvent thecoexistence with wh-phrases, there are examples in which matrix clauses are alsoquestions: Consider the use of du in C of (6).
I take an approach to move wh-phrases, using Karttunen’s theory of questions([8]). In [8], wh-phrases are treated in the same way as quantiﬁed expressions,which are introduced into a sentence through ‘rules of quantiﬁcation’ in Montague semantics ([11]). Thus, wh-phrases need to move higher than the sentence
they combine with.
The semantic object of the sentence which can be the complement of whphrases is called ‘proto-questions’ in [8]. In his theory, proto-questions are formedby an operator ‘?’; I assign the function of this operator to the question particle
ga in Aza-Irabu. Thus, the basic idea of forming wh-questions is to let wh-phrases
quantify into a set of propositions created by the particle ga.
To implement this idea, both wh-phrases and focus particles must move asin (8)6 , which is the LF representation of the example (1-c) and (2), repeatedhere as (9).
(8)

[ IP1 noo [ IP2 [ IP3 t8 =nu uti-taa ] {ga/du} ] ](9)

Noo=nu={ga/*du} uti-taa?
what=NOM=FOC fall-PST‘What did you drop?’

The trace of wh-phrases is interpreted as a variable with an index, which enables
ga to take a proposition as the argument as in (10).
(10)

6

(i)

a.
b.
c.

t8 utiM,w,g = λw .f all(x8 )(w )gaM,w,g = λp.λq[q(w) & q = p]gaM,w,g (t8 utiM,w,g ) = λq[q(w) & q = λw .f all(x8 )(w )]Aside from semantic interpretations, there is another evidence which advocates the
movement of focus particles. As shown in the following example, when a wh-phrase
resides within a syntactic island, ga cannot attach directly to that phrase and must
appear outside the island. This is straightforwardly explained if we assume movementof focus particles to the sentence periphery. Similar movement is assumed for focus
particles of Sinhala by [4] and [10].
[Ndza=nkai(=*ga) pi1 p1tu]=nu=ga
jamaQasa-ha-taa?
where=ALL(=FOC) go human=NOM=FOC many-ACOP-PST‘Wherei were there many [people that went ti ]?’- 113 -

Interpretation of Wh-words in Aza-Irabu

7

The result of applying ga to a proposition is a set of propositions as in (10-c),i.e. a proto-question, which will be combined with wh-phrases through Whquantiﬁcation Rule given in (11).
(11)

Wh-quantiﬁcation RuleIf α is a branching node with β ∈ PW H and γ ∈ PQ containing an
occurrence of ti for some integer i,then αM,w,g = {p|βM,w,g (λxi .[γM,w,g (p)])}

In Montague semantics, a set of phrases P for each category is deﬁned and rules
are applied to speciﬁc categories: The set of wh-phrases PW H and the set ofquestions PQ in this rule. Assuming that the semantic content of wh-phrases is
an existential quantiﬁer, we can correctly calculate the meaning of (9) with ga
attached.
(12)noo nu ga utitaaM,w,g={p|λP.∃x[P (x)](λy[λq[q(w) & q = λw .f all(y)(w )](p)])}={p|∃x[p(w) & p = λw .f all(x)(w )]}The denotation of (12) is a set of true propositions of the form ‘x fell.’Since du appears in declarative clauses in matrix contexts, it is reasonable toassume the semantics of du simply as in (13-a), which is applied to (10-a) as in(13-b).
(13)

a.
b.

duM,w,g = λp.pduM,w,g (t8 utiM,w,g ) = λw .f all(x8 )(w )Since the result is just a proposition, it fails to satisfy any part of the rule in(11): Hence, the ungrammaticality of (9) with du.
Let us now proceed to the subordinate clause. As noted in the previous section, the question constituted by ga can be embedded both by gara and tii (see(7)). Let us consider tii here as a simple complementizer whose semantic contribution is vacuous. Recall that gara, derived from ga, was regarded to function asa question marker. Here I assume the semantics of gara as an identity function
as in (15-a), i.e. a function from a set of propositions to a set of propositions.
These assumptions allow us to treat the embedded clauses marked by gara andtii uniformly as a set of propositions. An illustration of combining those embedded clauses with its matrix is given in (15), where gara is reduced to ‘ra’ for amorphophonological constraint mentioned in footnote 4.
(14)

Abaa
[noo=nu=ga
uti-taa]=ra sa-n.
1SG.TOP what=NOM=FOC fall-PST=Q know-NEG‘I don’t know what fell.’

(15)

a.
b.
c.
d.

garaM,w,g =λQst,t .Q
noo nu ga utitaa raM,w,g ={p|∃x[p(w) & p = λw .f all(x)(w )]}s1M,w,g =λQst,t .λxe .∃p ∈ Q[p(w) = 1 & doxw
x ⊆ p]
s1M,w,g (noo nu ga utitaa raM,w,g )(anM,w,g )= ∃p ∈ {p|∃x[p(w)& p = λw .f all(x)(w )]}[p(w) = 1&doxw
spkr ⊆ p]- 114 -

8

Tomohide Kinuhatae.

nM,w,g (abaa noo nu ga utitaa ra saM,w,g )= λq.¬q(∃p ∈ {p|∃x[p(w)&p = λw .f all(x)(w )]}[p(w) = 1&doxw
spkr ⊆ p])= ¬∃p ∈ {p|∃x[p(w)&p = λw .f all(x)(w )]}[p(w) = 1&doxw
spkr ⊆ p]

In (15-c), I follow [15] in the treatment of the predicate know (s1 in Aza-Irabu),
which takes a set of propositions and a subject, and returns true iﬀ the subject
believes a true proposition in that set.7 (15-d) is the result of s1 taking thecomplement and the subject, which is eventually in the scope of negation as in(15-e). If du appears in the clause marked by gara, as in the pattern of (7-c),the ungrammaticality is predicted, since a proposition like (13-b) cannot be theargument of gara in (15-a).
In problematic cases such as (16), wh-words must move out of the subordinate
clause, otherwise the semantic interpretation collapses due to the coexistence ofdu and wh-words in the same clause.
(16)

Abaa
[noo=nu=du
uti-taa]=tii=jas1dzi=du
1SG.TOP what=NOM=FOC fall-PST=QUOT=TOP know=FOC
bu1.
CONT
‘I know what fell.’

It is still not possible to apply Wh-quantiﬁcation Rule to (16) with the wh-phrase
dislocalted, since the matrix clause does not have a question denotation but isa declarative sentence. This forces us to posit a new rule which can combine
wh-words with some speciﬁc sorts of declarative sentences.
The basic idea behind the co-occurrence of wh-words and du is, as was argued
in the previous section, that while du denotes a proposition, wh-words are used tocover the term, for example, not to reveal it to the addressee. I embody this idea
in the notion of metalinguistic quantiﬁcation developed by [14]: His proposal isthat it is not only individuals but also expressions that can be quantiﬁed and thus
quantiﬁcation into quotational contexts is possible. The above idea is rephrased
using this notion as follows: While du expresses that the attitude holder knows
the proposition which constitutes the answer, wh-words are used to quantify theexpression of the focus of the answer to hide it.
In a model theoretic fragment which utilizes metalinguistic quantiﬁcation, a
new atomic type u, the type of expressions, must be added to the basic inventryof types e, t and s, and the model is modiﬁed accordingly. In addition, expressions
are also interpreted relative to Q, which assign entities in Du to variables overexpressions. Although [14] conceives that speciﬁc types of predicates directly takeexpressions, assuming, for example, separate lexical entries for the propositionalsay1 and the quotational say2 (cf. [12]):7

To simplify the discussion, I do not include exhaustivity in the interpretation ofembedded questions. See [6] and [2] for a method to induce exhaustive interpretationsin the Karttunen’s theory of questions.

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Interpretation of Wh-words in Aza-Irabu

(17)

a.
b.

9

say1 M,w,g,Q = λpst .λye .λw . report(p)(y)(w )say2 M,w,g,Q = λXu .λye .λw . utter(X)(y)(w ),I contend that, in Aza-Irabu, expressions are integrated into semantic compositions by way of tii, which leads us to assume separate entries for it: Tii1 forthe complementizer use and tii2 for the quotative use. Let < Φ ; α > an arbitrary linguistic object, where Φ is a phonological representation and α is a
syntactically structured object. Then, the function of each tii is described as in
(18).
(18)

a.
b.

tii1 (< Φ ; α >) = αtii2 (< Φ ; α >) = ΦNote that < Φ; α > itself does not belong to any type and thus is not an objectof meaning language.8 With the help of tii2 , the phonological content transforms
to ‘proto-expressions’, which will be the argument of quotative ndz12 (say2 ) by
the interpretation function  M,w,g,Q . I use   both for proto-expressions andexpressions which is a member of Du .
(19)

ndz12 M,w,g,Q (tii2 (<it’s sunny; [it [is [sunny  ]]] >)M,w,g,Q )= ndz12 M,w,g,Q ( it’s sunny M,w,g,Q )= λXu .λxe .λw .[utter(X)(x)(w)]( it’s sunny )= λxe .λw .[utter( it’s sunny )(x)(w )]Identifying the tii in (16) with tii2 gives us the following application.
(20)

tii2 (< noo nu du utitaa; [[noo ]du ]uti ] >)= noo nu du utitaa

In order for expressions like (20) to be the argument of s1 (know), the meaning of it must be ambiguated between the proposition-taking know and theexpression-taking know.
(21)

a.
b.

s11 M,w,g,Q = λpst .λye .[p(w) & doxwy ⊆ p]
M,w,g,Q= λXu .λye .∃pst [p(w) & doxw
s12 
y ⊆ p & p = Cont(X)]When s12 takes expressions X as in (21-b), the attitude holder believes the true
proposition p, which corresponds to the content of the expression X. Cont works
to make this correspondence based on the syntactic information α of < Φ; α >.9
The result of applying tii2 to a linguistic object constitutes a part of thesyntactic structure of the whole sentence. Since the wh-word noo is employed
here to quantify the expression, subscribed with u here, the phrase containing it
8[12] oﬀers a grammar which generates triples < Π ; Σ ; α : τ > where each member
represents phonological, syntactic and semantic content. He considers < Π ; Σ ; α :τ > itself also to be a member of Du , which is the crucial diﬀerence from theassumption adopted here.
9
If it is not possible to retrieve a propositional meaning from α, the function Cont is
undeﬁned.

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10

Tomohide Kinuhata

moves to a place where it can take a set of expressions as its argument. Thus the
LF representation of (16) looks like (22), in which the trace of the wh-phrase isindexed with an integer and a type ([14], [7]).
(22)
noou
abaa
 t<5,u> nu du utitaa  s1
Based on this LF representation, the meaning of (16) is appropriately calculated
as follows.
(23)

s12 M,w,g,Q (t<5,u> nu du utitaa M,w,g,Q )(anM,w,g,Q )= λXu .λye .∃p[p(w) & doxwy ⊆ p & p = Cont(X)](Y<5,u> nu du utitaa
)(spkr)= ∃p[p(w) & doxw
spkr ⊆ p & p = Cont(Y<5,u> nu du utitaa)]

Finally, we need a rule for metalinguistic quantiﬁcation to bind a variable
left in the proposition.
(24)

Metalinguistic (Wh-)Quantiﬁcation RuleIf α is a branching node with β ∈ PW H and γ ∈ Pt containing an
occurrence of t<i,u> for some integer i,then αM,w,g,Q = βM,w,g,Q (λX<i,u> .[γM,w,g,Q ])The denotation of the wh-phrase is an existential quantiﬁer as before, but diﬀers
from the previous version in that it quantiﬁes over expressions, being extracted
from inside expressions.
(25)

a.
b.

noou M,w,g,Q =λPut . ∃Xu .P (X)noou M,w,g,Q (λY<5,u> . abaa  t<5,u> nu du utitaa  s12 M,w,g,Q )= λPut . ∃Xu [P (X)](λYu . ∃p[p(w) & doxw
spkr ⊆ p & p = Cont(Y
nu du utitaa)])= ∃Xu ∃p[p(w) & doxw
spkr ⊆ p & p = Cont(X nu du utitaa)]

The formula in (25-b) predicts the use of du in the embedded clause since
it denotes a proposition p. Further, the account in this section predicts that duco-occurs with wh-words only within quotative clauses. This is because variablesof type u are created only by the quotative marker tii and metalinguistic quantiﬁcation by wh-phrases operates on this type of variables from the outside of
the clause.10
10

One might wonder whether the movement of wh-phrases is an indispensable part ofour analysis. Since wh-movement is required by the [8]’s treatment of wh-phrases as
quantiﬁers, it seems not impossible to get rid of the movement by adopting othertheories on questions ([5], [3] etc): For example, it is a viable option that noou translates to a variable over expressions in-situ and the existential closure takes place atthe top node. On the other hand, the change of the meaning of wh-phrases, i.e. the- 117 -

Interpretation of Wh-words in Aza-Irabu

411

Conclusion and further issue

In Aza-Irabu, focus particles agree with sentence types: Du with declaratives,
ru with yes/no interrogatives and ga with wh-interrogatives. In embedded questions, however, du can be used particularly when the speaker or the attitude
holder knows the answer of the question. In those examples, I considered theembedded clause to denote a proposition, a particular answer, and that whwords are used to hide the value of expressions. I materialized this idea in thenotion of metalinguistic quantiﬁcation.
Metalinguistic quantiﬁcation was ﬁrst proposed by [14] for the analysis of
wh-doublets in Japanese, which is similar to English such-and-such and so-andso. Wh-doublets in Aza-Irabu seem to have functions similar to Japanese ones,
appearing in quotational contexts as shown in (26).
(26)[Taru taru=ga niv-kam]=tii=du
c1mo=oidii
who who=NOM late-ACOP=QUOT=FOC heart=TOP out
bu-taa.
CONT-PST‘He complained that so-and-so is late.’

Diﬀerent from Japanese, wh-doublets in Aza-Irabu can be used in questions,
inducing plural interpretations (see [1] for extensive research of wh-doublets inYaeyaman): (27-a) is an example of direct questions and (27-b) is a plural version
of problematic examples.
(27)

a.

b.

Noo noo=nu=ga
uti-taa?
what what=NOM=FOC fall-PST‘What fell?’ (plural interpretation)Abaa[noo noo=nu=du
uti-taa]=tii=ja
1SG.TOP what what=NOM=FOC fall-PST=QUOT=TOP
s1dzi=dubu1.
know=FOC CONT‘I know what fell.’ (plural interpretation)

On the other hand, the use of wh-singlets in the context of (26) is not perfectaccording to the informants; they prefer proper nouns to wh-singlets in this context. But this is not a restriction imposed by the semantics of wh-singlets, but bypragmatics which requires explicit reference if there is only one individual. Thismeans that contextual adjustments make it possible for wh-singlets to function
like Japanese doublets. Though the precise condition of the use of wh-singlets
has not yet been fully clariﬁed, it is quite likely that the context in which theusual noo and noou , is essential. This is because any accurate analysis must accountfor the compatibility of du and wh-phrases in the subordinate clause and the incompatibility of them in the matrix clause simultaneously. Therefore, the explanation of
the incompatibility based on the meaning of du and wh-words automatically leads
to transforming the meanings of those items. This is exactly what is achieved in thispaper by the mechanism metalinguistic quantiﬁcation.

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12

Tomohide Kinuhata

speaker knows the proposition but is reluctant to inform it, c.f. (6)-E, is one ofthose adjusted contexts, which encourages the analysis of embedded questions
using metalinguistic quantiﬁcation.
Abbreviations
1sg = 1st person singular, 2sg = 2nd person singular, acc = accusative, acop
= adjectival copula, all = allative, conc = concessive, cond = conditional,
conj = conjecture, cont = continuous, cop = copula, evi = evidential, foc= focus, gen = genitive, neg = negation, nom = nominative, opt = optative,
pl = plural, pst = past, q = question, quot = quotation, sfp = sentence ﬁnal
particle, top = topic

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Language & Linguistic Theory 23. 1–51.
11. Montague, Richard. 1974. The proper treatment of quantiﬁcation in ordinary
English. In Richmond H. Thomason (ed.), Formal philosophy: Selected papers ofRichard Montague, 247–270. New Haven: Yale University Press.
12. Potts, Christopher. 2007. The dimensions of quotation. In Christopher Barker &Pauline Jacobson (eds.), Direct Compositionality, 405–431. Oxford: Oxford University Press.
13. Shimoji, Michinori & Thomas Pellard (eds.). 2010. An introduction to Ryukyuan
languages. Tokyo: Research institute for languages and cultures of Asia and Africa.
14. Sudo, Yasutada. 2013. Metalinguistic quantiﬁcation: Evidence from Japanese whdoublets. An unpublished manuscript. Paris; Institute Jean Nicod.
15. Uegaki, Wataru. 2015. Content nouns and the semantics of question-embedding.
Journal of Semantics. Advance Access published July 17, 2015.

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Interpretations of Embedded Expressives: A View fromthe Japanese Comparative Expressive Motto
Osamu Sawada⋆
Mie University 1577 Kurimamachiya-cho, Tsu city, Mie, 514-8507, Japan
sawadao@human.mie-u.ac.jp

Abstract. Recent studies of expressives have shown that when expressives likedamn are embedded in the complement of an attitude predicate, they can be either speaker-oriented or non-speaker-oriented (Amaral et al. 2007; Harris & Potts
2009). Amaral et al. (2007) and Harris and Potts (2009) have suggested that thisphenomenon is an instance of indexicality. In this paper, I will investigate theinterpretations of embedded expressives on the basis of new data in terms of theJapanese comparative expressive motto, and argue that the interpretation of theembedded expressive is not merely a matter of indexicality. More specifically, I
argue that (i) there can be a semantic shift from a conventional implicature to asecondary at-issue entailment at a clausal level in a non-speaker-oriented reading,and (ii) in some expressives, like the negative motto, a speaker-oriented reading
can arise only when there is an appropriate speaker-oriented modal in the mainclause.
Keywords: embedded expressives, motto, secondary at-issue entailment, projection via a modal support, consistency of a judge

1

Introduction

Potts (2005) has claimed that the meaning of expressives, such as bastard in (1), is aconventional implicature (CI) and that it is logically independent of “what is said”:(1) That bastard Kresge is famous.
opinion.)(Expressive/CI: Kresge is bad, in the speaker’s

However, recent studies have shown that when expressives are embedded in thecomplement of an attitude predicate, they can have either a non-speaker-orientation ora speaker-orientation (Amaral et al. 2007; Harris and Potts 2009; Tonhauser et al. 2013).
For example, it has been observed that while bastard in (2) is speaker-oriented, friggin’in (3) is construed as subject-oriented:⋆

I am grateful to Thomas Grano, Yusuke Kubota, Harumi Sawada, Jun Sawada, and the reviewers of LENLS for the valuable comments. Parts of this paper were presented at the International
Modality Workshop at Kansai Gaidai (2016), and I also thank the audience for their valuablecomments and discussions. This paper is based upon work supported by JSPS KAKENHI
Grant Number 26770140. All remaining errors are of course my own.

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(2) Sue believes that that bastard Kresge should be fired. (#I think he’s a good guy.)(Potts 2007)(3) (Context: The speaker likes mowing the lawn.) Monty said to me this verymorning that he hates to mow the friggin’ lawn. (Amaral et al. 2007)
Amaral et al. (2007) and Harris and Potts (2009) have informally suggested that
this phenomenon is an instance of indexicality. For example, Harris and Potts (2009)claimed that expressives (and appositives) are inherently underspecified for their orientation, and that there is a free variable for a judge (j) that is determined by context. Harrisand Potts (2009) further claimed on the basis of corpus and experimental evidence thatappositives and expressives are generally speaker-oriented, but certain discourse conditions can counteract this tendency (cf. Schlenker (2003, 2007) and Sauerland’s (2007)
semantic binding approach to a non-speaker-orientation).
In this paper, I will investigate the interpretation of embedded expressives on thebasis of new data in terms of the Japanese comparative expressive motto, and argue thatthe interpretation of the embedded expressive is not merely a matter of indexicality.
More specifically, as for the subject-orientation, I argue that (i) there can be a semantic
shift from a CI to a secondary at-issue entailment at clausal level in a non-speakeroriented reading. It will be shown that the semantic shift from a CI to a secondary
at-issue entailment is a general phenomenon and that it can also be observed in typicalexpressives.
As for the speaker-oriented reading, I will argue that in some expressives, like thenegative motto, a speaker-oriented reading can arise only when there is a modal in themain clause. I will argue that there is a specific type, a dependent projective content,
which requires consistency between at-issue and CI meanings including a judge. The
theoretical implication of this paper is that both semantic and pragmatic mechanisms
are involved in the interpretation of embedded expressives.

2

The Expressive Property of the Japanese Motto

2.1

The Degree and Negative Uses of Motto

Before investigating the interpretation of the expressive motto in an embedded context,
let us first discuss the meaning and use of the expressive motto in a non-embedded context. It has been observed in the literature that the Japanese comparative adverb motto
has two diﬀerent uses, namely a degree use and a negative/expressive use (Watanabe
1985; Sano 1998, 2004; Kinoshita 2001), as in (4):(4) Kono mise-no keeki-wa motto oishi-katta.
This store-GEN cake-TOP MOTTO delicious-PASTa. Degree reading: This store’s cake was {even/still far} more delicious than acontextual store’s cake.
b. Negative reading: This store’s cake was delicious. (Implied: It is not delicious now.)- 121 -

In the degree reading, the sentence is interpreted as an “elliptical” comparison. It conveys that although the given store’s cake and a contextual store’s cake were both delicious, the former was far more delicious. Thus, the degree motto has a positive meaning.
On the other hand, in the negative reading, motto conveys the speaker’s complaint
about the utterance context, i.e., the store’s cake is not delicious now. The phenomenon
we are going to focus on is this expressive (or negative) use. Let us consider the diﬀerence between the degree motto and the negative motto more closely.
2.2

The Meaning of the Degree Motto

The degree use of motto expresses an intensified comparison at the at-issue level and,in addition to this, there is a positive presupposition that the standard of comparison
satisfies the standard of an adjective (i.e., y is A). Consider the example in (5) with theexplicit standard yori PP: 1(5) Hanako-no keeki-wa Taro-no keeki-yori(-mo) motto oishi.
Hanako-GEN cake-TOP Taro-GEN cake-than-MO MOTTO delicious‘Hanako’s cake is {still far/even} more delicious than Taro’s cake’
We can analyze the meaning of sentence (5) as having two components, namely an
at-issue component and a presupposition component, as in (6):(6) The meaning of (5)a. At-issue: Hanako’s cake is much more delicious than Taro’s cake.
b. Presupposition: Taro’s cake is delicious.
We can then formalize the meaning of the degree motto as in (7), in which theunderlined part represents the presupposition component:(7) [[mottoDEGREE ]] = λg⟨d,⟨e,⟨i⟨s,t⟩⟩⟩⟩ λyλxλtλw : ∃d[d ≽ S tand ∧ g(d)(y)(t)(w)].
max{d|g(d)(x)(t)(w)} >!!max{d|g(d)(y)(t)(w)}
In the case of an elliptical degree reading, like that in (4b), a standard of comparison(the second argument) is implicit, so we need to posit a slightly diﬀerent lexical item
for the degree motto. However, essentially the same semantic mechanism is involved in
the case of the elliptical comparative (see Sawada (2014) for a detailed discussion).
2.3

The Negative Use of Motto is a CI/Expressive

Let us now consider the meaning of the negative motto, which is the main focus of thispaper. Sawada (2014 ) claims that the expressive/negative use of motto is an expressive
and that it conventionally implies that “the expected degree is much greater than acurrent degree,” as in (8):1Note that there is no negative reading in (5). If there is an explicit standard of comparison, we
cannot get a negative reading (Sawada 2014).

- 122 -

(8) Taro-wa (mukashi-wa) motto majime-da-tta.
Taro-TOP old days-TOP MOTTO serious-PRED-PAST
At-issue: Taro was serious.
Expressive (CI): The degree of seriousness of Taro in the past is much greaterthan the current degree. (Expected degree = the past degree.) (=> Taro is notserious now (conversational implicature)
Sawada (2014) then claims that the speaker’s negative attitude arises from the gap between the expected degree and the current degree (as a conversational implicature ).
The comparative meaning triggered by the negative motto is a CI because it is independent of “what is said” (Grice 1975; Potts 2005). In (8), the expressive meaning is notwithin the semantic scope of the past tense. Furthermore, the expressive motto can also
appear in an imperative, a conditional clause, or a modal sentence, but its expressive
meaning cannot be within the semantic scope of these operators. For example, in (9),the negative motto is clearly outside the scope of the imperative:(9) Motto hayaku hashi-re!
(imperative)
MOTTO fastrun-IMPERATIVE
a. Run even faster! (Degree reading)b. Run fast! The expected speed of running is much higher than the current
speed. (Implied: You are running slowly now.) (Negative reading)
Regarding the compositionality of the negative motto, Sawada (2014) claims thatthe negative motto is mixed content (McCready 2010; Gutzmann 2011) in that it has
both an at-issue meaning and a CI meaning, as shown in (10)(The left side of  is the
at-issue component and the right side of  is the CI component):2(10) [[mottoEXPRES S IV E ]] : ⟨Ga , ⟨ea , ⟨ia , ⟨sa , ta ⟩⟩⟩⟩ × ⟨Ga , ⟨ea , ⟨ia , ⟨sa , t s ⟩⟩⟩⟩ =λgλxλtλw.∃d[d ≽ S T AND ∧ g(d)(x)(t)(w)]λgλxλtλw.max{d|g(d)(x)(t)(w)} ≻!!max{d|g(d)(x)(t0 )(w0 )} (where t0 = current time, w0 = the actual world)
Roughly speaking, in the at-issue component, motto denotes that the degree associatedwith the gradable predicate is above a certain standard. In the CI component, it conventionally implies that the expected degree is far greater than the current degree.

3

Interpretations of Embedded Motto: Some Puzzling Facts

Let us now consider the interpretation of the embedded motto. Although previous studies have focused only on non-embedded cases of the negative motto, it has several puzzling properties in terms of its interpretation in an embedded environment.
2

Superscript c is a CI type and superscript a is an at-issue type (Potts 2005). Superscript s is a
type for a CI expression interpreted by a resource sensitive application (McCready 2010).

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3.1

Puzzle 1

First, the expressive meaning triggered by motto is interpreted as at-issue if it is embedded under an attitude predicate and has a subject orientation as in (11): 3(11) (Negative/expressive reading)Taro-wa motto isshoukenmei benkyoo-si-nakerebanaranai-to omo-tta.
Taro-TOP MOTTO seriously
study-do-must-that
think-PAST
At-issue: Taro thought that he must study hard.
Expressive (subject-oriented): Taro considered that the expected degree ofseriousness of his study was much greater than the “current degree in the past.”
The expressive meaning in (11) is at-issue because it is within the semantic scope of thepast tense; it relates to Taro’s past feeling. Notice, however, that the expressive meaning
triggered by motto is not within the semantic scope of the embedded deontic modal
nakerebanaranai ‘must.’ What does this mean?
3.2

Puzzle 2

A second puzzling characteristic of the embedded motto is that it can actually havespeaker-orientation if a deontic modal occurs in the main clause:(12) Taro-wa motto isshoukenmei benkyoo-si-nakerebanaranai-to
Taro-TOP MOTTO seriously
study-do-must-thatomou-bekida.
think-should
At-issue: Taro should think that he must study hard.
Expressive 1 (subject-oriented): For all worlds w” compatible with the rulein w0 and for all worlds w’ compatible with Taro’s beliefs in w”, the expected
degree of seriousness of Taro’s study is much greater than the current degree
for Taro in w’.
Expressive 2 (speaker-oriented/CI): The expected degree of seriousness of
Taro’s study is much greater than the current degree for me.
The above asymmetry between (11) and (12) clearly shows that in the case of the expressive motto, the determination of a perspective is not merely a matter of context.

4

The Empirical Diﬀerence between Speaker-Oriented and
Non-Speaker-Oriented Readings

How can we explain the above facts regarding the subject-oriented and speaker-oriented
readings? One might think that the speaker-oriented reading in the embedded motto3

Note that there is also a degree reading in (11), i.e. ‘Taro thought that he must study evenharder (than now).’ In the degree reading, there is a ‘positive’ presupposition that Taro has
already studied hard. This clearly contrasts with the negative reading. Because the main focus
is on the interpretation of the embedded expressive, we will not discuss the degree reading.

