Proceedings of the 19th International Workshop of
Logic and Engineering of
Natural Language Semantics 19 (LENLS19)

hosted by The Association for Logic, Language and Information(FoLLI)

Workshop Chair
Daisuke Bekki (Ochanomizu University)

Hybrid (Ochanomizu University | Online)
on 19 (Sat), 20 (Sun) November, 2022.
Hybrid (The University of Tokyo, Komaba 1 Campus | Online)
on 21 (Mon) November, 2022.

PrefaceThis 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 approaches to semantics and pragmatics
 Nonclassical logic and its relation to natural language (especially substructural, fuzzy,
categorical, and topological logic)
 Formal philosophy of language
 Scientiﬁc methodology and/or experimental design in linguistics
LENLS is being organized by an alliance of ”AI systems that can explain by language basedon knowledge and reasoning” project (Grant Number JPMJCR20D2), funded by JST CREST
Programs ”Core technologies for trusted quality AI systems.”

Workship Organizers/Program Committee
Daisuke Bekki (Ochanomizu University)
Alastair Butler (Hirosaki University)
Patrick D. Elliott (Heinrich-Heine University of Dusseldorf)
Naoya Fujikawa (University of Tokyo)
Yurie Hara (Hokkaido University)
Robert Henderson (University of Arizona)
Hitomi Hirayama (Keio University)
Magdalena Kaufmann (University of Connecticut)
Koji Mineshima (Keio University)
Elin McCready (Aoyama Gakuin University)
Yoshiki Mori (University of Tokyo)
David Y. Oshima (Nagoya University)
Katsuhiko Sano (Hokkaido University)
Osamu Sawada (Kobe University)
Ribeka Tanaka (Ochanomizu University)
Wataru Uegaki (University of Edinburgh)
Katsuhiko Yabushita (Naruto University of Education)
Tomoyuki Yamada (Hokkaido University)
Shunsuke Yatabe (Ochanomizu University)
Kei Yoshimoto (Tohoku University)iProgram and Table of Contents
1st Day: November 19 (Sat)9:50-10:00 Opening remark
10:00-11:30
The semantic markedness of the Japanese negative preterite: Non-existence of (positive) eventualities vs. existence of negative eventualities
David Yoshikazu Oshima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.1
Semantic Properties of the Japanese Emphatic Minimizer ’NP-no-kakera’ Based on the Modal
Base
Misato Ido . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.6
Slurs’ variability, emotional dimensions and game-theoretic pragmatics
Victor Carranza Pinedo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.11
13:30-15:00
Granularity in number and polarity eﬀects
Eri Tanaka and Kenta Mizutani . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.16
Formalizing argument structures with Combinatory Categorial Grammar
Shinnosuke Isono, Takuya Hasegawa, Kohei Kajikawa, Koichi Kono, Shiho Nakamura andYohei Oseki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.21
Detecting modality and evidentiality: Against purely temporal-aspectual analyses of the German semi-modal drohen
Shinya Okano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.26
15:30-17:00
Cumulative reading, QUD, and maximal informativeness
Linmin Zhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.31
Language politeness in social network
Liping Tang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.36

ii

The Absence of NOR in JapaneseHajime Mori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.41
17:00-18:00 Invited talkEnd-to-End Compositional Modelling of Vector-Based Semantics
Michael Moortgat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.46

2nd Day: November 20 (Sun)10:00-11:30
Measurement Theory Meets Mereology in Multidimensionality in Resemblance Nominalism
Satoru Suzuki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.47
Extraction pathway marking as proof structure marking
Yusuke Kubota and Robert Levine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.52
Constraining parse ambiguity with grammatical codes
Alastair Butler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.57
13:30-15:00
In Search of a Type Theory for Fuzzy Properties
Youyou Cong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.62
Logical Inference System with Text-to-Image Generation for Phrase Abduction
Akiyoshi Tomihari and Hitomi Yanaka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.67
Answers, Exhaustivity, and Presupposition of wh-questions in Dependent Type Semantics
Hayate Funakura . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.72
15:30-17:00
A Proof-Theoretic Analysis of Meaning of a Formula in a Combination of Intuitionistic andClassical Propositional Logic
Masanobu Toyooka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.77
Logic Operators and Quantiﬁers in Type-Theory of Algorithms
Roussanka Loukanova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.82

iii

Deriving formal semantic representations from dependency structures
Philippe de Groote . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.87
17:00-18:00 Invited talk
Modal Reasoning and Theorizing in Quantiﬁed Modal Logic
Kohei Kishida . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p.92

3rd Day: November 21 (Mon)
10:00-11:30
Factivity Alternation Types and Compensatory Prosodic Focus Marking
Chungmin Lee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.93
The Discourse Function of Aspect in Japanese
Kei Yoshimoto, Joseph Tabolt, Zhen Zhou, Hiromi Kaji and Tamami Shimada . . . . . p.98
Left-branching tree in CCG with D combinator
Masaya Taniguch and Satoshi Tojo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p.103
13:30-15:00
To be Canceled, or Not to be Canceled: Reconsidering the Caused Possession in the Dative
Alternation Experimentally
Junya Fukuta and Koji Shimamura . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .p.108
Events and Relative Clauses
Oleg Kiselyov and Haruki Watanabe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.113
Truthmaker Semantics for Degreeism of Vagueness
Shimpei Endo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.118
15:30-17:00
KK is Wrong Because We Say So
Simon Goldstein and John Hawthorne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.121
Three Rich-Lexicon Theories of Slurs: A Comparison
Dan Zeman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.126

iv

Contrafactives and learnability
David Strohmaier and Simon Wimmer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.130
17:00-18:00 Invited talk
Gricean Pragmatics According to Himself
Nayuta Miki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p.134
18:00-18:10 Closing remarkv

The semantic markedness of the Japanese negative preterite:
Non-existence of (positive) eventualities vs. existence of negative eventualities
David Y. Oshima (Nagoya University)[Introduction] In Japanese, the use of a negative preterite (past-perfective) clause is discourse-pragmaticallyconstrained, and oftentimes a negative nonpast-perfect (present-perfect) clause with -te IRU (where IRU is
a “nonperfective” auxiliary that may receive a wide array of interpretations, including progressive, resultative, and perfect; e.g. Kudo 2020) is used where a preterite clause is expected (Matsuda 2002, Yamashita
2014, Kusumoto 2016). To illustrate, the preterite in (1Ba ) sounds unnatural, conveying something to theeffect that the speaker could have hired a new nurse. A nonpast-perfect clause is not pragmatically loadedin the same way, as seen in (1Bb ) (Npfv = nonperfective, Npst = nonpast, Ger = gerund).
(1)(A and B are medical practitioners.)
A:
Anta no tokoro, sengetsu atarashii kangoshi yatotta?
you Gen place last.month new.Npst nurse
hire.Pst‘Did you hire a new nurse at your clinic last month?’
Ba : #E? Yatowanakatta yo. Nande? [preterite]
Intj hire.Neg.Pst
DPrt why(Huh? I didn’t hire anyone. Why?)
Bb : E? Yatotte naiyo. Nande? [nonpast perfect]
Intj hire.Ger NpfvAux.Neg.Npst DPrt why‘Huh? I didn’t hire anyone. Why?’
It can be said that, as long as negative clauses are concerned, the nonpast perfect rather than the preteriteis the default way to describe a situation in the past.
This work argues that the Japanese negative preterite invariably expresses the occurrence of a “negative eventuality”, whose existence and ontological nature have been extensively debated, rather than the
non-occurrence of eventualities. It will be furthermore argued that, while in general Japanese nonpasttensed clauses specify that the topic time (in Klein’s 1994 sense) is the present or a future time, this does
not necessarily apply to nonpast-perfect clauses, making it possible for a negative nonpast-perfect clause
to represent the non-occurrence of events in a past topic time.
[Background: Negative eventualities] It has been widely acknowledged in the literature that a negative
clause may describe the occurrence (existence) of a negative eventuality, rather than the non-occurrence(non-existence) of eventualities (Krifka 1989; de Swart 1996; Przepiórkowski 1999; Bernard & Champollion 2018; Fábregas & González Rodrı́guez 2020, among others). Among the most compelling piecesof evidence for “negative eventualities” are: (i) a negative clause can be a complement of a perception
verb like SEE, as in (2), (ii) a negative clause may occur in slots like “What happened is . . . ” and “. . . iswhat they did”, as in (3), and (iii) the content of a negative clause can be modified with a non-restrictive
relative clause, which may modify an eventuality but not a proposition, as seen in (4).
(2)
The police officer saw Chris not stop for the traffic light.
(3)
What happened next was that the consulate didn’t give us our visa.
(de Swart 1996:229)(4)a.
John kissed Mary, which {made her angry/*is shown by her blushed face}.
b.
John didn’t ask Mary to dance at the party, which made her angry.
(adapted from Przepiórkowski 1999:239)
Various proposals have been put forth on the semantics of a negative clause representing the occurrence of a negative eventuality. This work adopts Bernard & Champollion’s (2018) idea that each set of
eventualities P expressible with a clause nucleus has a negative counterpart, Neg(P ), which contains all1and only those eventualities which preclude—i.e., cannot co-exist in the same world with—every eventuality in P . The incompatiblity of an eventuality and its negative counterpart may be accounted for withan axiom along the lines (5), which mirrors the Law of Contradiction in classical logic.
(5)
Axiom of Negation: [∃e[e ∈ Neg(P )]] ↔ [¬∃e’[e’ ∈ P ]]When P is eventualities whereby Mary leaves, for example, Neg(P ) is something like eventualities
whereby Mary stays.
Bernard & Champollion (2018) assign a meaning along the following lines to English not; Subscript
E stands for “eventive”.
(6)
notE 7→ λPhv,ti [Neg(P )]The admittance of negative eventualities in the ontology makes it possible to develop reasonable semantic
representations for sentences like (2)–(4). It is an event of “anti-stopping” that is described to have been
seen by a police officer, it is an event of “anti-visa-issuance” that is described to “have happened”, andso forth.
Now, if a negative clause may describe a negative eventuality, does it always do so? Does, say,English not always represent something like (6), or can it represent the classical Boolean negation, i.e.
(7) where P stands for “propositional”, as well?
(7)
notP 7→ λp[¬p]
There is no consensus on this matter in the existing literature. With Przepiórkowski (1999), Higginbotham (2000), and Fábregas & González Rodrı́guez (2020), I maintain that clausal negation comes in
two varieties: propositional negation (7) and eventive negation (6). In a sentence like (8), negation occurs
in the complement of a perception verb and is forced to receive the eventive reading. In a sentence like(9), on the other hand, English not may, in theory, be either propositional or eventive.
(8)
Ken saw Mary notE dance.
‘There was a negative eventuality where Mary danced, and Ken saw it.’(9)
Mary did notP /E dance.
a.
‘There was no eventuality where Mary danced.’b.
‘There was a negative eventuality where Mary danced.’
A negative clause involving propositional negation can be said to express the non-occurrence of eventualities (NOE), and one involving eventive negation the occurrence of a negative eventuality (ONE). (a)and (b) are respectively paraphrases of the ONE and NOE readings of the sentence Mary did not dance.
I furthermore suggest that reference to a negative eventuality (corresponding to a dynamic event)—
i.e., the ONE reading of a (dynamic) negative clause—is highly constrained, and is available only when
the occurrence of a corresponding positive eventuality is or was expected or at least plausible.
It is commonly acknowledged that generally negative sentences are pragmatically more marked thantheir affirmative counterparts (Tian & Breheny 2019 and references therein). However, there seems to
be a significant difference in the degree of markedness between sentences with regular (propositional)
negation and ones with eventive negation.
In a context where there has been no expectation for Mary to take a picture, let alone a picture of an
eggplant, the negative sentence in (10a) would be a fairly strange thing to say. It nevertheless is judged
as a true statement, if indeed Mary did not take a picture of an eggplant. The second sentence in (10b),on the other hand, does not merely sound odder than that in (10), but seems not to be true. It is not clear
if it is even a false statement—it has a flavor of presupposition failure.
(10)a.
I observed Mary for three hours. She did not take a picture of an eggplant.
b.
I observed Mary for three hours. #I saw her not take a picture of an eggplant.

2

It seems thus sensible to suppose that the occurrence of eventive negation is much more constrained thanthat of propositional negation.
The logical translations of “Moe didn’t sing” on the NOE and ONE readings will look like (11a,b):(11)
Moe didn’t sing.
a.
¬∃e[TT < TU & At(e,TT) & sing(e) & Actor(e) = moe]]]] (NOE)b.
∃e[TT < TU & At(e,TT) & Neg(λe’[dance(e’) & Actor(e’) = ken])(e)] (ONE)TT and TU represent the topic time (Klein 1994) and the time of utterance, respectively. The logicalpredicate At (cf. Condoravdi 2002:70) is defined in (12). τ represents a temporal trace function (Krifka
1989:97), and “⊆” stands for temporal inclusion.
(τ (e) ⊇ t if e is stative(12)
At(e,t) =def
τ (e) ⊆ t otherwise[The markedness of the Japanese negative preterite] It has been observed in the descriptively oriented
literature on Japanese that the use of a negative preterite is rather strictly constrained, unlike its nonpast
counterpart (Matsuda 2002, Yamashita 2014, Kusumoto 2016). At the descriptive level, one may saythat a negative preterite conveys that the logical contrary of its propositional content—e.g. the speaker’s
hiring a new nurse in the case of (1)—was considered plausible by the speaker at some point prior to thediscourse. A puzzle, however, is what brings about such an effect.
I propose that the negation in a Japanese negative preterite is invariably eventive, so that, for example,
13 allows only the ONE reading.
(13)
Ken wa utawanakatta.
K. Top sing.Neg.Pst‘Ken did not sing.’a. #¬∃e[TT < TU & At(e,TT) & sing(e) & Actor(e) = ken]]]] (NOE)b. ∃e[TT < TU & At(e,TT) & Neg(λe’[dance(e’) & Actor(e’) = ken])(e)] (ONE)The plausibility requirement for a Japanese negative preterite straightforwardly follows from this supposition.
[The quasi-preterite interpretation of the nonpast perfect] The puzzle of the limited discourseconfigurational distribution of the negative preterite has a flip side: the unexpectedly wide distributionof the negative nonpast perfect. Assuming the Parsonsian (resultativity-based) account of the perfect(Parsons 1990), and adopting the premise that a nonpast tense specifies that the topic time is set to some
nonpast (present or future) time, the logical representations of the boldfaced parts of (14Ba,b ) should look
like (15a,b). The function RS maps an eventuality to its resultant state. ((14Bb ) in theory may receive theNOE reading, but here it is taken to receive the ONE reading.)(14)(A big soccer game was broadcast on TV the evening before.)
A: Kinoo, sakkaa mita?
yesterday soccer see.Pst‘Did you watch the soccer game yesterday?’Ba : Un, mita. Sono tame ni zangyoo
mo kotowatta nda.
yes see.Pst that purpose Dat overtime.work also resufe.Pst DAux Cop.Prs‘Yes, I watched it. I refused to work overtime for that purpose.’
Bb : Iya, mite nai.
Mitakatta kedo, zangyoo
ga atte.
no see.Ger NpfvAux.Neg.Npst see.want.Pst though overtime.work Nom exist.Ger

3

‘No, I did not watch it. I wanted to watch it, but I had to work overtime.’(15)a. ∃e[TT < TU & At(e,TT) & watch(e) & Actor(e) = Speaker & Undergoer(e) = the-game]b. ¬∃e[∃e’[TU ≤ TT & At(e,TT) & e = RS(e’) & watch(e’) & Actor(e’) = Speaker &Undergoer(e’) = the-game]]
It is counterintuitive, however, to suppose that the relevant parts of (14Ba,b ) are “about” different temporal
scenes; (14Bb ) strikes as being “about” the evening before—a past time—to the same extent as (14Ba )does.
I suggest that the Japanese perfect, expressed with IRU, optionally poses a restriction on the topic
time, specifying that it includes the time of the source event and is prior to the time of utterance; theunderline in (16) indicates optionality.
(16)
PERFECT Jpn = λP [λe[∃e’[TT < TU & At(e’,TT) & e = RS(e’) & P (e’)]]]
I furthermore suggest that the Japanese nonpast tense is semantically vacuous, and the use of a nonpast
tense conveys that the described eventuality occurs in a nonpast time only as an implicature arising fromthe absence of a past tense (cf. Sauerland 2002).
(17)
a. PASTJpn = λP [λe[TT < TU & At(e,TT) & P (e)]]b. NONPASTJpn = λP [P ]One piece of evidence for the semantic vacuity of the Japanese nonpast tense is that, when nonpast andpast features co-occur within the same complex predicate the whole predicate is interpreted as referring
to a past time (Plt = polite(ness)).
(18)
a. Mimasen
deshita.
see.Plt.Neg.Npst PltAux.Past‘(proi ) did not see (proj ). (Polite)’b. Aokatta
desublue.Pst.Pst PltAux.Npst‘(proi ) was hot. (Polite)’
The optional component of the semantics of the perfect leads to the two distinct representations of(19), one of which can be characterized as a “quasi-preterite” interpretation. The shaded part in (19a) is
a pragmatically enriched content.
(19)
Ken wa utatte inai.
K. Top sing.Ger Npfv.Npst
a. ‘Ken {has not/will not have} sung.’ [regular nonpast-perfect]¬∃e[∃e’[TU ≤ TT & At(e,TT) & e = RS(e’) & sing(e’) & Actor(e’) = ken]]b. ‘Ken did not sing.’ [quasi-preterite]¬∃e[∃e’[TT < TU & At(e’,TT) & e = RS(e’) & sing(e’) & Actor(e’) = ken]]
The proposed analysis has the additional merit of accounting for how an affirmative nonpast perfect clause
may receive a preterite-like interpretation (with a certain rhetorical effect), as seen in (20) (adapted froma Wikipedia article).
(20)
Korera ichiren noundoo
no tame ni, Gandhi wa tabitabi toogoku sareta.
these serial Cop.Attr movement Gen cause Dat G.
Top frequently imprison do.Psv.Pst
Tatoeba
1922-nen 3-gatsu ni wa, 2-nen-kan no
fufukujuu
undoo
nofor.example 1922-year 3-month Dat Top 2-year-for Cop.Attr disobedience movement Gen
tame ni, 6-nen-kan nochooekikei no hanketsu o ukete
iru.
cause Dat 6-year-for Cop.Attr imprisonment Gen judgment Acc receive.Ger NpfvAux.Prs

4‘Gandhi was frequently imprisoned for this series of movements. For example, on March 1922,
he was sentenced to six-year imprisonment for a two-year long civil disobedience movement.’[Why is -ta incompatible with the NOE negation?] A likely cause of the incompatibility of the pastmarker -ta with propositional negation is s its grammatical status/position. As discussed by Oshima(2014), -ta can sensibly be regarded as a particle separated from its host by a word boundary, unlike
nonpast suffixes like -(r)u. It is, on the other hand, a general fact in Japanese predicate morphology/syntax
that in a configuration where a predicate P is followed by an element E (e.g., an auxiliary), a component
within P cannot semantically outscope (a component within) E.
(21)
a. Shikaranaide ageru.
scold.NegGer BenAux.Npst‘(proi ) will benefit (proj ) by not scolding (proj/k )’ (Ben > Neg)b. Shikatte agenai.
scold.Ger BenAux.Neg.Npst‘(proi ) will not benefit (proj ) by scolding (proj/k )’ (Neg > Ben)It seems thus quite plausible that the word boundary blocks negation in the host to take scope over -ta,thereby inducing the differing scopal behaviors of the nonpast and past tense markers (Neg > Nonpast, Nonpast > Neg, Past > Neg, *Neg > Past). Under the current analysis of tenses and propositoinal/eventive negations, due to their semantic types, propositional negation (ht, ti) must be applied after
the closure of the eventuality variably, hence taking scope over the tense; eventive negation (hvt, vti), on
the hand, may take scope under the tense (hvt, vti). The impossibility of the “Neg > Past (-ta)” pattern
implies that the negation occurring in a preterite can only be eventive.
[References] Bernard, Timothée & Lucas Champollion (2018) Negative events in compositional semantics. Proceedings of SALT 28; Condoravdi, C. (2002) Temporal interpretation of modals: Modals
for the present and for the past. In: The construction of meaning CSLI Pubs.; Fábregas, A. & R.
González Rodrı́guez (2020) On inhibited eventualities. NLLT 38; Higginbotham, J. (2000) On eventsin linguistic semantics. In: Speaking of events, OUP; Krifka, M. (1989) The origins of telicity In: Eventsand grammar, Springer; Kudo, M. (2020) Tense and aspect in discourse In: Handbook of Japanese semantics and pragmatics, De Gruyter; Kusumoto, T. (2016) Kako no hitei jitai o arawasu shite inai ni
okeru hanashite no shinteki taido to hyoogen kooka; Area and Culture Studies 92; Matsuda, F. (2002)
Kakoji ni “. . . shita ka?” ni taisuru hitei no hentoo keishiki: Shite inai to shinakatta no sentaku ni kanshite. Nihongo Kyoiku 113; Oshima, D. (2014) On the morphological status of -te, -ta, and related forms
in Japanese: Evidence from accent placement; JEAL 23; Parsons, T. (1990) Events in the semantics ofEnglish: A study in subatomic semantics, MIT Press; Przepiórkowski, A. (1999) On negative eventualities, negative concord, and negative yes/no questions; Proceedings of SALT 9; Sauerland, U. (2002) The
present tense is vacuous Snippets 6; de Swart, H. (1996) Meaning and use of not . . . until J. of Semantics
13; Tian, Y. & R. Breheny. (2019) Negation. In: The Oxford handbook of experimental semantics andpragmatics, OUP; Yamashita, Y. (2014) Tensu no “ta” to asupekuto no “ta”. Journal of International
Student Center, Hokkaido University 8.

5Semantic Properties of the Japanese Emphatic Minimizer “NP-no-kakera”(1

Based on the Modal Base
Misato Ido (National Institute for Japanese Language and Linguistics)
1.

OutlineSawada (2022) suggested that a Japanese emphatic minimizer NP-no-kakera-mo (“ashred of NP”) is an NPI with an at-issue meaning in (1a) and the meaning of “expectation”
and “complain” as non-at-issue meaning as in (1bc).
(1) Ano-seijika-ni-wa(2

seijitsusa-no-kakera-monai.

that- politician-DAT-TOP sincerity-GEN-piece-even NEG“That politician does not have a shred of sincerity.”a. That politician does not have sincerity. (at-issue meaning)b. Politicians should have sincerity. (expectation)c. I am frustrated that that politician does not have sincerity. (complaint)
Sawada (2022) analyzes these non-at-issue meanings as conventional implicature (CI)with multidimensional meanings (Potts 2005, 2007; McCready 2010; Sawada 2018). I
demonstrate that Sawada’s descriptive generalization has systematic counterexamples
and provide a more accurate generalization behind the various meanings of NP-nokakera-mo nai (“not a shred of NP”). I propose an analysis that these non-at-issue
meanings are derived from the essential meaning of the NP-no-kakera-mo and negation,
so there is no need to set up the specific meaning such as “expectation” and “complain”
as CI. I demonstrate that the NP-no-kakera is an inherently modal expression and that it
refers to something that minimally exists in the modal base that is covertly introduced at
the non-at-issue level. In other words, kakera is a “thorough” minimizer (as it were), somuch so that it invokes minimization even in modal bases; hence, it produces expressive
effects that neutral minimization does not. I further argue that the non-at-issue meaning
based on a modal base, set by the contexts and knowledge of the attitude holder (typicallythe speaker), is neither a "typical" presupposition nor conventional implicature, which
Potts (2005) proposed originally.
2.

ProblemKakera literally means “piece” and is used as an emphatic negative polarity item (NPI)with the focus particle mo (“even”) as in (1). Sawada (2022) argues that non-at-issue
meanings as in (1bc) are CI because they are not negated by B’s replies in contexts suchas in (2).

6

(2)A: Ano-seijika-ni-wa(3

seijitsusa-no-kakera-mo nai.

“That politician does not have a shred of sincerity.”
B: Iya, sore-wa uso-da.
No that-TOP false-PRED“No, that’s false.”(Sawada 2022: 5)

Sawada’s (2022) analysis captures only one aspect of the whole set of empiricalproperties of NP-no-kakera. NP-no-kakera-mo nai (“not a shred of NP”) may express
neither “expectation” nor “complain” as in (3). The speaker in (3) seems to be favorably
impressed by “Taro's lack of rusticity.”(3) Hisabisani at-ta taro-wa inakakusasa-no-kakera-mo kanji-rare-nakat-ta.

(4In-ages meet-PST Taro-TOP provincialism-GEN-piece-even feel-PASS-NEG-PST“Taro, whom I hadn't seen in a while, didn't have an air of provincialism.”
Sawada (2022) claims that the NP in NP-no-kakera should be positive abstract nouns andthat examples such as (3) are peripheral examples in which kakera occurs with a negative
abstract noun. However, kakera with negative (or neutral) noun meanings are
systematically productive expressions, as we will see later. Thus, it is undesirable toregard them as peripheral examples.
3.

Proposal
I argue that the meaning of NP-no-kakera indicates something that minimally exists ina modal base in Kratzer’s (1981) sense. For example, (4), as it is, seems unnaturalcompared to (5), which contains a neutral NPI sukoshimo (“at all”). However, if we set
up a context that the speaker thinks that Yamada-kun must be an irresponsible person, asin (6), it becomes perfectly natural. This is because the context introduces the modal basethat Yamada should be irresponsible at least minimally in any possible world. Thisphenomenon indicates the validity of the direction of the analysis in this presentation.
(4)??Yamada-ni-wa musakininsa-no-kakera-mo

(5

nai.

Yamada-DAT-TOP irresponsibility-GEN-piece-even NEG“Mr. Yamada does not have an ounce of irresponsibility.”(5) Yamada-ni-wa musakininsa-ga

sukoshi-mo

(6

nai.

Yamada-DAT-TOP irresponsibility-NOM a-little-bit-even NEG“He has no irresponsibility at all.”(6) Kenkyuushitsu-de-no yamada-no(7

yoosu-ni-wa,

Laboratory-LOC-GEN Yamada-GEN appearance-DAT-TOP

7

hudan-no-kare-nomusekininsa-no-kakera-mo

kanji-rare-nai.

usual-GEN-he-GEN irresponsibility-GEN-piece-even feel-PASS-NEG“Yamada’s behavior in the lab shows no sign of his usual irresponsibility.”
NP-no-kakera-mo nai can express various modal meanings depending on the modalbase types that are set within the knowledge or situation assumed in the discourse.
(7) Deontic (e.g., Minimum properties a teacher must possess)(8

Tanaka-senei-ni-wa seijitsusa-no-kakera-mo nai. Komat-ta mono-da.
Tanaka-Mr.-DAT-TOP sincerity-GEN-piece-even NEG. Complain-PST-NMLZ-PRED“Dr. Tanaka does not have a shred of sincerity. It’s a problem.”(8) Bouletic or teleological (e.g., Minimum required to accomplish the project)(9

Kiryoku-no-kakera-mo nokot-tei-nakat-ta-node,purojyekuto-wo

Energy-GEN-piece-even leave-ASP-NEG-PST-because project-ACC

tatamu-koto-ni

shi-ta.