- 124 -

arises purely pragmatically because of the presence of the deontic modal bekida, i.e.,
speaker-orientedness pragmatically arises in addition to subject-orientedness. However,
the two tests set out below clearly show that both speaker-oriented and subject-oriented
readings exist in the logical structure.
First, if we add the discourse particle koo ‘like’ between the expressive motto andan adjective, the sentence only has a speaker-oriented reading, as in (13):(13) (The example with the discourse particle koo ‘like’)
Taro-wa motto koo sikkarisita ronbun-o kaka-nakerebanaranai-to
Taro-TOP MOTTO like solid
paper-ACC write-must-thatomou-bekida.
think-should
At-issue: Taro should think that he must write a solid paper.
Expressive (speaker-oriented, CI): The expected degree of solidness is muchhigher than the current degree for me.
In (13), the particle koo is used parenthetically to signal that the “speaker” is in the
middle of thinking about what an appropriate adjective would be. The function is similar
to that of the English like.
The second test regarding the distinction between a speaker-oriented and a subjectoriented reading is the insertion of the reflexive zibun ‘self.’ H. Sawada (1993) claims
that if a reflexive zibun occurs in the embedded clause, the perspective of the embedded
clause has to be the antecedent of zibun (i.e., the subject of the entire sentence). If weinsert the reflexive zibun in the embedded clause, only a subject-oriented reading ispossible, as in (14):(14) (The example with zibun ‘self’)Taro-wa motto jibun-wa sikkarisita ronbun-o kaka-nakerebanaranai-to
Taro-TOP MOTTO self-TOP solid
paper-ACC write-must-thatomou-bekida.
think-should
At-issue: Taro should think that he must write a solid paper. Expressive (subjectoriented): For all worlds w” compatible with the rule in w0 and for all words
w’ compatible with Taro’s beliefs in w”, the expected degree of seriousness of
Taro’s study is much greater than the current degree for Taro in w’.

5

Analyses

5.1

Subject-Oriented Reading of the Negative Motto: From a CI to a Secondary
Entailment

Let us now try to explain the first puzzle above. In the previous section, we observed thatwhen the negative motto is embedded under an attitude predicate, its meaning becomes
at-issue, as in (15):(15) (The negative motto = always subject-oriented)- 125 -Taro-wa motto isshoukenmei benkyoo-si-nakerebanaranai-to omo-ta.
Taro-TOP MOTTO seriously
study-do-must-that
think-PAST
At-issue: Taro thought that he must study hard.
Expressive (subject-oriented): Taro considered that the expected degree ofseriousness of his study was much greater than the “current degree in the past.”
The expressive meaning in (15) is at-issue because it is within the semantic scope of thepast tense; it relates to Taro’s past feeling. Notice, however, that the expressive meaning
triggered by motto is not within the semantic scope of the embedded deontic modal
nakerebanaranai ‘must.’ I propose that a semantic shift exists from a CI to a secondary
entailment, as set out in (16):(16) Shifting from a CI to a secondary entailment: A sentence S , which consistsof an at-issue meaning of type ta and a CI meaning of type tc (or type t s ), canshift into an at-issue product type ⟨ta × ta ⟩ if and only if, S is embedded underan attitude predicate and the judge of S is the attitude holder of the predicate(where the first ta is a primary entailment and the second ta is a secondary
entailment.)
The secondary entailment is at-issue but is not a primary at-issue meaning (Potts
2005). The embedded negative motto is an expressive and it conveys a subject’s attitude,
similar to the non-embedded negative motto.
The crucial point of this shift is that it applies at the root level of an embedded
clause. Before the semantic shift applies at the root of the embedded clause, the expressive behaves as a CI triggering expression and it cannot be scoped over by any logicaloperators. This idea is supported by the fact that in (6 ), motto is not within the semantic
scope of the embedded nakerebanaranai ‘must.’
Let us now analyze the meaning of the subject-oriented reading of (17), which is
ambiguous between the subject-oriented reading and the speaker-oriented reading.
(17) Hanako-wa kono mise-no keeki-wa motto oishi-katta-to
Hanako-TOP this store-GEN cake-TOP MOTTO delicious-PAST-that
omo-bekida.
think-should
At-issue: Hanako should think that this store’s cake was delicious.
Expressive 1 (subject-oriented, secondary at-issue): For all worlds w” compatible with the rule in w0 and for all worlds w’ compatible with Hanako’s
beliefs in w”, the expected degree of deliciousness of this store’s cake is muchhigher than the current degree for Hanako in w’.
Expressive 2 (speaker-oriented, CI): The expected degree of deliciousness of
this store’s cake is much higher than the current degree for me.
Inside the embedded clause, the negative motto behaves as a CI. The following
figure shows the logical structure of the embedded clause: 44

Technically, the meaning of the negative motto and at-issue elements are combined via mixed
application (McCready 2010; Gutzmann 2011):- 126 -

(18) The logical structure of the embedded clauseEmbedded S:∃d[d ≽ S T AND ∧ delicious(this store′ s cake)(past)(w0 ) = d]•max{d|delicious(this store′ s cake)(past)(w0 ) = d} >!!
max{d|delicious(thisstore′ scake)(t0 )(w0 ) = d} f or j

λw.∃d[d ≽ S T AND ∧ delicious(this store′ s cake)(past)(w) = d] w0
λw.max{d|delicious(this store′ s cake)(past)(w) = d} ≻!!
max{d|delicious(this store′ s cake)(t0 )(w0 ) = d} f or j

λtλw.∃d[d ≽ S T AND ∧ delicious(this store′ s cake)(t)(w) = d]λtλw.w.max{d|delicious(this store′ s cake)(t)(w) = d} >!!
max{d|delicious(this store′ s cake)(t0 )(w0 ) = d} f or jkatta ‘past’

λxλtλw.∃d[d ≽ S T AND ∧ delicious(x)(t)(w) = d]λxλtλw.max{d|delicious(x)(t)(w) = d} ≻!!
max{d|delicious(x)(t0 )(w0 ) = d} for jkono mise-no keeki‘this store’s cake’mottooishi‘delicious’
λgλxλtλw.∃d[d ≽ S T AND ∧ g(d)(x)(t)(w)]
λdλxλtλw.delicious(x)(t)(w) = dλgλxλtλw.max{d|g(d)(x)(t)(w)} ≻!!max{d|g(d)(x)(t0 )(w0 )}After the computation is complete, both the at-issue and CI meanings are gathered
via parse tree interpretation, as in (19):(19) Parsetree interpretation (McCready 2010)(cf. Potts 2005)
Let T be a semantic parsetree with the at-issue term α : σa on its root node,and distinct terms β1 : t{c,s} , ..., βn : t{c,s} on nodes in it. Then, the interpretationof T is the ⟨[[α : sigmaa ]], [[β1 : t{c,s} ]], ..., [[βn : t{c,s} ]]⟩ (Based on McCready2010: 32)
At this point, the speaker-oriented reading and the subject-oriented reading are thesame in terms of meaning, as shown in (20):(20) The final interpretation of the embedeed clause via parsetree interpretation⟨∃d[d ≽ S T AND ∧ delicious(this store′ s cake)(past)(w0 ) = d] : ta ,max{d|delicious(this store′ s cake)(past)(w0 ) = d} >!!max{d|delicious(this store′ s cake)(t0 )(w0 ) = d} f or ji : t s ⟩(i)α(γ)β(γ) : τa × υ s
αβ : ⟨σa , τa ⟩ × ⟨σa , υ s ⟩ γ : σa

- 127 -

However, after the parse tree interpretation, in the subject-oriented reading, the semantic shift from a CI to a secondary entailment applies, as shown in (21):(21) After the semantic shift from CI to a secondary entailment⟨∃d[d ≽ S T AND ∧ delicious(this store′ s cake)(past)(w0 ) = d],max{d|delicious(this store′ s cake)(past)(w0 ) = d} >!!max{d|delicious(this store′ s cake)(t0 )(w0 ) = d} f or ji ⟩ : ⟨ta × ta ⟩This meaning then interacts with the elements in the main clause. The figure in (22)shows the entire logical structure of sentence (17)(=subject-oriented reading):(22) Interpretation of the entire sentence (subject-oriented reading)t0
beki ‘should’
Hanako j
Embedded S λt[... motto j ...]

λw

omou ‘think’

The denotations of omou ‘think’ and beki ‘should’ are shown in (23) and (24):(23) The denotation of omou ‘think’
λp<s<i<t×t>>> λxλtλw∀w′ compatible with x′ s belie f s in w : p(w′ )(t) = 1(24) The denotation of beki ‘should’
λp<s<i<t×t>>> λt.∀w′′ compatible with the rules in w0 : p(t)(w′′ ) = 1 f or j
If we put everything together, we get the following meaning in (25) as a final meaning:(25) Final part of derivation (subject-oriented reading)
For all worlds w” compatible with the rule in w0 and for all worlds w’ compatible with Hanako’s beliefs in w”:⟨∃d[d ≽ S T AND ∧ delicious(this store′ s cake)(past)(w0 ) = d],max{d|delicious(this store′ s cake)(past)(w0 ) = d} ≻!!max{d|delicious(this store′ s cake)(t0 )(w0 ) = d} f or jT aro ⟩ at t0 in w′ = 1 f or, j (= speaker)
One might propose that the shifting from a CI to a secondary at-issue entailment occurs
at the lexical level. However, such an approach is problematic. As the above examples
show, the embedded motto behaves as a CI inside the embedded clause. This seems to
be natural, considering that it is the “expressive” feeling of a subject.
5.2

The Case of Subject-Oriented Reading in the English Expressives

The shift from a CI to a secondary entailment is pervasive in natural language and canalso be observed in typical embedded expressives. (26) clearly shows that the embedded
friggin’ is within the semantic scope of the past tense:- 128 -

(26) (Subject-oriented reading, friggin’ = Monty’s perspective)
Monty said to me two years ago that he hated to mow the friggin’ lawn, butnow, he doesn’t mind. (Subject-oriented reading)5
On the subject-oriented reading, friggin’ has to be within the scope of the matrix
tense. On the sequence-of-tense reading, which is the most salient, the time of Monty’s
speech corresponds with the time of Monty’s hating, i.e., the time at which Monty had
a negative attitude toward the lawn, as in (27).
(27) Monty said to me two years ago that he hated to mow the friggin’ lawn, butnow, he doesn’t mind. (embedded clause = past tense)
The important point, however, is that Monty’s attitude is an expressive; it relates toMonty’s attitude in the past. Thus, it is reasonable to consider that it is not a primaryat-issue.
The question arises as to how we might analyze the meaning of the embedded friggin’, as in (28), which is similar to (3). It seems that the interpretation of embedded
friggin’ becomes complicated if the embedded clause has present tense.
(28) (Subject-oriented reading, friggin’ = Monty’s perspective)
Monty said to me two years ago that he hates to mow the friggin’ lawn.
This is because this sentence has a “double access reading” (Ogihara 1996; Abush
1997, etc.), in which both a past situation and a present situation are relevant. Comrie (1985:115) has stated that (29b) is used “when the speaker is reporting a (real or
imaginary) illness which he believes still has relevance.”(29)

a. John said that he was ill.
b. John said that he is ill

This predicts that the expressive in (28) can be anchored to both the past and the present
if the embedded clause has present tense. This prediction is borne out. The expressive
friggin’ in (28) is obligatorily anchored both to the present and the past (i.e., obligatory
double access). This is supported by the fact the sentence in (30) sounds somewhat odd.
(30) ?? Monty said to me two years ago that he hates to mow the friggin’ lawn, butnow, he doesn’t mind. (embedded clause = present tense)
This fact is consistent with the hypothesis that subject-oriented embedded expressives
obligatorily give rise to the double access eﬀect when the embedded tense is present.
5.3

Speaker-Orientation of the Negative Motto: The Existence of Dependent
Projective Content

Let us now investigate the speaker-oriented reading of the embedded motto. The puzzle
was that the embedded negative/expressive motto can only be speaker-oriented if there
is a deontic modality in the main clause, as in (31):5Note that there is also a speaker-oriented reading in which the speaker has a negative attitude
toward the lawn.

- 129 -

(31) Hanako-wa kono mise-no keeki-wa mukashi-wa motto
Hanako-TOP this store-GEN cake-TOP old days-TOP MOTTO
oishi-katta-to
omo-tta.
(subject-oriented)
delicious-PAST-that think-PAST
At-issue: Hanako thought that this store’s cake was delicious.
Secondary at-issue: Hanako thought that the expected degree of deliciousness(i.e. the deliciousness in the past) was much higher than the current degree.)(32) Hanako-wa kono mise-no keeki-wa motto oishi-katta-to
Hanako-TOP this store-GEN cake-TOP MOTTO delicious-PAST-that
omo-bekida.
think-should (speaker-oriented/subject-oriented)
At-issue: Hanako should think that this store’s cake was delicious.
Expressive 1 (subject-oriented, secondary at-issue): For all worlds w” compatible with the rule in w0 and for all worlds w’ compatible with Hanako’s
beliefs in w”, the expected degree of deliciousness of this store’s cake is muchhigher than the current degree for Hanako in w’.
Expressive 2 (speaker-oriented, CI): The expected degree of deliciousness of
this store’s cake is much higher than the current degree for me.
This point is radically diﬀerent from a typical expressive like bastard. As we observed in the Introduction, bastard can be speaker-oriented even if there is no external
speaker-oriented element in the main clause, as in (33):(33) Sue believes that that bastard Kresge should be fired. (#I think he’s a good guy.)(Potts 2007)
How might we explain the “conditional” projective property of the embedded motto
shown in the previous section? I argue that the embedded motto is a dependent projective content. Namely, it can be speaker-oriented only when a deontic modal exists in themain clause because it requires that the judge of the motto is consistent with the judge
in the at-issue level. I posit such a constraint inside the lexical entry of motto, as in (34):(34) [[mottoEXPRES S IV E ]] : ⟨Ga , ⟨ea , ⟨ia , ⟨sa , ta ⟩⟩⟩⟩ × ⟨Ga , ⟨ea , ⟨ia , ⟨sa , t s ⟩⟩⟩⟩ =λgλxλtλw.∃d[d ≽ S T AND ∧ g(d)(x)(t)(w)]λgλxλtλw.max{d|g(d)(x)(t)(w)} ≻!!max{d|g(d)(x)(t0 )(w0 )} for j (where j is consistent with a judge in the at-issuelevel) (where t0 = current time, w0 = the actual world)
If there is no modal in the main clause, j of motto corresponds to the subject of
the sentence (the attitude holder). This is because the sentence merely describes the
subject’s thoughts. However, if there is a deontic modal in the main clause, motto canbe speaker-oriented because the modal bekida ‘must’ is a judge-sensitive expression(see also Stephenson (2007)), as shown in (37), and the judge variable of the embedded
motto can correspond to the judge of bekida:(35) [[bekida]] = λp⟨ia ,⟨sa ,ta ⟩⟩ λt∀w′ compatible with the rules in w0 : p(w′ )(t) = 1 for j- 130 -

Thus, motto can be anchored to either a speaker or a subject in the sentence withbekida. The following figure shows the logical structure of the entire sentence:(36) Interpretation of the entire sentence (speaker-oriented reading)t0

beki j‘should’

Hanakoomou ‘think’λw
λt

Embedded S

This store’s cake was delicious: ta•
The degree of deliciousness of the store’s cake is muchgreater than the current degree for j (=the speaker): t s

The following shows the final part of the derivation:(37) Final part of derivation (speaker-oriented reading)
For all worlds w”compatible with the rule in w0 and for all worlds w’ compatible with Hanako’s beliefs in w”: ∃d[d ≽ S T AND ∧ delicious(this store′ s cake)(past)(w0 ) = d] at t0 in w′ = 1) = 1 f or j speaker : ta•
max{d|delicious(this store′ s cake)(past)(w0 ) = d} ≻!!max{d|delicious(this store′ s cake)(t0 )(w0 ) = d} f or j speaker : t s
Note that the addition of the epistemic modality, such as kamoshirenai ‘may’ does
not help the embedded motto become speaker-oriented, despite the fact that it is also a
judge-sensitive expression (speaker-oriented), as is clear from (39):(38) [[kamoshirenai]] = λp⟨sa ,ta ⟩ .∃ w’ compatible with j’s knowledge in w0 : p(w’) =1 for j(39) Taro-wa motto isshoukenmei benkyoo-si-nakerebanaranai-to
Taro-TOP MOTTO seriously
study-do-must-thatomou-kamoshirenai.
think-should
At-issue: Taro may think that he must study hard.
Expressive (subject-oriented): For some worlds w’ compatible with Taro’s
knowledge in w0 , the expected degree of seriousness of Taro’s study is much

- 131 -

greater than the current degree for Taro in w’.
Why is it that the expressive motto cannot be speaker-oriented in (39). I would like topropose that this is because the meaning of the epistemic modality is not pragmaticallyconsistent with the expressive meaning of the negative motto. In the case of (32) the
deontic modality conveys a speaker’s complaint, and the negative motto also conveys a
judge’s complaint. Thus, proposing that the judge of motto and the judge of the deontic
modality are the same is natural. However, in the case of (33) no semantic consistency
exists between motto and the epistemic modality.

6

Conclusion and Theoretical Implications

In this paper, I investigated the interpretations of embedded expressives on the basisof new data, namely the Japanese comparative expressive motto, and argued that theinterpretation of the embedded expressive is not merely a matter of indexicality. Morespecifically, I argued that (i) there can be a semantic shift from a CI to a secondary
at-issue entailment at a clausal level in a non-speaker-oriented reading, and (ii) in some
expressives, like the negative motto, a speaker-oriented reading can arise only when
there is an appropriate speaker-oriented modal in the main clause.
What do these claims imply theoretically? I think that these claims theoreticallysuggest the interpretation of embedded expressives involves both semantic and pragmatic mechanisms. Harris and Potts (2009) contrast a configurational approach and acontextual approach and support the contextual approach:(40)

a. Configurational: The source of non-speaker-oriented readings of appositives and expressives is semantic binding: their content can be bound by
higher operators like attitude predicates, thereby shifting it away from the
speaker (Schlenker 2003, 2007; Sauerland 2007).
b. Contextual: The source of non-speaker-oriented readings of appositives
and expressives is the interaction of a variety of pragmatic factors. In general, these interactions favor speaker-orientation, but other orientations arealways in principle available, regardless of syntactic configuration (Potts
2007).

However, the phenomenon of the embedded negative motto suggests that both semantic and pragmatic factors are involved. In this paper, I proposed that there is asemantic shift from a CI to a secondary entailment in the interpretation of a subjectoriented reading. This is clearly non-contextual, but at the same time, it is not purely
semantic in that it maintains the meaning of subject-oriented expressives as secondary.
Furthermore, as for the speaker-oriented reading, I have proposed that there is a newclass of projective content, i.e. a dependent projective content. The new class of projective content is semantic in the sense that whether or not it can project depends on theexistence of a judge-sensitive element (i.e., a deontic modal in the case of motto). However, this dependency is also pragmatic in that the kind of external element it can support
as a projection of an embedded expressive is pragmatically determined by the extent to- 132 -

which the external element semantically matches with the CI meaning of motto. In thecase of the negative motto, an epistemic modal cannot support the projection of motto
because its meaning does not match with the CI meaning of the negative motto.
In this paper, I focused only on the Japanese motto and certain English expressives.
In future research, I would like to further investigate the interpretation of other embedded expressives and consider the variation from a broader perspective.

References
1. Abusch, Dorit. (1997). Sequence of tense and temporal de re. Linguistics and Philosophy 20:
1-50.
2. Amaral, Patricia, Craige Roberts, and Allyn Smith. (2007). Review of the logic of conventional implicatures by Chris Potts. Linguistics and Philosophy 30: 707-749.
3. Comrie, Bernard. (1985). Tense. Cambridge, UK: Cambridge University Press.
4. Grice, Paul. H. (1975). Logic and conversation. In Peter Cole and Jerry Morgan (eds.), Syntax
and Semantics, iii: Speech Acts, 43-58. New York: Academic Press.
5. Gutzmann, Daniel. (2011). Expressive modifiers & mixed expressives. Empirical Issues in
Syntax and Semantics 8, 123-141.
6. Harris, Jesee. A. and Potts, Christopher. (2009). Perspective-shifting with appositives and expressive. Linguistics and Philosophy 32: 523-552.
7. Kinoshita, Kyoko. (2001). Hikaku no fukushi motto ni okeru shukansei (The subjectivity inthe comparative adverb motto). Kokugogaku 52: 16-29.
8. McCready, Eric. (2010). Varieties of conventional implicature: evidence from Japanese. Semantics & Pragmatics, 3, 1-57.
9. Ogihara, Toshiyuki. (1996). Tense, attitudes and scope. Dordrecht, Netherlands: Kluwer Academic Publishers
10. Potts, Christopher. (2007). The expressive dimension. Theoretical Linguistics 33: 165-197.
11. Potts, Christopher. (2005). The Logic of Conventional Implicatures. Oxford: Oxford University Press.
12. Sano, Yukiko. (1998) Hikaku ni kakawaru teido fukushi ni tuite (On the comparison-relateddegree adverbs). Kokugogaku 195: 1-14.
13. Sano, Yukiko. (2004). Motto no hiteiteki youhou ni tuite (On the ‘negative usage’ of motto).
Nihongo kagaku 15: 5-21.
14. Sauerland, Uli. (2007). Beyond unpluggability. Theoretical Linguistics 33: 231―236.
15. Sawada, Harumi. (1993). Siten to syukansei (Viewpoint and Subjectivity). Tokyo: Hituzi
Syoboo.
16. Sawada, Osamu. (2014). An utterance situation-based comparison. Linguistics and Philosophy 37: 205-248.
17. Sawada, Osamu. (2010). Pragmatic Aspects of Scalar Modifiers. Doctoral Dissertation, University of Chicago.
18. Schlenker, Philippe. (2007). Expressive presuppositions. Theoretical Linguistics 33: 237―
245.
19. Schlenker, Philippe. (2003). A plea for monsters. Linguistics and Philosophy 26: 29―120.
20. Stephenson, Tamina. (2007). Judge dependence, epistemic modals, and predicates of personal taste. Linguistics and Philosophy 30: 487―525.
21. Tonhauser, Judith, David Beaver, Craige Roberts, and Mandy Simons. (2013). Toward ataxonomy of projective content. Language 89: 66-109.
22. Watanabe, Minoru. (1985). Hikaku no fukushi: ‘motto’ o tyuushinni (The comparative adverbs in Japanese: The case of motto). Bulletin of the Language Institute of Gakushuin University 8: 65-74.

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Non-canonical Coordination in theTransformational Approach
Oleg Kiselyov and Leo Tingchen Hsu
Tohoku University, Japanoleg@okmij.org

Abstract. Recently introduced Transformational Semantics TS formalizes, restraints and makes rigorous the transformational approach epitomized by QR and Transformational Grammars: deriving a meaning (inthe form of a logical formula or a logical form) by a series of transformations from a suitably abstract (tecto-) form of a sentence. TS generalizes
various ‘monad’ or ‘continuation-based’ computational approaches, abstracting away irrelevant details (such as monads, etc) while overcoming
their rigidity and brittleness. Unlike QR, each transformation in TS is
rigorously and precisely defined, typed, and deterministic. The restraintsof TS and the sparsity of the choice points (in the order of applying thedeterministic transformation steps) make it easier to derive negative predictions and control over-generation.
We apply TS to right-node raising (RNR), gapping and other instances of
non-constituent coordination. Our analyses straightforwardly represent
the intuition that coordinated phrases must in some sense be ‘parallel’,with a matching structure. Coordinated material is not necessarily constituent – even ‘below the surface’ – and we do not pretend it is. We
answer the Kubota and Levine challenge of analyzing RNR and gapping
without using directional types, yet avoiding massive over-generation.

1

Introduction

Non-canonical coordination, and in particular gapping, (2) provides
an unending stream of puzzles for the theory of semantics [8, 10]:(1) John gave a book to Mary and a record to Sue.
(2) I gave Leslie a book and she a CD.
(3) John gave a present to Robin on Thursday and to Leslie on Friday.
(4) Mrs. J can’t live in Boston and Mr. J in LA.
Interactions of coordination with scope-taking are particularly challenging: a competent theory needs to handle both narrow- and widescope reading of “a present” in (3) and the narrow- and wide-scope
coordination in (4).

- 134 -

Recently in [8, 9], Kubota and Levin put forward new analyses
of non-canonical coordination, applying hybrid categorial grammars
they have been developing. In contrast, the analyses in [6] use plain
old non-associative Lambek grammar. However, the main ideas of[6] are completely hidden behind thickets of complicated types andtheir interactions within a derivation. The intuition that coordinatedstructures must be parallel is thus lost in the details.
We present a new analysis of non-constituent coordination using
the more intuitive and less round-about framework TS (formerly
called AACG) [7], designed to take the ‘hacking’ out of tree-hacking.
TS lets us talk about QR and other transformations towards somesemantic form in a rigorous, formal, mostly deterministic way. We
remind of TS in §2.
Our analyses re-expose ideas from the earlier approach of [6], but
free them from the bondage of encoding. A notable feature of TS is
the absence of directional types. We use it to answer the challenge
posited by Kubota, Levin [10] and Moot (dubbed “the KLM problem” by Morrill): to analyze RNR within categorial-grammar–likeformalisms without directional types, while avoiding massive overgeneration.
One may categorize the various approaches to non-canonical coordination based on what exactly is being coordinated. Are complete
sentences being coordinated behind the scene, as in “John likes Bill”
and “Mary hates Bill” with “Bill” being later elided? Or perhaps sentences with holes are being coordinated, as in “John likes hypobj ”?
(as done in [6, 8, 9].) Or perhaps we regard “John likes” and “Maryhates” as constituents and coordinate as such (as in CCG). In thispaper we give another answer: we analyze “John likes and Mary hates
Bill” as the coordination of the complete clause “Mary hates Bill”
with the cluster “John” and “likes”. The types of the cluster components and their order guide the transformation that picks the neededmaterial from the clause “Mary hates Bill” to make the cluster thecomplete clause. The ‘picking transformation’ can be naturally supported within the existing setup of TS, using the same mechanism
used in [7] to analyze quantification and inverse linking. §3 makes
precise the intuition of ‘picking’ in the formal setting.
The structure of the paper is as follows. §2 reminds TS, in adifferent, clearer presentation. We then describe our approach to2- 135 -

coordination: transforming non-canonical one to the ordinary coordination of clauses. §4 discusses the related work that forms thecontext of our approach. The rigorous nature of TS makes it easierto carry analyses mechanically, by a computer. In fact, the analyses in the paper have been so programmed and executed. The
implementation, in the form of a domain-specific language embedded in Haskell – ‘the semantic calculator’ – is publicly available at
http://okmij.org/ftp/gengo/transformational-semantics/.

2

TS BackgroundTraditional Categorial Grammar approaches draw parallels between
proof systems and grammars: grammaticality is identified with theexistence of a derivation. It is rather challenging however to prove
the absence of a derivation, and to overview the space of possible
derivations in general.
TS (formerly, AACG) [7] in contrast pursues the computational
approach, harking back to Transformational Generative Grammars[2] of 1960s: Rather than trying to deduce a derivation, it tries to
induce the meaning (the logical formula) by applying a sequenceof precisely and formally defined transformations to a suitably abstract form of a sentence. The latter abstracts away the case and thenumber agreement, declination, etc. The transformations are deterministic; the order of their applications is generally not (there may
still be dependencies between particular transformations imposing
the order). The transformations are partial: the failure is taken as
ungrammaticality of the original sentence.
Formally, TS deals with term languages that represent typed finite trees. Each T-language is a set of well-typed terms built fromtyped constants (function symbols) c. Types are
Base types
υ
T-Types σ ::= υ | σ → σ
The set terms d is then inductively defined as: (i) each constantc of the type σ is a term; (ii) if c has the type σ1 → σ and d isa term of type σ1 , then c d is a term of type σ; (iii) nothing else
is a term. The set of constants and their types is a (multi-sorted)
algebraic signature; A T-language is hence a term language over thesignature, which defines the language.
3- 136 -

Table 1 shows three sample languages. TS with the single basetype string and numerous constants "John", "greet", "every", etc.
of that type describes the surface, “phonetic”, form of a sentence.
The constant - · - : string → string → string (usually written as an infix operation) signifies string concatenation. The language TA whosetypes are familiar categories represents the abstract form. TL is thelanguage of formulas of predicate logic, which describe the meaningof sentences. The (infinite) sets of constants varx , vary , . . . and the

υ
TS

TA

TL

c
·: string → string → string"John" , "greet" , "every" , . . . : string

string

John: N P
participant: N
greet: T V
cl: N P → V P → S
argp: T V → N P → V P
ppadv: V P → P P → V P
everyx , everyy , az : N → N P
varx , vary , . . . : N P
Ux , Uy , . . . , Ex , Ey , . . . : N → S → S

S, N P, N, V P, P P, T Ve, tconj , disj , . . . : t → t → t
john: eparticipant: e → t
greet: e → t → t
app: (σ1 → σ) → σ1 → σ∀x , ∃y : t → t
x, y, z, . . . : eTable 1. Signatures of various T-languages

corresponding Ux , . . . and Ex , . . . represent (to be) bound variables
and their binders. Unlike the conventional (lambda-bound) variables, they are not subject to substitution, α-conversion or captureavoidance. TL likewise has constants x, y, z, . . . of the type e andthe corresponding sets of constants ∀x , ∀y , . . . , ∃x , ∃y , . . . intended asbinders.
4- 137 -

As a way to introduce TS we show a sample analysis of quantification on “John greeted every participant”. The sample sentence inthe language TA has the form
cl john (argp greet (everyx participant))
The constant cl combines an N P and a V P into a clause. (Likewise,
argp attaches an argument to a verb and ppadv attaches a prepositional phrase as a VP complement.) Quantifiers are uniquely labeled
by x, y, z, etc. We assume it is the job of a parser to uniquely label
the quantifiers in the abstract form.
The meaning is derived by applying a sequence of transformations to a TA term. The transformation LU x gets rid of everyx , introducing varx and Ux instead. The transformation is context-sensitive.
Therefore, we first define a context C[], a term (tree) with a hole:C[] = [] | cl C[] d | cl d C[] | argp d C[] | ppadv C[] d | ppadv d C[]
where the meta-variable d stands for an arbitrary term. In words: acontext is the bare hole [] or a clause (the cl term) that contains ahole in the subject or the predicate, or a complemented VP with thehole in the head or the complement, etc. We further distinguish two
subsets of contexts Ccl [] and Cncl []:Ccl [] = cl Cncl [] d | cl d Cncl []Cncl [] = [] | argp d Cncl [] | ppadv Cncl [] d | ppadv d Cncl []Intuitively, Ccl [] is the smallest context that has a hole within aclause.
The transformation LU x can now be stated as follows
LU x [Ccl [everyx dr ]] 7→ LU x [(Ux dr )] LU x [Ccl [varx ]]The rule is written in the form reminiscent of extended top-downtree transducers: whenever a pattern, e.g., Ccl [everyx dr ], matches abranch of a tree, the branch is replaced with (Ux dr ) Ccl [varx ] andthe transformation is repeated on the branches. Here, dr is a patternvariable that stands for an arbitrary subterm (tree branch). That is,
Ccl [everyx dr ] on the left hand-side of the rule matches a tree that
contains, somewhere inside, a sub-expression of the form everyx dr (abranch headed by everyx ). On the right-hand side of the rule, Ccl [varx ]is the same tree in which everyx dr subterm has been replaced withvarx . If a tree does not match the left-hand side of any LU x clause,
the transformation is applied to the child branches.
5- 138 -

Our example matches the left-hand side of LU x immediately: drmatches participant and Ccl [] is john (argp greet []), The result of thetransformation(Ux participant) (cl john (argp greet varx ))is in effect the Quantifier Raising (QR) of “every participant”, but
in a rigorous, deterministic way. The intent of the new constants
should become clear: Ux is to represent the raised quantifier, andvarx its trace. Unlike QR, the raised quantifier (Ux participant) lands
not just on any suitable place. LU puts it at the closest boundary
marked by the clause-forming constant cl. LU , is type-preserving:it maps a well-typed term to also a well-typed term. Again unlike
QR, we state the correctness properties such as type-preservation.
The type preservation is the necessary condition for the correctnessof the transformations. Finally, to derive the meaning we apply thetransformation Lsem that produces the logical formula (a term in thelanguage TL )∀x app participant x ⇒ (greet x john)
by replacing john, etc. with the corresponding logical constants andUx with the universal quantifier.

3

Coordination in TS

We now apply TS to the analysis of (non-canonical) coordination.
As a warm-up, we take the non-problematic “John tripped and fell,”which is an example of the conventional VP coordination. We analyze
it differently, however, as ‘left-node raising’ so to speak, to introduce
the technique to be later used in right-node raising (RNR), argument
cluster coordination (ACC) and gapping 1 .
The abstract form of our example isandS,NP (cl john tripped) fell
The new constant andS,NP has the type S → N P → S. As common,
we assume a whole family of constants andX,Y of different types. The
1

We may even analyze NP coordination as a sort of RNR: after all, “John and Maryleft” can have the meaning of the conjunction of truth conditions of “John left”
and “Mary left”. Certainly, “John and Mary left” may also mean that “John andMary”, taken as a group, left. In the later case, the group can be referred as “they”.
Our analysis applies to the former (conjunction) case but not the latter. Hence weposit that ‘and’ is not only polytypic but also polysemic.

6- 139 -

constant andS,NP is not in the domain of Lsem . Therefore, to be able
to derive the logical formula, we have to transform it away. The
following transformation La does that:La [andS,NP (cl dN P dV P ) d] 7→
and La [(cl dN P dV P )] La [(cl d dV P )]The rule again is written in the form of extended top-down treetransducers: whenever the pattern on the left-hand side of a rulematches a branch of the tree, the branch is replaced with the righthand-side of the rule. Again, d with various indices are meta-variables
that stand for arbitrary subterms (tree branches). Applying the rule
to our TA term transforms it toand (cl john tripped) (cl john fell)
where and is the ordinary coordination, of the type S → S → S,
which can be given the meaning of propositional disjunction andwhich hence is in the domain of Lsem . The result is straightforward
to transform to a logical formula TL .
Our next example is a proper RNR: “John likes and Mary hates
Bill”, whose abstract form isand(NP,TV),S (john, like) (cl mary (argp hate bill))
We have added to TA tuples (d, d) and tuple types (σ, σ). The constant and(NP,TV),S has the type (N P, T V ) → S → S. Whereas (cl mary(argp hate bill)) is the complete sentence, (john, like) is certainly not.
It is not even a constituent; it is just a sequence of words: a cluster.
Since we added to TA tuples and new constants, we may need toextend our earlier transformation rules, specifically, Lsyn for transforming into the surface form of the sentence TS :Lsyn [and(NP,TV),S d1 d2 ] 7→ Lsyn [d1 ] · "and" · Lsyn [d2 ]Lsyn [(d1 , d2 )]7→ Lsyn [d1 ] · Lsyn [d2 ]Applying Lsyn to our TA clearly gives “John likes and Mary hates
Bill”. The ‘phonetic’ transformation is dull and uninteresting, incontrast to the higher-order phonetics of [8].
Let us derive the meaning, the TL formula, from the same TA
term. Before we can apply Lsem we need to transform away and(NP,TV),S ,which is not in the domain of the latter transformation. We extend
the La with a new clause:
La [and(NP,TV),S (d1 , d2 ) (cl d C[argp d4 d5 ])] 7→
and La [(cl d1 (argp d2 d5 ))] La [(cl d C[argp d4 d5 ])]7- 140 -

where d1 , d, d5 have to be of the type N P and d2 and d4 of thetype T V . The transformation is context-sensitive and type-directed.
It may be regarded as matching of (d1 , d2 ) against the complete
sentence (the second argument of and(NP,TV),S ). The matching is determined by the type of and(NP,TV),S . The parallel structure of thecoordination is clearly visible.
Analyses of RNR without directional types (e.g, using ACG) run
into trouble of over-generating “*John likes Bill and Mary hates”.
Although we can write the abstract form for that sentence as well:andS,(NP,TV) (cl john (argp like bill)) (mary, hate)
we do not provide the La transformation with the constant andS,(NP,TV) .
Since it remains uneliminated, Lsem cannot be applied and the meaning cannot be derived. In TS, transformations are partial and are notguaranteed to always succeed. The original sentence is considered
ungrammatical then.
The same transformation idea also works for gapping and argument cluster coordination (ACC). Take for example, “Mary liked
Chicago and Bill Detroit”, or, in the abstract form:
andS,(NP,NP) (cl mary (argp liked chicago)) (bill, detroit)
The transformational rule involving the constant andS,(NP,NP) thatpicks a suitable subterm that can relate two NPs from the left conjunctLa [andS,(NP,NP) (cl d C[argp d4 d5 ]) (d1 , d2 )] 7→
and La [(cl d C[argp d4 d5 ])] La [(cl d1 (argp d4 d2 ))]turns our TA term toand (cl mary (argp liked chicago)) (cl bill (argp liked detroit))with the clear meaning. The examples (1) and (2) of §1 are dealt withsimilarly. One may observe that the analysis of gapping is nearly thesame as that of VP coordination, used in the warm-up example.
The interaction of non-canonical coordination with quantification
is not much different from that of the ordinary coordination of twoclauses. For example, take (3) of §1, whose abstract form isandS,(PP,PP)(cl speaker (ppadv (ppadv (argp gave (ax present)) (to robin))(on thu)))(to leslie, on fri)contains two components to be eliminated by transformations: andS,(PP,PP)
and the QNP (ax present). The latter is to be handled by LE , which
8- 141 -

is analogous to LU but for the existential quantifier. The transformations La and LE can be applied in either order, which corresponds
to the wide- and narrow-scope–readings of (3). The narrow scope
happens when La goes first, producing
and(cl speaker (ppadv (ppadv (argp gave (ax present)) (to robin))(on thu)))(cl speaker (ppadv (ppadv (argp gave (ax present)) (to leslie))(on fri)))
In summary, we have presented the uniform analysis of both thecanonical and non-canonical coordination, reducing the variety of
coordination (VP, RNR, ACC, Gapping) to the choice of the coordinating constants andS,X or andX,S that adjoin material (often justa cluster of words) to a sentence. The transformation rules driven
by the constants pick the pieces from the sentence to complete thematerial to a clause. There remains a question of a general principle/pattern that governs the choice of the constants. For example,
the fact that in English the coordinated sentence appears on the rightfor RNR but on the left for ACC and Gapping boils down to the presence of and(NP,TV),S and andS,(NP,NP) and the absence of andS,(NP,TV)
and and(NP,NP),S . In contrast, one may say that this fact ‘falls out’ asa consequence of like-category coordination analyses in directional
categorial grammars. One may also say that the like-category coordination is itself a postulate, which does not come from any generalprinciple, but does have significant empirical justification. Like any
empirical principle, it has exceptions: unlike-category coordination,e.g., “John saw the facts and that Mary had been right”.
Since our TS approach is still new, we have not yet accumulated
enough empirical data to discern patterns and formulate postulates
that underlie the presence of coordination constants for some types
and their absence for others. For now, we leave the question open.

4

Related Work

Our transformational approach is rooted in Transformational Generative Grammars [2, 3], later carried into Minimalism [4]. Our abstract
form TA is similar to spell-out of Minimalism. However, whereas
spell-out is near culmination of a syntactic derivation for Minimalists, for us, it is just the beginning. We are not interested in how
9- 142 -

structure is created through a sequence of Merges from lexical selections. Rather, we consider our abstract form as given (by a parser)
and investigate its transformations into a semantic form. Our transformations are hence all covert.
Closely related to TS is the work of Butler [1], who also obtains a semantic representation as a result of a transformation froma parsed tree. Unlike us, he has applied his approach to a wealthof empirical data in many languages and has truly achieved wide
coverage. His transformations are rather complex and coarse, doing
many things at once, and not typed. One may view TS as an attempt
to re-engineer and understand Butler’s approach and decompose histransformations into elementary steps.
We are grateful to the anonymous reviewer for pointing out theanalysis of ACC and Gapping in [14].

(1) The interpretation of an elliptical construction is obtained by uniformly substituting its immediate constituents
into some immediately preceding structure, and computing
the interpretation of the results. [14, p. 162, (119)]We indeed share the underlying idea of picking and substituting of‘immediate constituents’ into the coordinated material (understoodat some level as an elliptical construction). The proposal of [14] remained rather informal; the present paper may be seen as an attempt
to formalize the idea.
There have been other attempts to solve the KLM problem without directional types (within the ACG-like formalisms). Kanazawa[5] proposes ‘regular constraints’ to prevent over-generation (which
recall structural constraints in Government and Binding). This amounts
however to duplication of lexical entries. The approach [13] reins in
the over-generation using subtyping. Either proposal can be classified as ‘proof search’ rather than computational like TS; in case of[13] with no guarantees that the proof search ever terminates (and,as the authors admitted, no good way to characterize the space ofavailable derivations and detect over-generation).
10- 143 -

5

Conclusions

We have demonstrated the transformational analyses of RNR and
Gapping. The analyses make precise various eliding schemas, demanding type preservation. The asymmetry of the type of and(NP,TV),Sand similar constants is what lets us answer the Kubota and Levine
challenge: how to prevent over-generation in analyses of RNR andgapping without directional types.
The idiosyncrasies of coordination are distilled to the ad hoc
choice of constants andXY . There are transformations for some types
XY but not for the others. There may be a pattern there. Collecting
the arbitrariness in one place might make the pattern easier to find.
It is interesting to consider interpreting the “sequence of words”
as a discontinuous sentence in the sense of Morrill [12].
Acknowledgments We thank an anonymous reviewer for many very
insightful and helpful comments. Numerous discussions with the participants of the workshop “New Landscapes in Theoretical Computational Linguistics” (Ohio State University, October 14-16, 2016)
are gratefully acknowledged.

11- 144 -

Bibliography[1] Alastair Butler. Linguistic Expressions and Semantic Processing - A Practical Approach. Springer, 2015.
[2] Noam Chomsky. Aspects of a Theory of Syntax. MIT Press,
Cambridge, Mass., 1965.
[3] Noam Chomsky. Lectures on Government and Binding. Foris,
Dordrecht, 1981.
[4] Noam Chomsky. The Minimalist Program. The MIT Press,
Cambridge, Massachusetts, 1995.
[5] Makoto Kanazawa. Syntactic features for regular constraints
and an approximation of directional slashes in abstract categorial grammars. In Kubota and Levine [11], pages 34–70.
[6] Oleg Kiselyov. Canonical constituents and non-canonical coordination - simple categorial grammar account. volume 9067
of Lecture Notes in Computer Science, pages 99–113. Springer,
2014.
[7] Oleg Kiselyov. Applicative abstract categorial grammars in fullswing. In Proc. LENLS 12. November 2015.
[8] Yusuke Kubota and Robert Levine. Gapping as like-category
coordination. In Denis Béchet and Alexander Dikovsky, editors,
Logical Aspects of Computational Linguistics: 7th International
Conference, pages 135–150. Springer, 2012.
[9] Yusuke Kubota and Robert Levine. Gapping as hypothetical reasoning. To appear in Natural Language and Linguistic
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[10] Yusuke Kubota and Robert Levine. Against ellipsis: Arguments
for the direct licensing of ‘non-canonical’ coordinations. Linguistics and Philosophy, 38(6):521–576, 2015.
[11] Yusuke Kubota and Robert Levine, editors. Proceedings for
ESSLLI 2015 Workshop ’Empirical Advances in Categorial
Grammar’. University of Tsukuba and Ohio State University,
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[12] Glyn Morrill, Oriol Valentı́n, and Mario Fadda. The displacement calculus. Journal of Logic, Language, and Information,- 145 -

20(1):1–48, 2011.
[13] Carl Pollard and Chris Worth. Coordination in linear categorial
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13- 146 -

From meaning representations to syntactic trees
Alastair Butler
National Institute for Japanese Language and Linguistics
ajb129@hotmail.com

Abstract. This paper describes a method for generating natural language, going from meaning representations to syntactic trees. Startingfrom Davidsonian predicate logic formulas, parse trees are built following the Penn Parsed Corpus of Modern British English, from which theyield (i.e., the words) can be taken. The novel contribution is to highlight
how arrangements of meaning content can inform decisions regarding theselection of language constructions.

1

Introduction

This paper pursues the idea that arrangement of information contained by ameaning representation can provide clues to drive a rule-based (pattern-action
driven) generation of natural language. Generation will follow a series of transformations to construct parse trees. Trees constructed will conform to the annotation scheme of the Penn Parsed Corpus of Modern British English (PPCMBE;
Kroch, Santorini and Diertani 2010), with the yield (i.e. terminal words) producing target sentences of English. The paper is structured as follows. Section 2
sketches the generation procedure with a simple example. Section 3 is the coreof the paper, detailing diﬀerent grammatical constructions and triggers for theircreation. Section 4 outlines an implementation. Section 5 is a conclusion.

2

A sketch of the generation procedure

This section describes generation of a canonical sentence with a transitive verb:(1) Girls see a boy.
A typical Davidsonian (Davidson 1967) meaning representation for (1) is asfollows:∃ EVENT[3] PERSON[2] PERSONS[1] (girls(PERSONS[1])∧ boy(PERSON[2])∧ see(EVENT[3], PERSONS[1], PERSON[2]))With information for argument roles (‘:ARG0’ and ‘:ARG1’) sourced from the arityof the ‘see’ predicate, the same content converted to a Penman representation(Matthiessen and Bateman 1991, Banarescu et. al. 2013) is as follows:- 147 -

( EVENT-3 / see:ARG0 ( PERSONS-1 / girls):ARG1 ( PERSON-2 / boy))

Already the Penman notation provides a base for growing tree structure:
EVENT-3see((((hhhh

:ARG0

:ARG1

PERSONS-1

PERSON-2

girls

boy

Clause structure is built by adjoining VBP, VP and IP layers to nodes beginningwith ‘EVENT’.
IP
VP
VBP
EVENT-3see((((hhhh

:ARG0

:ARG1

PERSONS-1

PERSON-2

girls

boy

Next, arguments of what has been made the main predicate are moved to populate the clause, with ‘:ARG1’ as the object inside VP, while ‘:ARG0’ creates the
subject outside VP.
IP(((((hhhhh
NP-SBJ
VP((((hhhh

PERSONS-1

VBP

NP-OB1girls

EVENT-3

PERSON-2

seeboyWith arguments in place, it is safe to remove the VP layer and type the clauseas IP-MAT (matrix clause), as well as add punctuation. Entity information isretained with BIND, which also contributes to the projection of noun part-ofspeech tags (N; singular vs. NS; plural).