Demo, kookai-wa shi-tei-nai.

close-NMLZ-DAT do-PST. But

regret-TOP do-ASP-NEG“I didn't have a shred of energy left, so I decided to fold the project. But I haveno regrets.”(9) Epistemic (e.g., The minimum psychological stress that Yoko has, as foreseen( 10

by speaker’s knowledge)
Yoko-wa daibutai-dekinchoo-no-kakera-mo

mise-zuni engishi-ta.

Yoko-TOP big-stage-LOC nervousness-GEN-piece-even show-NEG performe-PST

Watashi-ga shit-tei-ru
I-NOM

kanojyo-to-wa betsujin-no

yoo dat-ta.

know-ASP-PRS she-with-TOP different-person-GEN look PRED-PST“Yoko performed without a shred of nervousness on the big stage. She was likea different person from the one I knew.”(10) Stereotypical (e.g., Minimum properties that a typical politician possesses)( 11(Politicians are more or less arrogant, but) Tanaka-sensei-ni-wa koomansa-noTanaka-Mr.-DAT-TOP arrogant-GEN-

kakera-mo nai.

Dakara kare-wo sonkei-shi-tei-ru.

piece-even NEG. So

he-ACC respect-do-ASP-PRS“Politicians are more or less arrogant, but Mr. Tanaka has no trace of arrogance.
That is why I respect him.”
According to (7)-(10), the meaning of “expectation” and “complain” are not lexical
meanings of NP-no-kakera but they are implicature derived from the whole sentence only

8

when NP-no-kakera-mo is based on the deontic modal base. Specifically, when NP-nokakera denotes that something must exist at least minimally in the deontic modal base(e.g., the sincerity of the teacher in (7)), the whole sentence means that something thatshould be there at least minimally is not there. The meaning of “expectation” and“complain” are only the presumption that hearers make from the whole sentence, so theycannot be negated by the reply as in (2B). Similarly, the positive bias noted by Sawada
that NPs used in the NP-no-kakera tend to possess arises only as a matter of common
knowledge, in which many of the properties registered as something that minimally {mustexist/ should exist/exist to achieve the goal} in various modal bases are positive properties.
The crucual examples are those in (8)-(10). These examples are associated with modalbases that are not deontic. As showin in (8)-(10), in all these cases, a follow-up with asentencethat

contradicts

speaker’s

complaintisperfectly

natural,clearly

conterexemplifying Sawada’s claim in (1). When the NP-no-kakera denotes somethingthat minimally exists in the bouletic or teleological modal base (e.g., the motivation tocomplete the project in (8)), the whole sentence means that something that must be there
to achieve the goal is not there regardless of whether the speaker is dissatisfied. The same
applies to (9) and (10). NP-no-kakera denotes something that minimally exists in the
epistemic or stereotypical modal base, like the nervousness that Yoko must have on thestage, at least a little, in (9) or the arrogance that any typical politician has, at least a little,in (10).
4.

The semantic properties of the non-at-issue meaning of NPI kakera
Based on the discussion so far, I propose the meaning of NPI kakera as follows.

(11) Ano-seijika-ni-wa

seijitsusa-no-kakera-mo

nai. (=(1))

a. That politician does not have sincerity. (at-issue meaning)b. Politicians must be at least a little sincere (non-at-issue meaning)
I will show that this non-at-issue meaning based on the modal base is neither apresupposition nor CI (which at least Potts (2005) first proposed) from the viewpoints
that it can be a local (subject-oriented) interpretation when it is embedded in the attitude
verbs as in (12) (unlike CI such as a non-at-issue meaning in appositive) and and that itneeds no common ground between speaker and listener as in (13) (unlike a presupposition
such as a non-at-issue meaning in additive particles such as “too” and “also”).
(12) [The speakeri believes he speaks frankly.] Yoko-wa, kyoo-no watasii -no
Yoko-TOP today-GEN I-GEN

9

hatsugen-wa mezurashiku gomakashi-no-kakera-mo nai-to
speech-TOP unusuallyomot-ta-souda.

deception-GEN-kakera-even NEG-COMP think-PST-seem“Yoko did not seem to think there was not a shred of deception in what I said today, which
was unusual.”(13) A: Kenkyuushitsu-de-no yamada-no yoosu-ni-wa, hudan-no-kare-no musekininsa-

no-kakera-mo kanji-rare-nai.(=(6))“Yamada’s behavior in the lab shows no sign of his usual irresponsibility.”
B: Soo?

Somosomo, Yamada-wa hudan-kara musekinin-de-wa

Is-that-so? To-begin-with, Yamada-TOP usual-from irresponsible-COP-TOP

nai-toomou-yo.

NEG-COMP think-FP“Is that so? To begin with, I don't think Yamada is usually irresponsible.”5.

Conclusion and open question
This presentation approached the non-at-issue meanings of “expectation” and“complain” of NP-no-kakera analytically from the modal base. This argument suggeststhat some expression that needs to refer to the modal base yields the non-at-issue meaning
based on the attitude holder’s belief consequently. Finally, although I remain to clarify
the semantic status of the modal base that is covertly introduced at the non-at-issue levelas an open question in this presentation, I pointed out that this type of non-at-issue
meaning is not analyzed as a presupposition or CI at least in typical.
References
Kratzer, Angelika. 1981. The notional category of modality. Words, Worlds, and Contexts:
New Approaches in Word Semantics, ed. by Hans J. Eikmeyer and Hannes Rieser,
38–74. Berlin: Walter de Gruyter
McCready, Elin. 2010. Varieties of conventional implicature. Semantics & Pragmatics,
3(8):1–57.
Potts, Christopher. 2005. The Logic of Conventional Implicatures. Oxford: Oxford
University Press.
Sawada, Osamu. 2018. Pragmatic Aspects of Scalar Modifiers: The SemanticsPragmatics Interface. Oxford: Oxford University Press.
Sawada, Osamu. 2022. On the properties of expressivity and counter-expectation in theJapanese minimizer NPI kakera ‘piece’. Proceedings of the Linguistic Society ofAmerica. vol. 7, No 1, p. 5259.

10

Slurs’ variability, emotional dimensions andgame-theoretic pragmatics
Victor Carranza-Pinedo

1

Introduction

Slurs’ meaning is unstable. A slurring utterance like ‘Hey, F, where have you been?’(where F is a slur) may receive a wide array of interpretations depending on various contextual factors: the speaker’s identity, her relation to the addressee, her
intonation, etc (Davis and McCready, 2020). Standard semantic, pragmatic, andnon-content theories of slurs have issues to account for some or all kinds of variability observed. To solve this, I argue that slurs don’t convey emotional categories
such as ‘contempt’ but emotional dimensional qualities such as, e.g., ‘negative valence, neutral arousal, high dominance’. Then, I show how to translate this thesis
into a detailed game-theoretic model of slurs use and interpretation inspired by thework of Heather Burnett (2019). This new approach, called ‘Affective Meaning
Games’ (AFM), captures slur’s variation and situates pragmatic reasoning within
an independently motivated psychological account of emotional states.

2

Two types of variation

Slurs’ variation is janus-faced. On the one hand, speakers express a wide array of
emotions by using slurs. Typically, speakers can display negative emotions such as
contempt (i.e. ‘Fs are not allowed here’) or fear (i.e. ‘Fs are invading us’) by uttering
a slur. Yet, in other situations, when the slur is used among members of the target
group, the speaker is typically characterized as expressing positive emotions, suchas solidarity (i.e., ‘Hey, my F, I missed you’) or pride (i.e., ‘We should be proud of
being Fs’).
On the other hand, speakers elicit different degrees of offense by using slurs. Indeed,
slur’s offensiveness may vary i) across different lexical items, even when they are coreferential (e.g., ‘beaner’ may be considered more offensive than ‘greaser’), and ii)
across different uses of the same expression, depending on who uses it (e.g., uses of‘fag’ within the LGTB community are considered less offensive those performed byoutsiders) or how it is used (e.g., with a contemptuous or friendly intonation).
Importantly, it has been observed that the offense a slur elicits is independent of
the valence of the emotion it expresses (Popa-Wyatt and Wyatt, 2017). Uses of‘spade’ among hippies were often described as genuinely expressing appreciation11

or endearment towards Afro-Americans, but were nonetheless considered offensive(Nunberg, 2018). Conversely, uses of slurs that target dominant groups (e.g., ‘limey’)can express extreme contempt or disgust, but are nonetheless regarded as inoffensive.
Should we conclude that slurs’ offensiveness is independent of the emotions theyexpress? In what follows I argue that this is not the case. To wit, emotions are not
only understood as discrete emotional categories (e.g., ‘contempt’), but also as statesthat can be characterized using broad affective dimensions (i.e., valence, arousal, anddominance). In a nutshell, I argue that the degree of dominance expressed by a sluris at the root of its complex offensive profile.

3

Slurs and PAD theories of emotions

According to the classical view, each basic emotional category, such as happiness or
disgust, triggers a distinct set of behavioral and physiological responses. However,in the absence of evidence for a clear one-to-one correspondence between emotions
and bodily patterns (Barrett, 2017), researchers have proposed to characterize affective episodes using the following continuous, bipolar, and orthogonal dimensions(Mehrabian and Russell, 1974):• pleasure: evaluative dimension ranging from negatively valenced affective
states (e.g., sadness) to positively valenced ones (e.g., happiness).
• arousal: physiological dimension ranging from low mental alertness (e.g.,boredom) to high mental alertness (e.g., excitement).
• dominance: relational dimension ranging from the sensation of feeling submissive (e.g., frustration) to the sensation of feeling in control or powerful(e.g., anger).
This psychometric approach to affective phenomena, also known as the ‘PAD’ theoryof emotions, is widely applied in the analysis of affective episodes in a continuous
rather than discrete framework. Even though dominance is sometimes omitted(Russell and Pratt, 1980), it is at the basis of recent accounts of social perception
based on facial features (Oosterhof and Todorov, 2008) and, moreover, it allows us
to distinguish affective states such as anger and frustration, alertness and surprise,etc. (Mehrabian, 1996).
For example, anger and frustration are typically highly arousing, negative emotions,but differ in the extent to which the agent feels free to respond to the stimulus.
In the presence of individuals judged to be less dominant (e.g., a subservient), an
offense is more likely to elicit anger than frustration. In this sense, the dominanceexperienced by an agent is inversely proportional to the assessed dominance (i.e.,
physical strength, social status, aggressiveness) of the stimulus (Mehrabian and
Russell, 1974).
How are the states expressed by slurs related to the PAD dimensions?
1. Slurs typically express the speaker’s negative evaluation of the target group.
By saying ‘That building is full of Fs’ the speaker is more likely interpreted12

as expressing displeasure with respect to F’s target.
2. Slurs don’t seem to be statistically related to a particular degree of arousal.
Unlike other swear words (e.g., ‘fucking’), slurs don’t come as infelicitous incontexts where the speaker doesn’t feel intense emotions.
3. Finally, slurs typically express that the speaker feels superior with respect tothe target group. That is, slurs express that targets rank as low in worth withrespect to the speaker, who thereby presents himself as dominant.
The reasons why slurs convey a high degree of dominance are plausibly related tothe social context in which they circulate: slurs are typically coined and used bydominant groups, they often precede or co-occur with other types of aggression, etc.
Yet, it has been assumed that expressing that individuals are inferior is itself a formof negative evaluation (Jeshion, 2016). However, both dimensions can be distinguished: one can evaluate someone negatively without feeling that he is lesser, andone can feel that someone is lesser without evaluating him negatively. Thus, realizing that some emotions can be positive but nonetheless involve a threat of aggression(e.g., amusement at someone’s expense) allows us explain slurs’ offensiveness. Now,
I will translate this hypothesis (which we may call ‘the valence-dominance’ view)
into a formal theory of slur’s use and interpretation.

4

Affective Meaning GamesHow can we define emotional states in a theory of meaning? Inspired by Burnett(2019), I assume a structure ⟨Q, >⟩, where Q is a set of relevant affective qualities(e.g., positive pleasure or ‘[P+]’) and > encodes relations of incompatibility between
them (e.g., individuals cannot be in a [P-] and [P+] state simultaneously). Sinceslurs don’t correlate with a specific degree of arousal, this dimension is not included:(1)

Q = {[P+], [P-], [D-], [D+]}a. [P+] > [P-]b. [D-] > [D+]Based on ⟨Q, >⟩, we derive 4 types of affective states α: e.g., the [P-, D+] state,labeled contempt, etc. Importantly, these labels serve to assemble different discrete emotional categories. For example, contempt represents [P-, D+] states ingeneral (e.g., rage, hostility, etc.), and not only contempt:(2)

Affective states (aff):
aff affiliation amusement
α[P+, D-][P+, D+]anxiety[P-, D-]

contempt[P-, D+]

Then, we assume that for some slurring term F there is a co-referential non-slurring
alternative F*. How can we characterize the relation between F/F* and affective
states alpha? We have assumed that slurs are more strongly associated with [D+]13

states (e.g., amusement), etc. To capture such statistical regularities, I associate Fwith the probability distribution Pr(F |α), read as ‘the likelihood of uttering a slur
F given an affective state α’. Note that the alternative F* is associated with the
distribution Pr(F ∗ |α) = 1 - Pr(F |α):(3)Affective meaning of F and F*:
aff
Pr(F |α)Pr(F ∗ |α)affiliation amusement
0.3
0.6
0.7
0.4

anxiety
0.4
0.6

contempt
0.7
0.3

Finally, once S utters a slur F directed at a social group G, the L’s prior beliefs are
updated using Bayes rules. That is, by conditioning his prior beliefs ‘Pr(α)’ (i.e.,
the prior probability that S feels a with respect to G’) on F’s affective meaning.
Prior probabilities vary with respect to multiple factors: the speaker’s identity, hersocial relation to the addressee, etc. If S is catholic, we assume that he feels positive
about the Catholic church; if S and L are friends, that S doesn’t consider L as lesser,
etc. Even though such assumptions may be proven incorrect, social identities guide
how others think about others’ emotions (Appiah, 2010).
Thus, reasoning about the S’s potential emotions towards G can change the weighting of the affective states α in a given context, thus giving rise to the variation
observed in Section 2. For example, in a situation where L hears S using ‘spic’, andwhere S is not Latino, the slur will be interpreted as offensive. How can we explain
this? In this case, we assume that L doesn’t have any prior expectations about S’s
affective dispositions toward Latinos. Thus when S uses the slur, L’s prior beliefs
are updated by the affective meaning of ‘spic’. As a result, we obtain that L willinterpret S as more likely expressing contempt (cf. the fourth row):(4)

L’s beliefs after hearing S saying ‘spic’:
aff
Pr(α)
Pr(spic|α)Pr(α)·Pr(spic|α)Pr(α|spic)

affiliation amusement
0.25
0.25
0.3
0.6
0.075
0.150
0.15
0.3

anxiety
0.25
0.4
0.100
0.2

contempt
0.25
0.7
0.175
0.35

In contrast, in a situation where L knows that S is Latino, his utterance of ‘spic’
will be interpreted as non-offensive. In this situation, L will probably expect S tofeel [P+] and [D-] states towards Latinos, as it is implausible to feel members of
one’s group as worth of contempt (e.g., Pr(affiliation = 0.6). Thus, when L’s
prior beliefs are updated by the meaning of spic, we obtain that L will interpret S
as more likely expressing affiliation towards that group (e.g., Pr(affiliation|spic)= 0.537 > Pr(contempt|spic) = 0.104).
By taking into account formal models of emotions, and in particular PAD models,
into the calculation of the affective content conveyed by slurs, Affective Meaning

14

Games help us understand i) how slurs can be interpreted as expressing a widearray of emotions depending on listeners’ prior assumptions and ii) how the valencedominance hypothesis can serve to explain slur’s offensiveness.

References
Appiah, K. A. (2010). The ethics of identity. In The Ethics of Identity. Princeton
University Press.
Barrett, L. F. (2017). How emotions are made: The secret life of the brain. Pan
Macmillan.
Burnett, H. (2019). Signalling games, sociolinguistic variation and the constructionof style. Linguistics and Philosophy, 42(5):419–450.
Davis, C. and McCready, E. (2020). The instability of slurs. Grazer PhilosophischeStudien, 97(1):63–85.
Jeshion, R. (2016). Slur creation, bigotry formation: The power of expressivism.
Phenomenology and Mind, (11):130–139.
Mehrabian, A. (1996). Pleasure-arousal-dominance: A general framework for describing and measuring individual differences in temperament. Current Psychology, 14(4):261–292.
Mehrabian, A. and Russell, J. A. (1974). An approach to environmental psychology.
the MIT Press.
Nunberg, G. (2018). The social life of slurs. In New work on speech acts, pages
237–295. Oxford University Press.
Oosterhof, N. N. and Todorov, A. (2008). The functional basis of face evaluation.
Proceedings of the National Academy of Sciences, 105(32):11087–11092.
Popa-Wyatt, M. and Wyatt, J. L. (2017). Slurs, roles and power. PhilosophicalStudies, 175(11):2879–2906.
Russell, J. A. and Pratt, G. (1980). A description of the affective quality attributed
to environments. Journal of personality and social psychology, 38(2):311.

15

Granularity in number and polarity eﬀects
Eri Tanaka (Osaka University) and Kenta Mizutani (Aichi Prefectural University)
Introduction The recent literature on the polarity phenomena has revealed that vagueness and granularity have an impact on the polarity eﬀect (e.g., Solt (2018) on approximators such as approximately,
about, Goncharov and Wolf (2021) on some NP and minimizers). This work is yet another contribution
to this trend, reporting an unnoticed contrast between round and non-round numbers when associatedwith focus particles in Japanese. Our proposal is that non-round numbers compete with round numbers
in computing the (scalar) presupposition of these particles.
Data It has been documented in the Japanese traditional descriptive grammars that numerals marked byfocus particle mo ‘even’ cannot be in the scope of negation when the numeral it attaches to is a non-roundnumber, such as 48 (Ijima (1995)).
Consider first (1a), where a round number, 50, is used. (1a) is three-way ambiguous, as shown in(1b)-(1d). The first two interpretations diﬀer truth-conditionally: in a context where 200 people were
expected to come and 150 people came, and 50 people didn’t show up, (1c) is true but (1b) is not. They
also diﬀer in implication. In (1b), the number ‘50’ is taken to be a small number, while in (1c), it is taken
to be large. We call these two readings a small and a large readings (Nakanishi (2006)). (1b) and (1d)are the same truth conditionally, but they diﬀer in the implication: the latter implicates that ‘50’ is a large
number. We dub this reading ‘a large and fewer than n’ reading.
This ambiguity is explained when we assume a scope theory of even for Japanese mo, following Nakanishi(2006). Nakanishi (2006) assumes that mo is a propositional operator even if it is appended to a DP,
and the operator induces a scalar presupposition, where the prejacent is required to be the least likely
one among its alternatives (=(2a)). “Less likely” is here understood to be asymmetric entailment, as in(2b). The small reading results for (1b), because for the prejacent to be the least likely, the alternative
propositions need to refer to larger numbers. Likewise, in (1c) and (1d), the presupposition is satisfied ina context where propositions that include smaller numbers than ‘50’.
(1) a.

50-nin-mo ko-nakat-ta
50-CL-even came-���-����

(lit.) “Even 50 people didn’t come.”
b.

c.

d.

(2) a.
b.

mo > ¬ > 50:
Assertion: No more than 50 people came.
Implication: ‘50’ is a small number.

small readingmo > 50 > ¬:Assertion: There were 50 people who didn’t come.
Implication:‘50’ is a large number.

large reading¬ > mo > 50:
Assertion: No more than 50 people came.
Implication: ‘50’ is a large number.
J mo Kc,w = p. p(w) = 1, defined if 8q2 C[p 6=q ! p <likely q]
p is less likely than q iﬀ p entails q but not vice versa.

16large-and-fewer-than-n reading

(3) a.
b.
c.

(1b): mo [¬ 50 people came] is defined, if‘¬ 50 people came’ entails ‘¬ 51/52/53 . . . people came’(1c): mo [9x. people(x) ^|x| = 50 ^ ¬ x came] is defined, if‘9x.people(x) ^| x |= 50 ^ ¬ x came’ entails ‘9x. people(x) ^|x| = 49/48/47 . . . ^ ¬ x came’(1d): ¬ [mo [50 people came]] is defined, if‘50 people came’ entails ‘49/48/47 . . . people came.’

When we replace ‘50’ with ‘48’, as in (4), the small reading where [mo > ¬ > 48] is not available,
and only the large readings are available. Since in the aﬃrmative, the round vs. non-round contrast is notobserved (=(5), this happens when mo and numerals are intervened by ¬.
(4)

48-nin-mo ko-nakat-ta
48-CL-even came-���-����

(lit.) “Even 48 people didn’t come.”(5)*mo > ¬ > 48, OK ¬ > mo > 48, OK mo > 48 > ¬{50/48}-nin-mo kita.

{50/48}-CL-����

came.

”Even 50/48 people came.”
English also exhibits the same contrast between round and non-round numbers. English even behaves
diﬀerently from Japanese mo, in that even-sentences have only a small reading when they are in negative.
Thus when a non-round number is used in this context, the even-sentence sounds bizarre:(6) a.
b.

John didn’t even solve {50/#48} problems.
Not even {50/#48} people came.

Granularity in number We propose that the contrast observed between (1a) and (4) is the result of
the interaction between granularity in the interpretation of numerals and the semantics of mo. Following
Sauerland and Stateva (2007), we assume that granularity is a contextual parameter of interpretation. The
granularity function, grani , maps a number, n, to an interval [n 1/2⇥i  n  n + 1/2⇥i ], where irepresents the granularity level. In this interpretation, 50gran10 refers to the interval between [45-55]. The
maxim of Manner requires the shortest expression be used for each interval, which prevents ‘48’ fromdenoting the [45-55] interval (cf. Krifka (2009)).
(7) a.
b.

J 50 Kg = gran10 (50) = [45-55]

J 48 Kg = gran1 (48) = [47.5-48.5]

According to this interpretation, in a situation where ‘48gran1 people came’ is true, you can truthfully saythat ‘50gran10 people came’, but not vice versa. This entailment relation holds when a number interpretedwith a finer granularity function falls within the range of a number interpreted with a coarser granularity
function.
To state this relation, we first define the relative coarseness among granularity functions for numbers
as in (8), based on (Sauerland and Stateva 2007:233):

17

(8)

gran is finer than gran’ iﬀ
for all numbers n: max(gran(n))

min(gran(n)) < max(gran’(n)min(gran(n))(9) is now understood to be an asymmetric entailment relation between round and non-round number
sentences.
(9)

Let gran be finer than gran’ and n and m be variables for numbers.
For any number n, if there is a context c and a number m such that if gran(n) ⇢ gran’(m), then0
J (n) Kc,gran = 1 ! J [n/m] Kc,gran = 1, where does not contain ¬.

Polarity eﬀects explained Our proposal is that when the number contained in a proposition satisfies theproperty in (9), the number with a coarser granularity also counts as an alternative to the number with afiner granularity. Thus as shown in (11b), the proposition that contains 48gran1 invokes “¬[50gran10 peoplecame]” as its alternative, in addition to the propositions with the same granularity level. This additional
alternative does not aﬀect the scalar presupposition of mo when the prejacent is in aﬃrmative. In (10a),the prejacent ‘50 people came’ is evaluated against the alternatives with the same granularity level, andresults in a large reading for ‘50’. ‘48 people came’ in (10b) includes alternatives with gran1 , as wellas gran10 . The scalar presupposition of (10b) is satisfied because ‘50gran10 people came’ is entailed by‘48gran1 people came.’.
(10) a.
b.

mo > 50: 50gran1 people came entails 49, 48, 47 . . . people came.
The same holds for 50gran10 .
mo > 48:
48gran1 people came entails 47, 46, 45 . . . people came.
48gran1 people came entails 50gran10 people came.

(cf. (9))

When it comes to negative sentences, the interpretation with [mo > ¬ > n] requires [¬ (n)] to be theleast likely proposition. This asymmetric entailment relation cannot hold it n = 48, because ‘fewer than
48gran1 people came’ does not entail ‘fewer than 50gran10 people came’ (=(11b).
(11) a.

b.

mo > ¬ > 50:¬ [50gran10 people came] entails ¬ [60/70/80 . . . people came]¬ [50gran1 people came] entails ¬ [51/52/53 . . . people came]mo > ¬ > 48:¬ [48gran1 people came] entails ¬ [49/50/51 . . . people came]¬ [48gran1 people came] does not entail ¬ [50gran10 people came]In other words, if negation intervenes mo and a number, where a small reading obtains, the round vs.
non-round contrast results.
This analysis predicts that if a number does not have a contextually plausible round number, thenegation may take a wider scope than numerals. This prediction is borne out as shown in (12):(12)

How many students came to your seminar?
3-nin-mo ko-nakat-ta yo.
3-��-���� come-���-����

(lit.) “Even three students didn’t come.” (=Not even three students came.)

18Extension to contrastive topic wa
topic marker wa (Ijima (1995)):(13)

A similar contrast is observed when 50/48 is marked by contrastive{50/#48}-nin-wa kita

50/48-��-��

came“50/48 people came.”
Following Sawada (2008), We assume contrastive topic wa has a flipped scalar presupposition of mo,
as well as an anti-additive presupposition, as shown in (14):(14)

J wa Kc,w = p. p(w) = 1, defined if(i) 8p 2 C. p 6=q ! p is more likely than q.
(ii) 9q 2 C. p 6= q ^ ¬q

In this case, the aﬃrmative is not allowed because 48gran1 has to be more likely than 50gran10 , which is
not tenable.
(15) a.

b.

wa(50 people came) is defined if(i) [50gran1 people came] is entailed by [51, 52, 53, ... people came].
(ii) It is possible that ¬[]51 (or 52, 53...) people came]wa(48 people came) is defined if(i) [48gran1 people came] is entailed by [49, 50, 51 . . . people came].
(ii) [48gran1 people came] is entailed by [50gran10 people came].
It is possible that ¬ [49, 50, 51 . . . people came].

Remaining issues The current analysis could also make a prediction about a construction with an overt
approximator: with an overt approximator, yaku/oyoso 50-mo ‘even about 50’ still can be in the scope ofnegation. However, the only available interpretation is a reading where negation takes a narrower scope.
(16)

Yaku/oyoso 50-nin-mo kita/ko-nakat-ta.
about

50-��-mocame/come-���-����

(lit.) “Even about 50 people came/didn’t come.’*¬ > about 50, OK 50 ¬Solt (2018) argues that an overt approximator is a PPI. Her explanation is based on the competition
between a sentence with an overt approximator and a sentence with a bare number, and if the less simple
version is used in negative context, it produces an implication contradictory to the assertion.
(17)

Mary has/#doesn’t have approximately/about 50 sheep.

If this analysis is on the right track, the competition between an overt approximator and a bare numeralstill holds, which could lead to the PPI-hood of the former. We will discuss how our analysis may
accommodate this insight.
Conclusion This paper discusses a peculiar behavior of numerals associated with focus particles withrespect to polarity sensitivity. We show that the granularity of numerals has to be taken into consideration
when alternatives are computed to satisfy the presupposition.

19

References
Goncharov, Julie, and Lavi Wolf. 2021. Deriving polarity from granularity. In Proceedings of SALT,
volume 31, 683–702.
Ijima, Masahiro. 1995. Suuryooshi to wa to mo (quantifiers and wa and mo). In Tsukishima Hiroshi sensei
koki kinen kokugogaku ronshu (A Festschrift for Professor Hiroshi Tsukishima on his 70th birthday),
1041–1062. Kyuuko shoin.
Krifka, Manfred. 2009. Approximate interpretations of number words: A case for strategic communication. In Theory and evidence in semantics, ed. Erhard Hinrichs and John Nerbonne, 109–132. Stanford:
CSLI Publications.
Nakanishi, Kimiko. 2006. Even, only, and negative polarity in Japanese. Proceedings of SALT 16:138–
155.
Sauerland, Uli, and Penka Stateva. 2007. Scalar vs. epistemic vagueness: Evidence from approximators.
In Proceedings of SALT, volume 17, 228–245.
Sawada, Osamu. 2008. The Japanese contrastive Wa: A mirror image of EVEN. In Proceedings ofBerkeley Linguistic Society 34, 281–292.
Solt, Stephanie. 2018. Approximators as a case study of attenuating polarity items. In Proceedings of
NELS 48, ed. Sherry Hucklebridge and Max Nelson, 91–104.

20

Formalizing argument structures with
Combinatory Categorial Grammar*
Shinnosuke Isono1
Koichi Kono†1

Takuya Hasegawa†1
Shiho Nakamura†1

Kohei Kajikawa†1
Yohei Oseki1

1 University of Tokyo

This study proposes a formalization of the constructivist analysis of argument structure, often couched in terms of Distributed Morphology (hereafter DM; Halle & Marantz,
1993), in Combinatory Categorial Grammar (hereafter CCG; Steedman, 2000). This formalization paves way for a model of incremental processing of argument structure basedon constructivism. It also provides a novel account for the locality constraints on contextual allomorphy, which has been discussed extensively in the DM literature (Marantz,
2013a).
Most, if not all, analyses based on CCG encode the argument structure in the lexical
entry of a verb. The Japanese verb kowas- ‘break.transitive’, for example, would have thecategory S\NP\NP in such analyses. However, there is an alternative view on argument
structure called constructivism, often couched in terms of DM (Marantz, 2013b). Constructivism assumes that argument structure is composed in the syntax, rather than in thelexicon. In (1), for example, the√ internal and external arguments are introduced by v andVoice, respectively. The root KOWA does not project any arguments. Constructivism isable to explain why verbs can be used in novel argument structures,1 and why there is asystematic correspondence between syntactic positions and thematic roles (as formulated
in UTAH (Baker, 1988)). These facts remain mysterious if argument structure is lexicallyspeciﬁed.
(1)

John-ga kabin-o kowa-s-i-ta. ‘John broke the vase.’
VoicePDPJohn vP

Voice
-s-

DP √
KOWA vkabin
kowa- -Ø* We thank the anonymous reviewers for helpful comments to the earlier version of this abstract.
† Contributed equally.
1 An example of such novel argument structures is found in Clark and Clark (1979, p.803):

21

DM is suitable for constructivism since it captures both the hierarchical nature of argument structures and the irregularity of verbal morphology that reﬂects those structures.
While the former is captured by the single engine hypothesis (Arad, 2003), the latter iscaptured by late insertion. Late insertion means the morphological (and semantic) realization of terminal nodes is determined after the syntactic structure is built and can referto the syntactic context. In the Japanese
sentence (1), for example, Voice[+D] is realized√
as -s- in the context
of
the
root
KOWA ‘break’; the same head would be realized as -e√in the context of AK ‘open’ (Harley, 2008; Oseki, 2017). This analysis captures thefact that the transitivity morpheme in Japanese is separable from the root (cf. kowa-re‘break.intransitive’) but varies depending on the root.
While late insertion offers a nice account of contextual allomorphy, it is problematicwhen real-time use of language is taken into account since it presumes bottom-up structurebuilding (cf. Bresnan & Kaplan, 1982). This is the primary motivation of our study: if
we employ a surface-oriented grammar formalism that is compatible with incremental
processing, how can we capture the constructivist nature of argument structure? Note that
there is some psycholinguistic evidence that the decomposition of argument structure is
relevant for real-time processing, not just for competence grammar (e.g., Friedmann et al.,2008).
This consideration led us to the formalization of the constructivist analysis in CCG,
which is claimed to be compatible with incremental structure building (Steedman, 2000).
Although most analyses in CCG are non-constructivist as mentioned earlier, we will
demonstrate that the constructivist analysis can indeed be translated to CCG.
We focus on the Japanese verbal morphology as a test case since it has overt transitivity alternation morphology, as exempliﬁed earlier. As a preparation for the constructivist
analysis we ﬁrst present an analysis of verbal inﬂection in Japanese. Our analysis is basedon Bekki (2010) in many aspects, but decomposes moras into segments to draw a close
parallel with the DM analysis. A caution should be paid that in a surface-oriented approach, morphophonological rewriting rules like tob-da ‘ﬂy-PAST’ → ton-da cannot beassumed since a derivation must start from a string as it is observed. Instead, regularly alternating consonant like b/n in tob-u vs. ton-da must be separated from the nonalternating
part as inﬂectional consonant (Ic), as shown below:2
(2)

a.

to-

-b-

-u

V5stem /Icb \NPga V5stem \(V5stem /Icb ) Sterm|attr \V5stem<BV5stem \NPga

Sterm|attr \NPga
(i)

a.
b.
c.
d.
e.

<B

The factory horns sirened throughout the raid.
The factory horns sirened midday and everyone broke for lunch.
The police car sirened the Porsche to a stop.
The police car sirened up to the accident site.
The police car sirened the daylight out of me.

2 We assume that the inﬂectional consonant is already type-raised as V stem \(V stem /Ic ) in the lexicon, asb5
5
shown in (2a), rather than having a simple category Icb and then being type-raised in the derivation. This

follows the suggestion of one of the reviewers, who pointed out that the elimination of type-raising rules
from the grammar has desirable consequences concerning parsing and long-distance dependency.

22

b.

to-

-n-

-da

Veuph::d \(V5stem /Ic(b|m|n) )<B
Veuph::d \NPga

Sterm|attr \Veuph::dV5stem /Icb \NPga<B

Sterm|attr \NPgaOnce the analysis of inﬂection is established, we move on to the constructivist analysisof transitivity alternation (now with denotations):
(3)

kowa-

-Ø-

-s-

stem \NP\VRvi
Vbase::[1] \NP\R[1]V5::sbase::(v|vi|vii)
λ e.kowa(e) λ P.λ x.λ e.P(e) ∧ theme(x)(e) λ P.λ x.λ e.P(e) ∧ causer(x)(e)Vbase::vi \NPλ x.λ e.kowa(e) ∧ theme(x)(e)<stem \NP\NPV5::sλ x.λ y.λ e.kowa(e) ∧ theme(x)(e) ∧ causer(y)(e)<B

This analysis exempliﬁes how contextual allomorphy can be treated in CCG, and provides further insights about the nature of allomorphy. The choice of the transitivity morpheme is based on the classiﬁcation of the root (due to Jacobsen, 1992): kowa- belongs
to class 6 (written vi). The class feature is inherited to the constituent kowa-Ø- by thevariable [1]. The transitivity morpheme -s- then selects this feature, resulting in kowaØ-s-. Thus, in this analysis, contextual allomorphy is reduced to mere selection. The
allomorphs should be listed in the lexicon, as shown below.
(4)-s- ⊢ V stem \NP\Vbase::(v|vi|vii) : λ P.λ x.λ e.P(e) ∧ causer(x)(e)-as- ⊢ V stem \NP\Vbase::(vii|ix|x) : λ P.λ x.λ e.P(e) ∧ causer(x)(e)-os- ⊢ V stem \NP\Vbase::xi : λ P.λ x.λ e.P(e) ∧ causer(x)(e)..
.

(4) has repeated appearances of the same denotation and similar categories. It is apparently less elegant than the DM analysis, where the denotation needs to be written only
once (Voice[+D] ↔ λ P.λ x.λ e.P(e) ∧ causer(x)(e)). Yet we can achieve the same level ofabstraction in CCG as in DM by deﬁning a function à la Bekki (2010) that maps classfeatures to sufﬁx forms, as shown in (5). Then the set of lexical items in (4) are deﬁned
succinctly as (6).
-s- (c = v, vi, vii)
-as- (c = viii, ix, x)
de f(5)f (c) =
-os- (c = xi)




...
(6)

For any c ∈ dom( f ),f (c) ⊢ V stem \NP\Vbase::c : λ P.λ x.λ e.P(e) ∧ causer(x)(e)Another, more interesting implication of the selection-based account of allomorphy
concerns the locality constraints on the context for allomorphy. In the DM literature,23

it has been claimed that the choice of the allomorph to be inserted to a given terminal
node is conditioned by its local context (Marantz, 2013a). Speciﬁcally, linear (string) andstructural (semantic) adjacency seem relevant, although there are also cases where strict
adjacency is not required (Merchant, 2015). The selection-based approach we consider
here provides a natural explanation for these observations. In CCG, combinatory rules
can be applied only for linearly adjacent elements (Principle of Adjacency; Steedman,
2000, p.54). The principle would predict that contextual allomorphy is only sensitive tolinearly adjacent elements. But the feature inheritance mechanism we introduced earlier
circumvents this principle without any limit: in a string ABC, if B selects A and inherits
some of its features, and C selects B, then C is virtually sensitive to A’s features. Feature
inheritance is not just for (3) but necessary to deal with any adjunctions (e.g., adverbials)
that preserve the features of the target (e.g., agreement features on the verb). The realquestion is therefore how feature inheritance should be constrained. We propose a modiﬁed version of Principle of Categorial Type Transparency (Steedman, 2000, p.36):(7)

The Principle of Categorial Type Transparency, revised
For any constituent, the semantic type of its denotation, the morphological classof its pronunciation, and a number of language-speciﬁc directional parameter settings uniquely determine its category.

The intuition is that if B inherits features on A, they must be semantically or morphologically meaningful for the constituent AB. In the case of transitivity alternation (3), kowa-Ø
is allowed to inherit the morphological class feature of kowa- since -Ø- is the identity element and thus kowa-Ø is indistinguishable from kowa-. Conversely, a morphologicallyvisible element blocks inheritance of the morphological feature of the root. Thus, huka-m‘deep-en,’ tuyo-m- ‘strength-en,’ taka-m- ‘height-en,’ and so on, all show the same patternof transitivity alternation (-ar-/-e-). If the feature in question is semantic rather than morphological, what is relevant is not the morphological visibility of the intervening element
but rather the semantic congruity. This accounts for cases where contextual allomorphy
is able to see a ‘span’ of nodes, deﬁned in terms of extended projection, as discussed withGreek verbal conjugation by Merchant (2015).
As pointed out by both of our reviewers, a fundamental issue in constructivist analyses
is how to constrain the set of argument structures that are allowed with a particular root.
In the current framework, possible combinations of argument structures (i.e., argumentintroducing items) and roots are deﬁned by features that these items have. To capture
the fact that tabe- ‘eat’, for example, can be combined with the phonologically null transitive morpheme but not with a pronounced transitive morpheme or an intransitive morpheme, one can assign to the root tabe- some feature(s) to be selected by the appropriate
morpheme. One reviewer suggested that this would be just a ‘notational variant’ of the
projectionist analysis, where tabe- is inherently speciﬁed as V \NP\NP. We believe theconstructivist approach still has advantage over the projectionist one, since the decomposition captures the shared semantics between (i) alternating pairs such as kowa-s- andkowa-re-, and (ii) alternating and non-alternating (in)transitive verbs such as kowa-s- andtabe-.
In sum, we argue that a formalization of the constructivist analysis of argument structure in CCG, a grammar formalism compatible with incremental processing, is in factpossible, and even gives a natural explanation for why contextual allomorphy is subject

24

to locality constraints that seem to resist simple characterization by linear or structural
adjacency.

References
Arad, M. (2003). Locality constraints on the interpretation of roots: The case of Hebrew
denominal verbs. Natural Language & Linguistic Theory, 21, 737–778. https://doi.org/10.1023/A:1025533719905
Baker, M. C. (1988). Incorporation: A theory of grammatical function changing. University of Chicago Press.
Bekki, D. (2010). Nihongo bunpoo no keisiki riron [Formal theory of Japanese grammar].
Kurosio.
Bresnan, J., & Kaplan, R. (1982). Introduction: Grammars as mental representations oflanguage. In J. Bresnan (Ed.), The mental representation of grammatical relations(pp. i–lii). MIT Press.
Clark, E. V., & Clark, H. H. (1979). When nouns surface as verbs. Language, 55(4), 767–
811. https://doi.org/10.2307/412745
Friedmann, N., Taranto, G., Shapiro, L. P., & Swinney, D. (2008). The leaf fell (the leaf):
The online processing of unaccusatives. Linguistic Inquiry, 39(3), 355–377. https://doi.org/10.1162/ling.2008.39.3.355
Halle, M., & Marantz, A. (1993). Distributed morphology and the pieces of inﬂection. In
K. Hale & S. J. Keyser (Eds.), The view from Building 20: Essays in linguistics inhonor of Sylvain Bromberger (pp. 111–176). MIT Press.
Harley, H. (2008). On the causative construction. In S. Miyagawa & M. Saito (Eds.), The
Oxford handbook of Japanese linguistics (pp. 20–53). Oxford University Press.
https://doi.org/10.1093/oxfordhb/9780195307344.013.0002
Jacobsen, W. (1992). The transitive structure of events in Japanese. Kurosio.
Marantz, A. (2013a). Locality domains for contextual allomorphy across the interfaces. In
O. Matushansky & A. Marantz (Eds.), Distributed morphology today: Morphemesfor Morris Halle (pp. 95–115). The MIT Press.
Marantz, A. (2013b). Verbal argument structure: Event and participants. Lingua, 130,
152–168. https://doi.org/10.1016/j.lingua.2012.10.012
Merchant, J. (2015). How much context is enough? Two cases of span-conditioned stem
allomorphy. Linguistic Inquiry, 46(2), 273–303. https://doi.org/10.1162/LING a
00182
Oseki, Y. (2017). Voice morphology in Japanese argument structures [submitted manuscript].
https://ling.auf.net/lingbuzz/003374
Steedman, M. (2000). The syntactic process. MIT Press.

25

Detecting modality and evidentiality:
Against purely temporal-aspectual analyses of the German semi-modal drohen
Shinya Okano (Chiba University, Japan Women’s University)

Overview I argue that the semi-modal use of drohen `threaten’ in German, which roughlymeans that something undesirable is going to happen, has a modal as well as an evidential
component. In doing so, I argue against a purely temporal-aspectual analysis by Reis
(2007), and more generally, against reductionist attempts to apply analyses forprospective aspect (e.g. Bohnemeyer 2014, Bowler 2018) to every future-oriented item. I
propose a semantics of drohen which contains both modal and evidential components ina framework based on Mandelkern (2019) and Rullmann & Matthewson (2018).
Introduction The German verb drohen has a so-called semi-modal use, as is illustratedin (1). Syntactically, it takes an infinitival clause with zu placed in front of the infinitive
verb, and semantically it can be roughly paraphrased as ‘there is some sign that something
undesirable is going to happen’:(1) Das Wetter droht
schlechter zu
the weather DROH.PRES.3SG worse
to‘The weather threatens to become worse.’

werden.
become.INF

Future orientation The time at which drohen’s prejacent is supposed to hold must be
located after the evaluation time for the entire clause with drohen. This is shown by theinfelicity of (2), which is intended to convey the present residence of Nicole:(2) [I had expected that Nicole still lives in Amsterdam.]#Aber nun droht
sie auf einmal in Berlin zu wohnen.
but
now DROH.PRES.3SG she suddenly
in Berlin to
live.INF`#But now she suddenly threatens to live in Berlin.’ [Adapted from Colomo (2011: 239)]
Based on drohen’s future-orientation and its syntactic affinity to aspectual verbs such as
anfangen `begin’, Reis (2007) argues for a purely temporal-aspectual analysis of drohen
which involves reference to a phase preceding the prejacent eventuality (a “preparatory
process” in Moens & Steedman’s (1988) terms, which Bowler (2018) renames as “prestate”). I call this kind of analysis a purely prospective analysis (PPA). Below I point out
data which pose problems for PPAs.
Temporal-aspectual component alone is not enough. PPAs have difficulty with cases
which involve negation scoping over drohen. Under the assumption that a prospective
aspect operator denotes an existential quantifier over pre-states, as is proposed in recent
formal analyses such as Bohnemeyer (2014) and Bowler (2018), PPAs would predict thatthe existence of a pre-state of the prejacent eventuality could be denied by negation andthus sentences with negated drohen are incompatible with contexts which entail the26realization of the prejacent, because the realization of an eventuality implies therealization of its pre-state. However, this is not the case, as the felicity of (3) shows.
(3) Um 12:30 ist
auf dem Marktplatz plötzlich ein Paket explodiert,at 12:30 PERF.PRES on the market.place suddenly a packet explode.PP
obwohl bis zu diesem Zeitpunkt so etwas nie zu passieren gedroht hatte.
though till to thistimesuch something never to happen DROH.PP PERF.PAST`At 12:30 a package suddenly exploded in the marketplace, although until this time sucha thing had never threatened to happen.’
Given this difficulty with the negation data, I rebut PPAs and argue for an alternative
analysis which involves quantification over future times (relative to the evaluation time)
but does not posit a prospective component, i.e., quantification over pre-states.
Detecting modality: drohen’s prejacent must be epistemically possible. Drohen in thepresent tense shows an epistemic-modal-like behavior in that it is incompatible withpropositions which entail the negation of its prejacent, as (4) illustrates.
(4)#Der Damm droht
unter den Fluten zu brechen, aber er hält stand.
The dam DROH.PRES.3SG under the floods to
break
but it holds stand`#The dam threatens to burst under the floods, but it will hold.’ [Colomo (2011: 241)]
The parallel to epistemic modals is sharpened by the observation that the same effectholds in embedded environments (cf. Yalcin 2007 for epistemic might and must):(5)#Hans glaubt, dasseszuregnen droht,
Hans believes thatittorain.INF DROH.PRES.3SG
aber
er
glaubt auch, dass esnichtregnen wird.
but
hebelieves alsothat itnotrain.INF FUT.3SG`#Hans believes that it threatens to rain, but he also believes that it won’t.’
Furthermore, (6) shows that drohen’s modal force must be that of possibility: drohen canbe used even when the context makes it explicit that the chance of the prejacent’s holdingtrue is very small.
(6) [Bennu is an asteroid which might collide with the earth in 2182. The probability of
impact is calculated to be 0.037 percent.]
Bennu droht auf
die
Erde
zu
stürzen.
Bennu DROH.PRES.3SG onto
the
earth
to
crash
`Bennu threatens to crash to earth.’27

(Retrieved
fromhttps://www.berliner-zeitung.de/news/studie-so-koennte-eineatombombe-den-asteroiden-einschlag-verhindern-li.188246)
Taking also into the consideration the felicity of drohen p and drohen q (where q
contextually entails not p), I conclude that drohen has an epistemic modal component
whose modal force is that of possibility.
Detecting evidentiality: Epistemic possibility is still not enough. Given the data so far,it might seem that drohen is an epistemic possibility modal with future orientation.
However, a further observation points to an evidential component in the semantics ofdrohen: even if there is a salient agent whose epistemic state is compatible with theprejacent being realized in the future, that alone does not suffice for her to assert a drohenclaim. This is illustrated by the infelicity of (7), where it is contextually established that
Max entertains the possibility that the prejacent eventuality might happen but that there
is no concrete evidence for that. In contrast, in a context differing from (7) only in thatenough evidence for rainfall is established (e.g., dark clouds are quickly gathering over
Max and the wind is picking up) the sentence becomes felicitous. This is explainable if
the semantics of drohen requires the existence of some concrete evidence for the prejacent.
(7) [Context: Max is on the top of a mountain. Knowing that the weather in the mountains
can change quickly, he entertains the possibility that it might rain soon, even though it is
now completely cloudless and windless. He describes the situation as follows:]#Es droht
bald zu regnen.
it
DROH.PRES.3SG soon to
rain.INF‘It threatens to rain soon.’Analysis In my analysis, the modal and evidential components in the semantics of drohen
are captured as follows. For modality, to predict the incompatibility of drohen in thepresent tense (henceforth droh.PRES) with propositions implying the negation of its
prejacent, I adapt Mandelkern’s (2019) analysis of epistemic modals, which makes use of
the notion of local context. It is represented as a parameter κ of the evaluation index, but
unlike the original formulation, it is relativized to a time (see (8)), to capture the fact that
drohen can be semantically tensed. Drohen’s modality amounts to existential
quantification over a set of worlds which is obtained by applying the assignment function
g to the subscript that drohen carries (k in (9)), a world w, and a time t, representing somesalient individual’s (typically, the speaker’s) epistemic state at t and w. Drohen’s
definedness condition requires that such states be subsets of the set determined by its local
context (κ). Together with the truth-conditions of conjunction (10a), negation (10b), and28

attitudes like glaub-(en) ‘(to) believe’ (10c), droh.PRES p but not p is predicted to be acontradiction, whether embedded or unembedded: the conjunction (10a) requires thateach conjunct be interpreted with respect to a new information state function; it is almost
the same as the original function (κ in (10a)), with the only difference that it maps adesignated time in the index (t0’s in (10), which are by default identified with the time ofcontext tc) to a set of worlds where the other conjunct is true; Thus, as long as thisdesignated time is used to evaluate droh.PRES p but not p, droh.PRES p must be interpretedwith respect to a subset of an information state which supports the negation of theprejacent p, hence the contradiction. On the other hand, if some time other than thedesignated one in the index serves as an argument of the information state function, sucha flow of information from the other conjunct does not occur. This captures a further
observation that drohen in the past tense (droh.PAST), in contrast to droh.PRES, is
compatible with the negation of its prejacent, as is illustrated by (11), where the second
conjunct implies the negation of the prejacent of droh.PAST in the first conjunct (i.e., `The
child did not drown.').
Second, drohen’s evidentiality is reflected in the metalanguage expression ‘there is someconcrete evidence’ in the truth-condition (9b), which is intentionally left vague. Unlike
Murray (2017), which also posits a primitive evidence relation in the semantics ofevidentials, I do not posit an evidence holder, which is to cover cases with an unspecified
evidence holder.
(8) κ is a function from a time to an information state (a set of worlds).
(9) For any world w, time t, and tenseless proposition P (a set of world-time pairs),[[drohenk]]g,κ (w)(t)(P) is defined only if for any w' ∊ κ(t): g(k)(w')(t) ⊆ κ(t).
If defined, [[drohenk]]g κ (w)(t)(P) = 1 iff(a) there is some w' ∊ g(k)(w)(t) such that there is some t' > t such that P(w')(t') = 1 and(b) there is some concrete evidence at w, t for P’s holding true at some t’ > t.
(10) For any world w, tensed clause p, and clause P whose tense is abstracted over:𝑞𝑝a. [[p und/aber q]]g,tc,t0,κ (w) = 1 iff [[p]]g,tc,t0,𝜅𝑔,𝑡0 (w) = 1 and [[q]]g,tc,t0,𝜅𝑔,𝑡0 (w) = 1 whereφfor any world w and tensed clause φ, 𝜅𝑔,𝑡0 (t0) = κ(t0) ∩ {w’: [[φ]]g,tc,t0,κ (w’) = 1};
φ
for any time t other than t0, 𝜅𝑔,𝑡0 (t) = κ(t).
b. [[nicht p]]g,tc,t0,κ (w) = 1 iff [[p]]g,tc,t0,κ (w) = 0.
c. [[a glaub- P]]g,tc,t0,κ (w)(t) = 1 iff for any world w’ in Ba,w(t), i.e., the set of worlds
compatible with [[a]]g,tc,t0,κ’s belief at w, t, and any ‘subjective now’ t1 of [[a]]g,tc,t0,κat w,t:[[P]]g,tc,t1,Ba,w (w’)(t1).

29

(11)

Ein
zweijähriges
Kinddrohtezu
ertrinken,atwo-year-old
child
DROH.PST.3SG
to
drown
wurde
aber
von
Badegästen
gerettet.
PASS.PST.3SG
butbybathers
rescued`A two-year-old child threatened to drown but was rescued by bathers.’(Retrieved from https://www.westfalen-blatt.de/owl/kreis-herford/herford/herfordbadegaste-retten-kind-2-im-freibad-vor-dem-ertrinken-2588193)
Given these entries, I will present a compositional analysis of clauses with drohen as an
inflected element, based on Rullmann & Matthewson’s (2018) framework for the tenseaspect-modality interaction, along the line shown in (12). Furthermore, I argue that
drohen’s malefactive meaning (i.e. undesirability of the prejacent’s realization) is to betreated in a separated dimension, using a multidimensional framework such as Gutzmann(2015).
(12) Es droht zu regnen. `It threatens to rain.’ ⤳ (drohenk(PFV(it rain)))(PRES(ti))
a. [[it rain]]g,tc,t0,κ = λe.λw.rain(e)(w)b. [[PFV]]g,tc,t0,κ = λP.λi.λw.∃e(τ(e) ⊆ i & P(e)(w))
c. [[PRES(ti)]]g,tc,t0,κ = g(i) if g(i) = tc; undefined otherwise.
Selected References:
Bohnemeyer, J. (2014). Aspect vs. relative tense: the case reopened. Natural Language
Semantics, 32(3): 917–954.
Bowler, M. L. (2018). Aspect and evidentiality. Ph.D. thesis, University of California Los
Angeles.
Colomo, K. (2011). Modalität im Verbalkomplex. Ph.D. thesis, Ruhr-Universität Bochum.
Gutzmann, D. (2015). Use-Conditional Meaning: Studies in Multidimensional
Semantics. Oxford: Oxford University Press.
Mandelkern, M. (2019). Bounded modality. Philosophical Review 128(1): 1–61.
Moens, M. & Steedman, M. (1988). Temporal ontology and temporal reference.
Computational Linguistics, 14(2): 15–28.
Murray, S. E. (2017). The Semantics of Evidentials. Oxford University Press, Oxford.
Reis, M. (2007). Modals, so-called semi-modals, and grammaticalization in German.
Interdisciplinary Journal for Germanic Linguistics and Semiotic Analysis 12: 1-57.
Rullmann, H. & Matthewson, L. (2018). Towards a Theory of Modal-Temporal
Interaction. Language, 94(2): 281–331.
Yalcin, S. (2007). Epistemic modals. Mind, 116(464): 983–1026.