NP-SBJ

XXX

BIND

IP-MAThhh
VBP
NP-OB1.
```NS see*PERSONS* girls

BIND

N .

*PERSON*boy- 148 -

The resut leaves noun phrases bare if indeﬁnite, requiring further post-processing
to add an indeﬁnite determiner (a or an) when the noun head is singular.

3

Diﬀerent constructions

This section is concerned with how the make up of a meaning representation candetermine generation of particular English language constructions as PPCMBE
trees, with both meaning representation content and how the content is packaged
inﬂuencing output.

3.1

Passivisation

With ‘:ARG0’ missing, but ‘:ARG1’ present, the content of ‘:ARG1’ can be taken
to form the grammatical subject to create a passive clause with the verb tag
altered to VAG (passive participle) and BEP (present tense copula) added.
(2) A boy is seen.
∃ EVENT[2] PERSON[1] (boy(PERSON[1])∧ seen(EVENT[2], , PERSON[1]))( EVENT-2 / seen
:ARG1 ( PERSON-1 / boy))
IP-MAT``hhhh(((((h`NP-SBJ
BEP
VAN .

X

X

BIND

N

is

seen

.

*PERSON* boy

3.2

Expletive it

If there is no core argument (‘:ARG0’, ‘:ARG1’, or ‘:ARG2’) then expletive it shouldbe created to fulﬁl the grammatical subject role.
(3) It rained on Monday.
∃ EVENT[1] (past(EVENT[1])∧ rained(EVENT[1])∧ on(EVENT[1]) = DATE[Monday])- 149 -

( EVENT-1 / rained:MOD ( mod-1 / past):ON ( DATE-Monday / DATE
:name ( n-2 / name:op1 "Monday")))IP-MAT`
NP-SBJ-0
PRO

VBD
rained P

It

``
PP
hhh

.

NP
`.

on

``

BIND

NPR

*DATE*

MondayThis example also demonstrates creation of a PP adjunct from an ‘:ON’ argument,
as well as the eﬀect of past tense information altering the verb tag to VBD (pasttense verb).
3.3

Existential construction

Another variation in clause construction occurs when the predicate is the copulaand there is no ‘:ARG1’, but ‘:ARG0’ is present. This triggers creation of a there
subject that is coindexed with the contribution of ‘:ARG0’ captured as a nounphrase that immediately follows the copula.
(4) There is a happy laughing boy.
∃ ATTRIB[3] ATTRIB[2] EVENT[4] PERSON[1] (happy(ATTRIB[2])∧ laughing(ATTRIB[3])∧ is boy ATTRIBUTE(PERSON[1], ATTRIB[3])∧ is boy ATTRIBUTE(PERSON[1], ATTRIB[2])∧ copula is(EVENT[4], PERSON[1]))( EVENT-4 / copula is:ARG0 ( PERSON-1 / boy:ATTRIBUTE ( ATTRIB-3 / laughing):ATTRIBUTE ( ATTRIB-2 / happy)))
IP-MAT
h

hh
NP-1
hhh


NP-SBJ-1

BEP

EX

is

BIND

ADJ

ADJ

N .

There

*PERSON*

happylaughing

boy

3.4

.

Discourse

A discourse is created when there is content for two or more clauses that are
conjuncts of multi-sentence in the Penman representation.

- 150 -

(5) A boy is happy. He laughs.
∃ PERSON[3] EVENT[2] EVENT[4] PERSON[1] (boy(PERSON[1])∧ PERSON[3] = PERSON[1]∧ is happy(EVENT[2], PERSON[1])∧ laughs(EVENT[4], PERSON[3]))The presence of ‘PERSON[3] = PERSON[1]’ is a suﬃcient clue when converting fromthe Davidsonian predicate language formula to create conjuncts conjoined withmulti-sentence.
( CONJ-1 / multi-sentence:snt1 ( EVENT-2 / is happy:ARG0 ( PERSON-1 / boy)):snt2 ( EVENT-4 / laughs:ARG0 PERSON-1))

With there being multiple argument roles for the same entity in distinct conjuncts, it should be the ﬁrst instance in the Penman representation that is populated with information about the entity, e.g., (PERSON-1 / boy), while subsequentinstances are bare references, e.g., PERSON-1. It is such a bare reference that leads
to the creation of a pronoun with the generated output.
multi-sentence
IP-MAT``hhhh(((((h`NP-SBJ
BEP
VAG
.
XX

BIND

N

is

happy . BIND*PERSON* boy

3.5

IP-MAT

hhh
PP
((((h
NP-SBJ
VBP.
XXX*PERSON*

PRO laughs .
pro

VP coordination

A relation projecting a label beginning CONJ is a relation of coordination. Sucha relation name that is not multi-sentence is the foundation for forming intra
sentential coordination. If the content for arguments that become subjects isshared between conjuncts then VP coordination is established to share the same
subject.
(6) A boy is happy and laughs.
∃ EVENT[2] EVENT[3] PERSON[1] (boy(PERSON[1])∧ (is happy(EVENT[2], PERSON[1])CONJ and laughs(EVENT[3], PERSON[1])))- 151 -

( CONJ-4 / and:op1 ( EVENT-2 / is happy:ARG0 ( PERSON-1 / boy)):op2 ( EVENT-3 / laughs:ARG0 PERSON-1))
IP-MATX
NP-SBJ
XX

BIND

N

X

.

VP

((((hhhh
VP
CONJP .
P
PP


P

*PERSON* boy BEP
is

VAG

CONJ VP

happy

and

VBP
laughsIf there are other shared entities between conjuncts formed with the same argument role, then they are projected outside the VP layer, typically to the left,
but to the right when the argument role is ‘:ARG1’ and the verbs are active (alsohave ‘:ARG0’ arguments), or when the argument is heavy, e.g., containing many
terminal nodes.
3.6

Adverbial clause

A relation connecting two conjuncts projecting a label beginning CND (conditional) or CRD (coordinating relation) can trigger creation of an adverbial clausefrom the ﬁrst conjunct, with the relation name forming a subordinate conjunction (e.g., if , when, unless, although, because) that introduces the adverbial
clause to a containing clause formed with the content of the second conjunct.
(7) Because a boy is happy he laughs.
∃ PERSON[3] EVENT[2] EVENT[4] PERSON[1] (boy(PERSON[1])∧ PERSON[3] = PERSON[1]∧ CRD Because(is happy(EVENT[2], PERSON[1]),laughs(EVENT[4], PERSON[3])))

With the same entity ﬁlling arguments in distinct conjuncts, it is the instance inwhat forms the adjunct clause that is populated with information about the entity, while subsequent instances are bare references, with bare references leading
to the creation of pronouns.
( CRD-5 / Because:op1 ( EVENT-2 / is happy:ARG0 ( PERSON-1 / boy)):op2 ( EVENT-4 / laughs:ARG0 PERSON-1))- 152 -

IP-MAT(((((hhh
CRD
NP-SBJXXXVBP .
PP(((hhh
BIND
PRO laughs .
P
CP-ADV *(((hhhh*PERSON*pro
Because C
IP-SUB((((hhhh
0 NP-SBJ
`



BIND

*PERSON*

3.7

``

BEP

N is

VAG

happy

boy

Participal clause

Instead of an adverbial clause being created from the ﬁrst conjunct of a relation
projecting a label beginning CND or CRD, a participal clause with a controlled
subject is created when there is a bare reference for the ‘:ARG0’ of the ﬁrst
conjunct that is coreferential with the content of a core argument (‘:ARG0’, ‘:ARG1’or ‘:ARG2’) of the second conjunct.
(8) If laughing a boy is happy.
∃ EVENT[2] EVENT[3] PERSON[1] (boy(PERSON[1])∧ CND If(
laughing(EVENT[2], PERSON[1]),is happy(EVENT[3], PERSON[1])))( CND-4 / If
:op1 ( EVENT-2 / laughing:ARG0 PERSON-1):op2 ( EVENT-3 / is happy:ARG0 ( PERSON-1 / boy)))
IP-MATPP!PP!

(((((hhhh
CND
NP-SBJ BEP```P IP-PPL * BIND
IfVAG*PERSON*

N is

VAG

.

happy .

boylaughing

3.8

Purpose clauseA to-inﬁnitive clause can function as an adverbial expressing ideas of purpose
or outcome. The content for such a clause falls under a ‘:PRP’ tag, while insidethere is ‘:ARG0’ containing a bare reference coreferential with (i.e., controlled by)
the content of the ‘:ARG0’ of the containing clause. This creates a subjectless
IP-INF-PRP projection containing (TO to) and a non-ﬁnite verb (VB).

- 153 -

(9) To impress a girl a boy laughs.
∃ PRP[2] EVENT[4] EVENT[5] PERSON[3] PERSON[1] (boy(PERSON[1])∧ girl(PERSON[3])∧ is FACT THAT(PRP[2],impress(EVENT[4], PERSON[1], PERSON[3]))∧ laughs(EVENT[5], PERSON[1])∧ PRP(EVENT[5]) = PRP[2])( EVENT-5 / laughs:ARG0 ( PERSON-1 / boy)
:PRP ( EVENT-4 / impress:ARG0 PERSON-1:ARG1 ( PERSON-3 / girl)))
IP-MATh
IP-INF-PRP
((hh


((( 

TO

VB

To

impress

3.9hhh
NP-OB1```BIND

N*PERSON*

girlhh
NP-SBJ```BIND*PERSON*

VBP.

N laughs .

boy

Infinitive clause with long distance dependency

Content for an embedded to-inﬁnitive clause falls under a ‘:TOCOMP’ tag, while inside there is ‘:ARG0’ containing a bare reference coreferential with (i.e., controlledby) the content of the ‘:ARG2’ if present, or alternatively ‘:ARG1’ if present, or
alternatively ‘:ARG0’ of the containing clause. This creates a subjectless IP-INF
projection containing (TO to) and a non-ﬁnite predicate.
(10) What might the boy think to do?
∃ EVENT[3] EVENT[2] PERSON[1] (boy(PERSON[1])∧ QUEST(MD might(think TOCOMP(EVENT[2], PERSON[1],do(EVENT[3], PERSON[1], ENTITY[unknown])))))( EVENT-2 / think:domain-of ( QUEST-5 / QUEST):domain-of ( MD-4 / might):ARG0 ( PERSON-1 / boy)
:TOCOMP ( EVENT-3 / do
:ARG0 PERSON-1:ARG1 ( ENTITY UNK-6 / unknown)))- 154 -

CP-QUE
WNP-3.

IP-SUB

WPRO

MD

What

might

(((PP
NP-SBJ
VB` ``BIND*PERSON*

IP-INF ?
(((hhhN think

NP-OB1

TO VB

boy

*T*-3

to

do

The example also illustrates how a long distance dependency is established with‘:ARG1 (ENTITY UNK-6 / unknown)’ forming the foundation for an object nounphrase trace (NP-OB1 *T*-3) that is coindexed with a WH-phrase (WNP-3 (WPRO
What)) that is placed as the highest constituent of the ﬁrst commanding question
scope marker: ‘:domain-of (QUEST-5 / QUEST)’.
3.10

Embedded clause with long distance dependency

An embedded clause rather than a to-inﬁnitive is established with clause content
placed under ‘:THAT’. Inside the content for the embedded clause, bare references
form foundations for pronouns.
(11) What might the boy think that he will do?
∃ PERSON[3] EVENT[4] EVENT[2] PERSON[1] (boy(PERSON[1])∧ PERSON[3] = PERSON[1]∧ QUEST(MD might(think THAT(EVENT[2], PERSON[1],MD will(do(EVENT[4], PERSON[3], ENTITY[unknown]))))))( EVENT-2 / think:domain-of ( QUEST-8 / QUEST):domain-of ( MD-7 / might):ARG0 ( PERSON-1 / boy)
:THAT ( EVENT-4 / do:domain-of ( MD-6 / will):ARG0 PERSON-1:ARG1 ( ENTITY UNK-9 / unknown)))
CP-QUE
WNP-3.

IP-SUB

WPRO

MD

What

might

(((PP
NP-SBJ
VB```BIND*PERSON*

CP-THT ?

(((((hhhhh

N think C

boy

0

IP-SUBhh

NP-OB1*T*-3

NP-SBJ
X



BIND

*PERSON*

XX

hh

PRO

MD

VB

will

do

pro

The example again illustrates a long distance dependency established out of theembedding.

- 155 -

3.11

Embedded question

Having the scope marker for a question local to an embedded clause results inan embedded question.
(12) A boy wonders what he will do.
∃ PERSON[3] EVENT[4] EVENT[2] PERSON[1] (boy(PERSON[1])∧ PERSON[3] = PERSON[1]∧ wonders THAT(EVENT[2], PERSON[1],QUEST(MD will(do(EVENT[4], PERSON[3], ENTITY[unknown])))))( EVENT-2 / wonders:ARG0 ( PERSON-1 / boy)
:THAT ( EVENT-4 / do:domain-of ( QUEST-8 / QUEST):domain-of ( MD-7 / will):ARG0 PERSON-1:ARG1 ( ENTITY UNK-9 / unknown)))
IP-MAThh
NP-SBJ
X



BIND

X

N

VBP
wonders*PERSON* boy

hh
((

.

CP-QUE
((

WNP-3

C

WPRO

0

IP-SUBhh .

NP-OB1

what

*T*-3

NP-SBJ
X



BIND

*PERSON*

XX

hh

PRO

MD

VB

will

do

pro

Together with the examples of section 3.9 and 3.10, this demonstrates how it is
not enough for a meaning representation to mark a WH question with unknown
alone, but that scope marking the level of structure to place a fronted WH phrase
is also vital.

3.12

Small clause

Small clauses are embedded clauses that occur with the absence of a ﬁnite verb.
(13) A plan to laugh makes a boy happy.
Lack of a ﬁnite verb is reﬂected by creation of an unspeciﬁed ‘xxx’ predicate thatconnects the subject a boy to the happy attribute in the following Davidsonian
formula:- 156 -

∃ ATTRIB[5] PERSON[4] ENTITY[1] EVENT[2] EVENT[6] EVENT[3] (is plan TOCOMP(ENTITY[1],laugh(EVENT[2]))∧ boy(PERSON[4])∧ happy(ATTRIB[5])∧ makes TOCOMP(EVENT[3], ENTITY[1],xxx(EVENT[6], PERSON[4], ATTRIB[5])))( EVENT-3 / makes:ARG0 ( ENTITY-1 / plan
:TOCOMP ( EVENT-2 / laugh)):TOCOMP ( EVENT-6 / xxx:ATTRIBUTE ( ATTRIB-5 / happy):ARG0 ( PERSON-4 / boy)))This leads to projection of an IP-SMC embedded clause with neither verb nor
copula creation.
IP-MAT(((hhhh.
IP-SMChhh
X
NP-SBJADJP .
IP-INF makes!a```a!
VBP

NP-SBJ
XN

Plan TO

VB

BIND

to

laugh

*PERSON*N

ADJ

boy happy

This example also illustrates creation of a nominal with an inﬁnitive clauseembedding, triggered by ‘TOCOMP’.

3.13

Relative clause

The example of this section demonstrates creation of a relative clause with along distance dependency:(14) A boy that a girl says is happy laughs.
There is nothing overt to signal a relative clause with the Davidsonian formula:∃ EVENT[5] EVENT[4] EVENT[6] PERSON[3] PERSON[1] (girl(PERSON[3])∧ says THAT(EVENT[4], PERSON[3],is happy(EVENT[5], PERSON[1]))∧ boy(PERSON[1])∧ laughs(EVENT[6], PERSON[1]))But a base for realising a relative clause emerges with conversion to Penman
notation:- 157 -

( EVENT-6 / laughs:ARG0 ( PERSON-1 / boy:ARG0-of ( EVENT-5 / is happy:THAT-of ( EVENT-4 / says
:ARG0 ( PERSON-3 / girl)))))

A key requirement for Penman notation is to connect all content around a single
rooted node. This privileged node typically forms the main predicate of a clausefollowing generation. While not necessary for being a Davidsonian forumla, aconvention can be followed to place such a privileged predicate as the mostright-side predicate of the formula (so, ‘laughs’ of the example). Connection
to this single rooted predicate is possible by folding Penman material around‘inverse roles’ (signalled by ending a role name with ‘-of’). This acts to compact
to a local argument relation the long distance dependency of the relative clause.
Generation consists of unfolding the dependency, so taking ‘-of’ content andreintegrating the content with clausal embedded structure:
IP-MAT
hh

hhh

VBP

NP-SBJBINDN*PERSON* boy

CP-RELlaughs .

that NP-SBJ`

VBP((((((hhhhhh
WNP-1 C
IP-SUB(((((hhhhh
0CP-THT(((hhhh
N says C
IP-SUB((((hhhh

``



BIND

*PERSON*

3.14

.

girl

0 NP-SBJ

BEP

VAG

*T*-1

is

happy

It cleft

As a ﬁnal example, consider generation of an it-cleft:(15) It is the happy boy that laughs.
Such a cleft sentence leads to the presence of a copula predicate that connects‘ARG1’ content to an ‘ARG0’ entity that, with conversion to Penman notation, isgiven a dummy ‘ENTITY’ head and is inversed linked to material suﬃcient tocreate a clause:∃ ATTRIB[2] ENTITY[3] EVENT[5] EVENT[6] PERSON[1] (happy(ATTRIB[2])∧ is boy ATTRIBUTE(PERSON[1], ATTRIB[2])∧ laughs(EVENT[5], ENTITY[3])∧ copula is(EVENT[6], ENTITY[3], PERSON[1]))- 158 -

( EVENT-6 / copula is:ARG0 ( ENTITY-3 / ENTITY:ARG0-of ( EVENT-5 / laughs)):ARG1 ( PERSON-1 / boy:ATTRIBUTE ( ATTRIB-2 / happy)))

This leads to generation of the following tree:

NP-SBJ

BEP

PRO

is

It

IP-MAT((((
NP-OB1((((hhhh

BIND*PERSON*

ADJ.
CP-CLF(hhhh(((
N WNP-1 C
IP-SUB .
XX

happy boy0that NP-SBJ VBP*T*-1

laughsInternally, the it-cleft has the same structure as a relative clause, but externally,
it is a daughter of IP.

4

Implementation

A program available from http://www.compling.jp/generation implements thegeneration of this paper. The assumed engine to transform trees is provided
by tsurgeon (Levy and Andrew 2006). This works with tsurgeon scripts that
contain patterns with associated actions. Patterns describe tree structure withthe tree description language of tgrep (Pito, 1994) and actions transform thetree, e.g., moving, adjoining, copying or deleting auxiliary trees or relabelling
nodes. Alternative programs with similar functionality to transform trees withscripts are CorpusSearch (Randall 2009) and TTT (Purtee and Schubert 2012).

5

Conclusion

This paper has focused on the generation of various English language constructions from meaning representations. Starting as Davidsonian predicate language
formulas, a key step was conversion to Penman representations that formed thebasis for generation to proceed with successive tree structure changes.
Language generation raises the issue of how to choose between the many
ways a language oﬀers to present content. The novel contribution of this paper
has been to demonstrate how there can be a signiﬁcant role for meaning representations to play in inﬂuencing the selection of grammatical constructions withthe arrangement of information content, notably, handling distinctions of clausetype as well as the choice between discourse, or (VP) coordination, or projection of an adverbial clause, or participial clause, or creation of a small clause, or
inﬁnitive embedding, or ﬁnite embedding, or relative clause, or it-cleft. This has
simpliﬁed what it takes to create a generation component capable of rich, varied
and natural output that preserves meaning.

- 159 -

Acknowledgements This paper beneﬁtted from the comments of two anonymous reviewers, who are gratefully acknowledged. This research is supportedby the Japan Society for the Promotion of Science (JSPS), Research Project
Number: 15K02469.

References
Banarescu, L., C. Bonial, S. Cai, M. Georgescu, K. Griﬃtt, U. Hermjakob, K. Knight,
P. Koehn, M. Palmer, and N. Schneider. 2013. Abstract Meaning Representationfor Sembanking. In Proceedings of the 7th Linguistic Annotation Workshop andInteroperability with Discourse, pages 178–186.
Davidson, Donald. 1967. The logical form of action sentences. In N. Rescher, ed.,The Logic of Decision and Action. Pittsburgh: University of Pittsburgh Press.
Reprinted in: D. Davidson, 1980. Essays on Actions and Events. Claredon Press,
Oxford, pages 105–122.
Kroch, Anthony, Beatrice Santorini, and Ariel Diertani. 2010. The Penn-Helsinki
Parsed Corpus of Modern British English (PPCMBE). Department of Linguistics,
University of Pennsylvania. CD-ROM, second edition, (http://www.ling.upenn.
edu/hist-corpora).
Levy, Roger and Galen Andrew. 2006. Tregex and tsurgeon: tools for querying andmanipulating tree data structure. In 5th International conference on Language
Resources and Evaluation.
Matthiessen, Christian and John A Bateman. 1991. Text generation and systemicfunctional linguistics: experiences from English and Japanese. Pinter Publishers.
Purtee, A. and Lenhart K. Schubert. 2012. Ttt: A tree transduction language for syntactic and semantic processing. In EACL 2012 Workshop on Applications of Tree
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Randall, Beth. 2009. CorpusSearch 2 Users Guide. (http://corpussearch.sourceforge.
net/CS-manual/Contents.html).

- 160 -

Lambek Categorial Grammars
as Abstract Categorial Grammars
Philippe de Groote
Inria Nancy - Grand Est
FranceAbstract. This paper describes a construction that allows Lambek Categorial Grammars to be represented as Abstract Categorial Grammars. This construction is based
upon a family of combinators that allows Lambek lexical entries to be interpreted aslinear λ-terms.

1

Introduction

Abstract Categorial Grammars [2] (ACG, for short) differ from classical categorial grammars in an essential way: the ACG type system is based on a commutative logic (namely,
the implicative fragment of multiplicative linear logic [4]). For this reason, it has been argued that the way of encoding wh-extraction in an ACG corresponds to an uncontroledform of extraction, which results in syntactic overgeneration. In particular, an ACG could
not accomodate left and right peripheral extractions like a Lambek categorial grammar [9](LG, for short) does.
This claim about ACG and LG is certainly not true at the level of the string languages.
Indeed, Pentus’ theorem [11, 12] states that every Lambek grammar can be transformed
into a context-free grammar, and there is a canonical way of representing a context-freegrammar as an ACG [3].
The claim is not quite true eiteher at the level of the derivations since Kanazawa andSalvati have shown that Pentus’ construction preserve, in some sense, the derivations of
the original Lambek grammar [6]. As a consequence, given a Lambek grammar G, it ispossible to define an ACG that generates a set of λ-terms that correspond to the derivationof G. What is then needed in order to turn these derivations into strings is an appropriate
way of interpreting the lexical entries of the original Lambek grammar as linear λ-terms.
The main goal of this paper is to devise such an interpretation.

2

Mathematical preliminaries

Let A be a set of atomic types. The set TA of the simple types (built upon A) is inductively
defined according to the following rules:
TA ::= A | (TA → TA )The order of a simple type is inducively defined as follows:
1. ord(a) = 1, for a ∈ A;
2. ord(α → β) = max{ord(α) + 1, ord(β)}
A finite set of typed constants is called a higher-order signature. More formally, such ahigher-order signature consists of a triple Σ = hA, C, τ i, where:
1. A is a finite set of atomic types;
2. C is a finite set of constants;
3. τ : C → TA is a function that assigns to each constant in C a simple type in TA .

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Let Σ = hA, C, τ i be a signature. The order of Σ is defined to be max{ord(τ (c)) | c ∈ C}.
Let X be an infinite countable set of λ-variables. Given a higher-order signature Σ =hA, C, τ i, the set Λ(Σ) of the linear λ-terms built upon Σ is inductively defined as follows:
1. if c ∈ C, then c ∈ Λ(Σ);
2. if x ∈ X, then x ∈ Λ(Σ);
3. if x ∈ X, t ∈ Λ(Σ), and x occurs free in t exactly once, then (λx. t) ∈ Λ(Σ);
4. if t, u ∈ Λ(Σ), and the sets of free variables of t and u are disjoint, then (t u) ∈ Λ(Σ).
Λ(Σ) is provided with the usual notion of capture-avoiding substitution, α-conversion,
β-reduction, and η-reduction [1]. we take the relation of βη-equivalence as the notion ofequality between λ-terms.
Each λ-terms in Λ(Σ) may be assigned a simple type according to the following typesystem:−Σ c : τ (c)x : α −Σ x : α
Γ, x : α −Σ t : βΓ −Σ (λx. t) : (α → β)

Γ −Σ t : (α → β) ∆ −Σ u : αΓ, ∆ −Σ (t u) : β

Given two higher-order signatures Σ1 = hA1 , C1 , τ1 i and Σ2 = hA2 , C2 , τ2 i, a morphism Φ : Σ1 → Σ2 consists of an interpretation of the atomic types of Σ1 as types built
upon A2 together with an interpretation of the constants of Σ1 as linear λ-terms built upon
Σ2 . These two interpretations must be such that their homomorphic extensions commute
with the typing relations. More formally, a morphism between higher-order signatures,Φ : Σ1 → Σ2 , is defined to be a pair Φ = hη, θi such that:
1. η : A1 → T (A2 ) is a function that interprets the atomic types of Σ1 as simple types
built upon A2 ;2. θ : C1 → Λ(Σ2 ) is a function that interprets the constants of Σ1 as linear λ-terms built
upon Σ2 ;3. the interpretation functions are compatible with the typing relation, i.e., for any c ∈ C1 ,the following typing judgement is derivable:−Σ2 θ(c) : η̂(τ1 (c)),
where η̂ is the unique homomorphic extension of η.
Condition 3, in the above definition ensures that if x1 : α1 , . . . , xn : αn −Σ1 t : α, then:x1 : Φ(α1 ), . . . , xn : Φ(αn ) −Σ2 Φ(t) : Φ(α)
where, according to the context, Φ(·) denotes either the homomorphic extension of η or thehomomorphic extension of η.
Let Φ : Σ1 → Σ2 be a morphism between signatures. The order of Φ is defined to bemax{ord(Φ(a)) | a ∈ A1 }.
We end this section by introducing a few additional notations that will be useful inthe sequel of the paper. Let I = {i1 , . . . , in } be a totally ordered finite set of indices,and let (xi )i∈I (respectively, (αi )i∈I ) be a sequence of λ-variables (respectively, simple
types) indexed by I. We write (xi : αi )i∈I for the typing environment xi1 : αi1 , . . . , xin : αin .
Similarly, we write t[xi :=ui ]i∈I for the simultaneous substitution in the λ-term t of theterms ui1 , . . . , uin for the free variables xi1 , . . . , xin . In this latter case, we assume that theset of variables {xi | i ∈ I} corresponds exactly to the set of free variables of t.
Let Σ be an alphabet. As usual, we write Σ ∗ for the set of strings generated by Σ.
We use  to denote the empty string, and the infix operator ’+’ to P
denote string concatenation. Accordingly, given a sequence of strings (wi )i∈I , we write i∈I wi for the string
wi1 . . . win .
2- 162 -

Lambek Grammar
man : nwoman : nsome : np / nevery : np / nloves : (np \ s) / np
who : (n \ n) /(np \ s)whom : (n \ n) /(s / np)

Fig. 1.

3

Lambek categorial grammars

The classical notion of a Lambek categorial grammar is based on a deductive system known
as the associative Lambek calculus [9]. This calculus may be seen as a non-commutativefragment of implicative linear logic [4].
Let A be a set of atomic formulas. The syntax of the Lambek formulas (built upon A)
obeys the following formation rules:
FA ::= A | (FA \ FA ) | (FA / FA )where formulas of the form α \ β correspond to left-to-right implications (i.e., α impliesβ), and formulas of the form α / β to right-to-left implications (i.e., α is implied by β).
Lambek formulas are also called syntactic types.
The deduction relation is then specified by means of the following sequent calculus.
α −L αα, Γ −L β
Γ −L α \ βΓ, α −L β
Γ −L β / αΓ −L α ∆, β, Θ −L γ∆, Γ, α \ β, Θ −L γ

Γ −L α ∆, β, Θ −L γ∆, β / α, Γ, Θ −L γ

It should be stressed that the above system does not include any structural rule. In
particular, the non-commutativity of the systems is reflected by the absence of an exchange
rule. This, in turn, explains the presence of two different implications.
A Lambek categorial grammar (L-grammar, for short) is defined to be a quadruple
G = hΣ, A, L, si such that:
1. Σ is a finite set of terminal symbols;
2. A is a finite set of atomic types;
3. L : Σ −→ 2FA is a lexicon that assigns to each terminal symbol a finite set of types
built upon A;
4. s ∈ A is a distinguished type, called the initial type of the grammar.
A word a0 a1 . . . an ∈ Σ ∗ belongs to the language generated by G if and only if there exist
α0 ∈ L(a0 ), α1 ∈ L(a1 ), . . . αn ∈ L(an ) such that α0 , α1 , . . . αn −L s is derivable.
Figure 1 gives the lexicon of a Lambek categorial grammar that will serve as a running
example throughout this paper. According to this grammar, the sentence “every man who
3- 163 -

Context-free Grammar<np> → <np/n> <n>
 → <np> <(np\s) / np> <np><n> → <n> <(n\n)/(np\s)> <np\s><n> → <n> <(n\n)/(s/np)> <s/np><np\s> → <(np\s)/np> <np><s/np> → <np> <(np\s)/np><s/np> → <np> <(np\s) / np> <np/np><np/np> → <np/n> <n/np><n/np> → <n> <(n\n)/(np\s)> <(np\s)/np><(np\s)/np> → <(np\s)/np> <np/np><n> → man<n> → woman<np/n> → some
<np/n> → every<(np\s)/np> → loves<(n\n)/(np\s)> → who<(n\n)/(s/np)> → whomFig. 2.

loves some woman loves every woman” is grammatical because the following sequent isderivale:
np / n, n, (n \ n) /(np \ s), (np \ s) / np, np / n, n, (np \ s) / np, np / n, n −L s
The above example illustrates that when dealing with a categorial grammar, parsing
corresponds to proof-search. Consequently, a categorial parse structure amounts to a formal derivation. Then, using the Curry-Howard correspondance, it is possible to associate
a simply typed (actually, linear) λ-term to any derivation of the Lambek calculus. This isrealized by the following system:x : α −λL x : α
x : α, Γ −λL t : βΓ −λL λx. t : α \ βΓ, x : α −λL t : βΓ −λL λx. t : β / αΓ −λL u : α ∆ −λL t : α \ βΓ, ∆ −λL t u : βΓ −λL t : β / α ∆ −λL u : αΓ, ∆ −λL t u : β

According to Pentus’ theorem [11, 12], every Lambek grammar may be turned into an
equivalent context-free grammar. The context-free grammar of Figure 2, for instance, is agrammar that generates the same language as the Lambek grammar of Figure 1.
In the grammar of Figure 2, in agreement with Pentus’ construction, types of theLambek calculus are used as non-terminal symbols. In addition, every production rule
<α> → <α1 > . . . <αn > corresponds to a derivable sequent α1 , . . . , αn −L α whose
derivation, in turn, corresponds to a λ-term. These sequents, together with the λ-terms encoding their derivations, are given in Figure 3.
4- 164 -

Derivable sequentsx : np/n, y : n −λL x y : np
x : np, y : (np\s) / np, z : np −λL y z x : s
w : n, x : (n\n)/(np\s), y : np\s −λL x (λz. y z) w : n
w : n, x : (n\n)/(s/np), y : s/np −λL x (λz. y z) w : n
x : (np\s)/np, y : np −λL λz. x y z : np\sx : np, y : (np\s)/np −λL λz. y z x : s/np
w : np, x : (np\s) / np, y : np/np −λL λz. x (y z) w : s/np
x : np/n, y : n/np −λL λz. x (y z) : np/np
v : n, w : (n\n)/(np\s), x : (np\s)/np −λL λy. w (λz. x y z) v : n/np
w : (np\s)/np, x : np/np −λL λyz. w (x y) z : (np\s)/np

Fig. 3.

In some sense, a context-free grammar resulting from Pentus’ construction preserves
the parse structures of the original Lambek grammar. Consider, for instance, a sentence S
belonging to the language generated by the Lambek grammar of Figure 1. Accordingly,
there exists a derivation of S using the rules of the context-free grammar of Figure 2. Now,
using this context-free derivation together with the λ-terms given in Figure 3 it is possible tocompute the λ-term corresponding to the original derivation of S in the Lambek grammarof Figure 1. This has been shown by Kanazawa and Salvati [6].

4Abstract categorial grammars

Abstract categorial grammars have been introduced in [2]. Contrarily to the case of mostother notions of categorial grammar, they are based on a fully commutative logic.
Formally, an abstract categorial grammar is a quadruple G = hΣ1 , Σ2 , L, si where:
1. Σ1 and Σ2 are two higher-order signatures; they are called the abstract vovabulary andthe object vovabulary, respectively;
2. L : Σ1 → Σ2 is a morphism between the abstract vovabulary and the object vovabulary; it is called the lexicon;
3. s is an atomic type of the abstract vocabulary; it is called the distinguished type of thegrammar.
The abstract language generated by G, A(G), is defined as follows:
A(G) = {t ∈ Λ(Σ1 ) | −Σ1 t : s is derivable}
In words, the abstract language generated by G is the set of closed linear λ-terms, built
upon the abstract vocabulary Σ1 , whose type is the distinguished type s.
The object language generated by G, O(G), is then defined to be the image of theabstract language by the lexicon L:
O(G) = {t ∈ Λ(Σ2 ) | ∃u ∈ A(G). t = L(u)}The abstract language of an ACG may be thought of as its parse structures, and its object
language as the language it generates. Using this intuition, it is not difficult to show that
5- 165 -

Abstract Vocabulary
PROD 0 : <np/n> → <n> → <np>
PROD 1 : <(np\s) / np> → <np> → <np> → 

PROD 2 : <(n\n)/(np\s)> → <np\s> → <n> → <n>
PROD 3 : <(n\n)/(s/np)> → <s/np> → <n> → <n>
PROD 4 : <(np\s)/np> → <np> → <np\s>
PROD 5 : <(np\s)/np> → <np> → <s/np>
PROD 6 : <(np\s) / np> → <np/np> → <np> → <s/np>
PROD 7 : <np/n> → <n/np> → <np/np>
PROD 8 : <(n\n)/(np\s)> → <(np\s)/np> → <n> → <n/np>
PROD 9 : <(np\s)/np> → <np/np> → <(np\s)/np>
MAN : <n>
WOMAN : <n>
SOME : <np/n>
EVERY : <np/n>
LOVES : <(np\s)/np>
WHO : <(n\n)/(np\s)>
WHOM : <(n\n)/(s/np)>

Fig. 4.

Object Vocabulary
man : nwoman : nsome : n → np
every : n → np
loves : np → np → s
who : (np → s) → n → n
whom : (np → s) → n → n

Fig. 5.

every context-free grammar may be represented as an ACG [3]. Accordingly, the contextfree grammar of Figure 2 could be turned into an ACG. Now, using a similar construction
together with the result of Kanazawa and Salvati [6], one may take advantage of the λterms given in Figure 3 in order to devise an ACG that generates the λ-terms that encodes
the derivations of the original Lambek grammar of Figure 1. Lets call this ACG LDER.
Its abstract vocabulary and its object vocabulary are respectively given in Figure 4 andFigure 5. Its lexicon is defined by Figure 6 and Figure 7.
While every Lambek grammar is strongly lexicalized, LDER is not (in the sense thatsome of its abstract constants are interpreted by pure combinators). In [7, 8], Kanazawa andYoshinaka show that every second-order ACG (i.e., an ACG whose abstract vocabulary is
6- 166 -

Lexicon: type interpretation<n> := n<np> := np
 := s<n/np> := np → n<np/n> := n → np
<np/np> := np → np
<np\s> := np → s<s/np> := np → s<(np\s)/np> := np → np → s<(n\n)/(np\s)> := (np → s) → n → n<(n\n)/(s/np)> := (np → s) → n → n

Fig. 6.

Lexicon: term interpretation
PROD 0 := λxy. x y
PROD 1 := λxyz. x y z
PROD 2 := λwxy. w (λz. x z) y
PROD 3 := λwxy. w (λz. x z) y