30

Cumulative reading, QUD, and maximal informativeness
Linmin Zhang, NYU Shanghai (zhanglinmin@gmail.com)
0. Overview. Sentence (1) has a distributive reading and a cumulative reading (seeBrasoveanu 2013). This paper focuses on its cumulative reading and argues that (i)although cumulative reading involves multiple modified numerals, it does not involve
multiple independent maximality operators, but only one, and (ii) this maximality
operator is not mereology-based, but informativeness-based, with regard to a salient QUD.
(1)

Exactly threeu boys saw exactly fiveν movies.
Brasoveanu (2013)
Cumulative reading: The cardinality of all boys who saw any movies is 3, andthe cardinality of all movies seen by any boys is 5.

1. Background. According to Brasoveanu (2013)’s account for the cumulative readingof (1), the semantic contribution of modified numerals, exactly three (boys) and exactly
five (movies), is two-fold (or split, see also Bumford 2017): (i) First, (the indefinite
component of) modified numerals introduce (plural) discourse referents (drefs), x and y;(ii) Then, after all the relevant restrictions, i.e., boy(x), movie(y), and see(x, y), areadded onto these drefs, (the definite component of) modified numerals impose maximality
and cardinality tests, as delayed, post-suppositional evaluations (i.e., these tests areapplied at a global sentential level, and in particular, after the restriction see(x, y) is
applied). The maximality tests pick out the maximal boy-sum and movie-sum, and the
cardinality tests check the cardinalities of the maximal boy-sum and movie-sum (see (2)).
(2)[[(1)]] ⇔σxσy[boy(x) ∧ movie(y) ∧ see(x, y)]|{z}

the mereologically maximal x and y satisfying these restrictions∧ |y| = 5 ∧ |x| = 3|{z}cardinality tests(σ: maximality operator; for notation simplicity, cumulative closure is assumed.)
Crucially, as pointed out by Brasoveanu (2013), the two maximality operators need tobe applied simultaneously to guarantee that all ‘boy-seeing-movie’ events in the context
are taken into consideration. Otherwise, the reading (3) would be derived. Intuitively,(1) is true in the context of Fig. 1, but false in the context of Fig. 2, indicating that thepseudo-cumulative reading in (3) (true for Fig. 2) is unattested and needs to be blocked.
(3)

Unattested pseudo-cumulative reading of (1):σx[boy(x) ∧ σy[movie(y) ∧ see(x, y)] ∧|y| = 5] ∧|x| = 3{z}|
the mereologically maximal y{z}|
the mereologically maximal xi.e., The maximal plural individual x satisfying the restrictions (i.e., atomic members of x are boys and each of them saw some movies, and x saw a total of 5 movies
between them) has the cardinality of 3. ; True for Fig. 2! (see b2 ⊕ b3 ⊕ b4 andb1 ⊕ b2 ⊕ b4 , and there is no larger boy-sum satisfying these restrictions.)
movies m1 m2 m3 m4 m5 m6

movies m1 m2 m3 m4 m5 m6

boys

boys

b1 b2 b3 b4

Figure 1: The genuine cumulative
reading of (1) is true in this context.

b1 b2 b3 b4

Figure 2: The genuine cumulative
reading of (1) is false in this context.

31

2. An empirical challenge. As already pointed out by Krifka (1999), such a mereologicalmaximality-based analysis does not work for the case of (4). (4) is parallel with (1) inhaving a cumulative reading. However, Krifka (1999) argues that mereology-based maximality operators should pick out all land-owners and all the land, yielding an uninformative
sentence like In Guatemala, 100% of the population own 100% of the land : ‘What is
peculiar with (4) is that it wants to give information about the bias of a statistical
distribution . . . a picture of the skewing of the land distribution’. Evidently, this skewed
picture is due to the contrast between a small part of population and a large part of land.
(4)

In Guatemala, at most 3% of the population own at least 70% of the land.

Krifka (1999)’s discussion on (4) suggests that in accounting for the cumulative reading,(i) mereological maximality might only be a special case, and (ii) the multiple numericalexpressions that together contribute to the cumulative reading should be inter-connected(see also Brasoveanu 2013’s simultaneity in applying maximality operators).
3. Proposal. Obviously, sentences like (1) and (4) are used in different contexts, addressing different QUDs. Intuitively, (1) tells about an overall picture of film consumption
among boys and can serve as a felicitous answer to questions like how many boys saw how
many movies. Thus as analyzed by Brasoveanu (2013), when (1) is felicitously used, its
interpretation is based on the cardinality measurement of the mereologically maximal
relevant boy-sum and movie-sum (see the right-uppermost dot circled out in Fig. 3).
Then as argued by Krifka (1999), (4) addresses a degree QUD like how skewed wealthdistribution is in Guatemala. Thus when (4) is felicitously used, its interpretation is basedon the ratio between the percentage of owned land and the percentage of their owners
in the population, and the extreme value of this ratio (or gradient) is achieved at theleft-uppermost corner of the parallelogram (i.e., the point circled out) in Fig. 4.
The right-uppermost dot in Fig. 3 and the left-uppermost corner in Fig. 4 constituteextreme cases and represent maximal informativeness in addressing their respective QUD.
#movies5b2 ⊕ b3 ⊕ b4

extreme case of cardinalities

b3 ⊕ b4

4

land
100%

extreme case of gradientb2 ⊕ b43
b3 /b4 b2 ⊕ b3

2
1b2

population

#boys
1

2

3

4

100%

5

Figure 4: QUD: How skewed is wealthdistribution in Guatemala? The percentages of the population and their owned
land are within a parallelogram-like area.

Figure 3: QUD: How much is the overall
film consumption among boys? The cardinalities of some boy-sums and movie-sumsin the context of Fig. 1 are plotted as dots.

Thus I propose a new QUD-related, informativeness-based maximality operator and implement it within a dynamics semantics framework:
(5)

def

Mu1 ,u2 ,... = λm.λg. {h ∈ m(g)|¬∃h0 ∈ m(g). Gqud (h0 (u1 , u2 , . . .)) >info Gqud (h(u1 , u2 , . . .))}
(Type of m: g → {g}; Type of M: (g → {g}) → (g → {g}))32I assume meaning derivation to be a series of updates from an information state toanother, and an information state is represented as a function from an input assignment
function to an output set of assignment functions. As shown in (5), the operator Mu1 ,u2 ,...
works like a filter on information states. With the application of Mu1 ,u2 ,... , the discourse
referents (drefs, which are assigned to u1 , u2 , . . .) that lead to the maximal informativenessin resolving a QUD will be selected out. More specifically, to represent the resolution of a
QUD, the operator Gqud is applied on drefs and returns a value indicating informativeness.
In this sense, Gqud can be considered similar to a measurement function.
In addressing an overarching QUD like how high film consumption is among boys,
higher informativeness means higher consumption (e.g., with d1 > d2 , the consumption
level is d1 -high entails (i.e., is more informative than) the consumption level is d2 -high).
Thus the measurement of informativeness amounts to the measurement of cardinalitiesof plural drefs (see (6b)). Maximal informativeness is achieved when the mereologicallymaximal drefs (i.e., b2 ⊕ b3 ⊕ b4 and m2 ⊕ m3 ⊕ m4 ⊕ m5 ⊕ m6 in Fig. 1) are assigned (see(6c)). A step-by-step compositional analysis for (1) is thus shown in (6):


ν7→y
ν
u
u7
→
x
(6)
a. p = [ some boys saw some movies]] = λg . gmovie(y), boy(x), saw(x, y)b.

c.

Gqud = λx.λy.|x| + |y| (Simultaneously maximizing x and y is QUD-driven,
not stipulated. 
cf. Brasoveanu 2013.)

y = ιy.[movie(y) ∧ ∃x[boy(x) ∧ saw(x, y)]




|
{z
}






some
boys
saw
y






0
0
0
0


]
∧
∀y
=
6
y[movie(y)∧∃x[boy(x)∧saw(x,
y
)]
→
y
<
y]



{z
} 
|
 ν7→yyismereologicallymaximalu7→xMu,ν (p) = λg . g
x = ιx.[boy(x) ∧ ∃y[movie(y) ∧ saw(x, y)]




|
{z
}






x
saw
some
movies


0
0




∧
∀x
=
6
x[boy(x)∧∃y[movie(y)∧saw(x,
y)]
→
x
<
x]
]



{z
} 
|


x is mereologically maximal

d.

ν

u

[[(1)]]= [[exact 3 boys saw exactly 5 movies]] =
y = ιy.[movie(y) ∧ ∃x[boy(x) ∧ saw(x, y)] ν7→y0000∧∀y 6= y[movie(y ) ∧ ∃x[boy(x) ∧ saw(x, y )] → y < y] ]|x| = 3,u7→x
λg . g, ifx=
ιx.[boy(x)∧∃y[movie(y)∧saw(x,
y)]|y| = 5






0
0
∧∀x 6= x[boy(x) ∧ ∃y[movie(y) ∧ saw(x, y)] → x < x] ]In addressing how skewed wealth distribution is in Guatemala, higher informativenesscorresponds to higher skewedness, which means higher gradient (see Fig. 4). Thus themeasurement of informativeness amounts to the measurement of the ratio between thequantity of drefs (see (7b)). Maximal informativeness is achieved when the quantityof a dref y satisfying land(y) ∧ own(x, y) divided by the quantity of a dref x satisfying human(x) ∧ own(x, y) yields the maximal quotient (see (7c)). A step-by-stepcompositional analysis for (4) is thus shown in (7).
(7)

a.

u
p = [[somepopulation own someν land]]= λg .
b.

ν7→yg u7→xland(y), human(x), own(x, y)

Gqud = λx.λy.|y| ÷ |x|

33

c.

d.

















 ν7→yMu,ν (p) = λg . g u7→x
















u


hx, yi = hιx, ιyi such that land(y) ∧ human(x) ∧ own(x, y) ∧¬ ∃y 0 = y[land(y 0 ) ∧ own(x, y 0 )]


|
{z
}



y is the maximal land owned by x


0
0
0

∧ ¬∃x = x[human(x ) ∧ own(x , y)]



|
{z
}

x is the maximal owner of y∧∀x00 ∀y 00 [land(y 00 ) ∧ human(x00 ) ∧ own(x00 , y 00 )∧ ¬∃y 000 = y 00 [land(y 000 ) ∧ own(x00 , y 000 )]|{z}
y 00 is the maximal land owned by x00
000
00
000
000

00

|y |
|y|
∧ ¬∃x = x [human(x ) ∧ own(x , y 00 )] → |x|≥ |x
00 | ]|{z}x00 is the maximal owner ofy 00[[(4)]] = [[at most 3% of the population own at least 70%ν of the land]] =Mu,ν (p), if |x| ⊆ (0, 3%], |y| ⊆ [70%, 1]

4. Discussion. Under the current analysis, it is the contextually salient QUD (i.e.,
what interlocutors care about, their ultimate motivation behind uttering sentences) that
determines how informativeness is actually measured (see the implementation of Gqudin (6b) vs. (7b)), which further determines how the informativeness-based maximality
operator Mu1 ,u2 ,... filters on drefs (before the evaluation of the quantity of selected drefs).
The notion of informativeness-based maximality proposed here is in the same spirit as
but more generalized than the one proposed by Fintel et al. (2014) (which primarily aims
to account for the interpretation of the; see also Schlenker 2012). According to Fintel
et al. (2014), informativeness ordering is based on entailment relation (see (8)).
(8)

Fintel et al. (2014)’s notion of informativeness ordering: For all x, y of type α andproperty φ of type hs, hα, tii, x ≥φ y iff λw.φ(w)(x) entails λw.φ(w)(y).

Thus as shown in (9), depending on the monotonicity of properties, maximal informativeness corresponds to maximum or minimum values.
(9)

a.
b.

For upward monotone properties, maximal informativeness means maximum
values: e.g., given that 6 > 5.5, Mary is 60 tall entails Mary is 5.50 tall.
For downward monotone properties, maximal informativeness means minimum
values: e.g., given that m > n, n walnuts are sufficient to make a pan of
baklava entails m walnuts are sufficient to make a pan of baklava.

Compared to Fintel et al. (2014), the notion of QUD-based maximal informativenessdeveloped in the current paper is more generalized in two aspects.
First, in cumulative-reading sentences where multiple numerical expressions can be
involved, maximal informativeness does not directly correspond to whether the utterednumbers are considered maximum or minimum values. In example (4), as observed by
Krifka (1999), each of the numerical expressions (i.e., at most 3% and at least 70% ) alone
cannot be maximum or minimum values. It is how the combination of these utterednumbers contributes to resolve an implicit, underlying QUD that leads to the achievementof maximal informativeness.
Second, it is worth noting that under the scenario of Fig. 1, although exactly 3boys saw exactly 5 movies holds true, exactly 1 boy saw exactly 4 movies does not holdtrue (in Fig. 1, no boys saw more than 3 movies). Thus, it seems problematic to build
informativeness ordering directly upon the entailment relation between uttered sentences
and their alternatives (here derived by replacing uttered numbers with other numbers).

34


















However exactly 3 boys saw exactly 5 movies does indicate a higher film consumption (ora more prosperous situation) than the consumption level indicated by exactly 1 boy saw
exactly 4 movies, i.e., the uttered sentence indicates a higher informativeness in addressing
an underlying QUD than its alternatives. In this sense, by resorting to QUD, the current
proposal provides a more generalized perspective on informativeness.
An anonymous reviewer asks whether, beyond
5. More empirical coverage.
cumulative-reading sentences, there are other cases where informativeness is determined by
the immediate QUD rather than semantic entailment relation. Here is another case, which
I have discussed in Zhang (2022). As shown in (10) (this example is from Szabolcsi 2017),
under the given scenario, the use of an even-sentence is perfectly natural, but it challenges
the traditional analysis of even. First, the presuppositional requirement of additivity is
not met, because Eeyore was the only one who took a bite of thistles and spit them out.
In other words, the truth of the prejacent does not entail the truth of alternatives like X
spit thistles out (X is a member in the domain and different from Eeyore). Second, if noone other than Eeyore took a bite of thistles, it seems also questionable to claim that thelikelihood of the truth of the prejacent is lower than that of X spit thistles out.
In Zhang (2022), I propose a new degree-QUD-based analysis for the presuppositionof even. The use of even is always based on a contextually salient degree QUD (for (10),how prickly are those thistles). The prejacent of even (here Eeyore spit those thistles out)
provides information to resolve this degree QUD with an increasingly positive answer, andcompared with alternatives, this prejacent is also considered maximally informative inresolving this degree QUD (i.e., here Eeyore spit the thistles out is maximally informativein resolving the degree question how prickly are those thistles).
(10)

Scenario: Imagine Pooh and friends coming upon a bush of thistles. Eeyore(known to favor thistles) takes a bite but spits it out.
a. Those thistles must be really prickly! Even [Eeyore]F spit them out!
((10a) 6; Someone other than Eeyore spit thistles out.)

References
Brasoveanu, Adrian. 2013. Modified numerals as post-suppositions. Journal of Semantics 30:155–209.
https://doi.org/10.1093/jos/ffs003.
Bumford, Dylan. 2017. Split-scope definites: Relative superlatives and Haddock descriptions. Linguistics
and Philosophy 40:549–593. https://doi.org/10.1007/s10988-017-9210-2.
Fintel, Kai Von, Danny Fox, and Sabine Iatridou. 2014. Definiteness as maximal informativeness. In The
art and craft of semantics: A festschrift for Irene Heim, ed. Luka Crnič and Uli Sauerland, volume 2,
175 – 179. Cambridge, MA. https://semanticsarchive.net/Archive/jZiNmM4N/.
Krifka, Manfred. 1999. At least some determiners aren’t determiners. In The semantics/pragmatics
interface from different points of view , ed. K. Turner, 257–291. Elsevier. https://semantics.uchicago.
edu/kennedy/classes/w14/implicature/readings/krifka99.pdf.
Schlenker, Philippe. 2012. Informativity-based maximality conditions. Snippets 26:18–19. https://www.ledonline.it/snippets/allegati/snippets26007.pdf.
Szabolcsi, Anna. 2017. Additive presuppositions are derived through activating focus alternatives. In
The 21st Amsterdam Colloquium, 455–464. https://semanticsarchive.net/Archive/jZiM2FhZ/
AC2017-Proceedings.pdf.
Zhang, Linmin. 2022. The presupposition of even. https://ling.auf.net/lingbuzz/006760, to appear
in Proceedings of Semantics and Linguistic Theory (SALT ) 32.

35

Language Politeness in Social Network
Liping Tang
Waseda University
lipingsysu@gmail.com
Motivation
Language politeness (LP) has been an important topic in pragmatics. On the one hand,comparing to the non-politeness language form, the indirectness of LP brings more costs to the
speaker. On the other hand, from Grice’s theory of cooperate principle [2], interlocutors’ behaviorof LP can not be explained by Grice’s theory.
The pioneer work of politeness theory is developed by Brown and Levison [1]. According totheir politeness theory, politeness is a response to mitigate or avoid face-threatening acts such as
request or insults. Therefore, for each interlocutor, using LP is a way to balance the imposition
from raising a request and maintaining the relationship between the interlocutors. We call this
explanation of LP as the strategic explanation. Many works have intended to formalize language
politeness from the strategic point of view by models in game theory [11, 7, 13, 9].
On the other hand, contrary to the face-threatening theory, a lively area of politeness researchconcerns the conventional perspective of language politeness, which shades lights on the relationship between face and identity [10]. The honorific studies in Japanese language emphasize that
language politeness is more of a relation-acknowledging device rather than the strategic behaviorof saving faces [4]. In the Japanese culture, people are expected to act properly according to theirrelative position or rank with regard to other members of the group, and it is that relative position
that they want to maintain when they employ politeness strategies.
Although the strategic and conventional explanation of LP have different focuses on either as
an individual’s LP behavior or a conventional behavior in a society, they both shade lights on
the relationships between the interlocutors. By applying the social network theory [12, 3], we caneasily represent the social distance (by using distances between agents) and the social power (byusing the centrality notion in social network) between the interlocutors. Hence, Brown and Levison[1]’s three social variables (social distance, social power and imposition of a request) deciding thelevel of politeness can be modeled at the same time within the model of social network.
In addition, we can incorporate the conventional perspective of language politeness by considering an agents’ behavior as a function of his neighbors. Considering all those factors, we constructa language game of LP with players located in a social network. Then, we use simulation studies
to show how social structures affect the agents’ use of language politeness.
A communication game of LP in Social Network (CLPS)The CLPS is based on the communication game of language politeness in [6] with modifications.
Definition 1. The communication game of language politeness in social networks (CLPS) consistsof the following parts.
1. Players are located in a network g = {N, Eij } in which N is a set players, Eij represents theedges between the players i, j;

36

2. At each time t, if agents i and j are connected in g, then i and j play the game twice, oncei plays the role as the speaker and j be the hearer; and once j plays the role as the speaker
and i be the hearer;
3. The social distance between the speaker i and the hearer j is denoted as Dij = dij + Njn , inwhich dij represents the distance between two agents i and j in g. Njn represents the hearer
j’s number of neighbors within distance n in g, which characterizes the centrality of j in the
network.
4. A social imposition stij for the speaker i and the hearer j at time t is defined as stij =f t (Dij + r), which is a function of the social distance between the interlocutors and theimposition of a particular request r.
5. The speaker’s action is to choose certain politeness p ∈ R, representing the language politeness.
6. The hearer’s action is a function of the social imposition s given p, P (s|p) ∈ [0, 1].
7. The player i’s utility of choosing p is defined as Ui (p) = EUi − C(p), in which EUi =sij (D)P (s|p) and C(p) is a linear function of p, representing the cost of using p.
The model makes the following assumptions about the utility function.
1. The expected utility function EUS : R+ → R+ is supposed to be defined as a strictly
monotonously increasing function with a decreasing slope;
2. The cost function C(p) is defined as a strictly monotonously increasing function with respectC(p)to the politeness p. 1
𝐸𝑈 (𝑝)(p)
An utility function that has such properties can be simply defined as
𝐸𝑈 (𝑝)U(p)
𝐸𝑈 (𝑝)
(p)
1
if p > s;(p − s) 2
EUS (p) =(p)0,otherwise.
p(p)
!!

!"

S

!#

p*(r2) p*(r1)C(p) = β × p,

β∈R

p*(r0)(p)+

We use the following graphs to qualitatively describe the dynamics of the functions US , st (D)and related parameters.
s
s0(D)

!

𝐸𝑈 ! (𝑝)(p)"
𝐸𝑈 ! (𝑝)#(p)
𝐸𝑈 ! (𝑝)

C(p)
s1(D)

US
s2(D)

(p)
(p)

(p)

D
D’

p
*

2

*

1

p (s ) p (s )*0p (s )(b) dynamic of the social rates of imposition st when r=0(a) dynamic of the utility

Figure 1s
Figure 1(a)represents the dynamics (D)of the speaker’s utility function as the change of the politeness p and the social imposition s. It is obviously from the graph that as the social imposition
0

1

s (D)
Various works have shown that the indirectness
and the length of the phrases are positively correlated to thepoliteness ([5, 8]). That is the main reason that we make this assumption here.

1

s2(D)

D
D’

37

s decreases, the utility increases. The imposition of a request r is treated as a constant for eachutility curve. p∗ represents the optimal politeness for each utility curve. In Figure 1(b), we specify
the dynamics of the social rate of imposition st (D) as the times of communication t increases.
Given any fixed D0 , the rates of imposition decreases. Without loss of generality, in the graph, we
assume r = 0. For any fixed D0 , as communication enhanced, the social imposition decreases, i.e.
s0 (D0 ) > s1 (D0 ) > s2 (D0 ).
The players are playing the game repeatedly. At each round of the game, every agent plays theLP game with the agents they are connected in the network. More specifically, each agent sends
a request with respect to the social rate of the imposition s by sending a politeness phrase. After
repeated plays of the game, we explore the change of agents’ use of LP. In addition, we make twomodeling rules for the dynamic of the game.
Rule 1 (Punishment Rule): If an agent k rejects agent i’s (as a speaker) request, then agent
i has a positive probability to reject k’s request in the future.
Rule 2 (Dynamic Rule): The repetition of communication decreases the social rates of
imposition for a fixed request and a fixed social distance between the interlocutors. At the sametime, the speaker’s utility is increasing as the rates of imposition s decreases.
Depending on whether we consider the strategic perspective or the conventional perspective ofLP, we define two kinds of dynamics of the social imposition following Rule 2.
Dynamic 1(Strategic)s0ij = (Dij + r)
stij (D) =1∗ st−1ij (D)t−1U (1 − U t−1 ) ∗ st−1 (D),ij

if U t−1 ≥ 1;otherwise.

Dynamic 2 (Strategic + Conventional)
s0ij = (Dij + r)
stij (D) =




1

t−1
k∈Njn skjU ω((1 − U t−1 )st−1 (D) + (1 − ω) 1 P n st−1 ,
ij
kj
k∈N
Nj

ω(

st−1 (D)) + (1 − ω) N1j
t−1 ijP

jif U t−1 ≥ 1;otherwise.

in which ω ∈ [0, 1], The intuition for the second half of the formula is that the speaker isadopting the average of his neighbors’ social impositions.
Regarding Rule 1 (Punishment rule), we explore two versions for the hearer’s response.
Version 1: the hearer always responds positively to the speaker’s request.
Version 2: the hearer responds positively with probability pr to the speaker’s request, i.e.
there is (1 − pr) probability that P (s|p) = 0. Particularly, we assume that pr decreases as the
distance dij increases. We use pr1 represents the pr at the distance dij = 1.
We do four simulations with the following combinations in Table 1.

Version 1(always positive response)
Version 2(randomly positive response)

Dynamic 1
Strategic

Dynamic 2
Strategic+Conventionl

Simulation 1

Simulation 2

Simulation 3

Simulation 4

Table 138

Simulation results
Three different networks (ring, star, complete) with four agents are applied in the simulations.
Intuitively, the complete network characterizes the society or institute in which everyone is equallyconnected with respect to their distances and power differences. Hence, everyone in the complete
structure has the same contribution or influence to the community. On the contrary, the star
network represents the other extreme situation in which one person is the most powerful one(in terms of his centrality in the network), while other agents have the same importance in the
structure. Intuitively, the star structure represents a relatively hierarchical society. Moreover, thering structure represents a median situation with respect to the equality of the distance and thepower difference among the three networks.

(a) Ring(b) Star(c) Complete

We list two comparative results with respect to four simulations mentioned in Table 1. The first
result is to compare the degree of politeness through different networks with respect to each simulation (showing in Table 2).The second one is to compare different simulations within each network(showing in Figure 3). Overall, from all the simulations, we conclude the following observations.
(1) The level of politeness negatively correlates to the average of the utility.
(2) When the long distance communication has no cost, then the star shape network shows theleast average politeness and the highest utility in terms of the use of LP; The complete networkshows the highest level of politeness and the least average utility.
(3) When the long distance communication has a cost, i.e. long distance increases the failureof communication, then the complete network has the least LP and highest utility; while the star
shape network shows the opposite features.
(4) The dynamic differences in the models (three simulations) only has effects on certain network. In the current result, only the complete network is sensitive to the dynamics. Other two
networks have consistent comparative results through the four simulations.
Through the dynamic model within social networks, we can explore different quantitative features of using LP. The advantage of this method is that we can easily modify the model and extend
the model to any kind of network structures.
Simulation 1(strategic, positive response)
Simulation 2(strategic+conventional, positive response)
Simulation 3(strategic, random response)
Simulation 4(strategic+conventional, random response)

Ring
P0
U0
P0
U0
P0
U0
P0
U0

Star
P−
U+
P−
U+
P+
U−
P+
U−

Comp
P+
U−
P+
U−
P−
U+
P−
U+

Table 2: (P − and P + means the least average politeness and the highest average politeness among the three
networks. P 0 means the average politeness sits in the middle. U represents the average utility. Similar meanings
are assumed by using U + , U − and U 0 . Simulation results in this table are conducted by assuming ω = 0.5, pr1 =0.9, pr2 = 0.1)
39

(a) ring(b) star(c) complete

Figure 3: Comparison through simulations ( ‘Ring1 ” in the figure means that simulation 1 conducted on the ring
network, etc.)