PROD 4 := λxyz. x y z
PROD 5 := λxyz. x z y
PROD 6 := λwxyz. w (x z) y
PROD 7 := λxyz. x (y z)PROD 8 := λvwxy. v (λz. w y z) x
PROD 9 := λwxyz. w (x y) z
MAN := man
WOMAN := woman
SOME := some
EVERY := every
LOVES := loves
WHO := who
WHOM := whomFig. 7.

second-order) can be lexicalized. As a matter of fact, LDER is a second-order ACG because it derives from a context-free grammar. We may therefore lexicalize it. The resulting
ACG, which we call LDER lex shares with LDER the same object vocabulary (Fig. 5), thesame set of abstract atomic types, and the same type interpretation (Fig. 6). As for the new7- 167 -

Abstract Vocabulary (lexicalized grammar)
MAN : <n>
WOMAN : <n>
SOME : <n> → <np>
SOME0 : <n/np> → <np/np>
EVERY : <n> → <np>
EVERY 0 : <n/np> → <np/np>
LOVES : <np> → <np> → 

LOVES0 : <np> → <np\s>
LOVES1 : <np> → <s/np>
LOVES2 : <np/np> → <np> → <s/np>
LOVES3 : <np/np> → <(np\s)/np>
LOVES 4 : <(np\s)/np>
WHO : <np\s> → <n> → <n>
WHO 0 : <(np\s)/np> → <n> → <n/np>
WHOM : <s/np> → <n> → <n>Fig. 8.

Lexicon (lexicalized grammar)
SOME := λx. some x
EVERY := λx. every x
LOVES := λxy. loves x y
WHO := λxy. who (λz. x z) y
WHOM := λxy. whom (λz. x z) y
LOVES 0 := λxy. loves x y
LOVES1 := λxy. loves y x
LOVES2 := λxyz. loves (x z) y
SOME 0 := λxy. some (x y)EVERY 0 := λxy. every (x y)WHO 0 := λwxy. who (λz. w y z) x
LOVE 3 := λxyz. loves (x y) z
MAN := man
WOMAN := woman
LOVES 4 := λxy. loves x y

Fig. 9.

abstract constants togheter with their lexicalized interpretations, they are given in Figure 8
and Figure 9.
8- 168 -

5

From string to terms and vice versa

In the previous section, we have defined a lexicalized ACG that generates the λ-terms
corresponding to the derivations of a given Lambek grammar. It now remains to interpretthese λ-terms as strings. To this end, we first associate to each lambek type, α, a simple
type α, built over a type σ corresponding to a set of strings Σ ∗ :a = σ, for a atomic
α\β = α → β
β /α = α → βWe then define two families of combinators, Eα : σ → α and Pα : α → σ, which allow totransform a strings into a λ-terms and vice versa:
Ea w = w
Eα \ β w = λx. Eβ ((Pα x) + w)
Eβ / α w = λx. Eβ (w + (Pα x))
Pa t = tPα \ β t = Pβ (t (Eα ))Pβ / α t = Pβ (t (Eα ))We first state and prove two technical lemmas.
Lemma 1. For every string w and every type α, Pα (Eα t) = t.
tuProof. By induction on α.

Lemma 2. Let t be a λ-term in normal form such that (xi : αi )i∈I −λL t : α. Then, forevery set of strings {wi | i ∈ I}, the following properties hold:P
1. if t is a variable or an application, then t[xi := (Eαi wi )]i∈I = Eα ( i∈I wi );P
2. if t is an abstraction, then Pα t[xi := (Eαi wi )]i∈I = i∈I wi .
tuProof. By induction on t.
The following lemma is the main property of the operators Eα and Pα .

Lemma 3. Let t be a λ-term such that (xi : αi )i∈I −λL t : α. Then, for every set ofP
strings {wi | i ∈ I}, Pα t[xi := (Eαi wi )]i∈I = i∈I wi .
Proof. Let t0 be the normal form of t. The results follows from Lemma 2, Lemma 1, andthe fact that Pα t[xi := (Eαi wi )]i∈I = Pα t0 [xi := (Eαi wi )]i∈I .
tu
We then immediatelly obtain the following corollary.
Corollary 1. Let t be a λ-term such that (xi : αi )i∈I −λL t : a, whereP a is an atomic
type. Then, for every set of strings {wi | i ∈ I}, t[xi := (Eαi wi )]i∈I = i∈I wi .
tu9

- 169 -

Interpretation of the object constants
man := man
woman := woman
some := λx. some + x
every := λx. every + x
loves := λxy. y + loves + x
who := λxy. y + who + (x )
whom := λxy. y + whom + (x )Fig. 10.

6

From LG to ACG

Consider the signature given in Figure 5. It has been used as the object vocabulary of both
the ACGs LDER and LDER lex . We now use it as the abstract vocabulary of another ACG
whose lexicon, which is given in Figure 10, is obtained by applying the operator Eα . Let
us call this new ACG GEN string .
The fact that the object vocabulary of LDER lex is, at the same time, the abstract vocabulary of GEN string allows the two ACGs to be composed. Let us call the ACG resulting
from this composition CFG λ . The picture is then the following:•

Abstract Vocabulary (Fig. 4)
LDER lex

CFG λ


•

Object vocabulary (Fig. 5)
GENstring•

Strings

Now, in order to define an ACG that is strongly equivalent (up to a relabelling) to theoriginal Lambek grammar, we need a last transformation. Consider the ACG CFG λ , which
has been obtained by omposition. It is a second-order ACG whose lexicon is third-order.1
In fact, it is always possible to transform such an ACG in an object-language-equivalent
third-order ACG whose lexicon is second order. Roughly speaking, the construction consists in replacing each atomic abstract type α, whose interpretation is α := a → b, by a type
α1 → α2 (where α1 and α2 are fresh symbols) together with the interpretations α1 := aand α2 := b. Applying this transformation to CFG λ , we end up with a grammar whose abstract vocabulary and lexicon are respectively given in Figure 11 and Figure 12. The object
language of this ACG corresponds to the language generated by the original Lambek grammar. Its abstract language corresponds to the derivations of the original Lambek grammar(up to a relabelling, i.e., a homomorphism that is sending constants to constants).
1

It is third-order because the type of the strings is defined to be the second-order type o → o. Consequently, an apparently second-order interpretation such as <n/np> := σ → σ corresponds,
in fact, to a third-order interpretation (namely, <n/np> := (o → o) → o → o).

10- 170 -

Abstract Vocabulary (final grammar)
MAN : nWOMAN : n
SOME : n → np
SOME0 : (np 0 → n0 ) → np 1 → np 2
EVERY : n → np
EVERY 0 : (np 0 → n0 ) → np 1 → np 2
LOVES : np → np → s
LOVES0 : np → np 3 → s0
LOVES1 : np → np 4 → s1
LOVES2 : (np 1 → np 2 ) → np → np 4 → s1
LOVES3 : (np 1 → np 2 ) → np 5 → np 6 → s2
LOVES 4 : np 5 → np 6 → s2
WHO : (np 3 → s0 ) → n → n
WHO 0 : (np 5 → np 6 → s2 ) → n → np 0 → n0
WHOM : (np 4 → s1 ) → n → n

Fig. 11.
Lexicon (final Grammar)
MAN := man
WOMAN := woman
SOME := λx. some + x
SOME 0 := λxy. some + (x y)EVERY := λx. every + x
EVERY 0 := λxy. every + (x y)LOVES := λxy. y + loves + x
LOVES 0 := λxy. y + loves + x
LOVES 1 := λxy. x + loves + y
LOVES 2 := λxyz. y + loves + (x z)
LOVES 3 := λxyz. z + loves + (x y)LOVES 4 := λxy. y + loves + x
WHO := λxy. y + who + (x )
WHO 0 := λwxy. x + who + (w y )
WHOM := λxy. y + whom + (x )Fig. 12.

7

Conclusions and future work

We have demonstrated, on an example, how to represent a given Lambek grammar as an
ACG. The generality of our method is somehow ensured by the several mathematical results
11- 171 -

on which it is based: weak equivalence between Lambek grammars and context-free grammars (Pentus [11, 12]), representing context-free grammars as ACGs (Pogodalla and deGroote [3]), lexicalization of second-order ACGs (Kanazawa and Yoshinaka [7, 8]), preservation of the derivations (Kanazawa and Salvati [6]). In practice, however, it will not always
be the case that the resulting grammar will be strongly equivalent to the original Lambek
grammar. This potential problem is due to Pentus’ construction which produces highly redundant grammars exhibiting all the spurious ambiguities related to the associativity of theLambek calculus.
In this paper, we circumvented the spurious ambiguity problem by implicitely using aparticular instantiation of Pentus’ construction. This more specific construction seems tobe less general than Pentus’. Consequently, we do not know whether it can be put at workin every case. Investigating this question will be the subject of future work.

References
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3. Ph. de Groote and S. Pogodalla. On the expressive power of abstract categorial grammars:
Representing context-free formalisms. Journal of Logic, Language and Information, 13(4):421–
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ESSLLI 2015 Workshop ‘Empirical Advances in Categorial Grammars’ (CG 2015), pages 34–
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6. M. Kanazawa and S. Salvati. The string-meaning relations definable by lambek grammars andcontext-free grammars. In G. Morrill and M.-J. Nederhof, editors, Formal Grammar - 17th and18th International Conferences, FG 2012, Opole, Poland, August 2012, Revised Selected Papers,
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Logic, 62(2):648–660, 1997.
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de Groote and M.-J. Nederhof, editors, Formal Grammar - 15th and 16th International Conferences, FG 2010, Copenhagen, Denmark, August 2010, FG 2011, Ljubljana, Slovenia, August
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12- 172 -

Rich Situated Propositions: the ‘right’ objects
for the content of propositional attitudes?
Kristina Liefke and Mark Bowker?
Ludwig-Maximilians-University Munich??

Abstract We present a new account of linguistic meaning, Rich Situated Semantics [RSS], on which the meaning of sentential utterances issemantically rich and informationally situated. In virtue of its situatedness, the meaning of an utterance varies with the informational situationof the cognitive agent interpreting the utterance. In virtue of its richness, the meaning of an utterance contains information beyond the utterance’s lexically encoded information. The agent-sensitivity of meaning
solves a number of problems in semantics and the philosophy of language(cf. [15, 21, 25]). In particular, since RSS varies the granularity of utterance meanings with the interpreting agent’s information state, it solves
the problem of finding suitably fine- or coarse-grained objects for thecontent of propositional attitudes. In virtue of this variation, a layman
will reason with more propositions than an expert.
Keywords: Information-sensitivity · interpreter-dependence · propositional attitude contents · richness of NL meaning · situated semantics

1

Introduction

The same utterance of a (non-indexical) sentence has a di↵erent meaning todi↵erent interpreting agents. This is due to the fact that di↵erent agents havedi↵erent information about the sentence’s subject matter, which is used in theutterance’s agent-specific interpretation: Depending on the agent’s backgroundknowledge, the utterance of (1) in a particular context will be interpreted as an
informationally rich proposition (e.g. as a proposition which contains the information that the inhabitant of Gobbler’s Knob is a groundhog/that Punxsutawney Phil is a member of the largest existing marmot species) or as an informationally poorer proposition which does not contain this additional information.
(1)

Punxsutawney Phil is a groundhog.

?

We would like to thank two anonymous referees for LENLS 13, Robin Cooper, Manfred Krifka, Sebastian Löbner, and Markus Werning for their valuable comments andsuggestions. The research for this paper has been supported by the German Research
Foundation (via K. Liefke’s grant LI 2562/1-1) and by LMU Munich’s Institutional
Strategy LMUexcellent within the framework of the German Excellence Initiative.
??
Munich Center for Mathematical Philosophy, Faculty 10, Geschwister-Scholl-Platz 1,80539 Munich, Germany. {Kristina.Liefke, Mark.Bowker}@lrz.uni-muenchen.de- 173 -

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Kristina Liefke and Mark Bowker

Most formal theories of linguistic meaning (e.g. [8, 26, 27, 32]) restrict themeaning of sentential utterances to the utterances’ lexical information (for (1): tothe information that the referent of the name Punxsutawney Phil is a groundhog)
and delegate all other available information about the utterance’s subject matter to areas like pragmatics or psychology. However, Moltmann [25] has observedthat they thus seriously underspecify the content of propositional attitudes. As
a result, these theories cannot explain why an inference is valid for some agents,
but invalid for others. The disregard of agents’ information about the utterance’s
subject matter further conflicts with the (environ-)mental embeddedness of linguistic understanding that has been observed in Situated Cognition (cf. [6, 36]).
This paper solves the above problem by extending the meaning of sentential
utterances to other relevant information that is available to the interpreter ofthe utterance at the time of the interpretation. Below, we first sketch our new
account of linguistic meaning, called Rich Situated Semantics (in Sect. 2.2). We
then present the rigid granularity problem for the content of propositional attitudes (in Sect. 3) and show how Rich Situated Semantics solves this problem (inSect. 4). Section 5 identifies several other intensional phenomena that lend themselves to a rich situated semantic treatment. The paper closes with a discussionof the RSS-modelling of assertion and shared belief (in Sect. 6).

2Rich Situated Semantics

Rich Situated Semantics [hereafter, RSS] (cf. [20, 21]) is a novel account of linguistic meaning on which the content of (utterances1 of) declarative sentences issemantically rich and informationally situated. In virtue of its situatedness, themeaning of a sentence varies with the informational situation of the cognitive
agent interpreting the sentence. In virtue of its richness, the meaning of a sentence contains information beyond the sentence’s lexically encoded information.
Below, we first illustrate the richness and situatedness of sentence meanings
and identify a number of theories from linguistics, philosophy, cognitive and computer science that suggest this richness and/or situatedness (in Sect. 2.1). We
then specify the RSS-interpretation of sentences (in Sect. 2.2) and identify somenotable consequences of this interpretation with respect to linguistic entailment
and equivalence (in Sect. 2.3). The section closes with a definition of truth forRich Situated Semantics (in Sect. 2.4).
2.1

Illustration and inspiration for RSS

To familiarize the reader with the core idea of RSS, we illustrate richness andsituatedness of linguistic meaning through an example:
1

We hereafter sometimes use the expression ‘meaning of a sentence’, instead of ‘meaning of an utterance of a sentence’. This is merely a terminological shortcut. The reader is asked to keep in mind that sentences are uttered by a speaker (with certain
background information) in a spatiotemporal and communicative situation and are
directed at an addressee (with a certain, likely di↵erent, background information).

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3

Consider the interpretation of (1) by Alf, Bea, and Chris. Assume that, rePunxsutawney Phil (here: ‘Phil’), these agents have the following information:
Alf: Phil lives in Gobbler’s Knob.
Bea: Phil is celebrated each February 2nd.
Chris: Phil lives in Gobbler’s Knob; Phil is celebrated each February 2nd.
Since Rich Situated Semantics assumes the inclusion-in-linguistic-meaning of theinterpreter’s information about the sentence’s subject matter (here: Phil), therich interpretation of (1) at Alf’s informational situation contains the information that Phil is a groundhog who lives in Gobbler’s Knob. The rich interpretation of (1) at Bea’s and Chris’ situation contain the information that Phil is agroundhog who is celebrated each February 2nd (Bea) and that Phil is a groundhog who lives in Gobbler’s Knob and is celebrated each February 2nd (Chris).
Thus, (1) is interpreted by Alf as (1.i), by Bea as (1.ii), and by Chris as (1.iii):(1)

i. Phil is a groundhog who/and lives in Gobbler’s Knob.
[alternatively: Phil, who lives in Gobbler’s Knob, is a groundhog.]
ii. Phil is a groundhog who is celebrated each February 2nd.
iii. Phil is a groundhog who lives in Gobbler’s Knob and is celebrated
each February 2nd.

The non-identity of the rich interpretations of (1) at Alf’s, Bea’s, and Chris’
informational situation witnesses the situatedness of linguistic meaning in RSS.
The greater informativeness of the interpretation of (1) at any of the above situations in comparison to the sentence’s traditional, possible worlds-interpretation(which only contains the sentence’s lexical information) witnesses the richness of
linguistic meaning in RSS. In particular, the rich interpretation of (1) at Alf’s informational situation contains the information that Phil lives in Gobbler’s Knob,
which is not contained in the sentence’s the lexical information.
The situatedness of linguistic meaning is inspired by work in Situated Cognition and dynamic semantics ([10,44]). Dynamic semantics interprets sentences asstate transitions, i.e. as functions from information states to the result of updating these states with the sentence’s lexical information. Rich linguistic meanings
are found in Fregean theories of belief content ([5, 9]), in semantic descriptivism
and generalized quantifier theory ([3,9,41]), and in frame semantics ([2,22]). The
former assume that any adequate representation of belief contents involves themodes of presentation of the individuals the beliefs are about. The latter assume
that proper names be interpreted analogously to definite NPs, i.e. as sets of properties of individuals. Frame semantics represents utterance meanings by rich recursive feature structures that account for the content of mental concepts.
2.2

The RSS-interpretation of sentential utterances

To capture the situatedness of linguistic meaning, RSS interprets sentences asfunctions from interpreters’ informational situations to the sentences’ meanings
at these situations (i.e. to the sentences’ situated meanings). These functions are
objects of type s ! ↵ (abbr. s↵), where ↵ is the type for the sentence meanings.

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The richness of situated sentence meanings is captured via (characteristic
functions of) partial sets of situations (s.t. ↵ := st).2 Such sets are familiarfrom the representation of sentence meanings in generalizations of possible world
semantics, including some versions of situation semantics ([16, 27]). However, theset of situations that serves as the meaning of a sentence in RSS is generally
smaller than the set of situations that serves as the meaning of this sentence insituational generalizations of possible world semantics. This is due to the factthat – in addition to being restricted to situations in which the sentence is true –the RSS-set is further restricted to situations which contain the interpreting
agent’s information about the sentence’s subject matter. For example, while (1)as-received-by-Alf 3 is interpreted as (2) in situational possible world semantics,
it is interpreted as (3) in RSS. Below, i is a variable over situations, as reflectedin the superscript s. The formulas groundhog (phil )(i) and livesinGK(phil )(i) assert that Phil is a groundhog in i and that Phil lives in Gobbler’s Knob in i:is [groundhog (phil )(i)]

(2)

s

(3)

i [groundhog (phil )(i) ^ livesinGK (phil )(i)]To capture the informational imperfection of cognitive agents, we identify
situations with partial (i.e. informationally incomplete) spatio-temporal parts ofworlds4 in which the parts’ individual inhabitants may fail to have some of theproperties which they have at the relevant world-part. Situations in rich situatedsemantics are thus “partial specifications of some of the entities in the universe
with [their] properties” [24, p. 614]. They are obtained from worlds by reducing
the information about the world’s inhabitants to the information available tothe agent at the given point in time. As a result, situations are agent- and timespecific: the same agent may be in di↵erent situations at di↵erent points in time.
We assume that situated sentence meanings are partially (or selectively) rich,i.e. that they contain – next to the sentence’s lexical information – all and only information about the sentence’s subject matter that is available to the interpreterof the sentence at the time of the interpretation. As a result, RSS interprets any
sentence p as a function from informational situations i to sets of situations whose
members contain the lexical information of p together with all information from iwhich regards some individual about which p carries information. We hereafter
call such sentences aboutness-relevant. Sentences that carry information about
the same individuals are called aboutness-identical. The RSS-interpretation of p
is given in (4).
In (4), x is a variable over individuals. The formula 't ⇣ t asserts that
contains the information of ' (i.e. that is less partial/better defined than '),s.t. is true if ' is true and is false if ' is false (cf. [27, p. 50, 47]). ' ⇣ is2

One can increase the granularity of situated sentence meanings by analyzing theminstead as semantically primitive (i.e. non-analyzable) propositions (cf. [7,28,32,43]).
The development of hyperfine-grained RSS is left for another occasion.
3
In fact, the interpreter of the utterance is irrelevant in possible world semantics.
4
The inclusion of impossible worlds or situations (cf. [12, 34, 45]) captures the possibility of agents’ misinformedness or false belief. We here neglect impossible worlds.

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5

defined as ((' ^ ) _ ((' _ ) ^ ⇤)) = ', where ⇤ is the neither-true-nor-false
formula. The introduction of ⇣ is made necessary by our association of t withthe set of truth-combinations, the resulting existence of two orderings on thetype-t domain (i.e. a truth- and an approximation-ordering), and the referenceof the material conditional to the ‘wrong’ ordering for our purposes (i.e. to thetruth-ordering; on this ordering, is true if ' is true, but ' is false if is false).
The formula abt (xe )(q st ) asserts that q carries information about the referentof x. The behavior of abt is governed by a variant of the axioms from [29, p. 129](cf. [19, p. 120–121]). These axioms include the aboutness-relevance (with respect
to an individual) of atomic formulas that contain the designating constant for theindividual as a constituent, the closure of aboutness-relevant propositions under
non-contradictory conjunction,5 the closure of aboutness-relevant propositions
under disjunction (given that both disjuncts contain information about the subject matter), and the robustness of aboutness under semantic equivalence.
is j s [pst (j) ^ (8q st (q(i) ^ 9xe [abt (x)(q) ^ abt (x)(p)]) ⇣ q(j))](4)

To better understand the interpretation of sentences in Rich Situated Semantics, consider the rich meaning of (1) at Alf’s informational situation, alf (in (5)).

⌘

i [groundhog (phil )(i)^(8q (q ( alf ) ^ 9xe [abt (x)(q)^abt (x)(p)]) ⇣ q (i))] (5)i [groundhog (phil )(i)^(8q (q ( alf ) ^ abt (phil )(q)) ⇣ q (i))]

We assume for simplicity that alf only contains the information that Phil lives
in Gobbler’s Knob (cf. Sect. 2.1) and that Bea has red hair. Since only the first
informational item regards Phil, (1) will be RSS-interpreted at alf as (3).
We next identify a concrete candidate for the set of situations described by(3): Assume a universe consisting of four situations, alf , 1 , 2 , and 3 and two
individuals: Phil (abbreviated p) and Bea (abbreviated b). We assume that Phil
lives in Gobbler’s Knob (Kp) in alf , 1 , and 2 , that Bea has red hair (Rb) in
alf and 2 , and that Phil is a groundhog (Gp) in 1 , 2 , and 3 (cf. Fig. 1).
Kp
Gp

Rb

alf

5

Kp

Kp

|

Gp

Rb

1
2
{z
}
members of the set from (3)Gp

3Figure 1. The meaning-at- alf of (1).

To avoid the inclusion of information that does not regard the subject matter, we demand (contra Perry) that both conjuncts be aboutness-relevant. This also avoids the
problem of obtaining aboutness-‘relevant’ conjunctions by combining an aboutnessirrelevant proposition with a trivially aboutness-relevant verum, or with falsum.

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Kristina Liefke and Mark Bowker

Then, since the lexical information of (1) (i.e. Phil is a groundhog) and the Philrelevant information from alf (i.e. Phil lives is Gobbler’s Knob) are included onlyin 1 and 2 (and in none of the other situations), the meaning of (1) at Alf’s informational situation is represented by the set { 1 , 2 } (underbraced in Fig. 1).
2.3

Consequences of RSS

The RSS-interpretation of sentences has a number of important consequences forthe individuation of situated sentence meanings. In particular, since RSS updates
the available information about a sentence’s subject matter with the sentence’s
lexical information, it identifies the meanings of sentences at situations thatdi↵er only with respect to the inclusion of the sentence’s lexical information.
Consider the interpretation of (1) at Len’s informational situation in which Philis a groundhog and lives in Gobbler’s Knob. (We assume that this situation does
not contain any other information about Phil, s.t., as regards Phil, it is identicalwith 1 ): At this situation, (1) has the same meaning (i.e. { 1 , 2 }) as at alf .
Note that, although (1) has the same rich meaning at Alf’s and at Len’s informational situation, its utterance has a di↵erent e↵ect on Alf’s than on Len’s situation: While (1)-as-received-by-Alf updates Alf’s information about Phil with
the information that Phil is a groundhog (s.t. Alf’s information is extended tothe information from 2 ), it leaves Len’s informational situation unchanged. The
updating e↵ect of (1) on Alf’s Phil-specific information is witnessed by the factthat (the information associated with) the meaning of (1) at Alf’s informationalsituation (i.e. { 1 , 2 }) is properly contained in (the information associated with)the meaning of (6) at Alf’s situation (i.e. { alf , 1 , 2 }).
(6)

Punxsutawney Phil lives in Gobbler’s Knob.

As a result of its interpretations’ informational richness, RSS further identifies the meanings of di↵erent aboutness-identical sentences at situations whichcontain the sentences’ lexical information. Consider the interpretation of (1) and(6) at 1 : Since this situation already contains the lexical information of (1) and(6), these sentences have the same meaning (i.e. { 1 , 2 }) at this situation.
We will see in Section 4 that the identification-at-a-situation of the meaningsof di↵erent aboutness-identical sentences solves the problem of finding suitably
fine- or coarse-grained objects for the content of propositional attitudes. This
problem is described in Section 3. However, to prepare our discussion of assertion
in RSS (cf. Sect. 6), we first give a truth-evaluation procedure for RSS terms.
2.4

Truth-evaluation in RSS

We have described situated sentence meanings as the result of updating the available information about the sentence’s subject matter with the sentence’s lexicalinformation. As a result of this description, situated sentence meanings in RSS
contain much more information than sentence meanings in situational generalizations of possible world semantics. However, much of this information is irrelevant for the sentences’ evaluation. For example, it does (or should) not matter- 178 -

Rich Situated Propositions

7

for the truth of (1) whether Phil lives in Gobbler’s Knob. Since non-situatedsentence meanings (type (s(st))) do not have the ‘right’ type for truth-evaluableobjects (they do not yield a truth-value when applied to a world), we need toprovide a custom truth-evaluation procedure for sentences in RSS:
To evaluate the truth of a sentence in Rich Situated Semantics, we check
whether the world of evaluation w is a member of the union of the sentence’s richinterpretations at all informational situations. The resulting truth-definition is
given below. In this definition, we use denotation brackets, J · K, as a notational
device for non-situated RSS-interpretations of sentences:
Definition 1 (Truth at a world).
In Rich Situated Semantics, a sentence p
Sis true at a world w if w 2
JpK(), where JpK( ) is the rich interpretationsof p at the situation .

By taking the union of the rich interpretations, JpK( ), of p for each situation , we obtain the set of situations in which p is true. This set is a situational
generalization of the classical Lewisian proposition denoted by p. The rationale
behind the above strategy is as follows: Since we assume the existence of a situation for every consistent combination of information (including the ‘empty’ combination; cf. [31, 32]), the members of the above union will never share more thanthe lexical information of p (and of p’s presuppositions). Since we identify the
result of updating a situation’s information via incompatible information withthe empty set of situations6 , the members of this union will never share less thanthe lexical information of p. In particular, situations which contain the information that not-p will not contribute their information to the above union.
We next turn to the rigid granularity problem for the content of propositional
attitudes. This problem lies in the fact that most theories of finely-grained meaning assume a single, uniform level of granularity for belief contents. As a result of
this assumption, these theories cannot explain why an inference is valid for someepistemic agents (given their background knowledge), but invalid for others.

3

The Rigid Granularity Problem

To avoid predicting agents’ logical omniscience, many theories of formal semantics (e.g. [8, 27, 32, 43]) assume hyperfine-grained sentence meanings that havestricter identity-conditions than sets of possible worlds. The level of granularityof these meanings is chosen in accordance with speakers’ intuitions about synonymy (cf. [32, p. 553]). Since most speakers judge the meanings of many intensionally equivalent sentences (e.g. of (1) and (7)) to be non-identical, hyperfinegrained semantics distinguish the meanings of these sentences.7
6This is due to the fact that the available information about the sentence’s subject
matter at these situations will include the complement of the sentence’s lexical information. Since we have excluded impossible situations from our considerations (cf.
fn. 4), no situations contain both an item of information and its complement.
7
The identification of the meanings of these sentences in possible world semantics isdue to the fact that, in the actual world at the current time (cf. the adjective existingin (7)), groundhogs are the largest marmot species.

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Kristina Liefke and Mark Bowker(7)

Punxsutawney Phil is a member of the largest existing marmot species.

The success of these semantics is hampered by the fact that the above identity(or non-identity-)judgements are not shared by all speakers for all sentence-pairs.
This is due to the fact that speakers’ judgements about sentential synonymy are
influenced by their background information about the sentences’ subject matter.
Depending on their informational situation, speakers will thus identify or distinguish the meanings of the same sentences. Consider the case of (1) and (7): Since
she knows that groundhogs are the largest existing marmot species, a groundhog
expert (e.g. Eve in (8)) will identify the meanings of (1) and (7). Since he is unaware of this fact, a groundhog layman (e.g. Len in (9)) will treat the meaningsof (1) and (7) as distinct. Any reasoner who is familiar with Eve and Len’s levelof groundhog expertise (s.t. (s)he knows that Eve assigns the same meaning to(1) and (3), while Len assigns di↵erent meanings to these sentences), will conclude (8b) from (8a), but not (9b) from (9a). Since hyperfine-grained semantics