References[1] Penelope Brown, Stephen C Levinson, and Stephen C Levinson. Politeness: Some universals in language
usage, volume 4. Cambridge university press, 1987.
[2] Herbert P Grice. Logic and conversation. In Speech acts, pages 41–58. Brill, 1975.
[3] Kevin J.S.Zollman. Talking to neighbors: The evolution of regional meaning. Philosophy of Science, (1):69–85,
2005.
[4] Yoshiko Matsumoto. Reexamination of the universality of face: Politeness phenomena in japanese. Journal ofpragmatics, 12(4):403–426, 1988.
[5] Tony McEnery, Paul Baker, and Christine Cheepen. Lexis, indirectness and politeness in operator calls. In
New frontiers of corpus research, pages 53–69. Brill, 2002.
[6] Roland Muehlenbernd, Przemyslaw Zywiczynski, and Slawomir Wacewicz. The game theory of politeness inlanguage: A formal model of polite requests. 2019.
[7] Roland Mühlenbernd, Slawomir Wacewicz, and Przemyslaw Żywiczyński. Politeness and reputation in cultural
evolution. Linguistics and Philosophy, 44(6):1181–1213, 2021.
[8] Jan-Ola Östman. Testing iconicity: Sentence structure and politeness. Belgian Journal of Linguistics, 4(1):145–
163, 1989.
[9] Jason Quinley. Politeness and trust games. Student Papers Session, Proceedings of ESSLLI, 2011.
[10] Marina Terkourafi. The linguistics of politeness and social relations. Routledge Handbook of Linguistics, pages
221–235, 2015.
[11] Robert Van Rooy. Being polite is a handicap: Towards a game theoretical analysis of polite linguistic behavior.
In Proceedings of the 9th Conference on Theoretical Aspects of Rationality and Knowledge, pages 45–58, 2003.
[12] Elliott Wagner. Communication and structured correlation. Erkenntnis, 71(3):377–393, 2009.
[13] Erica J Yoon, Michael Henry Tessler, Noah D Goodman, and Michael C Frank. Polite speech emerges fromcompeting social goals. Open Mind, 4:71–87, 2020.

40

The Absence of N OR in JapaneseHajime Mori (The University of Tokyo)
morihajime6@gmail.com
1. Introduction
How we understand and express logical structure is a philosophically and linguistically importantquestion. This paper tries to answer this question partly, focusing on the lexicalization of binary
logical connectives in Japanese. Binary logical connectives are binary truth functions that take two
propositions as input and assigns one proposition or truth value as output. As far as we concern
Boolean truth values, there are 16 binary logical connectives including the conjunction operator∧ (or AN D), the disjunction operator ∨ (or OR), and the logical equivalence operator ↔ (orIF F ). Henceforth, for the sake of simplicity of expression, I would call them “logical connectives”
or “connectives.”
The lexicalization of binary logical connectives has been studied from the point of view ofboth cross-linguistically and individual-liguistically (Horn, 1972, 2011; Katzir and Singh, 2013;
Enguehard and Spector, 2021; Uegaki, 2022; a.o. for cross-linguistic studies: Mous, 2004; Bowler,
2015, for individual language studies.). These studies have rebealed that not all the connectives are lexicalizable in natural languages. Indeed, N AN D is never lexicalized in any natural
language. Also, some of the previous studies show that inventories of lexicalized connectives
vary among languages. For example, the inventory of logical connectives lexicalized in English
is {AN D, OR, N OR} though that in Warlpiri is {OR} (Bowler, 2015). However, no literature
has discussed which connectives are lexicalized in Japanese. The system of logical vocabulary inJapanese is diﬀerent from that of English. You can see it from the fact that there is no immediatetranslation for any of and, or, and nor in Japanese (cf. §§2, 3.2.1). Hence clariﬁcation of logicalvocabularies in Japanese leads to a deeper understanding of Japanese semantics.
In this paper, I propose that the binary logical connectives that are lexicalized in Japanese are
AN D and OR. In the next section, we brieﬂy observe that at least AN D and OR are lexicalized inJapanese. After showing the minimum candidate for the inventory of Japanese logical connective
items, I assert that no other logical connectives can be lexicalized in Japanese. Two facts support
this argument: one is a consequence in the cross-linguistic studies that the maximum logicalinventory lexicalizable in natural languages is {AN D, OR, N OR}, and the other is a theoretical
proof for the nonlexicalizability of N OR in Japanese. Lastly, I conclude this paper by overviewing
the whole discussion and the future outlook of this research.
2. Lexicalization of AN D and OR in Japanese
Although it is arduous to determine the perfect list of Japanese words corresponding to AN D
and OR, Japanese has obvious (pure) conjunction and disjunction items. As shown in (1-2), katsu
corresponds to AN D and aruiwa expresses OR 1 .
(1) Taro-ga hashiri katsu Hanako-ga arui-ta.
Taro-nom run
katsu Hanako-nom walk-past
Taro ran or Hanako walked.
(2) Taro-ga hashit-ta-ka aruiwa Hanako-ga arui-ta.
Taro-nom run-past-ka aruiwa Hanako-nom walk-past1

aruiwa in (2) is optional though ka is necessary to express the disjunctive sentences. For this reason, one might
say that ka is what expresses disjunction and aruiwa is only used to emphasize disjuctiveness. However, I assume
here that aruiwa is the logical word because it is controversial whether ka is merely a disjunction operator (Uegaki,2018). Whichever it is, the statement that Japanese has a lexical item corresponding to OR is true; hence thediscussion does not essentially change.
41

Taro ran or Hanako walked.
Though slightly arguable, the particle mo is also an expression of AN D in Japanese.
(3) Taro-mo Hanako-mo hashit-ta.
Taro-mo Hanako-mo run-past
Taro and Hanako ran.
The property mo has and and doesn’t is that a single mo cannot create conjunctive sentence.
It needs to appear as a pair as it does in (3). However, it is cross-linguistically prevalent that
conjunction operators need to appear as a pair (Mitrović and Sauerland, 2014, 2016). Furthermore,sentences such as (3) do mean the conjunctive proposition in the absence of other logical-operatorlike words, which is the most obvious reason that mo is one of the lexical items corresponding to
AN D.
3. N OR in Japanese
As we have seen in the previous section, AN D and OR are lexicalized in Japanese. Thus, let
J the set of all connectives that are lexicalized in Japanese and C the set of all connectives (i.e.
|C| = 16), we have (∗):(∗) {AN D, OR} ⊆ J ⊆ C.
In this section, I try to give linguistic evidence that J = {AN D, OR}.
3.1. Universally unlexicalizable items
A number of studies have advocated that logical connectives beyond the Aristotelian square (i.e.
connectives other than AN D, OR, N OR, and NAND) and NAND cannot be lexicalized in natural
languages (Horn, 1972, 2011; Katzir and Singh, 2013; Carcassi and Sbarbolini, 2021; Enguehardand Spector, 2021; Uegaki, 2022; a.o.). Moreover, Carcassi and Sbarbolini (2021), Enguehardand Spector (2021), and Uegaki (2022) give independent accounts for the empirical fact that theinventories of logical connectives lexicalizable in a natural languages are only {AN D, OR, N OR},{AN D, OR}, {AN D}, and {OR}. These results allow us to strengthen the condition (∗) to (∗∗):(∗∗) {AN D, OR} ⊆ J ⊆ {AN D, OR, N OR}.
That is, at this point, the candidates for J are narrowed down to two inventories: {AN D, OR}
and {AN D, OR, N OR}.
3.2. Absence of N OR in JapaneseThen, the question is whether Japanese has a word for N OR or not. Simple observations suggest
the negative answer to this question. A sentence which expresses a N OR proposition does notcontain any speciﬁc word that only appears in sentences of N OR propositions.
(4) Taro-ga utai-mo
odori-moshi-nakat-ta.
Taro-nom singing-mo dancing-mo do-neg-past
Taro neither sang nor danced.
However, since the system of Japanese logical words is not simple, this mere observation does not
conﬁrm the absence of a lexical item with meaning of N OR in Japanese. It might be the case that
there actually is a word corresponding to N OR, but that it is hard to identify the word just as no
Japanese word is the immediate translation of and. Therefore, to conclude that J = {AN D, OR},
we need to prove that N OR is not lexicalized in Japanese. In the following subsections, I showthat N OR is not lexicalizable in Japanese, which is suﬃcient for the proof for the nonexistence of
the N OR item in Japanese.
423.2.1How N OR is expressed in Japanese

As we have seen above, the most natural way to express N OR proposition is the negation ofmo-conjunction sentence. Furthermore, Japanese sentences such as (4) can only be interpreted as
N OR sentences. This is, I would argue that, the primary reason why N OR is not lexicalized inJapanese. On the contrary, the negation of a conjunctive sentence is ambiguous in two senses inEnglish.
(5) Taro-ga utai-mo
odori-moshi-nakat-ta.
Taro-nom singing-mo dancing-mo do-neg-pasta. ¬sing(T aro) ∧ ¬dance(T aro) ≡ sing(T aro) N OR dance(T aro)b. ∗ ¬(sing(T aro) ∧ dance(T aro)) ≡ sing(T aro) N AN D dance(T aro)
(6) John did not dance and sing.
a. ¬sing(John) ∧ ¬dance(John) ≡ sing(John) N OR dance(John)b. ¬(sing(John) ∧ dance(John)) ≡ sing(John) N AN D dance(John)
This ambiguity in English leads to the necessity of a lexical item corresponding to N OR and theunambiguity in Japanese does not.
The unambiguity of the negation of mo-conjunction is supported by a theory if we admita proposal by Mitrovic̀ and Sauerland(2014, 2016). According to them, there are two types ofconjunctions. One conjoins two type t entities and the other conjoins two type ⟨e, t⟩ objects.
English and is an example of the former and Japanese mo is the latter sort of conjucntion. Thierproposal is that latter sort of conjunctive items have the meaning shown in (7).
(7) JmoK ≡ λSλT.S ⊆ T
Following this deﬁnition, the meaning of the sentence (5) is calculated as follows.
T(8) ¬do(T aro, {S|{singing, dancing} ⊆ S})(8) cannot mean but that John did neither singing nor dancing, which is equivalent to N OR
proposition. This is why (5b) is ungrammatical.
3.2.2

The informativeness/complexity trade-oﬀ model by Uegaki(2022)

Uegaki(2022) proposes an account for why only limited connectives are lexicalizable in natural languages. His proposal is based on the informativeness/complexity trade-oﬀ model, which is proposedby Kemp, et al. (2018) and succeeds in explaining the cross-linguistic pattern of lexicalization ofseveral kinds of content words. If an inventory of concepts is lexicalizable, it is Pareto eﬃcientconcerning informativeness and complexity. According to Uegaki(2022)’s formulation, complexity indicates how simply the connectives in the inventory can be expressed only by ¬, ∧, and ∨.
The more blocks are needed to express a connective, the higher complexity it has, and vice versa.
Informativeness is the summation of the degree of the likelihood that the use of the connective
accurately conveys the speaker’s intention under the scalar implicature. For example, the use of
AN D exactly expresses the possible world where the propositions coordinated by AN D are both
true, while the use of OR is ambiguous in that it cannot be determined which of disjuncts is true.
3.2.3

The formal deﬁnition of the informativenessThe complexity of an inventory of logical connectives does not vary among languages. However, theinformativeness might change depending on languages. In this subsection, I modify the deﬁnitionof the informativeness so that the grammar of individual language is reﬂected.
First, let me explain the formal deﬁnition of informativeness given by Uegaki(2022). Following
Fox(2007), scalar implicature of a proposition is given as (9). Based on the deﬁnition (9), the scalar
implicature of a logical connective is deﬁned as (10). For example, let L = {AN D, OR, N OR} and43

for arbitrary propositions p and q, w1 be the world where both are true, w2 be the world where
only p is true, w3 be the world where only q is true, and w4 be the world where neither is true.
Then, JLK = {{w1 }, {w1 , w2 , w3 }, {w4 }}, and JORK+
L = {w1 , w2 , w3 } ∩ {w2 , w3 , w4 } ∩ {w1 , w2 , w3 } ={w2 , w3 }.
T(9) a. ScalarImp(p, A) := p ∩ {p′ |p′ ∈ IE(p, A)}T
b. IE(p, A) := {A′ ⊆ A|A′ is a maximal subset of A s.t. p ∩ {p′ |p′ ∈ A′ } ̸= ∅}(10) JcK+
L := ScalarImp(JcK, JLK),
where JcK is the set of worlds where the proposition p c q is true and JLK := {JcK | c ∈ L}.
Utilizing the deﬁnition given by Steinert-Threlkeld(2019), he then gives the deﬁnition of informativeness I of an inventory of logical connectives L and the utility function u as follows. His
calculation assumes that for every world w, the probability of the world P (w) = 41 . Also, it is′assumed that P (c|w) = n1 where n = |{c′ ∈ L | w ∈ Jc′ K+L }| if such c exists (otherwise P (c|w) := 0)+
and P (w|c) = m1 where m = |JcK+
L | if w ∈ JcKL (otherwise P (w|c) := 0).
PPP(11) I(L) := w∈W P (w) c∈L P (c|w) w′ ∈W P (w′ |c)u(w, w ′ )
1 if w = w′ ,′ :=(12) u(w, w )0 otherwise.
As we have seen in the previous section, in some natural language, the negation of a connective
might have a unique interpretation, depending on the grammar of the language. If a connective is
uniquely represented in another way, there is no use to lexicalize the connective. This hypothesis
can be realized by modiﬁcation of utility function, using the set of grammatical connectives GC.
The diﬀerence between (12) and (13) is that (13) excludes the case that lexicalization of a connectivethat can already be expressed by another (simple) expression has utility.
(13) Given an individual language L,
1 if w = w′ and {w} ∈/ GC LL′ :=a. u (w, w )0 otherwise,


′
′
′
+ c ≡ c -N EG or c ≡ N EG-c for some c ∈ L,
L :=
b. GC
JcKL
,
and there is a unique expression of c in L
where ϕ N EG-c ψ ≡ (¬ϕ) c (¬ψ) and ϕ c-N EG ψ ≡ ¬(ϕ c ψ).
Replacing u in (11) with u′ , we ﬁnally can calculate the informativeness relativized to an individual
language L, which I denote by I L .
3.2.4

Application of I L to Japanese

The purpose of deﬁning the informativeness relative to a speciﬁc individual language was to prove
the absence of N OR in Japanese. As I mentioned before, the candidates for J are {AN D, OR}
and {AN D, OR, N OR}. Thus, to determine the inventory lexicalized in Japanese, it is enough to
calculate I J PN ({AN D, OR}) and I J PN ({AN D, OR, N OR}).
For the process of calculation, we ﬁrst need to determine GC J PN . The observation I made in§3.2.1 conﬁrms the following result. 2+(14) GC J PN = {JN EG-AN DK+
L } = {JN ORKL }Following the deﬁnition in (9-14), we obtain the following result:

J PN
Precisely speaking, what the observation in §3.2.1 veriﬁes is no more than that {JN EG-AN DK+⊆
L } ⊆ GC
{c-N EG|c = AN D, OR, N OR} ∪ {N EG-c|c = AN D, OR, N OR}. However, it can be easily veriﬁed that whether
J PN{JN EG-AN DK+
or not, I J PN ({AN D, OR}) = I J PN ({AN D, OR, N OR}); thus the same conclusion.
L } ⊊ GC
2

44

(15)

a. I J PN ({AN D, OR}) = 0.5.

b. I J PN ({AN D, OR, N OR}) = 0.5.

Since I J PN ({AN D, OR}) = I J PN ({AN D, OR, N OR}) and the complexity of {AN D, OR, N OR}
is larger than that of {AN D, OR}, the inventory {AN D, OR, N OR} is no more Pareto eﬃcient.
Therefore, {AN D, OR} is the only candidate for J ; that is, J = {AN D, OR}.
4. Conclusion
In this paper, I showed that the inventory of logical connectives lexicalized in Japanese, J , isequal to {AN D, OR}. First, {AN D, OR} ⊆ J because katsu and aruiwa correspond to AN D
and OR. Then, J ⊆ {AN D, OR, N OR} by the cross-linguistic regulation on lexicalization ofconnectives. Lastly, J ⊆ {AN D, OR} is derived from the fact that N OR proposition is unambiguously expressed by the negation of mo-conjunction sentence in Japanese. The unambiguity of
the negation of mo-conjunction sentence leads to the unnecessity of a lexical item corresponding to
N OR. This process was formalized within the informativeness/complexity model, by relativizing
the informativeness to Japanese. Therefore, the conclusion is that J = {AN D, OR}.
5. Future research
The idea of the informativeness relativized to an individual language can be trivially applied
to other languages including Warlpiri and Iraqw, whose logical connectives are researched byBowler(2015) and Mous(2004), respectively. If this idea accounts for all of those reported cases,
it might lead to another theory of typology. That is, if two languages share the grammaticalproperties relevant to a speciﬁc domain of vocabulary, then the vocabulary of the domain should
also be shared.
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Updates and Boolean Universals. Ms. Universaity of Amsterdam.
Enguehard, Émile &Spector, Benjamin. (2021). Explaining gaps in the logical lexicon of natural languages: A
decision-theoretic perspective on the square of Aristotle. Semantics and Pragmatics. 14. Fox,
D. (2007). “Free Choice and the Theory of Scalar Implicatures.” In: Sauerland, U., Stateva, P.
(eds) Presupposition and Implicature in Compositional Semantics. Palgrave Studies in Pragmatics,
Language and Cognition. Palgrave Macmillan, London. Han, CH & Romero, M. (2004).
Disjunction, Focus, and Scope. Linguistic Inquiry - LINGUIST INQ. Horn, L. R. (1972). On
The Semantic Properties Of Logical Operators In English. Horn, L. R. (2011). Histoire d’∗ O:
Lexical pragmatics and the geometry of opposition. New Perspectives on the Square of Opposition,
Jean-Yves Béziau and Gillmann Payette, editors, 393-426. Bern: Peter Lang.
Katzir, R.,& Singh, R. (2013). Constraints on the lexicalization of logical operators. Linguistics andPhilosophy, 36(1). 1-29. Kemp, C, Yang Xu, and Terry R. (2018). Semantic Typology
and Eﬃcient Communication. Annual Review of Linguistics 4(1),109-128. Mitrović, M., &Sauerland, U. (2016). Two conjunctions are better than one. Acta Linguistica Hungarica, 63(4),471-494. Mitrović, M., & Saureland, U.. (2014). Decomposing coordination. Proceedingsof the 44th North East Linguistic Society (NELS), 39-52. Mous, M. (2004). The grammar of
conjunctive and disjunctive coordination in Iraqw. Coordinating Constructions. Shimoyama, J.
(2011). Japanese Indeterminate Negative Polarity Items and Their Scope. Journal of Semantics.
28. 413-450. Steinert-Threlkeld, S. (2019). “Quantiﬁers in natural language optimize thesimplicity/informativeness trade-oﬀ”. Szabolcsi, A (2015). What do quantiﬁer particles do?
Linguistics and Philosophy 38 (2):159-204. Uegaki, W. (2018). A uniﬁed semantics for theJapanese Q-particle ka in indeﬁnite, questions and disjunctions. Glossa: a journal of generallinguistics 3(1), 14. Uegaki, W. (2022). The Informativeness/Complexity Trade-Oﬀ in the
Domain of Boolean Connectives. Linguistic Inquiry 202245

End-to-End Compositional Modelling of Vector-Based Semantics
Michael Moortgat, Utrecht University
Compositionality models the syntax-semantics interface as a structure-preserving map relating
syntactic categories and derivations to their counterparts in a corresponding meaning algebra.
In a categorial grammar, the syntactic categories take the form of types with a combinatorics
governed by an appropriate type logic. The target meaning algebra for a distributional semantics
has vector-based representations of word meanings obtained from data as its basic building blocks.
An end-to-end view on compositionality here requires not only the elementary word embeddings
to be obtained from data, but also the categories/types and their internal composition so thatneural methods can then be applied to learn how the steps of a syntactic derivation can besystematically mapped to operations on the data-driven word representations.
In this talk, I report on the results of a five year project that aimed to realize such an end-to-end
approach. The project focuses on Dutch data, but given that a categorial grammar essentiallyboils down to a type lexicon, its methods are readily adaptable to other languages. Key ingredients are an extended type logic that relies on the expressivity of categorial modalities to simultaneously capture dependency and function-argument structure; a geometry-aware approachto constructive supertagging that fully exploits the internal category structure, and a proofnetbased neural parser that associates raw text with programs for meaning assembly expressed aslinear lambda terms. In all, the project highlights the benefits of a balanced neural-symbolic
approach to NLP.

References[1] Konstantinos Kogkalidis and Michael Moortgat. Geometry-aware supertagging with heterogeneous dynamic convolutions. CoRR, abs/2203.12235, 2022.
[2] Konstantinos Kogkalidis, Michael Moortgat, and Richard Moot. Æthel: Automatically extracted typelogical derivations for Dutch. In Proceedings of The 12th Language Resources
and Evaluation Conference, LREC 2020, Marseille, pages 5257–5266. European Language
Resources Association, 2020.
[3] Konstantinos Kogkalidis, Michael Moortgat, and Richard Moot. Neural proof nets. In
CoNLL2020, Proceedings of the 24th Conference on Computational Natural Language Learning, pages 26–40. Association for Computational Linguistics, 2020.
[4] Michael Moortgat, Konstantinos Kogkalidis, and Gijs Wijnholds. Diamonds are forever.
Theoretical and empirical support for a dependency-enhanced type logic. In R. Loukanova,
P. Lumsdaine, and R. Muskens, editors, Logic and Algorithms in Computational Linguistics
2021, Studies in Computational Intelligence. Springer, (to appear).
[5] Richard Moot. Type-logical investigations: proof-theoretic, computational and linguistic aspects of modern type-logical grammars. Habilitation à diriger des recherches, Université Montpellier, November 2021.

46

Measurement Theory Meets Mereology in Multidimensionality in Resemblance
Nominalism (Extended Abstract)
Satoru Suzuki (Komazawa University)
1. Motivation
The problem of particulars and universals is one of the most essential problems in the formal
philosophy of language in the sense that it consists in a crossroads of ontology and semantics: When we translate a natural language into a first-order (modal) language, (though it isa problem which formal language we should adopt in this translation), the semantic problem
as to which entity we should choose as the semantic value of a symbol in the model of firstorder modal logic depends crucially on the ontological problem as to which ontology we should
adopt. According to Rodriguez-Pereyra (2015), there are at least two kinds of Nominalism:
one that maintains that there are no universals and the other that maintains that there are
no abstract objects like classes, functions, numbers and possible worlds. On the other hand,
Realism about universals is the doctrine that there are universals, and Platonism about abstract objects is the doctrine that there are abstract objects. The doctrines about universals
and the doctrines about abstract objects are independent. According to Rodriguez-Pereyra(2015), Nominalisms about universals can be classified into at least eight types: (I) Trope
Theory, (II) Predicate Nominalism, (III) Concept Nominalism, (IV) Ostrich Nominalism, (V)
Mereological Nominalism, (VI) Class Nominalism, (VII) Resemblance Nominalism, and (VIII)
Causal Nominalism. Resemblance Nominalism in general is confronted with at least sevenproblems: (i) Imperfect Community Problem, (ii) Companionship Problem, (iii) Mere Intersections Problem, (iv) Contingent Coextension Problem, (v) Necessary Coextension Problem,(vi) Infinite Regress Problem, and (vii) Degree of Resemblance Problem. As Rodriguez-Pereyra(2015) argues, according to Resemblance Nominalism, it is not because things are scarlet thatthey resemble one another, but what makes them scarlet is that they resemble one another.
Resemblance relations are primitive and the properties of a thing are defined by resemblance
relations. Resemblance Nominalism reifies neither resemblance relations nor accessibility relations in themselves. Suzuki (2020) proposes, in terms of measurement theory, a first-ordermodal resemblance logic MRL that can furnish solutions to all of the problems (i)-(vii). Yi(2014) raises a version of degree of resemblance problem. Yi (2014, pp.622-625) argues asfollows:(1) Carmine resembles vermillion more than it resembles triangularity.
(2) is a resemblance-nominalistic formulation that expresses what makes (1) true:(2) Some carmine particular resembles some vermillion particular more closely than
any carmine particular resembles any triangular particular.
Rodriguez-Pereyra defines the degree of resemblance as follows (Rodriguez-Pereyra (2002,p.65)):
Definition 1 (Degree of Resemblance) The particulars resemble to the degree n iff theyshare n sparse properties (About a sparse property, refer to Rodriguez-Pereyra (2002, pp.5052).).
Under Definition 1, (2) compares the maximum degrees of resemblance. But (2) is false because
a possible carmine particular completely resembles a possible triangular particular. For thesame particular might be both carmine and triangular. Rodriguez-Pereyra (2015) responses to
Yi by replacing (2) by (3):(3) Some carmine particular resembles some triangular particular less closely than
any carmine particular resembles any vermillion particular.
Again under Definition 1, (3) compares the minimum degrees of resemblance. RodriguezPereyra (2015, p.225) argues that (3) is true because the minimum degree to which a carmine
particular can resemble a triangular particular (degree 0) is smaller than the minimum degree
to which a carmine particular can resemble a vermillion particular (a degree greater than 0).
Yi (2018, p.796) criticizes this Rodriguez-Pereyra’s response by arguing that it rests on a false
47