assume the same level of granularity of content for all agents interpreting a sentence, they cannot distinguish between the validity of these two inferences.
(8)

a. Eve knows that Phil is a groundhog.
Tb. Eve knows that Phil is a member of the largest existing marmot
species.
T(9)

a. Len knows that Phil is a groundhog.
Tb. Len knows that Phil is a member of the largest existing marmot
species.
F

In particular, since hyperfine-grained semantics dinstinguish the meanings of (1)and (7), they will counterintuitively predict the invalidity of (8). Since traditional(coarse-grained) possible world semantics identifies the meanings of (1) and (7),it will counterintuitively predict the validity of (9).

4Rich Situated Attitudes

Rich Situated Semantics solves the above problem by varying the granularity of
meanings with the informational situation of the sentence’s interpreter. This ispossible since RSS identifies the meanings of di↵erent aboutness-identical sentences at situations which contain the sentences’ lexical information (cf. Sect. 2.3).
4.1

Solving the rigid granularity problem

Since Eve’s informational situation, eve , contains the lexical information of (1),(6) and (7), RSS identifies the meanings of (1) and (7) at this situation (i.e. (11)).
Since Len’s situation does not contain the lexical information of (7) (s.t. this sentence is interpreted as an update on Len’s information about Phil), RSS distinguishes the meanings of (1) (i.e. (3)) and (7) (i.e. (11)) at Len’s informationalsituation. With respect to the relevant subject domain, a layman will thus reason
with more sentence meanings than an expert.

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9

is [groundhog (phil )(i) ^ largestmarmot (phil )(i)](10)
is ([groundhog (phil )(i) ^ largestmarmot (phil )(i)] ^ livesinGK (phil )(i)) (11)The variation of sentences’ semantic granularity with the epistemic agent’s
informational situation explains the intuitive validity of the inference from (8)and the intuitive invalidity of the inference from (9). However, this explanation
presupposes the reasoner’s familiarity with Eve and Len’s level of expertise about
Phil (cf. Sect. 3). Reasoners who are not familiar with the two agents’ level of subject expertise (s.t. they are, in particular, unaware of Eve’s coarse-grained interpretation of (1) and (7)) will not be able to make the inference from (8).8
To capture the dependence of (8) on the reasoner’s awareness of the agent’s
subject expertise, we stipulate the following: when they occur in the complementof epistemic verbs like know, sentences are interpreted as sets of situations whose
members only encode the agent’s information about the sentence’s subject matter of whose availability to the agent the reasoner is aware. For the occurrence of(1) from (8a), this set is specified in (12). There, r is a variable for the reasoner.
aware (q( eve )(r)) ^ abt (phil )(q)] ⇣ q (i))]
i [groundhog (phil )(i) ^ (8q [aware(12)

We illustrate the reasoner-dependence of epistemic inferences by means of an
example: Compare the interpretation of (8a) and (8b) by two reasoners, Dan andFred, who have di↵erent degrees of familiarity with Eve’s knowledge about Phil.
In particular, Dan knows that, in Eve’s situation, Phil is a groundhog, belongs to
the largest existing marmot species, and lives in Gobbler’s Knob. Fred only knowsthat Phil is a groundhog in this situation. The complements of the occurrences ofknow from (8a) and (8b) are then interpreted as (11) by Dan and as (2) (cf. (8a))
and (10) (cf. (8b)) by Fred. Since only Dan is, thus, aware of Eve’s semantic identification of (1) and (7), only he will be able to make the inference from (8).
The above suggests the distinction between two types of validity: validity relative to an agent’s informational situation (here called situational validity) andtraditional validity (called validity simpliciter ). The two types are defined below:
Definition 2 (Situational validity). An inference is valid relative to the informational situation of some specific reasoner9 (or is valid-at- ) if the interpretation-at- of the inference’s premise(s) entails the interpretation-at- of theinference’s conclusion.
Definition 3 (Validity simpliciter). An inference is valid simpliciter if, at all
informational situations , the interpretation-at- of the inference’s premise(s)entails the interpretation-at- of the inference’s conclusion.
The condition from Definition 3 corresponds to requiring the entailment of the
traditional, possible worlds-interpretation of the conclusion by the traditional,
possible worlds-interpretation of the premise(s).
89The ability of (8b) to extend the reasoner’s knowledge depends on this unfamiliarity.
We assume as above that the reasoner’s information contains information about theepistemic agent’s level of subject expertise.

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Kristina Liefke and Mark Bowker

The di↵erent types of validity are illustrated by (8) and (13):(13)

a. Eve knows that Phil is a groundhog and lives in Gobbler’s Knob. T
b. Eve knows that Phil is a groundhog.
T

Since the situated interpretation of (8a) does not entail the situated interpretation of (8b) at some situations (e.g. 1 ), (8) is not valid simpliciter.
4.2

Consequences of situating attitudes

As a result of its rich situated interpretation of epistemic complements, RSS also
predicts the validity of inferences between epistemic reports like (14), whose complements are not intensionally equivalent.
(14)

a. Eve knows that Phil is a groundhog.
Tb. Eve knows that Phil lives in Gobbler’s Knob. T

The validity of these inferences may be justified by the reasoner’s familiarity
with the epistemic agent’s level of subject expertise: A reasoner (e.g. Dan) who isaware of the agent’s degree of informedness about the interpreted sentence’s subject matter will follow the agent in identifying his/her situated interpretation of
the complements of know from (14a) and (14b). However, intuitively, inferences
like (14) have a di↵erent kind of validity from inferences like (8).
To block inferences of the form of (14), we modify the meaning of the epistemically embedded occurrence of (1) from (12) to the set of situations whose members only encode the information contained in the complement’s lexical information of whose availability to Eve the reasoner is aware. This modification restricts
the set of validly substitutable complements of epistemic verbs like know to CPsthat are traditionally entailed10 by the CP. For the complement of know from(8a), this is achieved by (15). There, w ranges over possible worlds.
8w[groundhog (phil )(w) ! q(w)] ^ aware (q( eve )(r))) ^(15)i(8q [(8w[groundhog
abt (phil )(q)] ⇣ q (i))
Consider Dan’s interpretation of the complements from (14a) and (14b). Following (15), these complements are interpreted as (10) (cf. (14a)) and (16) (cf. (14b)).
Since the set of situations denoted by (10) is not contained in the set of situations
denoted by (16), the inference from (14) is no longer valid on this interpretation.
is [livesinGK (phil )(i)](16)

The interpretation from (15) is in line with the understanding of propositional
knowledge as focusing on a particular item of the agent’s subject-relevant information (at a given point in time), rather than as surveying all of this information(at this point in time). It di↵ers from most attitude treatments by extending
propositional knowledge to the union of the sentence’s lexical information andthe available aboutness-relevant information of its traditional entailments.
Our previous considerations may have made it seem as if our interpretationof epistemic complements was only an ad hoc move to prevent counterintuitive
10

Entailment is here defined in terms of (subset) inclusion of sets of possible worlds.

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11

inferences of the form of (14). This is not the case. To the contrary: Since di↵erent
verbs have di↵erently strict requirements on the substitution of their complements (with some verbs like remember allowing the substitution of other than the
traditionally entailed complements, see below), only RSS enables a modular account of complement restriction.
Consider the substitution properties of the complements of the verbs say verbatim, know, and remember : While know allows the substitution of its complement by sentences with the same subject matter to which the complement istraditionally equivalent (cf. the intuitive support for (8)), say verbatim does notallow such a substitution (cf. the intuitive support against (17), below). In contrast to the class of ‘substitutable’ complements of the verb know, the class of
substitutable complements of the verb remember extends beyond the complement’s traditional equivalents. The substitution-generality of the complement of
remember is witnessed by the intuitive support for the inference from (18).11 It
is reflected in the neutrality of remember between taking a CP or an NP as its
complement (cf. [13, 14, 37]) and by the semantic acceptability of the result of
replacing the CP complement of remember by the CP’s subject NP. This acceptability is illustrated in (19).12(17)

a. Eve said verbatim that Phil was a groundhog. T
b. Eve said verbatim that Phil was a woodchuck. F(18)

a. Dan remembered that Phil was nibbling at a dandelion. T
b. Dan remembered that Phil was endearing.
T(19)

a. Dan remembered [cp that Phil was endearing].
b. Dan remembered [np Phil].

5

Other Applications of RSS

We have shown above that Rich Situated Semantics solves the rigid granularity
problem for the content of propositional attitudes. Our presentation of RSS suggests that this semantics can also be used to explain several other intensional
phenomena. In particular, RSS helps solve some familiar problems of intensionality that have recently resurfaced in the philosophy of language. These include
the cognitive accessibility problem for propositions (cf. [15, 25, 42]), the problemof rational illogical belief (cf. [4, 38, 40]), and the substitution problem for theobjects and contents of propositional attitudes (cf. [1, 23, 25, 33]). Respectively,
they regard the difficulty of most mainstream theories of semantic content to explain how communicative agents can grasp abstract propositions, how rational
11

This inference assumes that the complements of the two occurrences of remember
from (18) describe the same remembered situation. The intuitive validity of this typeof inference is discussed in detail in [21, Sect. 4, 5].
12This substitution is not, in general, acceptable for other NP/CP-neutral verbs likeknow, which (assuming the same lexical entry) only accept abstract NP complements.

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12

Kristina Liefke and Mark Bowker

agents can jointly believe superficially contradictory propositions,13 and how thecontents of propositional attitudes (as denoted by the CP complements of epistemic verbs) di↵er from the objects of these attitudes (as denoted by the comple).
ments’ NP nominalizations of the form the proposition that
RSS solves these problems by incorporating the interpreting agents’ information about the sentences’ subject matter into the meaning of these sentences.
This information corresponds to the agents’ mode of presentation of the subject
matter (cf. [5,9,39]). In RSS, an object’s mode of presentation is thus represented
by the set of situations (type st) in which the object has the properties that theagent associates with it.14 Since rich situated sentence meanings depend on theinformation of the sentence’s interpreting agent, they are cognitively accessible.
The ability of agents to interpret di↵erent occurrences of the same NP w.r.t. theirinformational situations at di↵erent times further explains the possibility of rational illogical belief. The non-substitutability of CPs by their NP nominalizations in many contexts is explained by the situated (rich) interpretation of embedded CPs and the non-situated (poor) interpretation of their nominalizations.
The RSS-solution to the substitution problem is presented in [21, Sect. 7.3].
The detailed description of a rich situated solution to the remaining problems
will be given in a sequel to this paper.

6

Assertion and Shared Belief

Our previous discussion has focused on applications of RSS that exploit the availability of rich situated sentence meanings. However, there are a number of linguistic phenomena whose explanation makes recourse to informationally poor(er)and/or non-situated meanings. These include the use of sentences to state facts(cf. Sect. 2.4), to update the receiver’s current informational situation (or theconversational common ground), to give reasons (for actions and for facts), and to
express shared belief. These uses are illustrated in (20) to (23):(20)

Eve asserted that Phil belonged to the largest existing marmot species.

(21)
(22)

Eve taught Len that Phil belonged to the largest existing marmot species.
Since Phil is a member of the largest existing marmot species, he weighs
more than eight pounds.

(23)

Len believes some of the things Eve believes, including the proposition
that Phil is a groundhog.

In particular, Eve may utter (7) to communicate the fact that Phil belongs to thelargest existing marmot species (rather than some other fact she knows aboutPhil) (cf. (20)) and to update Len’s information about groundhogs by this fact
(rather than by some other fact about Phil that is new to Len) (cf. (21)). Similarly, she may utter (7) to give a reason for Phil’s high weight (cf. (22)). Many
13

These include Pierre’s simultaneous belief that London is pretty and that London is
not pretty (cf. [18])14This contrasts with the standard formal semantic representation of modes of presentation as sets of the object’s properties (type (et)t) (cf. [3, 17, 26]).

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Rich Situated Propositions

13

other sentences which receive an identical interpretation at her informational situation (e.g. (6)) would not serve this purpose. An analogous observation holds forthe belief report from (23), which does not warrant the substitution of (1) by (7).
The above examples illustrate the need to distinguish a sentence’s lexical information from its rich situated interpretations. Remarkably, this distinction isreadily available in Rich Situated Semantics: Section 2.4 has already identified astrategy for obtaining the lexical information of richly interpreted sentences. This
strategy is supported by the fact that non-situated sentence meanings only contain the sentences’ lexical information. It is complemented by the possibility ofanalyzing informational situations as situations that represent the discourse participants’ common ground and by the resulting possibility of replacing interpreters’ situating informational situations by common ground-representing situations. Such a replacement will be relevant in the explanation of justification andshared belief. The detailed treatment of the above phenomena is left as a projectfor future work.

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[18]- 186 -

A Nonmonotonic Sequent Calculus with Downward Conditional
Shuhei Shimamura1
1

Nihon University, Part-Time Lecturer, Chiba, Japanshuhei.shimamura0803@gmail.com

Abstract. I present a nonmonotonic sequent calculus equipped with a newlogical connective, the downward conditional. This system is motivated by
the philosophical idea of logical expressivist inferentialism, and has thefollowingtwo

distinctive

features.

First,

thesystemextendsanonmonotonic material consequence relation over an atomic language into
the one over a logically complex language, demonstrating how the usageof logically complex formula can be determined by the usage of atoms.
Since this extension is conservative, the resulting consequence relation is
also nonmonotonic. Second, the system shows how logically complex
language thus extended enables us to explicate the underlying material
consequence relation. Especially, the new connective of downward
conditional, together with the regular conditional, allows the consequence
set of an arbitrary premise set to codify all the consequences of every other(finitely different) premise set. In this sense, the system is monadological.
Keywords: Sequent Calculus, Nonmonotonicity, Downward Conditional,
Monadologicality.

1

Philosophical Motivations

Logical inferentialism claims that the meaning of a bit of logical vocabulary can be explained by
specifying its inferential role, namely the rule according to which it should be used in the
logical inference. My system 1 can be regarded as a product of such a logical inferentialist
project. However, this system is also motivated by commitment to two distinctive philosophicalideas: Semantic inferentialism and logical expressivism, both of which are developed by Robert
Brandom. Although these two ideas may not be as familiar to logicians as logical inferentialism,
I believe that they shed new and interesting light on the characteristic role that the logicalvocabulary plays in our inferential practice. Furthermore, they are the key to understand thetwo distinctive features of the system presented here, nonmonotonicity and the downward
1

This paper owes a great deal to collaborative work of the research group of Prof. Robert Brandom.

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conditional, neither of which has drawn too much attention in literature.
First, semantic inferentialism can be understood as a radical generalization of logicalinferentialism. Semantic inferentialists claim that not only the meaning of a bit of logicalvocabulary, but also the meaning of a bit of non-logical vocabulary are determined by its
inferential role, namely the way it ought to be used, in connection with other relevant
expressions, in our everyday inferential practice. 2
It is crucial that “inferential practice” here is not understood narrowly as the logical one, butmore broadly as “the game of giving and asking for reasons” in general. In this use-theoretic
approach to the meaning, an inference that contributes to determine the meanings of given
expressions is called a material inference (in contrast to a formal inference), since the correctnessof that inference does not depend on its logical form, but rather on the way those expressions
appear in it. For example, “A match is struck. Therefore, the match lights.” is an instance of suchmaterial inferences, since this inference is correct not because it has a certain logical form, but
because “match,” “is struck,” and “lights” are arranged in this particular manner in it. As
exemplified in this instance, material inferences are often defeasible, and therefore
nonmonotonic. Thus, for semantic inferentialists, who do not limit attention to the formal
inference, monotonicity is no longer a feature of inference that can be assumed across the board.
In a word, for semantic inferentialists, the meaning of a given expression, whether it is logicalor not, is determined by the role that it plays in the inferential practice broadly understood. At
this stage, however, one may wonder on what basis semantic inferentialists demarcate some bitsof vocabulary as logical from those that are not, and what is the relationship between thoselogical and non-logical bits of vocabulary. Logical expressivism is an answer to such ademarcation problem concerning logicality.
According to logical expressivism, there are two essential criteria to demarcate the logicalvocabulary from the non-logical one. First, the ability to use logical vocabulary can bealgorithmically elaborated from the ability to use non-logical vocabulary. In other words, if one
already knows how to use a set of non-logical vocabulary in the underlying material inferential
practice, one need not acquire any extra ability in order to come to know how to use a set of
logical vocabulary. Second, the distinctive role that bits of logical vocabulary play is to express or
explicate those material-inferential rules that we implicitly follow when we use bits of
non-logical vocabulary. 3 To put differently, logical vocabulary enables us to explicitly talk about,within the object language, what we implicitly follow.
These background philosophical commitments put three substantial constraints on the logicalsystem pursued in this paper. First, a material consequence relation that (according to2

3

Strictly speaking, this is what Brandom (1994) calls strong inferentialism.
See Brandom (2008).

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inferentialists) makes non-logical sentences mean what they mean should not be assumed to be
monotonic. Second, the consequence relation between logically complex sentences must be
determined by systematically extending this underlying nonmonotonic material consequence
relation. The connective rules conducting this work demonstrates how the ability to use
non-logical vocabulary can be algorithmically elaborated into the ability to use logical one.
Third, for the inferential roles of logical connectives thus fixed to count as explicating theunderlying material consequence relation, such extension of the underlying nonmonotonic
material consequence relation must be (at least) conservative. Consequently, the extended
consequence relation must also be nonmonotonic. Thus, logical inferentialist expressivists needa nonmonotonic logical system.
For logical expressivists, the paradigm of the logical (i.e., explicating) vocabulary is theconditional, since it enables us to codify (or “talk about”), within the object language, material
implications if a new proposition is added to a premise set (i.e., Γ|~ A→B iff Γ, A|~ B, where A ∉Γ). In this paper, I enrich the current inventory of logical vocabulary with a new type ofconditional――what I call the downward conditional ( －→). The expressive role of this new
conditional is the mirror image of that of the regular conditional: The downward conditional
enables us to codify (“talk about”) material implications if a proposition is subtracted from (asopposed to added to) a premise set (i.e., Γ|~ A－→B iff Γ－{A}|~ B, where A∈Γ). 4

2

Merits

It may seem that the downward conditional is a rather exotic connective from the standard
truth-conditional semanticist viewpoint. However, from the logical expressivist inferentialist
viewpoint, it is a natural counterpart of the regular conditional. Furthermore, there are at least
two merits of having the downward conditional in our system. First, the downward conditional
massively increases the expressive power of the system. Suppose Γ is given as a premise set.
The regular conditional only allows us to talk about, within the object language, consequencesof a premise set (finitely) bigger than Γ (i.e., Γ|~ U1→…→(Un→A) iff Γ, U1, …, Un |~ A, where U1,…, Un ∉ Γ). With the downward conditional in hand, however, we become able to talk about

consequences of an arbitrary premise set (finitely) different from Γ (i.e., Γ |~ D1－→…－→(Dm－→(U1→…→(Un→A)…)…) iff Γ－{D1,…, Dn}, U1, …, Um |~ A, where D1,…, Dn∈Γ and U1,…, Um ∉Γ). In this respect, each premise set is like Leibnizian monad in that “complete knowledge of it(i.e., knowledge of its consequences) gives us complete knowledge of every other (finite

4

The idea of the downward conditional and the related idea of Monadologicality (to be explained below)
are originally suggested by Bob Brandom. My contributions mainly consist in the technical results I am
presenting here.

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different) premise set (i.e., its consequences).” 5
Apart from this logical expressivist inferentialist viewpoint, however, the downward
conditional can also have wider appeal of its own. This unorthodox conditional enables us toformalize a special class of inferences that are made by setting aside, in contrast to obtaining orassuming, a certain set of knowledge. Although they have not drawn much attention fromlogicians, inferences of this class play an essential role in several philosophical arguments.
Cartesian Skepticism (Descartes 1641/2008) and The Veil of Ignorance (Rawls 1971) are two of
the most prominent examples. More casual instances can be found in everyday mind-reading
practice. One example of them is the false belief task, in which one is required to infer from theviewpoint of another who lacks a piece of knowledge that one already has. Note that such
downward inferences cannot be easily assimilated to those inferences already expressible by theregular logical vocabulary. After all, setting aside a premise A is different from either assuming￢A or assuming A∨￢A.

3

The Base Consequence Relation

To begin with, let L0－ be a set of atomic formula. From the perspective of logical expressivist
inferentialism, the material implication and incoherence among atomic formula can be regarded
as a given basis on which a semantic explanation can be built. Using “⊥,” these material
implication and incoherence can jointly be considered as a consequence relation from P(L0－) to
L0, where L0 = L0－∪{⊥}. Such a consequence relation can be represented in either of thefollowing manners:
Γ0 |~0 p; or p∈Cn0(Γ0),
where if p≠⊥, the premise set Γ0 implies A, while if p = ⊥, the premise set is incoherent. As
discussed before, the material implication is supposed to be defeasible. Therefore, Weakening is
not imposed as a structural rule in our system. 6 On the other hand, since the premise of theconsequence relation is treated as a set, Contraction and Permutation are automatically built
into the structure.
At this base level of material consequence relation, only the following two general properties
5
6

I borrow this phrase from a remark given by Ulf Hlobil, a member of our study group.
One might wonder if there is any connection between our system and relevance logic, since relevancelogicians also give up Weakening. However, our reason to abandon Weakening (i.e., the deferability ofthe base material consequence relation) is independent from the consideration of relevancy. Furthermore,
our base consequence relation is not supposed to be sensitive to the relevancy between the premise andconclusion, since as mentioned just below, we stipulate that Reflexivity holds with respect to an arbitrary
premise.

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are stipulated to hold:
Coherence of the empty set:

It is not the case that Ø|~0 ⊥.

Reflexivity at the base:

For any Γ0⊆L0－ and p∈L0－, Γ0, p|~0 p.

Let us call those material consequence relations that satisfy these conditions proper. Hereafter,
we only deal with proper material consequence relations.
In the next section, it is shown how a given material consequence relation |~0 over the atomic
language L0 can be extended into the consequence relation |~ over the logically complex
language L according to our sequent calculus. In course of this construction, the following threeaims are pursued. First, Reflexivity should be preserved at the logically extended consequence
relation.
Preservation of Reflexivity:

For any Γ⊆L－ and A∈L－, Γ, A|~ A.

Second, for our extension of a given material consequence relation to count as logical at all, it
has to conserve the original material consequence relation.
Conservativeness:

For any Γ0⊆L0 and p∈L0, Γ0|~ p, if and only if Γ0|~0 p.

Finally, the main purpose of enriching our logical vocabulary with the downward conditional is
to make the logically extended consequence relation monadological in that the consequence set ofa given premise encodes all the information about arbitrary consequences of that premise’s
finite neighborhood.
Monadologicality:

For any Γ, Γ’⊆L－ such that Γ－Γ’ and Γ’－Γ are finite, and for any
A such that Γ’ |~ A, there is some B such that Γ |~ B iff Γ’ |~ A.

These three properties are proved in section 5.

4

The Construction

First, the syntax of our logically complex language is given as follows:
Syntax. L =df. L－∪{⊥}, where L－ is inductively defined as follows:(1) For any p∈L0－, p∈L－, where L0－ is the set of atomic formula;- 191 -

(2) For any A, B∈L－, (￢A), (A→B), (A－→B), (A&B), (A∨B)∈L－;(3) These are the only formula that belong to L－.
Next, let us call our sequent calculus the “nonmonotonic paired-conditional” system (“NMPC”
for short). As semantic inferentialists, we regard the base material consequence relation as given.
Such a given base forms the axioms of NMPC.
Axioms of NMPC.
Ax:

If Γ|~0 A, then Γ|~ A is an axiom.

These axioms are only concerned with the consequence relation over the atomic language. Thisconsequence relation is extended to the one over the logically complex language by way of thefollowing sequent rules:
Rules of NMPC.
Γ－{A} |~ A→B――――――――― CCP

Γ, A |~ B―――――――――― CP

Γ, A |~ B

Γ－{A}|~ A→B

Γ, A |~ A－→B

Γ－{A} |~ B――――――――― CCP－
Γ－{A} |~ B
Γ |~ A――――――――――― CP－
Γ, A |~ A－→B
Γ, A |~ ⊥――――――――― LN――――――――― RNΓ ,￢A|~ ⊥Γ |~ ￢A

Γ－{A&B}, A, B |~ C―――――――――――― L&Γ－{A, B}, A&B |~ C
Γ－{A→B}, A, B |~ B―――――――――――― wwLC
Γ－{B}, A, A→B |~ B

Γ|~ A

Γ|~ B

――――――――――― R&
Γ|~ A&B
Γ－{A－→B}, A, B |~ C―――――――――――――― wLC－
Γ－{A, B}, A－→B |~ C

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Γ－{A∨B}, A |~ C

Γ－{A∨B}, B |~ C

Γ|~ A―――――――――――――――――――― LV―――――― RV1

Γ－{A, B}, A∨B |~ C

Γ|~ A∨B

Γ|~ B―――――― RV2
Γ|~ A∨B

Γ, A|~ A―――――――― COW
Γ, A, B |~ A
The extended consequence relation is defined as the minimal set of those sequents derivablefrom the axioms via the rules as specified above.
Three comments are in order here. First, it would be noticed that CCP and CCP－ aresimplifying rules. As seen in the next section, the presence of these simplifying rules somehow
complicates the proof of Conservativeness in NMPC. However, these rules cannot be dropped
since, as also seen below, they play essential role in the proof of Monadologicality of NMPC.
Second, Cut is neither an explicit nor admissible rule of NMPC. This is because in thepresence of CP and CCP, Cut collapses into Weakening (or Monotonicity). 7 Thus, thenonmonotonic base consequence relation cannot be conservatively extended by NMPC with
Cut.
Finally, the so-called Deduction Theorem (hereafter DT) is not a metatheorem of NMPC.
What holds instead is only the following slightly weaker version of DT: Γ, A |~ B iff Γ－{A}|~ A→B. The reason to give up full DT is the following. To ensure DT, CCP and CP must be
strengthened into the following CCP* and CP*:Γ |~ A→B――――――――― CCP*
Γ, A |~ B

Γ, A |~ B――――――――― CP*
Γ|~ A→B

With CCP*, however, the following illicit derivation 8 can be made with the help of thedownward conditional:
Γ, A, B |~ C――――――――――――――――― CP－
Γ, A, B, A&B |~ A&B －→C――――――――――――――――― CP78

See Hlobil (2016, sec. 4.3).
This was originally pointed out by Daniel Kaplan, another member of our study group.

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Γ, A, B |~ A&B →(A&B －→C)――――――――――――――――― L&
Γ, A&B |~ A&B →(A&B －→C)――――――――――――――――― CCP*
Γ, A&B |~ A&B －→C――――――――――――――――― CCP－
Γ |~ CThus, Conservativeness is violated. In fact, this derivation is a typical instance of how easily