assumption: the minimum degree to which a carmine particular can resemble a vermillion
particular is greater than 0. For, on Rodriguez-Pereyra’s notion of resemblance, a carmine
particular cannot resemble a vermillion particular unless they share a sparse property, butthey might not share any such property. No doubt this argument by Yi needs examining indetail, but we can safely say that the main culprit of this Rodriguez-Pereyra-Yi Problem isDefinition 1 on which both (2) and (3) are based. We consider this problem to be a problem
of multidimensionality (such three dimensionality as carminity, vermillionity and triangularity)that requires quantitative (numerical) representations because we cannot have computational
method of aggregation only in terms of qualitative resemblance relations. When we considered
this problem, we realized that the model of MRL was not able to deal appropriately with themultidimensionality of this type of problem. The aim of this talk is to revise MRL so thatthe revised first-order modal resemblance logic RMRL can solve Rodriguez-Pereyra-Yi Problem
in terms of measurement-theoretic multidimensional representation (cf. Suppes et al. (1989)). Measurement theory makes it possible that qualitative resemblance relations can represent
quantitative (numerical) functions, whereas it is not designed to explicate the parthood between
a particular and its parts (referred to for determining the raking on a resemblance relation).
So, in the construction of the multidimensional model of RMRL, we would like to connect
measurement-theory with mereology (cf. Varzi (2019)) that can explicate the parthood between
a particular and its parts referred to for determining the raking on a resemblance relation.
The punch line of Resemblance Nominalism is the reducibility of universals into resemblance
relations. The point of formalizing Resemblance Nominalism in RMRL is to avoid the circularity
in this reduction into which it tends to slide. In this talk, we try to give a solution to RodriguezPereyra-Yi Problem by defining in RMRL the degree of unresemblance (Definition 9 ), insteadof using Definition 1 (on which both (2) and (3) are based) that is the main culprit of thisproblem so that, in the multidimensional comparison of unresemblance of (1), the weighted
sum of the degrees of unresemblance of carmine particulars to triangular particulars may begreater than that of carmine particulars to vermillion particulars. In so doing, RMRL obtains thecapacity to deal with multidimensionality in general beyond Rodriguez-Pereyra-Yi Problem. In
the semantics of RMRL, a resemblance relation is primitive and the degree of unresemblance isdefined in Definition 9 by it via Representation Theorem (Theorem 1) and Uniqueness Theorem(Theorem 2).
2. Measurement Theory Meets Meleology in RMRL
We define the language L of revised first-order modal resemblance logic RMRL:
Definition 2 (Language) Let V denote a class of individual variables, C a class of individual constants, and P a class of one-place predicate symbols. Let ⩽F denote a four-placeresemblance predicate symbol indexed by F . When n ≥ 2, let ⩽F1 +⋯+Fn denote a four-placeresemblance predicate symbol indexed by F1 , . . . , Fn . The language L of RMRL is given by thefollowing BNF grammar:t ∶∶= x ∣ a
ϕ ∶∶= F (t) ∣ t1 = t2 ∣  ∣ ¬ϕ ∣ ϕ ∧ ψ ∣ (t1 , t2 ) ⩽F (t3 , t4 ) ∣ (t1 , t2 ) ⩽F1 +⋯+Fn (t3 , t4 ) ∣ ◻ ϕ ∣ ∀xϕ,
where x ∈ V , a ∈ C , and F1 , . . . , Fn ∈ P. ⊺, ∨, →, ↔, <F , <F1 +⋯+Fn , ◇ and ∃ are introducedby the standard definitions. (t1 , t2 ) ⩽F (t3 , t4 ) means that t3 does not resemble t4 more than t1
resembles t2 with respect to F -ness. When n ≥ 2, (t1 , t2 ) ⩽F1 +⋯+Fn (t3 , t4 ) means that t3 does
not resemble t4 more than t1 resembles t2 with respect to F1 -ness and . . . and Fn -ness. The setof all well-formed formulae of L is denoted by ΦL .
Remark 1 (Modal Part of RMRL) The modal part of RMRL is necessary only to solve (vi)Contingent Coextension and (v) Necessary Coextension Problems, and so does not directly relate
to multidimensionality that is the main topic of this talk.
Here we would like to introduce such measurement-theoretic concepts as scale types, representation and uniqueness theorems, and measurement types on which the argument of this talkis based. First, we classify scale types in terms of the class of admissible transformations ϕ.
When ϕ is an positive affine transformation, we call a corresponding scale an interval scale.
Second, we state about representation and uniqueness theorems that give a solution to two48

main problems with measurement theory: the representation problem: justifying the assignment of numbers to objects, and the uniqueness problem: specifying the transformation up
to which this assignment is unique, respectively. Third, we classify measurement types. We
call the representation (d1 , d2 ) ≾ (d3 , d4 ) iff ∣f (d1 ) − f (d2 )∣ ≤ ∣f (d3 ) − f (d4 )∣, for any object
d1 , d2 , d3 , d4 , absolute difference measurement. By using some measurement-theoretic conceptsof Krantz et al. (1971) and Suppes et al. (1989), we prepare the following six steps to constructa model M of RMRL: First Step: a step to prepare an absolute difference structure for thesemantics of ⩽F and ⩽F1 +⋯+Fn . We resort to an absolute difference structure in order to solve
the problems of Resemblance Nominalism. Krantz et al. (1971, pp.172-173) define an absolutedifference structure:
Definition 3 (Absolute Difference Structure) Suppose D is a nonempty class of objectsand ≾ a quaternary relation on D. (D, ≾) is an absolute difference structure iff the following
six axioms are satisfied: Weak Order, Absoluteness, Betweenness, Weak Monotonicity, Solvability, and Archimedean Property defined in Krantz et al. (1971, pp.172-173) (Due to space
limitations, we cannot describe the details.).
We define a property class as a maximal resemblance class in terms of resemblance relation andabsolute difference structure:
Definition 4 (Property Class) A ⊊ D is a property class iff (D, ≾) is an absolute difference
structure and for any d1 , d2 , d3 ∈ A and for any d4 ∈ A, (d1 , d2 ) ≺ (d3 , d4 ) ( Maximality), where(d1 , d2 ) ≺ (d3 , d4 ) ∶= (d3 , d4 ) ≾/ (d1 , d2 ), for any d1 , d2 , d3 , d4 ∈ D.
Second Step: a step to prepare a basic multidimensional structure for ⩽F1 +⋯+Fn . We preparea basic multidimensional comparison structure, called a factorial proximity structure:
Definition 5 (Factorial Proximity Structure) (D, ≾) is a proximity structure iff the following conditions are satisfied for any d1 , d2 ∈ D: ≾ is a weak order. (d1 , d1 ) ≺ (d1 , d2 ) wheneverd1 ≠ d2 . (d1 , d1 ) ∼ (d2 , d2 ) ( Minimality). (d1 , d2 ) ∼ (d2 , d1 ) ( Symmetricity). The structure iscalled factorial iff D ∶= ⨉ni=1 Di , where for any di ∈ Di (1 ≤ i ≤ n), an n-tuple d1 ⋯dn ∈ D.
Third Step: In order to make each dimensional factor the absolute value of a scale difference,a factorial proximity structure (D, ≾) should satisfy Betweenness, Restricted Solvability, and
the Archimedean Property defined in Suppes et al. (1989, pp.180-181). Fourth Step: Inorder to represent the sum of dimensional factors, a factorial proximity structure (D, ≾) should
satisfy Independence and the Thomsen Condition only for the dimensionality n = 2.
Definition 6 (Additive Difference Factorial Proximity Structure) When n ≥ 2 and thefactorial proximity structure (D(∶= ⨉ni=1 Di ), ≾) satisfies Betweenness, Restricted Solvability,
the Archimedean Property, Independence, and the Thomsen Condition only for n = 2 defined in
Suppes et al. (1989, pp.180-182), we call it an additive difference factorial proximity structure.
Fifth Step: The ontological status of an n-tuple d1 ⋯dn ∈ D in Definition 5 is not clear. So
in order to describe the parthood between a particular and its parts referred to for determining
the raking on a resemblance relation, we would like to introduce mereology:
Definition 7 (Mereology) A mereological parthood function P (Varzi (2019, p.14)) is afunction from D × D to ℘(D 2 ) satisfying the following properties: For any d ∈ D, P (d, d)( Reflexivity). For any d1 , d2 , d3 ∈ D, if P (d1 , d2 ) and P (d2 , d3 ), then P (d1 , d3 ) ( Transitivity).
For any d1 , d2 ∈ D, if P (d1 , d2 ) and P (d2 , d1 ), then d1 equals d2 ( Antisymmetry). For any
d1 , d2 ∈ D, a mereological proper parthood function P P (d1 , d2 ) is such a function that P (d1 , d2 )and d1 does not equal d2 . For any d1 , d2 ∈ D, a mereological overlap function O(d1 , d2 ) is sucha function that there exists d3 ∈ D such that P (d3 , d1 ) and P (d3 , d2 ). For any d1 , d2 ∈ D, if
P P (d1 , d2 ), then there exists d3 ∈ D such that P (d3 , d2 ) and not O(d3 , d1 ) ( Supplementation)(Varzi (2019, pp.51-52)). For any d1 , d2 , d3 ∈ D, a mereological product function S(d3 , d1 , d2 )is such a function that P (d4 , d3 ) iff P (d4 , d1 ) and P (d4 , d2 ), for any d4 ∈ D. For any d1 , d2 ∈ D,then there exists d3 ∈ D such that S(d3 , d1 , d2 ) ( Product Existence). For any d1 , d2 ∈ D, wedefine d1 ⊗ d2 as the uniquely existential object bearing the relation S with d1 and d2 , in symbols,d3 S(d3 , d1 , d2 ) (Varzi (2019, pp. 51-52).
ι

49

Final Step: By connecting measurement-theoretic concepts with mereological concepts, wedefine a mereorogical additive difference factorial proximity structured model M of RMRL:
Definition 8 (Mereorogical Additive Difference Factorial Proximity Structured Model)
The mereological additive difference factorial proximity structured frame of RMRL is a structure F ∶= (W , R, D, {≾F }F ∈P , {D≾F }F ∈P , P, {≾F1 +⋯+Fn }F1 ...,Fn ∈P ), where W is a non-empty
class of worlds, R a binary accessibility relation on W , D a non-empty class of particulars,{≾F }F ∈P a non-empty class of such quaternary relations ≾F on D that (D, ≾F ) is an absolutedifference structure and ≾F satisfies Maximality of Definition 4, {D≾F }F ∈P a non-empty classof D≾F which is a non-empty class of the parts of particulars referred to for determining the
ranking on ≾F and which postulates that there exists a unique object being a part of a particularbelonging to D, P a mereological parthood function from (D ∪ ⋃F ∈P D≾F ) × (D ∪ ⋃F ∈P D≾F )to ℘((D ∪ ⋃F ∈P D≾F )2 ), {≾F1 +⋯+Fn }F1 ...,Fn ∈P a non-empty class of such quaternary relations≾F1 +⋯+Fn on DF1 +⋯+Fn defined as a non-empty class of such ⊗ni=1 di where di ∈ D≾Fi (1 ≤ i ≤ n)that (DF1 +⋯+Fn , ≾F1 +⋯+Fn ) is an additive difference factorial proximity structure. A function
I is an interpretation of F if I assigns to each a ∈ C and each w ∈ W some object that isa member of D that satisfies Transworld Identity: for any w, w′ , I(a, w) = I(a, w′ ), andassigns to each four-place resemblance predicate symbol ⩽F and each w ∈ W such a quaternary
relation ≾F , and assigns to each four-place resemblance predicate symbol ⩽F1 +⋯+Fn and eachw ∈ W such a quaternary relation ≾∗F1 +⋯+Fn that it is defined as follows: when, for any
d1 , d2 , d3 , d4 ∈ D, di+4 P (di+4 , d1 ), di+5 P (di+5 , d2 ), di+6 P (di+6 , d3 ), di+7 P (di+7 , d4 ) ∈ DFi aresuch uniquely existential objects that they satisfy P ( , d1 ), P ( , d2 ),P ( , d3 ),P( ,d4 )respectively, (d1 , d2 ) ≾∗F1 +⋯+Fn (d3 , d4 ) iff ⊗ni=1 di+4 P (di+4 , d1 ), ⊗ni=1 di+5 P (di+5 , d2 )) ≾F1 +⋯+Fn(⊗ni=1 di+6 P (di+6 , d3 ), ⊗ni=1 di+7 P (di+7 , d4 )). By Definition 4, a property class I(F, w) isdefined as a maximal resemblance class in terms of a resemblance relation ≾F : A ⊊ D
is a property class I(F, w) iff (D, ≾F ) is an absolute difference structure and for any
I(a, w), I(b, w), I(c, w) ∈ A and for any I(d, w) ∈ A, (I(a, w), I(b, w)) ≺F (I(c, w), I(d, w)).
The mereological additive difference factorial proximity structured model of RMRL is a structure
M ∶= (W , R, D, {≾F }F ∈P , {D≾F }F ∈P , P, {≾F1 +⋯+Fn }F1 ...,Fn ∈P , I).
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Remark 2 (Not Absoluteness But Conditionality) The mereological additive difference
factorial proximity structured model of RMRL does not require that ≾F should absolutely satisfy
such conditions above as Betweeness and the Archimedean Property and so on, but requires that
if ≾F satisfies them, then Theorems 1 and 2 below can be proven.
Remark 3 (Nominalism about Universals) M is nominalistic both about such universals
as properties and about ≾F and R neither of which are reified, whereas it is Platonistic
about such abstract objects as classes, functions, numbers and possible worlds. As RodriguezPereyra (2015) observes, Realism/Nominalism about universals is independent of Platonism/Nominalism about abstract objects.
Remark 4 (Non-Circularity of Resemblance Relation) Since a resemblance relation ≾Fdepends not on a property class I(F, w) given by ≾F but on a predicate symbol F , where
I(F, w) is the semantic value of F . In this sense, ≾F is not circular.
Remark 5 (Reducibility and Resemblance Nominalism) M
nominalistic in the sense that I(F, w) is reducible to ≾F .

isresemblance-

When an assignment s to x ∈ V is defined in such a way that the extended assignment s̃ agrees
with I in the assignment to a ∈ C , we can provide RMRL with the satisfaction definition relative
to M (i.e., (M, w) ⊧ ϕ[s]), define the truth at a world in M (i.e., (M, w) ⊧ ϕ) by means ofsatisfaction, and then define validity (i.e., ⊧ ϕ) (Due to space limitations, we cannot describe thedetails.). By Definition 8, we can prove the following representation and uniqueness theoremsfor ≾F1 +⋯+Fn by modifying the proof method of Theorem 8 of Suppes et al. (1989, p.185):
Theorem 1 (Representation) If (W , R, D, {≾F }F ∈P , {D≾F }F ∈P , P, {≾F1 +⋯+Fn }F1 ...,Fn ∈P , I) isa mereological additive difference factorial proximity structured model of RMRL, then there exist
functions f≾Fi (1 ≤ i ≤ n) ∶ D≾Fi → R≥0 and monotonically increasing functions g≾Fi (1 ≤ i ≤ n) ∶R≥0 → R≥0 such that for any d1 , d2 , d3 , d4 ∈ D,(d1 , d2 ) ≾∗F1 +⋯+Fn (d3 , d4 )50niff (4)−f≾Fi ( di+5 P (di+5 , d2 ))∣)
∑i=1 g≾Fi (∣f≾Fi ( di+4 P (di+4 , d1 ))
n
g
(∣f
(
d
P
(d
,
d
))
−
f
(
d
P
∑i=1 ≾Fi ≾Fi i+6
i+6 3
≾Fi
i+7 (di+7 , d4 ))∣),
di+4 P (di+4 , d1 ), di+5 P (di+5 , d2 ), di+6 P (di+6 , d3 ), di+7 P (di+7 , d4 ) ∈ D≾Fi .

≤
where

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Theorem 2 (Uniqueness) The f≾Fi are interval scales and the g≾Fi are interval scales witha common unit.
We define the degree of unresemblance and its weight in terms of Theorems 1 and 2:
Definition 9 (Degree of Unresemblance and Its Weight) For any d1 , d2 ∈ D, the degreeof unresemblance with respect to ≾Fi is defined by ∣f≾Fi ( di+4 P (di+4 , d1 ))−f≾Fi (( di+5 P (di+5 , d2 ))∣of (4), and its weight is defined by g≾Fi of (4), where the existence and uniqueness of f≾Fi andg≾Fi are guaranteed by Theorems 1and 2 respectively.
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3. Conclusion
Suppose that Cx ∶= x is carmine, V x ∶= x is vermillion, T x ∶= x is triangular, and (x, y) <C+V +T(z, w) ∶= x resembles y more than z resembles w with respect to carminity and vermillionity andtriangularity. Then the RMRL-logical form of (1) is ∀x∀y∀z((Cx ∧ V y ∧ T z) → (x, y) <C+V +T(x, z)). Its semantic value (satisfaction condition) is given by the following proposition that
follows from Theorem 1:
Proposition 1 (Solution to Rodriguez-Pereyra-Yi Problem by RMRL)(M, w) ⊧ ∀x∀y∀z((Cx ∧ V y ∧ T z) → (x, y) <C+V +T (x, z))[s]iff there is no d1 , d2 , d3 ∈ D such that d1 ∈ I(C, w) and d2 ∈ I(V, w) and d3 ∈ I(T, w) and suchthat there exist f≾C ∶ D≾C → R≥0 and f≾V ∶ D≾V → R≥0 and f≾T ∶ D≾T → R≥0 and g≾C , g≾V , g≾T ∶R≥0 → R≥0 such that (g≾C (∣f≾C ( d4 P (d4 , d1 )) − f≾C ( d5 P (d5 , d2 ))∣) + g≾V (∣f≾V ( d6 P (d6 , d1 )) −f≾V ( d7 P (d7 , d2 ))∣) + g≾T (∣f≾T ( d8 P (d8 , d1 )) − f≾T ( d9 P (d9 , d2 ))∣)) ≥ (g≾C (∣f≾C ( d4 P (d4 , d1 )) −f≾C ( d10 P (d10 , d3 ))∣) + g≾V (∣f≾V ( d6 P (d6 , d1 )) − f≾V ( d11 P (d11 , d3 ))∣) + g≾T (∣f≾R ( d8 P (d8 , d1 )) −f≾T ( d12 P (d12 , d3 ))∣)).
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We have the following conclusion: When we choose as the weight-assignment functions
such functions g≾C , g≾V , g≾T that the value of g≾T is much greater than those of g≾C andg≾V , Proposition 1 can give a solution to Rodriguez-Pereyra-Yi Problem by Definition 9 interms of giving the satisfaction condition of (1) in RMRL so that the weighted sum of thedegrees of unresemblance of carmine particulars to triangular particulars may be greaterthan that of carmine particulars to vermillion particulars, instead of using Definition 1 thatis the main culprit of this problem. In so doing, RMRL obtains the capacity to deal withmultidimensionality in general beyond Rodriguez-Pereyra-Yi Problem.
Acknowledgements
The author would like to thank two anonymous referees of LENLS19 for their very helpfulcomments.
References[1]Krantz, D.H. et al.: Foundations of Measurement, vol. 1. (1971). [2] Rodriguez-Pereyra, G.:
Resemblance Nominalism (2002). [3] Rodriguez-Pereyra, G.: Nominalism in Metaphysics, Stanford Encyclopedia of Philosophy (2015). [4] Rodriguez-Pereyra, G.: Resemblance Nominalism
and Abstract Nouns. Analysis 75, 223-231 (2015). [5] Suppes, P., et al.: Foundations of Measurement, vol. 2. (1989). [6] Suzuki, S.: Measurement-Theoretic Foundations of Resemblance
Nominalism. Proceedings of LENLS 17 (2020). [7] Yi, B.U.: Abstract Nouns and Resemblance Nominalism. Analysis 74, 622-629 (2014). [8] Yi, B.U.: Nominalism and Comparative
Similarity. Erkenntnis 83, 793-803 (2018). [9] Varzi, A.: Mereology, Stanford Encyclopedia of
Philosophy (2019).

51

Extraction pathway marking as proof structure marking
Yusuke Kubota (NINJAL) and Robert Levine (Ohio State University)
Data Extraction pathway marking is a phenomenon, exhibited by typologically diverse languages including Chamorro (Chung 1982), French (Kayne and Pollock 1978), Icelandic (Zaenen 1983) and Irish (McCloskey 1979), whereby the ‘movement pathway’ that establishes thelinkage between the filler and the gap site is marked morpho-syntactically. The relevant data canbe illustrated most clearly by the choice of different complementizers in Irish reported in McCloskey (1979). For expository convenience, we illustrate the relevant empirical patterns hereby a pseudo-language called Iringlish, which is like Irish in having the relevant distinction of
two complementizers but otherwise identical to English in all other respects.
In Iringlish, the complementizer goN is used when no extraction takes place:(1)

I thought goN [he would be there].

(2)

I said goN [I thought goN [he would be there]].

Iringlish has another form of complementizer aL, which is restricted to clauses that contain an
undischarged gap site inside itself (possibly in an embedded position):(3)a. the man aL [would be there]b. the man aL [I thought aL [ would be there]]c. the man aL [I said aL [I thought aL [ would be there]]]Note that, as shown in (4), a substructure inside a relative clause which itself doesn’t contain an
extraction pathway is marked by goN:(4)

the man aL [thought goN [he would be there]](4) shows that the aL/goN distinction cannot be analyzed as a surface constraint on complementizer choice which merely indicates whether the environment in question is ‘inside’ or ‘outside’of a relative clause. Rather, the complementizer reflects the existence of an extraction gap inside
the clause it marks.
In Iringlish, extraction pathway marking needs to be consistently encoded in all clause boundaries that are part of the extraction pathway. Thus, the following two examples are ungrammatical:(5)a. *the man aL [I said aL [I thought goN [b. *the man goN [I said aL [I thought aL [would be there]]]would be there]]]In (5a), the lowest clause is marked by goN despite containing a gap. In (5b), the highest clausein the relative clause still contains an unbound gap to be matched with the filler (i.e., the headnoun in the relative clause). In both cases, the right form is aL, and the string is ungrammatical.
Problem In Type-Logical Grammar, the linkage between the gap and the filler in long-distancedependencies is mediated by a general inference rule of hypothetical reasoning. In particular, inHybrid Type-Logical Grammar (Hybrid TLG), which exploits prosodic lambda binding due toOehrle (1994) in modelling both overt and covert ‘movement’ phenomena, extraction is modelled
by a single chain of hypothetical reasoning. Specifically, Kubota and Levine (2020) utilize an
operator which embeds an empty string in the gap position for the analysis of overt movement,
building on an idea suggested in Muskens (2003). In this approach, (6) is analyzed as in (7).

52

(6)

The guy who John thinks Mary said Bill criticized

(7)

guy;
guy;
N

.

[ϕ0 ; x; NP]1
..
.
said;
bill ◦ criticized ◦ ϕ0 ;say;criticize(x)(b); S
VP/S
mary;
said ◦ bill ◦ criticized ◦ ϕ0 ;m;say(criticize(x)(b)); VP
NP
thinks;mary ◦ said ◦ bill ◦ criticized ◦ ϕ0 ;think;
say(criticize(x)(b))(m); S
VP/S
john;
thinks ◦ mary ◦ said ◦ bill ◦ criticized ◦ ϕ0 ;j;think(say(criticize(x)(b))(m)); VP
NP
john ◦ thinks ◦ mary ◦ said ◦ bill ◦ criticized ◦ ϕ0 ;λσ.who ◦ σ(�);think(say(criticize(x)(b))(m))(j); S
λP λQλu.
1�1→
�I
λϕ0 .john ◦ thinks ◦ mary ◦ said ◦ bill ◦ criticized ◦ ϕ0 ;Q(u) ∧ P (u);λx.think(say(criticize(x)(b))(m))(j); S�NP
(N\N)�(S�NP)
who ◦ john ◦ thinks ◦ mary ◦ said ◦ bill ◦ criticized ◦ �;λQλQ(x) ∧ think(say(criticize(x)(b))(m))(j); N\Nguy ◦ who ◦ john ◦ thinks ◦ mary ◦ said ◦ bill ◦ criticized ◦ �;
λx.guy(x) ∧ think(say(criticize(x)(b))(m))(j); N

Here, a relative clause is analyzed as a sentence missing an NP type gap, of type S�NP (step �
1 ),which is then given as an argument to the relative pronoun that has a higher-order semantics andprosody of the appropriate type to form a noun modifier of type N\N. The process that derives
a gapped sentence of type S�NP is mediated by a general rule of hypothetical reasoning for thevertical slash �. A hypothesis of type NP is posited in the object position of the most deeply
embedded verb, and this hypothesis is withdrawn at step �
1 to yield S�NP.
Crucially, in this analysis, the identification between the filler and the gap is mediated by asingle chain of hypothetical reasoning. But this type of analysis seems to face a major challengein view of the extraction pathway marking data reviewed above, since intermediate steps ofderivation do not reflect the fact that extraction has ‘taken place’, unless one can directly inspect
the structure of the proof tree. The two intermediate clauses are just plain S (and not S�NP)
in this analysis, so, there is no way for the complementizer to ‘know’ that they contain a gap.
The standard view in CG research dictates that derivations of linguistic signs just reflect thehistory of proof (of well-formedness of the derived sign), and that such proof histories are notreal reified ‘objects’ that rules and constraints of the grammar can directly make reference to.
But then, there is no obvious locus for the identification of the notion ‘connectivity pathway’. In
fact, considerations of just this sort have led Kubota and Levine (2020) to propose an analysis
which essentially simulates the ‘cyclic movement’ analysis in Transformational Grammar and
the Principles and Parameters (P&P) framework (Kayne and Pollock 1978, among others) via achain of hypothetical reasoning steps within a type-logical setup.
Solution We call into question the ban on making reference to structures of proofs, and propose
a new analysis of extraction pathway marking in Hybrid TLG. The idea that derivation trees are
not linguistic representations seems to stem from the tradition of Montague Grammar (where
the idea that analysis trees do not have a similar status as syntactic trees in transformational
grammar was repeated emphasized), and it was incorporated into the linguistic foundations ofType-Logical Grammar essentially as an unquestionable (or rather, unquestioned) given.
But once one starts looking outside of linguistics, one immediately notices that studying thestructures of proofs formally is one of the central topics in the proof-theoretic/type-theoretic

53

studies of formal logic including the Lambek calculus. In what follows, we show that an elegant
and straightforward analysis of the extraction pathway marking patterns immediately becomes
available once we incorporate a modest amount of technology from this literature into the underlying formal system of Hybrid TLG.
To facilitate the ensuing discussion, we introduce here an alternative notation of linguistic derivations in which a derivation/proof can be written as a proof term, building on the socalled Curry-Howard Isomorphism (Howard 1969). Essentially, one can establish a one-to-one
correspondence between proofs and lambda terms by taking elimination steps to correspondto function application and introduction steps to correspond to lambda abstraction. Formally,
this syntactic lambda calculus distinguishes three types of function application (app/ , app\ , andapp� ), and three types of lambda abstraction (λ/ , λ\ , and λ� ), corresponding to the three slashes
in the underlying logic of Hybrid TLG.
As an illustration, consider the derivation (9) for (8).
(8)

John read every book.

(9)1
λϕ2 λσ.σ(every ◦ ϕ2 );book;john; read; read; TV [ϕ1 ; x; NP] /E
λP. P ;
book;read ◦ ϕ1 ; read(x); VP
j; NPS�(S�NP)�N
N\E
�E
john ◦ read ◦ ϕ1 ; read(x)(j); S
λσ.σ(every ◦ book);
�I1
λϕ1 .john ◦ read ◦ ϕ1 ; λx.read(x)(j); S�NP
book ; S�(S�NP)
�E
john ◦ read ◦ every ◦ book; book (λx.read(x)(j)); S

A

A

A

The derivation in (9) can be rewritten as a proof term as in (10), with the lexicon in (11).¹(10) EVERYS�(S�NP)�N (BOOKN )(λ� x.READTV (xNP )(JOHNNP ))READTV = read; read; TV
EVERYS�(S�NP)�N = λϕ2 λσ.σ(every ◦ ϕ2 ); λP.
JOHNNP = john; j; NP
BOOKN = book; book; N

P ; S�(S�NP)�N

A

(11)

Note that Slash Elimination in (9) corresponds to function application in (10) and Slash Introduction in (9) corresponds to lambda abstraction in (10) (the types of application are not specified
in (10) since they are all uniquely recoverable given the lexicon in (11)).
The proof term notation of derivations introduced above enables a concise formulation of
the extraction pathway marking patterns exhibited by the Iringlish (or Irish) reviewed above. We
illustrate this point with a fragment of Iringlish with the lexicon in (12).², ³¹Note that here, the variable xNP is a variable in the syntactic logic and is thus unrelated to the x in the semantic
component of the hypothesis in (9); we use the same variable letter only for expository ease.
²An anonymous reviewer for LENLS 2022 reminds us that the notion of ‘only defined in’ in (12) requires somecomment. As noted by the reviewer, in a proof system, the well-formedness/typability of terms is usually defined
via the typing rules that refer to the recursively specified formation rules for terms, and the side condition in (12)is not statable in terms of such typability conditions (which only look at local structures of terms). What we intend
by the ‘only defined in’ clause in (12) is that, on top of the usual recursive specification of the purely formal wellformedness/typability conditions of proof terms in the logic, there is a separate filtering component for linguisticallywell-formed terms which inspects the forms of proof terms directly to determine whether they correspond to actual
linguistic signs in the language. The point here is that the analysis of the phenomenon becomes much simpler if we
assume this additional filtering component (whose technical formulation is trivial) rather than to encode its effect
explicitly in the recursive typing definitions of terms.
³We assume that pronouns like I in (12) are (indexical) variables in the semantic component, but that at the levelof syntactic proof terms they are constants. Thus, the presence of the pronoun in the embedded clauses in (13) does
not affect the choice of the complementizers aL and goN.

54

(12) WBTNP\S = would ◦ be ◦ there; λx.exist(x, there); NP\SMANN = man; man; N
INP = i; zsp ; NP
THOUGHT(NP\S)/S� = thought; thought; (NP\S)/S�SAID(NP\S)/S� = said; said; (NP\S)/S�
ALS�/S = aL; λp.p; S� /S
where for any α, AL(α) is defined only if fvX+wh (α) �= ∅�GONS�/S = goN; λp.p; S /S
where for any α, GON(α) is defined only if fvX+wh (α) = ∅REL(N\N)�(S��NP+wh) = λσ2 λϕ2 .ϕ2 ◦ σ2 (�); λP λQλy.Q(y) ∧ P (y); (N\N)�(S� �NP+wh )The key components of this analysis are the restrictions imposed on aL and goN that refer to thestructures of the terms given as their (first) arguments. fvΦ is the standard, inductively defined
function that returns all free variables contained in a term, except that it filters the output of thegeneral purpose fv to type Φ.
The derivation for (13) with the lexicon in (12) goes as in (14).
(13)

the man aL [I said aL [I thought aL [(14)

REL(N\N)�(S��NP+wh) (λ� x.ALS�/S (SAID(NP\S)/S�

would be there]]](ALS�/S (THOUGHT(NP\S)/S�(ALS�/S (WBTNP\S (xNP+wh )))(INP )))(INP )))Note that each token of aL applies to a clausal complement containing a free NP+wh variable.
(15)

a. ALS�/S (WBTNP\S (xNP+wh ))b. ALS�/S (THOUGHT(NP\S)/S� (ALS�/S (WBTNP\S (xNP+wh )))(INP ))c. ALS�/S (SAID(NP\S)/S� (ALS�/S (THOUGHT(NP\S)/S� (ALS�/S (WBTNP\S (xNP+wh )))(INP )))(INP ))In each of these cases, fvX+wh (α), where α is the underlined term, returns {xNP+wh }.
The ungrammaticality of the examples in (5) also follows immediately in this approach. In
the case of (5a), goN is used instead of aL in the subproof corresponding to (15a). This violates
the constraint fvX+wh (α) = ∅ on the first argument of goN. Similarly, in (5b), goN replaces the firstaL in the subproof corresponding to (15c). Here again, the relevant ‘no unbound +wh hypothesis’constraint on goN is violated.⁴
It is important to note that extraction and apparently similar phenomena such as Right-nodeRaising are distinct, the former treated via the vertical slash � in Hybrid TLG and the latter via
the Lambek slash /. Support for such an assumption comes from the complementizer choices
in Irish. To see this point, note the following examples from McCloskey (2011):(16)

a. Níor chualas gur
leag
nó gurmharaigh na tramannanior heard [S1] C-[PAST] knock-down or C-[PAST] kill
the trams
duine ar bith ariamh.
person any ever

⁴A reviewer wonders whether an alternative, simpler analysis would be possible in which clauses containing
a gap are distinguished from clauses that don’t contain a gap in terms of the syntactic category. For example, theformer could be S�NP and the latter could be S. This doesn’t work in Type-Logical Grammar, since nothing forcesintermediate S’s to be analyzed as S�NP In which a long-distant linkage between the gap and the filler is mediatedby a single instance of hypothetical reasoning as in (7). In order to make it obligatory that all intermediate gapcontaining S’s are analyzed as S�NP, some sort of additional constraint needs to be imposed. Kubota and Levine(2020) indeed opt for such a solution, using the clause-level indexing mechanism due to Pogodalla and Pompigne(2012) to simulate the ‘cyclic movement’ analysis in the P&P literature. A detailed comparison of this analysis andthe present proposal which allows direct access to the structures of proof terms is beyond the scope of the present
paper.

55

‘I never heard that the trams ever knocked down or killed anyone.’b. B’facthas dom go mbíodh, agus go bhfuil fós, na Beanna Arda mar phluid
seemed to-me that used-to-be and that are still the cliffs high like blanket
mhór thart orm.
great around on-me‘It seemed to me that the Great Cliffs had been and were still like a great blanket
around me.’
The two conjoined clauses are of type S/(TV\S) in (16a) (and similarly for (16b)). Crucially,such expressions do not contain a free variable corresponding to the Right-node Raised expression in the syntactic proof term, since the Right-node Raised expression in question is alreadybound by /. Thus, our approach, together with independently motivated assumptions about
Right-node Raising (see, e.g., Kubota and Levine (2020)), correctly predicts the occurrence of
goN instead of aL in (16).
Discussion Our proposal boils down to the claim that extraction pathway marking falls out
immediately as a linguistic marking of proof structure once we make one tiny step into thedomain of proof theory. This of course comes at the price of breaking the dogma that proofs are
not representational objects. But this move is worth making, for both technical and conceptual
reasons. Technically, the step we have made is literally a tiny step, as it merely involves making
reference to the notion of free variables in a typed lambda calculus, something that is alreadyneeded in formally interpreting lambda terms (for example, in the semantic domain).
Conceptually, our analysis of extraction pathway marking is arguably the simplest one amongits competitors, but to see this point, we need to compare it with its major alternatives. Outside ofcategorial grammar, the consensus in the syntactic literature is that extraction pathway marking
motivates a particular type of analysis of extraction in which the linkage between the filler andthe gap is mediated by a chain of local dependencies: cyclic movement in P&P and the local
inheritance of the SLASH feature in HPSG. These approaches have in common the property thatthey both explicitly encode the local chain of extraction pathway via some dedicated theoreticalmechanism (such as a chain of locally restricted movement operations in P&P and the feature
inheritance mechanism for the set-valued SLASH feature in HPSG).
Our analysis captures the same empirical patterns as these alternatives, but it differs fromthese more familiar proposals in that it involves nothing more than the independently needed
formal modelling of gap inheritance via hypothetical reasoning. We think that the picture thatemerges from this reconceptualization of extraction pathway marking is illuminating. Under
this proof-theoretic perspective on the syntax of natural language, the phenomenon of extraction
pathway marking is something that is naturally expected to be available in natural language, by
making absolutely minimum assumptions about the nature of the extraction phenomenon itself.
References Chung, S. 1982. Unbounded dependencies in chamorro grammar. LI 13:39–77. //Howard, William A. 1969. The formulae-as-types notion of construction. // Kayne, R. and J.-Y.
Pollock. 1978. Sylistic inversion, successive cyclicity and Move NP in French. LI 9:595–621.
// Kubota, Y. and R. Levine. 2020. Type-Logical Syntax. MIT Press. // McCloskey, J. 1979.
Transformational Syntax and Model-Theoretic Semantics. Reidel. // McCloskey, J. 2011. The
shape of Irish clauses. In A. Carnie, ed., Formal Approaches to Celtic Linguistics, 143–178.
Cambridge Scholars Publishing. // Muskens, R. 2003. Language, lambdas, and logic. // Oehrle,
R. 1994. Term-labeled categorial type systems. L&P 17(6):633–678. // Zaenen, A. 1983. On
syntactic binding. LI 14(3):469–504.

56

Constraining parse ambiguity with grammatical codes
Alastair Butler
Hirosaki University

1

Introduction

This paper introduces a wide coverage “toolkit” for obtaining full parse analysis for English
sentences. Parse results produced with the toolkit follow the annotation scheme of agrammatically analysed corpus, the Treebank Semantics Parsed Corpus (TSPC; Butler
2022). The primary goal for the toolkit has been to empower students who are medium toadvanced learners of English to experience techniques of language analysis, and so acquire
skills e.g., relevant for exploiting the online TSPC resource with its forty thousand treesof analysed data.
Originally the toolkit involved adding post-processing to statistical and neural basedparser systems. This gave wide coverage and a way to manage ambiguity: rely on a“best” guess. However, this also gave unpredictable parsing errors, requiring students tobe sensitive to miss-analysis. Also, these systems produce only a bare parse: There is nofunction information (e.g., subject), and there are no zero elements (e.g., relative clausetraces). Attempts to add this information with post-processing were error prone in nonobvious ways, a task that essentially requires a full parse to do better! A further issue wasthat use of the original toolkit was simply too passive: There were no opportunities forstudents to influence the parse result with their own ideas about analysis.
These limitations motivated a re-orientation of the toolkit around a logic based grammar approach: The underlying automatic parser is now a Definite Clause Grammar (DCG;
Pereira and Warren 1980) of the XSB Tabling Prolog system (Swift and Warren 2022).
A DCG consists of a Prolog program for parsing content given with a difference list as
input. A DCG is written as phrase structure rules, possibly enhanced with extra logicalparameters that can be used to accumulate structure from the parse or pass on other kindsof unifiable values such as selection criteria and records of long distance dependencies. The
XSB implementation is particularly notable for allowing phrase structure rules to include
left recursion without infinite loops arising from rule evaluation, achieved by remembering
what was already evaluated with a technique called tabling.
The availability of this parsing engine creates new challenges: to increase grammar
coverage for unconstrained English input, and more crucially to deal with proliferations ofparse ambiguity.
Classroom experience suggests creation of word analysis is something students can dowell. In contrast, creation of full parse structure is a major challenge. Partly this is a matterof familiarity: Students are already well drilled into identifying nouns, verbs, adjectives,and adverbs. But familiarity aside, creation of a full parse can remain a hard task that
depends on a comprehensive understanding of language grammar and word interactions.
There are also practical issues of dealing with a format for full parse representation.
The idea that is the focus of this paper is to unite the strengths of both human word
analysis and an automatic system that creates structure from disambiguated word information. This paper is concerned with the word information that should be provided andthe impacts of this information on subsequent depictions of derived analysis.

57

2

Utterance layers

To give an idea of how grammar rules are written, let’s consider the utterance layer, whichis the topmost structural layer of a parse. Prolog code (1) defines a phrase structure rulethat will create a structure with IP-MAT (declarative matrix clause) as the topmost node.
(1)
utterance(Id,n(Id,IP,none)) -->clause_top_layer(statement,[],IPL-L1),
punc(final,L1-[]),{
IP =.. [’IP-MAT’|IPL]}.

To succeed, (1) needs content to parse from an input list of items where all but the last item
satisfies a call of clause_top_layer with statement word order and no inherited displaced
items ([]), and where the last list item will be an instance of final punctuation (identified
with punc).
In (2), two Prolog calls are made. The first call has tphrase_set_string to establish
a list of items to parse. The second call has parse to question whether the establishedparse list has content to satisfy an utterance rule with ’ex2’ to match Id of (1), while thesecond parameter for accumulating overall parse information is kept hidden internally tothe parse call. If parse succeeds then all parse results are pretty printed as bracketed tree
output.
(2)| ?- tphrase_set_string([’PRO’(’He’),’VBP;˜I’(smiles),’PU’(’.’)]),
parse(utterance(’ex2’)).
( (IP-MAT (NP-SBJ (PRO He))
(VBP;˜I smiles)(PU .))(ID ex2))
yes

The pretty print from (2) shows the return of a structure with IP-MAT as the topmost node,from which it follows that rule (1) completed successfully.
As can be seen from rule (1), tree structure is built layer upon layer at parse time,with difference lists accumulating already built compound terms of the same layer. The
difference list for a given layer is closed by the empty list ([]) and has the layer name added
as the head list item to then be converted to a compound term with the univ operator, asin (3).
(3)| ?- IP =.. [’IP-MAT’,’NP-SBJ’(’PRO’(’He’)),’VBP;˜I’(smiles),’PU’(’.’)].
IP = IP-MAT(NP-SBJ(PRO(He)),VBP;˜I(smiles),PU(.))
yes

58

3

Motivating word analysis and the human touch

A word can only occur in contexts that are compatible for its contribution and this willin turn constrain the context available to other words of the same sentence and in thisway much potential ambiguity is eliminated. Ambiguity surfaces when word analysis is
not sufficiently sensitive to its context of occurrence and/or doesn’t affect the contexts forother words.
Identifying word class significantly constrains potential ambiguity. Thus a noun can beeither (i) the head word of a noun phrase or (ii) the modifier of a same level head word of
a noun phrase. Identifying a word as a finite verb leads to the projection of a clause layer.
With this background and with the word information of (4) (where N=noun, VBD=pasttense lexical verb, and PU=punctuation), we might therefore expect the parse results ofeither (5) or (6).
(4) Word_N word_N word_VBD ._PU
(5)

(IP-MAT (NP-SBJ (N Word)(N word))(VBD word)(PU .))(6)(IP-MAT (NP-109 (N Word))(NP-SBJ (N word))(VBD word)(NP-OB1 *ICH*-109)(PU .))For (5), the first noun is a modifier of the second noun which heads a single noun phrasethat is the clause subject. For (6), each noun is the head of its own noun phrase, withthe first noun phrase a displaced item that is indexed linked to take object function, while
the second noun phrase is the clause subject. Which parse needs to apply hinges on theselection requirements of the verb: if the verb is intransitive, then (5) is valid structure; if
the verb is mono-transitive taking a noun phrase object, then (6) is valid structure.
The extra selection information of the verb can be given as a student oriented task withgrammatical codes: I=intransitive verb, and Tn=mono-transitive verb with noun phrase
object, so that (7) will unambiguously lead to the parse of (5), while (8) will unambiguously
lead to the parse of (6).
(7) Cheese_N pizza_N smells_VBD;˜I ._PU
(8) Cheese_N pizza_N needs_VBD;˜Tn ._PU

4

Word class and grammar codes

The use of word class information and grammatical codes to feed parse analysis leads tothe need for a system of word classes and a system of grammatical codes. The systems
employed are both part of the annotation scheme for the TSPC, which in turn builds onword class analysis from BNC-Consortium (2005), and grammatical codes from Hornby(1975) and Cowie (1989).
The grammar code system is a cashing out of types of verb complementation foundin English sentences, which are in turn associated with word sense definitions in Cowie(1989). For example, among its sense meanings, the verb smell can be a transitive verb
with noun phrase object (code: Tn) with the word sense of (9), or it can be an intransitive
verb and so lack complements (code: I) with the word sense of (10).
(9) [Tn] notice (sth/sb) by using the nose: Do you smell anything unusual?

59

(10) [I] have an unpleasant smell: Your breath smells.
Note how it is only word sense (10) that is compatible with (7) above.

5

Insight beyond the parse tree

While creating parse trees that conform to the TSPC annotation scheme is already revealing of language properties, there is opportunity to go further in regard to obtaining
insight from analysis: The parse trees can be fed to the Treebank Semantics evaluation
system (Butler 2021). This system can process (multiple) constituency tree annotations
as input and return a logic-based meaning representation as output. As a recent development, it is now especially helpful to see created meaning representations as dependency
graphs to make visually apparent connections that the design of the annotation capturesin combination with the Treebank Semantics calculation.
As an example, consider word analysis for the minimal pair of (11) and (12), withdifferences hinging on the grammatical codes assigned to was.
(11) The_D job_N was_BED;˜equ_Vg cleaning_VAG;˜Tn a_D dog_N ._PU(12) The_D boy_N was_BED;˜cat_Vg cleaning_VAG;˜Tn a_D dog_N ._PU
Differences are slightly magnified in terms of the labels of constituents resulting from thecreation of parse trees, seen in (13) and (14), but the differences are hardly dramatic:
Labels aside, the structural bracketing is identical.
(13)
(14)
(IP-MAT (NP-SBJ (D The)
(IP-MAT (NP-SBJ (D The)(N job))(N boy))(BED;˜equ_Vg was)
(BED;˜cat_Vg was)
(IP-PPL-PRD2 (VAG;˜Tn cleaning)(IP-PPL-CAT (VAG;˜Tn cleaning)(NP-OB1 (D a)(NP-OB1 (D a)(N dog)))(N dog)))(PU .))(PU .))

Yet differences are greatly magnified with the dependency analysis of (15) and (16).
(15)

The equ_Vg code marks was as an equative verb and so a main verb to establish equivalenceof the subject (arg0) with the subject predicative (prd2) which has content made up of thepresent participle cleaning and its object (arg1) argument.

60

(16)

The cat_Vg code marks was as a catenative verb that provides past progressive aspect forthe main verb which is the present participle as seen from the subject (arg0) linking.

6

Conclusion

To sum up, this paper proposes an approach for wide coverage parsing of English thatinvolves phrase structure rules for a DCG parsing engine to build up parse informationfrom rich word information. The rich word information is made with markers of word classand grammatical codes. Grammatical codes given to verbs double as partial indicators ofword sense.
The word information can be supplied by students who are learning about grammaticalanalysis. This takes away the mundane crunching tasks of reaching parse analysis whileleaving the task of providing the essential information (word class and grammatical codes)
to determine the directions a parse takes. These are the really hard decisions of parsingthat computers are still not very good at making, but this is exactly the information thathumans excel at giving and are representative of in-depth insight into language competency,
so skills language learners need to master. Also, ambiguity is not eliminated from parse
results by some “best” guess. In fact, all results are returned, only with tree structure
where spurious ambiguity is eliminated.
The use of human supplied codes gives similarity with discriminant-based treebanking(Oepen and Lonning 2006) or “bits of wisdom” (Basile et al. 2012) approaches that include
human supplied constraints to guide wide coverage syntactic/semantic parsing. Arguably,
the grammatical codes given to verbs considered in this paper are of special interest because
they have an extra purpose too: They form information to disambiguate word sense linked
to an existing dictionary resource. That is, the grammatical codes are themselves key
insights into word sense that are independently of value for English language learners toknow.
References
Basile, V., J. Bos, K. Evang, and N.J. Venhuizen. 2012. Developing a large semantically annotated corpus.
In Proceedings of the 8th Int. Conf. on Language Resources and Evaluation.
Butler, Alastair. 2021. Meaning representations from treebanks. The Treebank Semantics Web Site.
http://www.compling.jp/ajb129/ts.html.
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Tech. rep., Stony Brook University.

61

In Search of a Type Theory for Fuzzy Properties
Youyou Cong
Tokyo Institute of Technology

1Introduction

Types are a powerful language for describing properties. Over the past decades, researchers have used
this aspect of types for diverse purposes, ranging from verification of compilers [14, 1], to representation of discourses [21, 2], and to formalization of music theory [24, 7]. These applications are all
built on top of the Curry-Howard isomorphism, which relates types to propositions and programs toproofs.
Typically, whether an object satisfies a property is considered as a binary question. For instance,in the case of compiler verification, we say that a compiler is correct only if it is fully consistent with
its specification. Such binary questions can be easily translated to a type inhabitation problem e : τ,
where e is a term representing the object we are interested in, and τ is a type encoding the property
we would like e to satisfy.
On the other hand, there are cases where satisfaction of a property is not a binary question. As an
example, in music composition, a choice of a note may be regarded better or worse than other choices,rather than absolutely good or bad. Such fuzzy properties require a type system with some form ofgrades, telling us to what extent an object satisfies a certain property.
In this abstract, we describe our work in progress on developing a type theory for expressing fuzzy
properties. We begin by comparing three existing type systems with the notion of grades, designed forprogramming [3], linguistics [8], and music [6]. We then compare these systems to effect systems andcoeffect systems, which allow grades-like annotations to appear in restricted places. It is our belief
that a general theory of fuzzy properties will open up new possibilities of transferring knowledge
between different areas.

2

Existing Type Systems with Grades

2.1

Precision Types

Boston et al. [3] implement precision types in the context of approximate computing, a technique for
improving performance by allowing imprecise results. In their type system, types are annotated with aprobability, representing how precise a value is. As an example, consider the square function below.
@Approx(0.8) int square(@Approx(0.9) int x) {@Approx(0.8) int xSquared = x * x;
return xSquared;}

In the signature of square, we state that the input x is precise with probability 0.9, and that we would
like the output to be precise with probability 0.8. Assuming multiplication always produces a precise

62result, we can deduce that the product xSquared is precise with probability 0.9 × 0.9 = 0.81. This
probability is greater than the expected probability 0.8, hence the function is judged well-typed. Thus,
using the information of probabilities, we can soundly reason about the quality of a computation.

2.2

Probabilistic Type Theory

Cooper et al. [8] formulate a probabilistic type theory for natural language semantics, with a special
focus on the modelling of human cognition and learning. In their framework, typing judgments may
be associated with a probability, representing how likely a situation is true. For instance, in a situation
where we have two strawberries, three red apples, and five oranges, we have the following judgments:p(a : Apple) = 0.3
p(a : Red) = 0.5
p(a : Apple | a : Red) = 0.6
The first judgment states that an object a is an apple with probability 0.3, and similarly, the second
one states that a is red with probability 0.5. The last judgment states that a is an apple with probability
0.6 given that a is red. Using judgments like these, we can learn classifiers of situations and compute
semantic values of sentences.

2.3

Weighted Refinement Types

Cong [6] proposes weighted refinement types as a means to formalize the rules of counterpoint, a styleof composition where one composes a melody against another melody. In a weighted refinement type,every refinement predicate is paired with a weight, representing the reward one gets by choosing aninterval satisfying that predicate. To illustrate this idea, we give a partial encoding of the rules forcomposing first-species counterpoint [10] as well as two example compositions.
CP : TypeCP = List (Pitch * Interval) < (isImperfect @ 30), (isDissonant @ -100) >cp1 : CP @ 150
cp1 = [(c, per8), (d, maj6), (e, min6), (f, maj3),(e, min3), (d, maj6), (c, per8)]
cp2 : CP @ -10
cp2 = [(c, per8), (d, per5), (e, min3), (f, aug4),(e, min6), (d, maj6), (c, per8)]

The type CP defines counterpoint as a list of pitch-interval pairs refined by two predicates, each ofwhich is coupled with a reward. The rewards encode the guidance that imperfect intervals (i.e., thirds
and sixths) are preferred and dissonant ones (i.e., seconds, fourths, and sevenths) should be avoided.
By summing up the rewards, we can discuss the theoretical correctness of counterpoint compositions.
In the above example, cp1 is considered more correct than cp2 as it has more imperfect intervals andno dissonant ones.

3

Generalizing Existing Type Systems

We have seen three type systems that express fuzzy properties by means of grades. Now we turn toour research question: How can we design a general theory that subsumes these type systems? We do63not have a concrete answer yet, but we conjecture that the resulting theory would share similarities
with effect systems and coeffect systems.
Effect systems [16, 18] are a framework for tracking what side effect a program causes to the environment. Representative applications of effect systems include checked exceptions in Java, which
prevent runtime errors, and the Cats Effect library of Scala, which guarantees resource safety of asynchronous programs. In an effect system, computation types or typing judgments carry an annotation,such as a set of exceptions and reading/writing actions. These annotations are similar to the precisionon the output type of the square function from Section 2.1, and the probability on the judgments ofsituations from Section 2.2.
Dual to effect systems, coeffect systems [5, 20] are a framework for tracking what resource a
program demands of the environment. Notable applications of coeffect systems include boundedvariable use, where each variable can only be used a certain number of times, and secure information
flow, where high-security data are guaranteed not to leak. In a coeffect system, variable bindings
carry an annotation, such as the usage bound and security level. These annotations are similar to theprecisions on the input types of the square function from Section 2.1.
There also exists a concept, called graded modal types [19], that allows simultaneous handling ofeffects and coeffects. In a system with graded modal types, computations and variable bindings may
both carry an annotation, representing either a side effect or a resource demand. These annotations
might subsume the three domain-specific annotations we saw in the previous section, including the
rewards on refinement predicates we saw in Section 2.3 if we emulate refinement types as dependentpair types [23].
Once we have a uniform theory for fuzzy properties, we can explore opportunities to transfer
techniques developed in one area to another area. For instance, in a type system that counts variable
use, we can exploit the usage information to guide synthesis of programs [4]. By replacing variable
use with probabilities, we could potentially extend this idea to perform proof search for sentences.
As a different example, in a type system with probabilistic judgments, we can model incremental
interpretation of utterances [12]. By viewing probabilities as rewards, we could possibly apply thisidea to model incremental composition of music.

4

Related Work

The idea of grading types first appeared in the effect system of Lucassen and Gifford [16], which isdesigned for finding scheduling constraints of parallel computations. Since then, researchers havedeveloped various graded type systems that enable fine-grained reasoning of programs [15, 22, 26],
as well as techniques that make those systems usable in practice [25, 27].
In the context of natural language, grades have been used to give a compositional account of“linguistic effects”, including anaphora resolution [11] and scope ambiguity [13]. We are however
not aware of similar work on “linguistic coeffects”, although there exist linguistic applications [9, 17]of a special case of coeffects (namely linear logic).
The recent years have also seen implementations of type systems with grades, such as Granule [19], Idris 2 [4], and Lambda VL [26]. These languages cannot currently express the three typesystems discussed in Section 2, although their future versions may be able to do so by allowing the
programmer to define custom grades.

5

Conclusion

We described our ongoing work on developing a general type theory for expressing fuzzy properties.
As a first step toward this goal, we observed three type systems featuring grades and identified their64

similarities to effect and coeffect systems. We hope our work will stimulate interesting discussions
across research communities and lead to better understanding of fuzzy phenomena.

Acknowledgments
The author is grateful to the anonymous reviewers for their encouraging comments. This work was
supported by JST, ACT-X Grant Number JPMJAX210C, Japan.