adding the downward conditional to an otherwise unproblematic set of connectives wrecks theresulting consequence relation. Adding the downward conditional to a system is a balancing act.
Although it significantly enhances the expressive power of the system, it often interacts withother connectives of the system in such surprising manners that illegitimate derivations such as
the one just presented are possible. For analogous reasons, the bottom premise of CP－ and the
top premise of CCP－ must be Γ－{A} instead of Γ. The reason for which the top and bottom
premises of the left rules need those set subtraction clauses as presented above is also similar.

5

Properties of the system

To begin with, NMPC preserves Reflexivity of the base consequence relation.
Proposition 1. Given that for any Γ0⊆L0－and any p∈L0－, Γ0, p|~0 p, then for any (finite) Γ⊂L－and any A∈L－, Γ, A|~ A.
Proof. By induction on the complexity of the formulae on the right-hand side, A. Base case isimmediate from Reflexivity at the base and COW. Induction step is also straightforward.

■

Next, let us show that NMPC is a monadological system. All that is needed to prove thisproperty are the right and simplifying rules concerning the paired conditionals. First, noticethat CP and CCP assures that Γ’s implying B→A (i.e., Γ|~ B→A, where B ∉ Γ) “says” that Γ
together with B implies A (i.e., Γ, B |~ A) in that the former implication holds if, and only if thelatter holds. The situation is analogous with respect to the downward conditional. In thepresence of CP－ and CCP－, Γ’s implying C－→A (i.e., Γ|~ C－→A, where C ∈ Γ) “says” that Γ

minus C implies A (i.e., Γ－{C} |~ A) in that the former implication holds if, and only if the lather
holds. 9 Putting these two points together, Γ’s implying C－→(B→A) (i.e., Γ|~ C－→(B→A, where
9Note that the right-to-left direction of this biconditional claim would not hold, if it were not for CCP－.

- 194 -

B ∉ Γ and C ∈ Γ) “says” that Γ minus C together with B implies A (i.e., Γ－{C}, B |~ A), and soon and so forth. To generalize this point, any implication of any premise set that is reachable by
way of adding/subtracting a finite number of members to/from Γ can be encoded as animplication of Γ by using these two conditionals jointly and iteratively.

Proposition 2. For any Γ, Γ’⊂L such that Γ－Γ’ and Γ’－Γ are finite, Γ |~ D1－→…－→(Dm－→(U1→…→(Un→A)…)…) iff Γ’ |~ A, where A≠⊥, D1,…, Dn∈Γ－Γ’, and U1,…, Um∈Γ’－Γ.
Proof. Note that ((Γ－{D1,…, Dn})∪{U1,…, Um} = Γ’. The left-to-right direction is straightforwardfrom repeated applications of CCP－and CCP. The right-to-left direction is also immediate fromrepeated applications of CP and CP－.

■

Finally, let us turn to the proof of Conservativeness of NMPC. Although the details of thefollowing proof can seem complicated, the intuitive idea underlying it is relatively simple.
Consider an arbitrary proof tree T whose root is Γ0|~ p, where Γ0⊆L0－ and p∈L0. We want toshow that T does not violate Conservativeness――that is, (C) T’s leaves already contain Γ0|~ p.
If p = ⊥, the root can only come by Ax, and (C) immediately follows. However, if p ≠ ⊥, theroot can come by COW, CCP or CCP－. If the root comes by CCP－ (one of our notorious
simplifying rules!), the penultimate sequent is Γ0, A|~ A－→p (where A ∉ Γ0). Since A can be ofarbitrary complexity, apparently such a sequent can come by any of our left rules (except LN)
along with the three rules as mentioned above. This makes it appear that we would have toconsider intractably various types of cases in climbing up T to its leaves. The truth is, however,that no left rule can be applied on T, given that the left-hand side of T’s root contains noconnective (i.e., Γ0⊆L0－) (as shown by Lemma 2). This is because (roughly put) left rules always
adds one more connective to the left-hand side of the snake turnstile, while in NMPC the total
number of connectives on the left never decreases in descending a path of a proof tree as long as
the formulae on the right-hand side remains the same (as shown by Lemma 1). This observation

For instance, consider the following derivation, where p, q∈L0－.
q, p |~ p――――――――――――― wLC－
q－→p |~ p――――――――――――― CP－q, q－→p |~ q－→p――――――――――――― wwLC
q, q→(q－→p) |~ q－→p
If the right-to-left direction of the biconditional at issue holds, it must be the case that q→(q－→p) |~ p.
However, this sequent cannot come by any rule other than CCP－.

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simplifies the situation, and thereby allows us to specify a common form and properties sharedby any possible sequents occurring in T (see Lemma 3). Based on those form and properties, we
can show that every proof tree that concludes an atomic sequent satisfies (C) (Proposition 3).
To flesh out this proof sketch, we need to introduce a few definitions and prove several
lemmas. To begin with, let us define the summation of complexity of a set Γ, Σc(Γ), as follows:
Definition 1. Let Γ be {B1,…, Bm}.

𝑚Σc(Γ) =df �i=1 c(Bi), where c(Bi) is the complexity of Bi.

In other words, Σc(Γ) is the total number of the connectives contained in Γ. Note that it is often
the case that Σc(Γ)≠c(Γ), where c(Γ) is the complexity of Γ (i.e., the number of the connectives ofthe most complex formula(e) in Γ). Using this concept, our first lemma is articulated as follows:
Lemma 1. Suppose that Γ|~ A is proved at height h of a path of a proof tree, where A≠⊥, andΓ’|~ A is proved at height h’ of the same path, where h’≧h. Then, Σc(Γ’)≧Σc(Γ).
Proof. By induction on the remainder of the height of Γ’|~↑X A minus the height of Γ|~↑X A.
Base case is immediate. In induction step, all the cases except for the ones in which the rootcomes by CP, CP－, CCP, or CCP－ are also straightforward. In the latter exceptional cases,
however, the penultimate sequent does not share the same formulae on the right with the root.
In such cases, we have to divide up the proof path into several sections so that the top andbottom sequents of each section share the same formulae on the right. Such divisions are madeby appealing to the fact that CP[/CP－] is the only rule to introduce a regular[/downward]
conditional to the right. Then, we apply the induction hypothesis to the top and bottom
sequents of each section.

■

Next, let B be an arbitrary nested paired conditional, An [－]→(An-1[－]→…[－]→(A1[－]→p)…).
Two variants of the summation of complexity can be defined with respect to B: First, thesummation of complexity of all those Ai that are upward antecedents of B, ΣcuANT (B); second, thesummation of complexity of all those Aj that are downward antecedents of B, ΣcuANT (B).
Definition 2. let B be An [－]→(An-1[－]→…[－]→(A1[－]→p)…).
𝑛ΣcuANT (B) =df. �i=1 c uANT (B, i),c(Ai): Ai is an upward antecedent of B0

: otherwise.

where cuANT(B, i) =- 196 -

𝑛

ΣcuANT (B) =df. �i=1 c dANT (B, i),c(Ai): Ai is a downward antecedent of B0

: otherwise. 10

where cdANT(B, i) =Based on Lemma 1 and Definition 1 and 2, the following lemma is provable, which plays anessential role in showing that no left rule can be applied in any proof trees that could lead toany violation of Conservativeness.
Lemma 2. Suppose Γ |~↑X B is provable, where B = An [－]→(An-1[－]→…[－]→(A1[－]→p)…) and p∈
L0. Then, Σc(Γ) + ΣcuANT(B)≧ΣcdANT(B).
Proof. By induction on the complexity of the nested paired conditional on the right, B. In bothbase case and induction step, two cases are divided depending on whether the main connectiveof B is the regular or downward conditional. In those cases, Lemma 1 and (again) the fact that
CP[/CP－] is the only rule to introduce a regular[/downward] conditional to the right plays anessential role in showing the claim. I leave it to the reader to fill in the details of the proof.

■

We are in a position to prove the lemma specifying the form and properties shared by any
sequents consisting of a proof tree suspectable of the Conservativeness violation. Before that,
however, let us define one last thing. Let B be Am[－]→(Am-1 [－]→…(A1[－]→p)…) again. ∪/－(Γ, B)
is defined as the set obtained by adding to Γ those upward antecedents of B and subtractingfrom Γ those downward antecedents of B in the order from m to 1. That is:
Definition 3. Let B be Am[－]→(Am-1 [－]→…(A1[－]→p)…).
(…((Γ∪[/－]{Am})∪[/－]{Am-1})∪[/－]…)∪[/－]{A1} : m≠0∪/－(Γ, B)

11

=df
Γ

:m=0

10 More rigorously, cuANT(B, n) and cdANT(B, n) are inductively defined as follows. (Base case) If An is themain upward antecedent of B, then cuANT(B, n) = c(An); otherwise, cuANT(B, n) = 0. If An is the main
downward antecedent of B, then cdANT(B, n) = c(An); otherwise, cdANT(B, n) = 0. Let Bn-1 be An-1 [－]→(An-2[－]→…[－]→(A1[－]→p)…). (Induction step) If An-m is the main upward antecedent of Bn-m, then cuANT(B, n－m) =c(An-m); otherwise, cuANT(B, n－m) = 0. If An-m is the main downward antecedent of Bn-m, then cdANT(B, n－m)
= c(An-m); otherwise, cdANT(B, n－m) = 0. Let Bn-(m+1) be An-(m+1) [－]→(An-(m+2)[－]→…[－]→(A1[－]→p)…).
11 Here is the more rigorous definition of ∪/－(Γ, B): If B = p, then ∪/－(Γ, B) = Γ; otherwise, ∪/－(Γ, B) = S0,
where S0 is inductively defined as follows. (Base case) If Am is the main upward antecedent of B, then Sm-1 =Γ∪Am. If Am is the main downward antecedent of B, then Sm-1 = Γ－Am. Let Bm-1 be Am-1 [－]→(Am-2[－]→…[－]→(A1[－]→p)…). (Induction step) If Am-n is the main upward antecedent of Bm-n, then Sm-(n+1)= Sm-n∪Am-n. If Am-n
is the main downward antecedent of Bm-n, then Sm-(n+1) = Sm-n－Am-n. Let Bm-(n+1) be Am-(n+1) [－]→(Am-(n+2)[－]→…[－]→(A1[－]→p)…).

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Now, let us prove the crucial lemma.
Lemma 3. Suppose that T is a proof tree whose root is Γ0|~ p, where Γ0⊆L0－ and p∈L0－. Also
suppose that T’s height is n. For any height n－m (1≦m≦n－1), any sequent of T has thefollowing form (F) with the following two properties (P1 and P2):F:Γm|~ Bm, where Bm = Al[－]→(Al-1 [－]→…(A1[－]→p)…) and 0≦l≦m.

P1:

Σc(Γm) + ΣcuANT(Bm) = ΣcdANT(Bm).

P2:

Either ∪/－(Γm, Bm) = Γ0; or ∪/－(Γk, Bk)⊂Γ0 and p∈Γm.

Proof. By induction on the distance from the root, m (where n－m is the height of the sequent
whose form and properties are at issue). Base case is immediate. Induction step is alsostraightforward. Note that in induction step Lemma 2 plays a crucial role in assuring that thesequent whose form and properties are at issue can never come by any of our left rules.

■

It immediately follows from Lemma 3 that NMPC is conservative.
Proposition 3. Γ0|~ p, if and only if Γ0|~0 p, where Γ0⊆L0－ and p∈L0.
Proof. (The right-to-left direction) Straightforward from Ax and the fact that NMPC only extends(i.e., never shrinks) the consequence relation. (The left-to-right direction) Take an arbitrary proof
tree T whose root is Γ0|~ p, where Γ0⊆L0－ and p∈L0. We want to show that Γ0|~0 p. Now,
either p = ⊥ or not.
(Case 1) p = ⊥. Since Γ0⊆L0－, Γ0|~ ⊥ cannot come by any rules. Thus, Γ0|~ ⊥ is an axiom. By
Ax, Γ0|~0 ⊥.
(Case 2) p≠⊥. Take an arbitrary leaf of T (i.e., an arbitrary sequent at height 0 of T). By Lemma
3, any sequent of T has the following form with the following two properties:F:Γm|~ Bm, where Bm = Al[－]→(Al-1 [－]→…(A1[－]→p)…) and 0≦l≦m.

P1:

Σc(Γm) + ΣcuANT(Bm) = ΣcdANT(Bm).

P2:

Either ∪/－(Γm, Bm) = Γ0; or ∪/－(Γm, Bm)⊂Γ0 and p∈Γm.

However, no sequent with a complex formulae on the right-hand side can appear at height 0.
Thus, by (F), that given leaf must have the form: Γm|~ p. By (P2) and Definition 3, either it is thecase that Γm = Γ0, or that Γm⊂Γ0 and p∈Γm.
(Case 2.1) Γm = Γ0. Γ0|~ p at height 0. Since this can only come by Ax, Γ0|~0 p.

- 198 -(Case 2.2) Γm⊂Γ0 and p∈Γm. Γm|~ p at height 0. Since this can only come by Ax, Γm|~0 p. SinceΓm⊂Γ0, p∈Γm, and it is stipulated that Reflexivity holds at the base consequence relation, Γ0|~0
p.

6

■

Conclusion

As a semantic inferentialist, I started with a nonmonotonic material consequence relation overan atomic language. As a logical expressivist, I then presented a sequent calculus that maps thisbase consequence relation onto the one over a logically complex language. My sequent calculus
preserves Reflexivity of the base consequence relation, and extends the base consequence
relation conservatively. Thus, the resulting logically complex consequence relation is also
nonmonotonic. Furthermore, the newly introduced logical connective of the downward
conditional, along with the regular conditional, makes it possible to explicate, within the object
language, an arbitrary consequence of an arbitrary premise set that is finitely reachable from agiven premise set. In this sense, the extended consequence relation is monadological.

References
Brandom, R. B. (1994), Making it Explicit, Cambridge, MA: Harvard University Press.
Brandom, R. B. (2008), Between Saying and Doing: Towards an Analytic Pragmatism, Oxford:
Oxford University Press.
Descartes, R. (1641/2008), Meditations on First Philosophy: With Selections from the Objections andReplies, trans. Michael Moriarty, Oxford: Oxford University Press.
Hlobil, U. (2016), “A Nonmonotonic Sequent Calculus for Inferentialist Expressivists.” In Pavel
Arazim and Michal Dančák (eds.) The Logica Yearbook 2015, pp. 87-105, College
Publications: London.
Rawls, J. (1971), A Theory of Justice, Cambridge, MA: Harvard University Press.

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Game semantics andvagueness in natural language
Matthias F.J. Hofer?
Vienna University of Technology

Abstract We take up the challenge to define a procedure to systematically evaluate natural language statements involving vagueness, as is thecase for “About half days are nice.”, which is quantified with a vague proportional quantifier and applied to a vague predicate in the scope. Our
approach is embedded into an analytic game semantic framework, which
extends Giles’s game for Lukasiewicz logic by means of a randomization
operator and the Baaz-Delta operator.

Keywords game semantics; fuzzy logic; rough sets; natural language
processing; vagueness; proportional quantifiers; vague predicates; multiagents1

Introduction

The modeling of vague proportional quantifier expressions, like “about half”, “almost all”, and “at least about a third”, and of vague predicates, like “tall”, and“nice” is a great challenge taken up by many different researchers and communities, amongst which we have linguists [3,20,29], philosophers [7,32], computerscientists [1,28], and logicians [3,12,26]. There is a huge amount of literature on
fuzzy quantification, summarized in the recent survey article [6], while there isalso a whole monograph of Glöckner [15] about this topic. Also, recent developments in the field of mathematical fuzzy logic [4,5] contribute to the matterby addressing it using a game semantic framework, extending Giles’s game (Ggame) for Lukasiewicz logic by a randomization operator [11]. We here intend
to further extend the existing analytic game semantic framework, while following the systematic approach of Liu and Kerre [24], who proposed to split the
problem of generalized quantification into four steps in the following way1 :Type I: the quantifier as well as the scope predicate is crisp;
Type II: the quantifier is crisp, but the scope predicate is vague;
Type III: the quantifier is vague, but the scope predicate is crisp;
Type IV: the quantifier as well as the scope predicate are vague.
?
1

Supported by FWF projects I1897-N25 and W1255-N23
Crisp here means binary, hence zero or one valued.

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A Type I statement may have the form “All mothers are women.”, or “Thereis a man being more than 180 cm tall.”, where the quantifiers, ∀ and ∃, are crisp
in the sense that they only take one of the two classical truth values, zero or one.
The same goes for the properties, “being a mother”, “being a man/woman”, or“being more than 180 cm tall”, which are also taken as crisp, as each object
either completely fulfills them or not. A general Type IV statement may then beof the more complicated form “Almost all beaches in Thailand are beautiful.”,where the quantifier “almost all” and the scope predicate “beautiful” are vague.
To be able to model vague proportional quantifiers and vague predicatesadequately, we show how we can define a projection operator, also known as
Baaz-Delta, in our game semantic setting. Then Type I quantification alreadybecomes a rather easy task, and we continue to show how vague predicates can beconceived (Type II), following particular ideas from rough set theory or analytic
philosophy [27] respectively, where vague concepts depend on several crisp ones,
as well as on finitely many agents. We then define vague proportional quantifiers,
again following ideas from rough set theory, namely the one of granular levels[37]. This goes back to Zadeh [38], and has attracted some intensified attention
during the last years [21,35]. We apply this idea in the context of proportional
quantification (Type III) and eventually combine it with our vague predicates,
to arrive at the general level of Type IV proportional quantification.
The rest of the paper is organized as follows: Section 2 illustrates the core
aspects of the used game semantic framework and defines an important new
operator, the so called Baaz-Delta. Section 3 to Section 6 follow exactly the hierarchy of Liu and Kerre, which means each section corresponds to a respective
type of quantification, as introduced just above. Section 7 summarized the contribution and describes what has to be done to further augment the presented
material.

2

Giles’s game and extensions thereof

Giles’s game for Lukasiewicz logic is a two player zero sum game of perfect
information, where the players are called P (proponent) and O (opponent). In
contrast to the more classic Hintikka game [17], in Giles’s game it is possible forboth players to have asserted multisets of formulas at each state of the game.
This feature results from the following implication rule [13]:Game Rule 1 (R→ ) If P asserts F → G then O may attack by asserting F ,obliging P to assert G.
In this way, any game state of the form [F1 , ..., Fn | G1 , ..., Gm ], where theF 0 s are O’s and the G0 s are P’s asserted formulas, gets decomposed into a stateof the form [A1 , ..., An | B1 , ..., Bm ], where the A0 s are atoms which O eventually
has to take responsibility for and the B 0 s are those for which P has to accountfor [4]. Taking up responsibility for an assertion of an atom means to accept
having to pay 1e to the opponent player in case the atom is evaluated to falsewith respect to a given interpretation I (over a finite domain U = {a1 , . . . , an },- 201 -

with n ∈ N) and risk value assignment hiI , i.e. for every atomic formula A welet hAiI be its risk and have vI (A) = 1 − hAiI . Hence, the final risk, from P’s
perspective, of a game is computed as:
hA1 , ..., An | B1 , ..., Bm i =X

hBi iI −1≤i≤m

X

hAj iI(1)1≤j≤n

Note that the truth function corresponding to the previous game rule matches
the well known truth function of Lukasiewicz implication:vI (F → G) =min(1, 1 − vI (F ) + vI (G))
The negation of a formula F , defined as (F → ⊥), introduces role switch of
the players, and the following rule for strong conjunction the principle of limited
liability [4]:Game Rule 2 (R& ) If P asserts F &G, then, if O attacks, P has to either
assert F as well as G, or else ⊥.
Again, the truth function turns out to correspond to the known one,vI (F &G) =max(0, vI (F ) + vI (G) − 1),since, as well as O need not attack a by P asserted formula, also P canhedge her/his loss of asserting more than one formula, due to the definition of
the game rule. Here, with the strong conjunction rule it is stated explicitly that
P can assert ⊥ instead of both F and G, in case they are both wrong. It beunderstood that this so called principle of limited liability always be in placethroughout this paper, although it sometimes remains implicit [4]. We can givea characterization of strong Lukasiewicz logic via G-games as follows:
Theorem 1 ([10]). For every atomic formula A let hAi be its risk and let I
be the L-interpretation given by vI (A) = 1 − hAi. Then, if both, P and O, playrationally, any game starting in state [|F ] will end in a state where P’s final risk
is 1 − vI (F ).
To generalize the game for Lukasiewicz logic L, the authors of [12] introduce
the following randomizing quantifier rule in contrast to the ones for the existential and universal quantifiers, where either the proponent or the opponent canchoose a witnessing constant2 c:
Game Rule 3 (RΠ ) If P asserts ΠxF (x) then P has to assert F (c) for a randomly picked c.
2

For simplicity we identify objects from the domain with unique constants from a set
called U again.

- 202 -

It’s truth function turns out to be the following:vI (ΠxF (x)) =Pc∈U vI (F (c))|U|

= P ropx F (x)We now also define the so called Baaz-Delta [5], which leaves the truth valueof F unchanged in case it is 1, and projects all others to 0, in the following way3 :Game Rule 4 (R∆ ) If P asserts ∆F then, O attacks by choosing j ∈ N, obliging P to assert F j .
Theorem 2. A L(R∆ )-sentence ∆F , for a L formula F is evaluated tovM (∆F ) = x in an interpretation I iff every G-game for ∆F augmented byrule (R∆ ) is (1 − x)-valued for P under risk value assignment h·iI , i.e. P has astrategy to make his/her final risk at most 1 − x, and O has a strategy to makeP’s final risk at least 1 − x.
Now we define the following Nabla operator, which, complementary to the
Baaz-Delta, leaves the truth vahlue of F unchanged in case it is 0, and projects
all others to 1:
Definition 1. ∇F := ¬∆¬F
The Delta and Nabla Operator can be seen as tools to create discontinuous
truth functions, as they represent the limiting case of what can be expressed inordinary Lukasiewicz logic, where all truth functions are continuous.

3

Type I - the most simple case

As mentioned in the introduction we follow the systematics of Liu and Kerre,and start with defining crisp quantifiers which may only be applied to crisp
predicates. We define them in terms of our game semantic connectives, which
we have introduced in section 2. The resulting quantifiers behave as expected,
in the sense that they are merely a reformulation of respective quantifiers as
they are well known from the literature [15]. The main difference here, fromthe more computational approaches there, is that we embed quantifiers into an
analytic framework, instead of just giving the computation rule without relatingthem to a logical theory. The proportional quantifiers which we define in thepresent section evaluate for a given number k ∈ [0, 1] to true if and only if the
proportion of elements of the domain that fulfill the scope predicate is k, and tofalse otherwise, and we denote them by Q[=k] :Definition 2. ∀k ∈ [0, 1] we define Q[=k] xF̂ (x) = ∆(ΠxF̂ (x) ↔ k)F̂ denotes a crisp formula, i.e. one composed of crisp (zero or one valued)
predicates only. For formulas F and G, F ↔ G is defined as (F → G)&(G → F ),and for k ∈ [0, 1], k denotes the truth constant with the value k.
3

For a formula F , F j means F & . . . &F , j times.

- 203 -

Note that we can express the universal and existential quantifier as follows:∀xF (x) = ∆(ΠxF (x))∃xF (x) = ∇(ΠxF (x))
So far, we can deal with statements like “Exactly half (of the elements of thedomain) are students.”, or “Exactly a third (of the elements of the domain) aremothers.”, as being a student, or mother respectively, is supposed to be a crisp
property fully possessed or not by each element of the domain. The next section
is devoted to augmenting the framework to predicates which need not be of this
kind.

4

Type II - adding vague predicates

Vague predicates, such as “tall”, “friendly“, or “nice” are notoriously difficult tomodel, since it is not objectively determinable what makes competent speakers
judge objects to possess such a property. There is a lot discussion to be found in
the literature, coming from the linguists side [19] as well as from the side of fuzzy
logicians, who also give models for vague predicates [23]. We here relate our approach to the so called rough set theory [31], and show how we can model vague
predicates within our game semantic framework, following the convincing ideasof rough sets theorists [27,36], as well as those of linguists and philosophers[19,32], or computer scientists [14], still staying in a neat and uniform logicalframework. We understand vague predicates as dependent on a finite number ofcrisp predicates in the following way:
Definition 3. A vague predicate Pv comes with a set {Pv1 , . . . , Pvmv } of crisp
predicates, where mv ∈ N, such that for all c ∈ U it holds:vI (Pv (c)) = mrv iff vI (Pvi (c)) = 1 for r indices i ∈ {1, . . . , mv }As game rule this may be formalized as follows:
1
Game Rule 5 (RP) If P asserts Pv (c), then, if O attacks, i ∈ {1, . . . , mv }v
gets picked randomly, and then P has to assert Pvi (c).

The corresponding truth function can be determined as:vI (Pv (c)) =Pmv

i
i=1 vI (Pv (c))

mv

For simplicity we here assume the influence of each crisp predicate to be thesame, hence describe an unweighted scenario. The weighted case can be achieved
through allowing for multiple occurrences of the same indices in the sample spaceof the above rule.

- 204 -

Hence, a vague atomic formula evaluates to (completely) true if and only if
all crisp atomic formulas relevant to the vague one evaluate to true. In this sense
we can talk about lower approximations in the sense of rough set theory. Also, avague atomic formula evaluates to a truth value greater than zero, if and only ifthere is a crisp atomic formula relevant to it, which is true. This corresponds tothe idea of an upper approximation with regard to rough set theory. Only in caseall relevant crisp atomic formulas evaluate to false, the respective vague atomic
formula also evaluates to false. We use the following example, to illustrate how
this translates into the language of the theory:

4.1

ExampleWe define the vague predicate “tasty” through the following three crisp predicates: “salami”, “mushrooms”, and “garlic”. Let two universes U1 , U2 consist of
100 meals each. For the first universe we have 50 portions of pasta, and 50 pizzas,
all of which have salami, mushrooms, and garlic on top (hence are tasty!). For
the second universe we have 50 portions of pasta, 17 pizzas with only salami andgarlic on top, and 33 with only mushrooms on top. Now consider the following
statement:“Exactly 50% of all meals are (fully) tasty pizzas.”
With respect to U1 this statement is true, since the defining properties of“tasty” are (completely) fulfilled for exactly half of the elements of the domain.
Formally, we express this statement as:∆(Πx(∆Ptasty (x) ∧ pizza(x)) ↔ .5))
With respect to U2 we still have exactly 50% meals with a tasty-value greaterthan zero, hence we can still evaluate the following statement as true:“Exactly 50% of all meals are kind of tasty pizzas.”
This “kind of” - hedge gets formally represented in the following way:∆(Πx(∇Ptasty (x) ∧ pizza(x)) ↔ .5)
We are now able to express a much greater range of statements, namely
also those that involve vague predicates. Also we have described the vague linguistic hedge “kind of”, which in natural language expresses a certain kind of
uncertainty of the speaker regarding the definition of vague predicates present
in some utterance. This becomes possible through the nabla operator ∇, definedin section 2, using the delta operator ∆.

- 205 -

4.2

Definitions and a multi agent extension

Definition 4. crisp quantifier, vague predicates/formulas
Q[=k] x(fully F (x)) := ∆(Πx(∆F (x)) ↔ k)
Q[=k] x(kind of F (x)) := ∆(Πx(∇F (x)) ↔ k)
We can also think of the following generalization of the just described approach to vague predicates. Instead of taking one particular set of crisp predicates, relevant to describe the meaning of a vague predicate, we may think ofmany different such, reflecting the fact that different agents (competent speakers) may have different reasons to judge an object as “nice”, or the like. This
more general setting can be achieved through simply changing the respective
game rule in the following way, after we fixed some notation, to be able to referto different agents:
Definition 5. An agent aj , j ∈ {1, . . . , ma } comes with a set of crisp properties
mv,a
1
{Pv,a
, . . . , Pv,aj j } for any vague predicate Pv , mv,aj ∈ N, such that for all c ∈ U
jit holds:
ri
iff vI (Pv,a(c)) = 1 for r indices i ∈ {1, . . . , mv,aj }vI (Pv,aj (c)) = mv,ajj

The general game rule for vague predicates now is:2
Game Rule 6 (RP) If P asserts Pv (c), then, if O attacks, j ∈ {1, . . . , ma }v
gets picked randomly, followed by a random pick of i ∈ {1, . . . , mv,aj }, and theni(c).
P has to assert Pv,aj

Similarly to the first rule for vague predicates, we can determine the truth
function as:
Pma PmvvI (Pv (c)) =j=1i
i=1 vI (Pv,aj (c))

ma mv,aj

Again, the weighted case, where the influence of crisp predicates to a vague
one, or of the different agents to the evaluation respectively, is not uniform, caneasily be achieved, as described above.

5

Type III - vague proportional quantifiers, crisp scope

Modeling vague proportional quantifiers is, as argued by Liu and Kerre [24], as
well as by Glöckner and others [12,15], best performed in a step by step manner,
first focusing on the quantifiers and only later showing how they can then be
applied to formulas which may involve vague atomic subformulas. Following thisapproach, we now develop a way of evaluating vague proportional quantifiers,
again being inspired by rough set theory, while staying in our game semantic- 206 -

framework. Hence, in this section, we assume the scope formulas of quantifiers
to be crisp again, and only combine it all together in the next section.
An idea going back to Zadeh [38], being carried out much in recent years,
is granular computing [2]. The idea is to attach a level of granularity to certain
scenarios, hence making objects indistinguishable with respect to some (equivalence) relation. This idea, applied to vague concepts [37], is here extended tovague quantification in an seemingly obvious way. We apply the simple idea oftolerance intervals around some crisp value. Take the quantifier expression“about
half”, which can be associated to several such, e.g. [37, 5%, 62, 5%], [45%,55%],[49,5%,50,5%], or others. We can partition the unit interval in many different
ways, where each partitioning then corresponds to some level of granularity.
Having several such levels, we can talk about a granular hierarchy [21,35,37].
However, following everyday experience, we propose the following systematicrefinement procedure:– 3-partitioning: This can be associated to the common classification into three
categories, e.g. “‘small”,“medium”, and “large”– partitioning intervals: [0, 13 ), [ 31 , 23 ), [ 23 , 1]
– 5-partitioning: Five categories, say “tiny”, “small”, “medium”, ‘large”, “huge”– partitioning intervals: [0, 15 ), . . . , [ 45 , 1]
– 7-partitioning: E.g. “almost none”,“few”,“several”,“about half”,“most”,“many”,“almost all”
– partitioning intervals: [0, 17 ), . . . , [ 67 , 1]
– tenner-partitioning: (About) 0%, 10%, 20%, . . . , 90%, 100%
1
1
3
19
19
– partitioning intervals: [0, 20), [ 20, 20), . . . , [ 17
20 , 20 ), [ 20 , 1]

– fiver-partitioning: (About) 0%, 5%, 10%, 15%, . . . , 90%, 95%, 100%
1
1
3
39
39
– partitioning intervals: [0, 40), [ 40, 40), . . . , [ 37
40 , 40 ), [ 40 , 1]

– oner-partitioning: (About) 0%, 1%, 2%, 3%, . . . , 98%, 99%, 100%
1
1
3
199
199
– partitioning intervals: [0, 200), [ 200, 200), . . . , [ 197
200 , 200 ), [ 40 , 1]

– decimal place-partitioning: (About) 0%, 0.1%, 0.2%, 0.3%, . . . , 99.8%, 99.9%, 100%
1
1
3
1999
1999
– partitioning intervals: [0, 2000), [ 2000, 2000), . . . , [ 1997
2000 , 2000 ), [ 2000 , 1]

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All these classifications are, of course, somehow freely defined, and may hence
be changed accordingly. To describe the semantics of some vague proportional
quantifier Q, we need to fix a finite number of such levels of granularity, say
GL1 , . . . , GLmQ , mQ ∈ N, with respect to which we can evaluate respective
statements. In the present case, for statements “about half (of the domain elements) fulfill property F̂ ” we then have acceptance intervals4 as follows:– [ 13 , 23 ) (3-partitioning)– [ 52 , 35 ) (5-partitioning)– [ 73 , 47 ) (7-partitioning)– [45, 55) (tenner-partitioning)– [47.5, 52.5) (fiver-partitioning)– [49.5, 50.5) (oner-partitioning)– [49.95, 50.05) (decimal place-partitioning)
Definition 6. A granularity level GL corresponds to a partitioning of the realunit
SmGLinterval [0, 1] into finitely many disjoint intervals z1 , . . . , zmGL , such that
i=1 zi = [0, 1].
Definition 7. A vague proportional quantifier Q comes with a set {GL1 , . . . , GLmQ }of granularity levels, where mQ ∈ N, such that each such level has an unique acceptance interval for Q, i.e. for all granularity levels GLi , i ∈ {1, . . . , mQ } thereis exactly one interval zQ,GLi of the corresponding partitioning such that it holds:+−vI (QGLi xF̂ ) = (∆(ΠxF̂ (x) → zQ,GL))&(∆(zQ,GL→ ΠxF̂ (x)))
ii

If for some fixed vague proportional quantifier Q and granularity level GL,+−
zQ,GL is the acceptance interval for Q, we set zQ,GL, zQ,GLthe upper, and lower,boundary of the interval. QGLi denotes the quantifier Q restricted to one particular granularity level GLi .
As a game rule, we can express this definition in the following way:[≈k]Game Rule 7 (RGL,III )If P asserts QxF̂ (x), then, if O attacks, i ∈ {1, . . . , mQ } gets chosen randomly,+−
and then P has to assert (∆(ΠxF̂ (x) → zQ,GL))&(∆(zQ,GL→ ΠxF̂ (x))).
i
i
The corresponding truth function is the following:
PmQvI (QxF̂ (x)) =+−i=1 vI (∆(ΠxF̂ (x)→zQ,GLi )&∆(zQ,GLi →ΠxF̂ (x)))

mQ

The range of statements we can express now includes all those that startwith quantifier expression like “about halt”, “about a third”, or “almost all”. Bymeans of combining them, using the logical ∨ connective, we can also express
4

I.e. the statement is true if P ropx F̂ (x) is an element of this acceptance interval.

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statements like “at least about half”, or “at most about a third”, by simply
linking respective statements together. This allows for an even wider range ofquantification than Type II, and hence augments the applicability enormously, as
many natural language statements in real life are of this form. In a last remaining
step we combine Type II and Type III quantifiers, and end up with the final
Type IV quantifiers, which are able to systematically evaluate statements thatare vaguely quantified and have vague scope formulas at the same time.

6

Type IV - combining it all together

As our game semantic framework analytically decomposes formulas down toatomic subformulas, quantification, with respect to formulas potentially build byvague atomic subformulas, is rather straightforward, as we only need to combine
game rule 5 (or 6) with game rule 7 of the present paper. Hence we may restate
game rule 7 with the only adjustment of dropping the hat of the formerly crisp
scope formula F̂ .
[≈k]Game Rule 8 (RGL,IV )If P asserts QxF (x), then, if O attacks, i ∈ {1, . . . , mQ } gets chosen randomly,+−))&(∆(zQ,GL→ ΠxF (x))).
and then P has to assert (∆(ΠxF (x) → zQ,GLii
If we want to determine the truth function, we find the following:
PmQvI (QxF (x)) =+−i=1 vI (∆(ΠxF (x)→zQ,GLi )&∆(zQ,GLi →ΠxF (x)))

mQ

We are now able to evaluate all sort of vague statements, like “At most abouta third (of all domain elements) are nice”, or “Almost all (of the domain elements) are friendly or tall”, as long as the involved quantifiers and predicatesare well defined. This allows for a great deal of flexibility, and particularly embeds evaluations into a neat logical machinery, which is fully linked to recent
developments in the field of mathematical fuzzy logic [4].

7Conclusion and Outlook

We defined a way to systematically evaluate natural language statement within
an analytic game semantic framework. Our approach follows the hierarchy prescribed by Liu and Kerre and focuses first on crisp quantification with respect
to crisp scope formulas. In a next step we described how vague predicates canbe defined and show how we can quantify over vague formulas. We then introduce granular levels and define vague proportional quantification based on thisnotion, followed by a final step, where we combine all together.
The presented procedure may be extended into at least two important directions. One of them is the multi arity of quantifiers, as usually natural language statements are at least binary, as is the case with “Almost all children are- 209 -

friendly.”. It has been pointed out [15] that even higher arities may be of importance, and hence we focus on this aspect in ongoing work. Another important
augmentation is introducing non-proportional quantifiers, which may depend on
intensional matters, like “many” and “few” [22,25]. These are closely linked tomodal logics, but may also be integrated into the present game semantic setting
as we will show in future work.

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Proper names in interaction
Robin Cooper
University of Gothenburg
We present an argument that proper names need to be treated from a dialogicalperspective and that they present a simple argument that language is a dynamic
system in a state of flux with changes taking place in the language during thecourse of a dialogue. The analysis illuminates the nature of dialogue information
states as well as their relation to other cognitive components such as long term
memory and representation of the visual scene.
The analysis also gives us an interesting perspective on some old philosophicalpuzzles such as Kripke’s discussion of the Paderewski example. In addition it
points to a natural language technology in which we think in terms of linguistic
systems in a state of flux and have transparent representations of information
states that can be reasoned about and modified.

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Conversation as a Game
Yasuo Nakayama
Graduate School of Human Sciences, Osaka University
nakayama@hus.osaka-u.ac.jp

Abstract. Paul Grice (1975) proposed a theory of implicature that consists of acooperative principle and four maxims, namely maxim of quality, maxim ofquantity, maxim of relation, and maxim of manner. Sperber and Wilson
2(1986/1995 ) replaced these maxims by a single principle of relevance; theprinciple says that the speaker tries to be as relevant as possible in the circumstances. In this presentation, we propose to interpret a conversation as a gameand analyze conversational games based on a modification of Dynamic Normative Logic (DNL). In other words, we show how to analyze scalar implicaturesand conversational implicatures based on a framework that is an extension of
DNL.
Keywords: Paul Grice, dynamic normative logic, scalar implicature, conversational implicature

1 Dynamic Normative Logic and Its Modification
Dynamic Normative Logic (DNL)1 is based on Logic for Normative Systems(LNS) that was proposed in Nakayama (2010, 2011). We extend LNS and
define BOD Logic. We use not, &, or, , and  as meta-language expressions of logical connectives.
Definition 1. We use cons() as an abbreviation of " is consistent". Let
BB, OB, and DB be consistent sets of FO-sentences (i.e. sentences of FirstOrder Logic). Let BOD = BB, OB, DB. BB, OB, DB are called belief base,obligation base, and desire base, repeatedly. BOD is called a BOD- system.
(1a) [Deductive closure] Cn() = { :  deductively follows from }.
(1b) [Belief] BBOD  def Cn(BB).
(1c) [Possibility] MBOD  def cons(BB{}).
(1d) [Obligation] OBOD  def(cons(BBOB) & Cn(BBOB) & not (Cn(BB))).
(1e) [Prohibition] FBOD  def OBOD .
(1f) [Permission] PPB p def (cons(BBOB{})& not (Cn(BB))).
1

DNL was proposed in Nakayama (2014, 2015).

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(1g) [Desire] DBOD  def(cons(BBDB) & Cn(BBDB) & not (Cn(BB))).
(1h) BOD is consistent def (cons(BBOB) & cons(BBDB)).
(1i) mod(T) = def {M : M is a FO-model of T}.
Now, we show how to read main BOD-sentences.
(2a) BBOD : It is believed in BOD that .
(2b) MBOD : It is believed to be possible in BOD that .
(2c) OPB  : It is obligated in BOD that .
(2d) FPB  : It is forbidden in BOD that .
(2c) PPB  : It is permitted in BOD that .
(2f) DPB  : It is desired in BOD that .
The following list shows some of valid BOD-sentences2.
Proposition 1. The following sentences are main meta-logical theorems of
BOD Logic. Here, we assume BOD = BB, OB, DB.
(3a) (XBOD (  ) & XBOD )  XBOD , when X = B or X = O or X = D.
(3b) (XBOD (  ) & BBOD )  XBOD , when X = O or X = D.
(3c) (XBOD x1... xn ((x1,..., xn)  (x1,..., xn)) & BBOD (a1,..., an) & not
BBOD (a1,..., an))  XBOD (a1,..., an), when X = O or X = D.
(3d) (XNS x1... xn ((x1,..., xn)  (x1,..., xn)) & BNS (a1,..., an) &not BNS (a1,..., an))  XNS Q(a1,..., an), when X = F.
Proof. See Nakayama (2015).
In DNL, the belief base can be updated. Here, we also update BOD-systems
and call this framework BOD-DL. A BOD-system in BOD-DL involves information about its stage. We write a BOD-system of BOD-DL as follows:
BOD(n) = BB(n), DB(n), DB(n). Note that any BOD-system in BOD-DL is a
BOD-system in BOD Logic.

2 Ascription of Propositional States and Mutual Belief
In BOD-DL, we can imitate ascriptions of propositional states. We introduce
the following notations.

2'Valid' means here the validity in the First-Oder Logic (FOL).

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Definition 2.
(4a) BB(A>B,n) =def { : A believes in stage n that B believes that }.
(4b) OB(A>B,n) =def { : A believes in stage n that B believes that it is obligated that }.
(4c) DB(A>B,n) =def { : A believes in stage n that B desires that }.
(4d) A BOD-system for ascription of A to B in stage n is defined as in Definition 1: BOD(A>B,n) = BB(A>B,n), OB(A>B,n), DB(A>B,n).
(4e) BBOD(A>B,n) , OBOD(A>B,n) , and DBOD(A>B,n)  are defined in the same
way as in Definition 1.
(4f) KBOD(A>B,n)  def (BBOD(A>B,n)  & BBOD(B,n) ).
(4g) KDBOD(A>B,n)  def (DBOD(A>B,n)  & DBOD(B,n) ).
We read KBOD(A>B,n)  as follows: A knows in stage n that B believes that .
Similarly, KDBOD(A>B,n)  means that A knows in stage n that B desires that .
We assume that a common belief implies a corresponding mutual belief ingeneral.3
Assumption 1. Let G = {A, B}. Suppose that X = B or X = O or X = D.
Then, the following sentence holds.
XBOD(G,n)  (XBOD(A,n)  & XBOD(B,n)  & XBOD(A>B,n)  & XBOD(B>A,n) ).
We can easily show the following proposition.
Proposition 2. Here, we assume that BOD = BB, OB, DB and that Assumption 1 is satisfied. Then, the following BOD-sentences hold.
(5a) BBOD(G, n)   (KBOD(A>B, n)  & KBOD(B>A, n) ).
(5b) DBOD(G, n)   (KDBOD(A>B, n)  & KDBOD(B>A, n) ).

3 Gricean Implicatures in BOD-DLThe Gricean Theory consists of a cooperative principle and four maxims.
(6a) [Cooperative Principle] Contribute what is required by the acceptedpurpose of the conversation.
(6b) [Maxim of Quality] Make your contribution true; so do not convey
what you believe false or unjustified.
(6c) [Maxim of Quantity] Be informative as required.
3

We consider that O(G,n)  and D(G,n)  are kinds of mutual beliefs. For example, we read O(G,n)
 as "It is a common belief in G that it is obligated that ".

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(6d) [Maxim of Relation] Be relevant.
(6e) [Maxim of Manner] Be perspicuous; so avoid obscurity and ambiguity,and strive for brevity and order.
In this paper, we propose to interpret Gricean conversational rules as normative requirements for communication partners (see Assumption 2). Furthermore, we interpret the Gricean rules as rules in a conversational game. To
express the maxim of quality within BOD-DL, we need a relation of being
more informative.
Definition 3.
(7a) [Definition of Proper Subset] X  Y def (X Y & X Y).
(7b) more-informative(X, Y) def(cons(X) & cons(Y) & Cn(Y)  Cn(X)).
(7c) more-informative[T](, ) def
not (Cn(T)) & more-informative(T{}, T{}).
Usually, we use more-informative with respect to a belief base, for example
more-informative[BB(H)](, ). Note that the degree of information depends on
the belief state of communication partners. If S knows that  (thus,Cn(BB(S))), then  is not informative for S. However, the same information  can be informative for H, when H does not know that  (thus, not(Cn(BB(H))).
Let tr be a translation function that translates a FOL-sentence into an English sentence. Now, we can interpret Gricean maxims as obligations.
Assumption 2. Here, we assume that OB(S) denotes the obligation base ofa speaker. In this paper, we call S a Gricean speaker, when OB(S), BB(S)and BB(S>H) satisfy conditions (8a), (8b), and (8c). We call H a Gricean
hearer, when OB(H>S), BB(H>(H>S)), and BB(H>S) satisfy conditions(8d), (8e), and (8f). Furthermore, we call A a Gricean communication partner, when A is both a Gricean speaker and hearer. We consider a Griceancommunication partner as a player of Gricean conversational games.
(8a) [Maxim of Quality 1 for S] (  say(S, tr())) is an element of
OB(S).
(8b) [Maxim of Quality 2 for S] (say(S, tr())  ) is an element of BB(S).
(8c) [Maxim of Quantity for S] If more-informative[BB(S>H)](, ), then (( )  say(S, tr())) is an element of OB(S).
(8d) [Maxim of Quality 1 for H] (  say(S, tr())) is an element of
OB(H>S).
(8e) [Maxim of Quality 2 for H] (say(S, tr())  ) is an element of- 216 -

BB(H>S).
(8f) [Maxim of Quantity for H] If more-informative[BB((H>(S>H))](, ), then((  )  say(S, tr())) is an element of OB(H>S).
Based on BOD-DL, the following sentences follow from these requirements.
Proposition 3. If both S and H are Gricean communication partners, thenthe following BOD-sentences hold.
(9a) BBOD(S)   FBOD(S) say(S, tr()). (If S believes , then it is forbidden
for S to say tr().)(9b) BBOD(S) say(S, tr())  BBOD(S) . (If S believes that S says tr(), then S
believes .)(9c) more-informative[BB(S>H)](, )  (BBOD(S) (  )  OBOD(S) say(S,tr()). (If S believes that  is more informative for H than , then [if S
believes (  ), then it is obligated for S to say tr().)(9d) BBOD(H>S)   FBOD(H>S) say(S, tr()). (If H believes that S believes ,then H believes that it is forbidden for S to say tr().)(9e) BBOD(H>S) say(S, tr())  BBOD(H>S) . (If H believes that S believes thatS said tr(), then H believes that S believes .)(9f) more-informative[BB((H>(S>H))](, )  (BBOD(H>S) (  )  O BOD(H>S)
say(S, tr()). (If H believes that S believes that  is more informativefor H than , then [if H believes that S believes (  ), then H believes that it is obligated for S to say tr().)(9g) [Secondary Implicature4 for Belief] (not BBOD(S>H,n)  & (BBOD(S>H,n) or BBOD(S>H,n) ))  BBOD(S>H,n) .
Proof. (9a) follows from (8a) and Definition 1, and (9b) follows from (8b)and Definition 1, and (9c) follows from (8c) and Definition 1. (9d), (9e),and (9f) can be proved in the same way as (9a), (9b), and (9c). (9g) follows
from the classical logic. Q.E.D.
The maxim of relation and the maxim of manner can be also interpreted as
obligations for communication partners. In this paper, we do not intensively
discuss them, because they are heavily context sensitive.

4 Scalar Implicature
The scalar implicature is a paradigmatic application case for the Gricean theo4

The notion secondary implicature is proposed by Sauerland (2004). Here, we modified
Saueland's definition. For Sauerland's approach, see Section 4.1 of this paper.

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ry. In this section, we briefly summarize a standard neo-Gricean approaches to
Scalar Implicatures, then propose to reformulate this problem as a game.

4.1 Sauerland's Approach to Scalar Implicatures5
In this section, we summarize Sauerland's approach to scalar implicatures.
Sauerland's approach is based on an approach of Gazdar (1979: pp. 57-61).6
Gazdar's approach is applicable only to sentences  where an expression froma quantitative scale  appears not in the scope of any logical operator. Thiscan be expressed as  = f(). If  is not the maximal item on its quantative
scale Q, then Gazdar's mechanism predicts an implicature for . Let  be theword hat is following  on Q. In this case, the mechanism forms  = f() byreplacing  with  in . Finally, the scalar impliactures of  generated are
all expressions B for any  that can be derived in the way just described.7
Sauerland (2004) points out that Gazdar's approach predicts many wrong
results when it is applied to logically complex sentences. Furthermore,
Soames (1982: p. 521) and Horn (1989: p. 432) criticize the epistemic commitment assigned to the implicature by Gazdar's proposal. Both Soams andHorn argue that it only follows from Gricean maxims of conversation that  isuncertain, rather than that  is certainly false as Gazdar claims. In other worse,
they point out that what is implicated is B rather than B. B would
follow from B, only if some additional knowledge is assumed, for example
the knowledge that B  B holds (Sauerland 2004: p. 383).
Following Soams and Horn, Sauerland (2004) distinguish two kinds of
implicatures. A primary implicature is expressed by B and a secondary
implicature is expressed by B. These two implicatures are characterized asfollows.
(10a) If  is an element of the set of scalar alternatives of  and (  )and not (  ), then B is a primary implicature of .
(10b) If B is a primary implicature of  and B is consistent with the
conjunction of  and all primary implicatures of , then B is a secondary implicature of .
As Sauerland (2004) points out, to reason for a secondary implicature B,
one has to assume that the speaker obeys all the Gricean maxims to derive
5This section is added after a suggestion given by one of the reviewers.
See Sauerland (2004: p. 368f).
7
In Gazdar's original text, K is used to express that the speaker is certain that  is false.
We replaced K by B, because our paper uses B to express that the speaker is certain that  is false.
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B and furthermore that B  B or some stronger assumption is justified.
Spector (2007: p. 228f) points out that Sauerland's definition of the set of
scalar alternative is ad hoc. To identify implicatures of (p or q) Sauerland introduced two binary connectors cL and cR such that (p cL q) is equivalent to p,and (p cR q) is equivalent to q, and then stipulates the following scale: or,and, cL, cR. In the next section, we show that our approach does not need any
such special binary connectors to explain implicatures of disjunctions.

4.2 Interpretation of Scalar Implicartures within BOD-DL
Let us begin with a simple example of quantification.
(11) S says to H in stage 1: "Some athletes smoke".
We translate this sentence as x (athlete(x)  smoke(x)) and abbreviate it assome-smoker. Here, we assume that both S and H are Gricean communication
partners. After S's assertion of (11), we set: BB(S, 2) = BB(S, 1){say(S,tr(some-smoker))} & BB(H>S, 2) = BB(H>S, 1){say(S, tr(some-smoker))}.
Thus, we have KBOD(H>S, 2) say(S, tr(some-smoker)). Then, because of (9b) and(9e), KBOD(H>S, 2) some-smoker. Here, we abbreviate x (athlete(x) smoke(x)) as all-smoker. Because (all-smoker  some-smoker) is a FOLtheorem, according to (7c) and (9f), B(H>S, 2) all-smoker  O(H>S, 2) say (S, "Allathletes smoke"). However, because H knows that S makes statement (11), not
B(H>S,1) all-smoker. Thus, H believes that S's statement (11) implicates that S
does not know whether all athletes smoke. This is a primary implicature of(11). In a case that H is sure that S believes all-smoker or S believes allsmoker, because of (9d), we obtain B(H>S,2) all-smoker. This is a secondary
implicature of (11).
Now, let us take S's statement of (p or q) as an example. Let us assume thatboth S and H are Gricean communication partners in sense of Assumption 2.
We set: BB(S, 2) = BB(S, 1){say(S, (p or q))} & BB(H>S, 2) = BB(H>S, 1)
{say(S, (p or q))}.Then, because of (9e), we can easily show that (BBOD(H>S,2)(p  q). Furthermoe, according to (9f), not BBOD(H>S,2) p & not BBOD(H>S,2) q &not B(H>S,2) (p  q)). Now, suppose B(H>S,2) p. Then, it holds: B(H>S,2) q, because ((p  q)  p)  q) is a FOL-theorem. Then, according to (9f),
OBOD(H>S,2) say(S, tr(q)). However, because S did not say tr(q), we can infer
not BBOD(H>S,2) p. In the same way, we obtain not BBOD(H>S,2) q. This means,
not BBOD(H>S,2) p and not BBOD(H>S,2) q are primary implicatures, but p and qcannot be secondary implicatures of (p or q). We have these consequences
purely from the Gricean maxims. In this inference, we had no need for special
connectors such as cL or cR, which Sauerland (2004) introduced to block the- 219 -

possibility of secondary implicatures of p and q. As a result, only (p  q)
can be a possible secondary implicature. When BBOD(H>S,2) (p  q) or
BBOD(H>S,2) (p  q), we obtain BBOD(H>S,2) (p  q), which is a primary
implicature of saying "(p or q)".
As the next example, let us consider the following sentence from Sauerland(2004: p. 374).
(12) It's not the case that Paul ate all of the eggs.
This sentence can be expressed as x (egg(x)  ate(Paul, x)), which isabbreviated as not-all-eggs. We abbreviate x (egg(x)  ate(Paul, x)) as noegg and x (egg(x)  ate(Paul, x)) as some-eggs. Note that it holds in FOL:
no-egg  not-all-eggs. Now, we assume that the speaker S of (12) and thehearer H are Gricean communication partners in sense of Assumption 2 andthey know this each other. We set: BB(S, 2) = BB(S, 1){say(S, (12))} &BB(H>S, 2) = BB(H>S, 1) {say(S, (12))}. In this situation, because of (9e)and (9f), we obtain: (BBOD(H>S,2) not-all-eggs & not BBOD(H>S,2) no-egg). Thus,
not BBOD(H>S,2) no-egg is a primary implicature. In the case, in which(BBOD(H>S,2) no-egg or BBOD(H>S,2) no-egg) holds, because of (9f), we have
BBOD(H>S,2) no-egg. Then, we obtain BBOD(H>S,2) some-eggs, because (no-egg
 some-eggs) is a FOL-theorem. Here, BBOD(H>S,2) some-eggs is a secondary
implicature of (12).
To obtain the same result, Sauerland (2004: p. 373f) distinguishes an upward and downward operating environment. As we have just seen, to explain
implicatures in combination of negation and quantifiers, we do not need suchan additional assumption.

4.3 Implicatures and Background Knowledge
As Sauerland (2004: p. 309) points out, Chierchia's criticism against Gricean
approaches can be defended by taking a background knowledge into consideration. See the following example of Sauerland (2004: p. 309).
(13a) Every student wrote a paper or made a classroom presentation.
(13b) x (student(x)  (wrote-paper(x)  made-presentation(x)).
Sauerland (2004: p. 310) suggests that implicatures of (13a) depends on our
background knowledge. Without special background knowledge, the following sentence seems to be implied by (13a).
(14a) Not every student wrote a paper and made a classroom presentation.
(14b) x (student(x)  (wrote-paper(x)  made-presentation(x)).

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Indeed, according to our approach, because of (9f) and (9g), (14b) can be accepted as a secondary implicature, when (14b) or its negation is believed. According to Chierchia (2004: p.50), (13a) has the following implicature in aneutral context.
(15a) Every student wrote either a paper or made a presentation but not
both.
(15b) We abbreviate x (student(x)  (wrote-paper(x)  madepresentation(x)) as nobody-both.
(15c) (13b)  nobody-both..
However, when we already know that some of students made both assignments, implicature of (15c) is canceled (Chierchia 2004: p.50).
These observations by Sauerland (2004) and Chierchia (2004) can be supported by our approach. Suppose that two persons H1 and H2 try to interpret
sentence (13a) asserted by speaker S. We assume that these three persons are
Gricean communication partners in sense of Assumption 2. H1 does not knowthat some of students made both assignments, but H2 knows it (We abbreviatex (student(x)  wrote-paper(x)  made-presentation(x)) as both). Furthermore, we assume that H1 believes that all students are lazy and they do not doanything more than required (We abbreviate this statement lazy-students).
Thus, we have: lazy-studentsCn(BB(H1, 1)) & bothCn(BB(H2, 1)) & not(bothCn(BB(H1, 1))) & not (lazy-studentsCn(BB(H2, 1))). We assume:lazy-studentsCn(BB(H1>S, 1)) & bothCn(BB(H2>S,1)) & not(bothCn(BB(H1>S, 1))) & not (lazy-studentsCn(BB(H2>S, 1))).
Firstly, we analyze a belief update of BB(H1>S). We set: BB(H1, 2) =BB(H1, 1){(13b)} & BB(H1>S, 2) = BB(H1>S, 1) {(13b)}. We assume:(lazy-students  (13b))  nobody-both. Thus, BBOD(H1>S,2) ((13b)  nobodyboth). Because (nobody-both  (14b)) is a FOL-theorem, we obtain:
BBOD(H1>S,2) (14b). Here, (14b) is not an implicatue but a result of a straightforward inference from belief base of H1 at stage 2.
Now, let us analyze a belief update of BB(H2>S). We set: BB(H2, 2) =BB(H2, 1){(13b)} & BB(H2>S, 2) = BB(H2>S, 1){(13b)}. Here, we have:
BBOD(H2>S,2) ((13b)  both). Because (both  nobody-both) is a FOL-theorem,
it holds: BBOD(H2>S,2) (15c). When (BBOD(H2>S,2) (14b) or BBOD(H2>S,2) (14b)),
we obtain BBOD(H2>S,2) (14b), which is a secondary implicature of (13b).
This analysis shows that an analysis of implicature should be combinedwith a presupposition of background knowledge and that BOD-DL is an appropriate framework for dealing with this problem.

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5 Conversation as a GameLet us analyze some examples of short conversations. The following example
is an example of speaker implicature from Davis (2014: sect. 1). In this example, Alan asks a question and Barb answers to it.
(16a) Alan: Are you going to Paul's party?
(16b) Barb: I have to work.
Let us assume that Alan and Barb are Gricean communication partners.
Here, we introduce some additional assumptions which deal with expressions
about desire.
Assumption 3. When S and H are Grican communication partners, then thefollowing conditions are satisfied.
(17a) [Maxim of Quality 2 for Ss Desire] (say(S, tr(DBOD(S, k) ))  ) is
an element of DB(S, k).
(17b) [Maxim of Quality 2 for Hs Attribution of Ss Desire] (say(S,tr(DBOD(S, k)) )  ) is an element of DB(H>S, k).
From Assumption 3, we have the following consequence.
Proposition 4. If both S and H are Gricean communication partners and
BOD(S, k) and BOD(H>S, k) are consistent, then the following BODsentence hold.
(18) (not BBOD(S, k)  & not BBOD(H>S, k)  & KBOD(H>S, k) say(S, tr(DBOD(S, k) )))
 KDBOD(H>S, k) .
Proof. We assume that both S and H are Gricean communication partners
and BOD(S, k) and BOD(H>S, k) are consistent. Then, (18) follows from(1b), (1g), (4f), (4g), (17a), and (17b). Q.E.D.
In BOD-DL, this conversation can be interpreted in the following way.
Here, we assume A = Alan, B = Barb, and G = {A, B}. We assume that both
Alan and Bob are Gricean communication partners.
The goal of this game is for Alan to know if Barb wants to go Paul's party.
This goal can be formally expressed as follows: k (KDBOD(A>B,k) goingparty(B) or KDBOD(A>B,k) going-party(B)). In the first stage of the conversation, Alan does not know Barb's intention. Therefore, Alan asks question (16a). Here, we assume that Alan and Barb mutually believe thatwork and participation in a party are incompatible, formally: BBOD(G, 1)- 222 -

x (work(x)  going-party(x)). Then, from (5a) and Assumption 1, we
have: KBOD(A>B,1) x (work(x)  going-party(x)). After Barb's answer(16b), we obtain: KBOD(A>B, 2) say(S, tr(DBOD(B, 2) work(B))). Then, according to Proposition 4, we have: KDBOD(A>B, 2) work(B))). Because we can assume that not BBOD(A>B,1) going-party(B)) & not BBOD(B,1) goingparty(B)), we obtain: KDBOD(A>B, 2) going-party(B))). So, Alan knows instage 2 that Barb is not going to the party. This game ends here, because
the initial goal has been achieved.
Next, let us consider the following conversation from Sperber and Wilson(1995: p. 34).
(19a) Peter: Do you want some coffee?
(19b) Mary: Coffee would keep me awake.
The goal for Peter is serving coffee for Mary, if she wants. So, Peter needs toknow if Mary wants some coffee. There are two possible implicatures of (19b)depending on their mutual belief. We assume P = Peter, M = Mary, and G = {P,M}. We assume that Peter and Mary mutually believe that being awake is incompatible with sleeping: BBOD(G,1) x (awake(x)  sleep(x)). Then, from
Assumption 1 and Proposition 2, we have: KBOD(P>M,1) x (awake(x) sleep(x)). Furthermore, we assume that P and M are Gricean communication
partners.
[Case 1] Peter and Mary mutually believe that Mary wants to be awake:
DBOD(G,1) awake(M). Then, from Assumption 1 and Proposition 2, we
have: KDBOD(P>M, 1) awake(M). After Mary's answer (19b), DBOD(G, 2) say(M,(19b)). Then, because of (4f), (9b), (9e), and Proposition 2, we have:
KBOD(P>M, 2) (drink(M, coffee)  awake(M)). Here, we assume that Peterand Mary mutually believe that drinking coffee is the best method for being awake in this situation. 8 Then, we have: DBOD(P>M,2) drink(M, coffee).
So, Peter ascribes to Mary a desire of drinking coffee. Thus, Peter willserve Mary coffee and the game ends.
[Case 2] Peter and Mary mutually believe that Mary wants to sleep: D(G,1)
sleep(M). Then, from Proposition 2, we have: KDBOD(P>M,1) sleep(M). After
Mary's answer (19b), Peter ascribes to Mary the belief that coffee keeps
her awake as described in case 1. Then, in BOD-DL, we can prove: not
DBOD(P>M,2) drink(M, coffee). This is a primary implicare of (19b). Becauseof the maxim of relation, it is likely that (DBOD(P>M,2) drink(M, coffee) or
8Here, we will need a formulation of abductive reasoning. However, for lack of space, we skip
this part.

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DBOD(P>M,2) drink(M, coffee)). When this assumption holds, we obtain:
DBOD(P>M,2) drink(M, coffee)). This is a secondary implicature of (19b).
So, Peter ascribes to Mary the desire of not-drinking coffee. The gameends here.
In this way, we can analyze conversational implicatures based on BOD-DL.

6 Concluding Remarks
Grice's theory of implicature is too strict and has many counterexamples (SeeDavis 2014: sect. 6~9). Our approach is more flexible than Gricean approachand it is more precisely defined than the Relevance Theory. In a ceremony,people use official expressions, because games of a ceremony have an official
meaning. As this example shows, there are different types of conversational
games. A game can contain some sub-games and these sub-games can containother sub-games. In this way, we can introduce conversational structure inorder to analyze complex conversations.9

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