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and Systems - Volume 8410, page 351–370, Berlin, Heidelberg, 2014. Springer-Verlag.
[6] Youyou Cong. Towards type-based music composition. In The 11th International Workshop on
Trends in Functional Programming in Education, TFPIE 2022, 2022.
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7th ACM SIGPLAN International Workshop on Functional Art, Music, Modeling, and Design,
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66

Logical Inference System with Text-to-Image Generation for Phrase
Abduction
Akiyoshi Tomihari⋆ and Hitomi Yanaka ⋆⋆The University of Tokyo, Tokyo, Japan

1

Introduction

ference system to provide an RTE classification model.
However, they used a commercial API to retrieve images from limited image databases, which makes it difficult to obtain knowledge for arbitrary combinations ofphrases.
In the present study, we aim to improve the performance of ccg2lambda1 [16], a higher-order inference
system that automatically conducts natural deduction
proofs on compositional semantics of natural language
based on Combinatory Categorial Grammar (CCG) [22]parsers. For this purpose, we use the state-of-the-art
text-to-image generation model DALL-E [20], which
automatically generates images from text prompts andhas a high zero-shot performance. We obtain candidates
for phrase correspondences from CCG syntactic trees
and semantic representations, and we recognize themas such when the images generated by DALL-E fromeach phrase are similar. Our primary contribution inthis study is providing a new way of solving RTE problems by using a text-to-image generation model to acquire phrasal knowledge. We improve the performanceof ccg2lambda on the SICK dataset [15] compared witha previous logical approach with knowledge injection,
especially in terms of recall.

Recognizing Textual Entailment (RTE), that is, predicting whether a given premise sentence entails a hypothesis sentence, is regarded as a key task in Natural Language Processing (NLP) applications such as question
answering and machine translation [6]. A variety of approaches to RTE have been proposed, including logical approaches [9, 10, 12, 19] and machine learning approaches [13, 14].
In this study, we focus on logical approaches, which
succeed in obtaining semantic representations of sentences with linguistically challenging phenomena suchas generalized quantifiers and comparatives. Logical approaches tend to achieve high precision (the number ofcorrectly predicted entailment labels divided by the total number of predicted entailment labels) and low recall (the number of correctly predicted entailment labels
divided by the total number of entailment labels in all
premise–hypothesis pairs given to a system). However,logical inference systems cannot correctly predict entailment labels when the systems do not have the background knowledge necessary to prove that a premise entails a hypothesis. This is one of the main reasons for the
low recall of such systems [3]. Previous logical inference systems [17, 25] have attempted to overcome thischallenge by using knowledge databases such as WordNet [18] and the Paraphrase Database [7] to insert axioms as background knowledge during a proof. However, those systems still lacked the knowledge necessary
for completing the proof.
Simultaneously, the recent development of deeplearning has stimulated research on vision and language, including visual question answering [1], image
captioning [11], and visual entailment [24]. Vu et al.
[23] provided a visually grounded version of RTE problems. The work that is most closely related to ours isthat by Han et al. [8]. They used visual denotations toacquire phrasal knowledge and combined this knowledge with textual and logic features from a logical in⋆⋆⋆

2

Methodology

2.1

System Overview

We consider RTE problems consisting of premise–
hypothesis pairs annotated with three relations: entailment (yes), contradiction (no), and neutral (unknown).
Our system is based on ccg2lambda with SPSA, an RTE
system in which the on-demand injections of lexical
knowledge are guided by a natural deduction theorem
prover [17]. First, the system parses premise and hypothesis sentences into CCG syntactic trees with the
C&C parser [5] and obtains semantic representations
from them. Second, the system uses Coq [2] to prove
whether a premise entails a hypothesis. Coq is an interactive natural deduction theorem prover that is fully au-

tttttttttttttttttomy@g.ecc.u-tokyo.ac.jp
hyanaka@is.s.u-tokyo.ac.jp

167

https://github.com/mynlp/ccg2lambda

Fig. 1: Overview of the proposed method with an example. We try to prove whether a premise entails a hypothesis
by SPSA (Selector of Predicates with Shared Arguments). If SPSA outputs unknown, we perform phrase extraction,
image generation by DALL-E, and image evaluation. Depending on the result of the image evaluation, we either
insert an axiom and continue the proof by SPSA or finish the proof with the result unknown when we do not obtainany axioms.
2.2

tomatic with several built-in theorem-proving routines
called tactics. In a proof process, SPSA monitors theproof and finds unprovable sub-goals that have the same
arguments with logical predicates of premises. Then,SPSA retrieves word-to-word relations from external
knowledge databases such as WordNet, inserts corresponding axioms on demand, and proves the entailment
relations again. Note that the original SPSA is limited
to capturing word-to-word relations since it does not retrieve phrase-to-phrase relations. We enable the captureof phrase-to-phrase relations by combining a text-toimage generation model and ccg2lambda with SPSA.
The proof process for our system is as follows:

Phrase Extraction

We define a phrase as any part of a sentence corresponding to an NP (noun phrase), a VP (verb phrase), a PP
(prepositional phrase), or an S (sentence) in CCG syntactic trees. When the result of SPSA is unknown, we
sample one sub-goal from the unprovable sub-goals. We
use CCG syntactic trees and semantic representations to
extract phrases from premise–hypothesis pairs. We extract all phrases from the premise, but we extract only
the minimum phrase that contains the unprovable subgoal in the semantic representation from the hypothesis.
2.3

1. Try to prove whether a premise entails a hypothesis by SPSA. If the result is yes or no,then finish the proof. If not, then go to 2.
2. Try to insert a phrasal axiom using DALL-E(Sections 2.2 and 2.3). If a phrasal axiom isinserted, then return to 1. If not, then finish
the proof with the result unknown.

Image Generation and Evaluation

Using DALL-E, we generate two images from each extracted phrase. For each premise phrase, we calculate its
phrase distance from the hypothesis phrase. We define
the phrase distance between a phrase p from a premise
and a phrase h from a hypothesis asdis(p, h) =1 X
min {ωc f (ip , ih ) + ωg f (gray(ip ), gray(ih ))},
|Ip | i ∈I ih ∈Ihp

Steps 1 and 2 are repeated for all the unprovable subgoals.

p

(1)

where
68

– Ip and Ih are sets of images generated fromp and h,– gray(i) is a grayscale image converted fromthe image i,– ωc and ωg are parameters of the phrase distance for which 0 ≤ ωc , ωg ≤ 1, and ωc +
ωg = 1, and– f is a function from AugNet [4] for calculating the distance between two images.

precision was the over-generation of axioms, which resulted in more wrong axioms being inserted with alarger threshold.
ωc ωg Threshold Acc.
1
15
83.53
0
0.3 0.7 17
83.58
0.5 0.5 16
83.67
0.7 0.3 16
83.79
1
0
15
83.32
83.13
SPSA
Han et al. (2017)84.29
No axioms
76.65

The reasoning behind this definition is that if p entails
h (i.e, if h necessarily follows from p), then each image
in Ip is likely to be similar to at least one image in Ih .
The motivation here is to express the non-symmetrical
relation between the premise phrase and the hypothesis
phrase.
If the minimum distance between the premise phrase
and the hypothesis phrase is shorter than a certain
threshold, then we recognize the closest phrase pair asa phrase correspondence, insert the phrasal axiom, andcontinue the proof by SPSA. The phrasal axiom is formulated by converting the corresponding semantic representations of both phrases into a Coq script. If we do
not obtain any phrase correspondences, then we do notinsert any phrasal axioms and finish the proof with theresult unknown.

3

Prec.
95.05
94.69
94.63
95.43
94.62
96.95
90.24
98.90

Rec.
63.61
63.58
63.68
63.30
63.61
62.65
71.08
46.48

F-measure
76.17
76.08
76.13
76.11
76.07
76.08
79.52
63.24

Table 1: RTE results on the SICK dataset. We select thethreshold that maximizes the F-measure per parameter
pair.

Analysis Figure 2 shows images generated by DALLE from extracted phrases. Figures 2a and 2b show examples of our system producing positive results, where
similar images were successfully generated by DALLE from a similar concept being conveyed with different phrases. In all the examples that were proved correctly with phrasal axioms, we observed that only one
phrasal axiom was inserted by our system. Figures 2c
and 2d show examples of our system producing a negative result. An axiom for the phrase correspondence
Two men → Two people was correct, and proper images
were generated by DALL-E. However, the distance between the two images, calculated by f in (1), was long,so our system did not insert the axiom for the phrase
correspondence. This shows that the phrase distance (1)alone cannot fully capture the relationship between thetwo images, and it even leads to a substantial number of
errors. In future work, we will compare different textto-image generation models and distance metrics to see
if they can improve our method of comparing images.
Another interesting direction for future work is to calculate phrase distances by using text embedding vectors
in vision-and-language models.

Experiment

Experimental Setup We used the SemEval-2014 version of the SICK dataset [15]. This dataset contains
problems involving lexical and logical phenomena.
Thus, lexical and phrasal knowledge is needed to correctly predict the entailment labels of these problems.
The SICK dataset contains problems with train/trial/test
splits of 4500/500/4927 premise–hypothesis pairs anda yes/no/unknown label distribution of .29/.15/.56. We
used the test set to evaluate our system. We changed theparameters ωc and ωg in (1) and the thresholds of thephrase distance, then compared the results.
Results Table 1 shows our experimental results. The
thresholds are the integer values from 15 to 20 withwhich we obtained the highest accuracies on 100 randomly chosen examples from the SICK dataset. We
obtained higher accuracy (83.79 versus 83.13), recall(63.30 versus 62.65), and F-measure (76.11 versus
76.08) over the SPSA baseline when ωc = 0.7 andωg = 0.3, although the precision was lower than thebaseline (95.43 versus 96.65). One reason for the low

Figures 2e and 2f show another example of our system producing a negative result. These were generated
from phrases containing negation, and they do not sufficiently express the meanings of those phrases. A recent study [21] reported on the challenges involved in
expressing visual denotations of negation, and how todeal with these types of phrases is an issue that remains
to be solved.
69

(a) A man is trekking in (b) The man is hiking in
the woods
the woods(c) Two men(d) Two people(e) No man(f) Nobody is standing in
the ocean

Fig. 2: Images generated by DALL-E from phrases. (a) and (b) are from the problem with ID 3184 (Premise: A
man is trekking in the woods., Hypothesis: The man is hiking in the woods., Gold: yes, SPSA: unknown, Oursystem: yes, Phrase correspondence of the inserted axiom: A man is trekking in the woods → The man is hiking in
the woods). (c)-(f) are images that led to incorrect results. (c) and (d) are from the problem with ID 9613 (Premise:
Two men are seated on a camel and another camel is in the foreground., Hypothesis: Two people are seated on acamel and another camel is in the foreground., Gold: yes, SPSA: unknown, Our system: unknown). (e) and (f) are
generated from phrases containing negation.

4

Conclusion

2. Y. Bertot and P. Castéran. Interactive Theorem
Proving and Program Development. Springer
Berlin, Heidelberg, 2004.
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Representations for Textual Inference. CSLI Publications, 2014.
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statistical parsing with ccg and log-linear models.
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Springer Cham, 2013.
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and H. Laga. A comprehensive survey of deeplearning for image captioning. ACM Computing
Surveys (CsUR), 51(6), 2019.
12. H. Hu, Q. Chen, K. Richardson, A. Mukherjee, L. S. Moss, and S. Kuebler. MonaLog: alightweight system for natural language inference

In this paper, we introduced a new phrase abduction mechanism for the higher-order inference system
ccg2lambda that uses the state-of-the-art text-to-image
generation model DALL-E. Our experimental results
demonstrated that our system can obtain knowledge
from long phrases. Our system improved the recall andaccuracy of ccg2lambda on the SICK dataset comparedwith the previous system, which used the word-level abduction mechanism SPSA. In this paper, we measured
phrase distances by calculating the distances between
images generated by a text-to-image generation model,
allowing us to use the image information for phrasal
knowledge. We can also measure phrase distances by
calculating the distances between embedding vectors ofphrases in a vision-and-language model, but we leavethis for future work.

5

Acknowledgements

We thank the two anonymous reviewers for their helpful comments and suggestions. We also thank Tomoya
Kurosawa and Tomoki Sugimoto for their helpful advice on writing our paper. This work was supported byJST, PRESTO Grant Number JPMJPR21C8, Japan.

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model and inner-attention. CoRR, abs/1605.09090,
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R. Bernardi, and R. Zamparelli. A SICK cure forthe evaluation of compositional distributional semantic models. In Proceedings of LREC. European
Language Resources Association (ELRA), 2014.
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system. In Proceedings of ACL System Demonstrations. Association for Computational Linguistics,
2016.
17. P. Martı́nez-Gómez, K. Mineshima, Y. Miyao, andD. Bekki. On-demand injection of lexical knowledge for recognising textual entailment. In Proceedings of EACL: Volume 1, Long Papers. Association for Computational Linguistics, 2017.
18. G. A. Miller. Wordnet: A lexical database for english. Commun. ACM, 38(11), 1995.
19. K. Mineshima, P. Martı́nez-Gómez, Y. Miyao, andD. Bekki. Higher-order logical inference with compositional semantics. In Proceedings of EMNLP.
Association for Computational Linguistics, 2015.
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for Computational Linguistics, 2018.

71

Answers, Exhaustivity, and Presupposition of wh-questions in Dependent Type Semantics
Hayate Funakura∗†
1

Introduction

In this paper, I analyze direct and embedded who/which questions in Dependent Type Semantics (DTS) [1, 2], which is one of the frameworks of proof-theory-based natural language
semantics. This study presents a diﬀerent strategy than the preceding analysis of interrogatives
in DTS [15]. The previous study [15] explains the relationship between questions and answers
in terms of inferences since DTS is a semantic framework based on proof-theoretic semantics.
My analysis inherits this idea, but diﬀers in the following respects:• I do not assume lexical ambiguity in wh-words.
• I analyze wh-words itself as a lambda abstractor, and let empty operators play the roleof existential quantiﬁers.
The second point in particular may seem eccentric, but it paves the way to a proper analysis
of embedded interrogatives in DTS1 .
In the current method, various levels of answers to a question are predicted to be answers
based on a single semantic representation (for short, SR) of the interrogative (Section 3.1),
whereas most existing approaches account for multiple levels of answers by assuming the ambiguity of interrogatives. In addition, the analysis will also be extended to embedded interrogatives (Section 3.2), which were not yet addressed in [15]. I will attempt to solve several puzzles
regarding presupposition and exhaustivity of wh-complements, satisfying general requirementsin the semantics of clause-embedding predicates.
Section 2 presents the issues that this analysis seeks to resolve. Then, in Section 3, I sketch
the analysis and how it solves the problems in Section 2.
2

Empirical facts taken into account

2.1

Possible answers

There are at least three levels of answers to a question: mention-some (MS), weak exhaustivity(WE), and strong exhaustivity (SE). As an example, consider the following question:(1) Who danced?

(Situation: John and Susan danced, and Mary didn’t dance.)

The answers to this question, corresponding to each level, are as follows.
(2) a.
b.
c.

John danced.
John and Susan danced.
John and Susan danced, and Mary didn’t dance.

(Mention-some answer)(Weakly exhaustive answer)(Strongly exhaustive answer)

Traditional methods [7, 5] and subsequent analyses based on them attempt to capture
these answer levels by assuming the ambiguity of interrogatives. In the proposed analysis,
which treats declaratives and interrogatives in a uniﬁed way and formulates answerhood by thegeneral relation of entailment, the variety of answers is captured based on a single SR of eachinterrogative.
∗

Graduate School of Human and Environmental Studies, Kyoto University
I sincerely thank Takashi Sakuragawa, Matthew de Brecht, Nayuta Miki, Mizuki Yamamoto, and two
anonymous reviewers for their insightful discussions and comments.
1
This treatment also gives the possibility of realizing the propositional-answer-oriented approach (e.g. [6, 7,
5, 3]) and the constituent-answer-oriented approach (e.g. [14, 8, 4]), which are often thought to be essentiallydiﬀerent, in a single theory.
†

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2.2

Exhaustivity and presupposition in embedded contexts

2.2.1

Exhaustivity

Question-embedding sentences are ambiguous about exhaustivity. In principle, each propositionin (4) is possible as a paraphrase of (3) (let V be some responsive predicate)2 .
(3) Annie V who danced.
(4) a.
b.
c.

(Situation: John and Susan danced, and Mary didn’t dance.)

There is a p ∈ {John danced, Susan danced} s.t. Annie V p. (Mention-some reading)
For all p ∈ {John danced, Susan danced}, Annie V p.
(Weakly exhaustive reading)
For all p ∈ {John danced, Susan danced}, Annie V p and Annie V Mary didn’t dance.
(Strongly exhaustive reading)

As a related example, the following inference is valid only under the strongly exhaustive readingof John knows who danced.
(5) a.
b.
c.

John knows who danced. (Strongly exhaustive reading)Mary didn’t dance.
∴ John knows Mary didn’t dance.

A natural application of the DTS-based analysis of interrogatives [15] to that of factive
predicates [12] yields the following analyses.
(6) a.
b.

John knows who danced. (Mention-some reading)know(j) ((x : e) ⊕ d(x)) (@)(7) a.
b.

John knows who danced. (Strongly exhaustive reading)know(j) ((x : e) → d(x) ] ¬d(x)) (@)

The SR in (7-b) does not capture the inference pattern as in (5). In addition, the following
sentences are predicted to be semantically equivalent.
(8) a.
b.

John knows who danced. (Strongly exhaustive reading)John knows that each person either danced or did not dance.

Also, with regard to the mention-some reading (6), the following inferences are not predicted
unless some additional axiom is added.
(9) a.
b.

John knows who danced. (Mention-some reading)∴ John knows of at least one person that he or she danced.

I will give a diﬀerent strategy, which accounts for these inferences (sketched in Section 3.2).
2.2.2

Presupposition

Factive predicates that take wh-complement trigger the existential presupposition.
(10) a.
b.

John knows who danced. presupposes Someone danced.
John knows which student danced. presupposes Some student danced.

It is also widely known that factive predicates trigger the factive presupposition.
(11) John knows that Sue smokes. presupposes Sue smokes.
2

In addition to the interpretations listed in (4), the intermediate exhaustive readings are also said to bepossible [9]. This reading is not considered in this study. I thank the two anonymous reviewers for pointing this
out.

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2.2.3

Uniformity of the responsive predicates’ meaning

So far, I have observed exhaustivity and presupposition in the embedding context. I aim toaccount for them under a uniform meaning of each responsive predicate. That is, it is notassumed that responsive predicates have diﬀerent meanings when taking a declarative andwhen taking an interrogative. The following examples support this treatment:(12) Alice knows/realized/reported that Ann left and Bill knows/realized/reported which othergirls left.
[13, p. 2](13) Alice knows who danced and that John hosted the dance party.
These suggest that a single predicate can simultaneously take a declarative complement andan interrogative one.
In this paper, I introduce three unpronounced operators to derive each interpretation in (4).
The issue of ambiguity regarding exhaustivity is reduced to the choice of these operators. Also, I
show that an existing analysis of factive predicates [12], motivated by the factive presupposition
as in (11), can be applied to the existential presupposition as in (10).
3Proposal

3.1

Direct questions

I analyze who/which questions as Σ-types3 . In the analysis here, wh-words create an abstract,which is fed to the unpronounced operator ∅Q I deﬁne in (14)4 . ∅Q is assumed to appear at
the beginning of wh-interrogatives.
(14) ∅Q := Swq /(Swqa /N P ) : λp.(x : e) × p(x)The followings are examples of the analysis.
(15) a.
b.

Who danced?
(x : e) × d(x)(16) a.
b.

Which student danced?
(x : e) × (s(x) × d(x))

The mapping from (15-a) to (15-b) is obtained by the following CCG derivation tree:∅Q
Swq /(Swqa /N P ) :(17)
λp.(x : e) × p(x)Whodanced(Swqa /N P )/(S\N P ) : λp.λx.p(x)S\N P : λx.d(x)>Swqa /N P : λx.d(x)>S : (x : e) × d(x)SRs of embedded interrogatives are derived by operators diﬀerent from ∅Q (see next section).
To sum up, the strategy of this study is that a wh-interrogative, by itself, has only one SR (i.e.,an abstract), and various SRs are derived from this5 .
[]x:A
is called a Σ-type. To save space, this is often written as (x : A) × B(x). ForB(x)a more formal exposition of DTS, see [1, 2, 11].
4This study adopts Combinatory Categorial Grammar (CCG; [10]) as a syntactic component and thereforedeﬁnes lexical entries in the following form: PF := CCG category : SR
5
The part of the derivation of (17) that does not involve ∅Q embodies the idea of the so-called categorial
approach: an interrogative is itself an incomplete object, and when combined with a characteristic short answer
it becomes a sentence.
3

A term of the form74

In the preceding analysis [15], answerhood is deﬁned via entailment. Adding (19-a) and(19-c) to this original deﬁnition, we deﬁne answerhood as follows:(19) Let SQ , and SA be types, and K be a global context. SA is an answer to SQ iﬀ (19-a) andeither of (19-b) and (19-c) hold.
a. K, a : SA ⊬ ⊥ true
b. K, a : SA ` SQ truec. K, a : SA ` ¬SQ true
Under (19), the answers mentioned in Section 2.1 are predicted to be answers.
Furthermore, the proposed analysis allows us to capture the impossibility of self-denial of
the existence of a positive answer (except in the case of a rhetorical question). For example,
the following utterance would sound contradictory.
(20) #Nobody danced. Who danced?
This mini-discourse has the following SR, which leads to contradiction, as expected.
[

]

(21)

(x : e) → ¬d(x)(x : e) × d(x)3.2

Embedded questions

Here I deﬁne three unpronounced operators whose SRs map a direct question into an embedded question. It is assumed that one of these will appear at the beginning of interrogativecomplements. The ambiguity about exhaustivity is due to the choice of these operators.
(22) ∅MS := S\(S/S̄)/(Swqa /N P ) : λF.λV.(x : e) × V (F (x))[(23) ∅WE := S\(S/S̄)/(Swqa /N P ) : λF.λV.
(x : e) → F (x) → V (F (x))
V ((y : e) × F (y))[]

F (x) → V (F (x))
 (x : e) →¬F (x) → V (¬F (x))(24) ∅SE := S\(S/S̄)/(Swqa /N P ) : λF.λV. 
V ((y : e) × F (y))

] 



Then, the SR of the sentence (25) can be obtained in three ways, depending on the choice ofthe operators. Each SR is shown in (26). The lexical item of know is that given in [12, p. 404]
concerning declarative complements.
(25) John knows who danced.
(26) a.

Mention-some reading:[John knows ∅MS who danced 7→(18) a.
b.
c.

x:eknow(john) (d(x)) (@1 )]Who danced?
John.

Who
danced(Swqa /N P )/(S\N P ) : λp.λx.p(x)S\N P : λx.d(x)>Swqa /N P : λx.d(x)Swqa : d(j)John
NP : j>In words, the proposed analysis can probably be converted to a categorial theory by detaching null operatorsfrom the lexicon. In this sense, the current analysis is potentially a hybrid theory. Further discussion on thispoint is warranted, but I will leave that for another occasion.

75

b.

Weakly exhaustive reading:(x : e) → d(x)2)([→ know(john)(d(x))(@
])


y:e
John knows ∅WE who danced 7→ know(john)
(@1 )
d(y)

c.

Strongly exhaustive reading:[d(x) → know(john)(d(x))(@2 ) (x : e) →¬d(x) → know(john)(¬d(x))(@3 )([])
John knows ∅SE who danced 7→ y:e
know(john)
(@1 )
d(y)

] 






These SRs lead us to predict that the inference (5) is only valid under the SE reading. With
respect to the existential presupposition, it is predicted that for the SR of John knows who
danced to be well-formed, the SR of Someone danced must be true. These are desirable results.
4

Limitations and future work

It is necessary to extend the analysis to more types of interrogatives. Besides, it is not evident
whether the proposed analysis can predict which predicates will select which exhaustivity. In
addition, comparisons with other frameworks, such as inquisitive semantics [3], are also required.
These will be discussed on another occasion. As mentioned in several footnotes, this theory
can also embody a question-as-function view (e.g. [14, 8, 4]) that is adept at dealing with nonsentential answers. This feature is desirable, given that the strengths of theories with sentential
answers in mind and theories with non-sentential answers in mind are complementary and thata question-answering task needs to be able to handle both types of answers. This perspective
has not been discussed in this paper and is needed in the future.
References[1] D. Bekki. “Representing anaphora with dependent types”. In: International conference on logical aspects ofcomputational linguistics. Springer. 2014. [2] D. Bekki and K. Mineshima. “Context-passing and underspeciﬁcation in dependent type semantics”. In: Modern perspectives in type-theoretical semantics. Springer, 2017.
[3] I. Ciardelli, J. Groenendijk, and F. Roelofsen. Inquisitive Semantics. Oxford University Press, 2019. [4]J. Ginzburg. “Abstraction and ontology: Questions as propositional abstracts in type theory with records”.
In: Journal of Logic and Computation 15.2 (2005). [5] J. A. G. Groenendijk and M. J. B. Stokhof. “Studies
on the Semantics of Questions and the Pragmatics of Answers”. PhD thesis. Univ. Amsterdam, 1984. [6] C.
Hamblin. “Questions in Montague English”. In: Foundations of Language 10.1 (1973). [7] L. Karttunen. “Syntax and semantics of questions”. In: Linguistics and philosophy 1.1 (1977). [8] M. Krifka. “For a Structured
Meaning Account of Questions and Answers”. In: Audiatur Vox Sapientia. A Festschrift for Arnim von Stechow (2001). [9] B. Spector. “Exhaustive interpretations: What to say and what not to say”. In: Unpublishedpaper presented at the LSA workshop on Context and Content. 2005. [10] M. Steedman. The syntactic process. MIT press, 2000. [11] R. Tanaka. “Natural Language Quantiﬁcation and Dependent Types”. PhD thesis.
Ochanomizu University, 2021. [12] R. Tanaka, K. Mineshima, and D. Bekki. “Factivity and presupposition
in dependent type semantics”. In: Journal of Language Modelling 5 (2017). [13] W. Uegaki. “The semantics
of question-embedding predicates”. In: Language and Linguistics Compass 13.1 (2019). [14] A. Von Stechow.
“Focusing and backgrounding operators”. In: Discourse Particles: Descriptive and Theoretical Investigations
on the Logical, Syntactic and Pragmatic (1991). [15] K. Watanabe, K. Mineshima, and D. Bekki. “Questionsin Dependent Type Semantics”. In: Proceedings of the Sixth Workshop on Natural Language and Computer
Science. Gothenburg, Sweden: Association for Computational Linguistics, May 2019.

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A PROOF-THEORETIC ANALYSIS OF MEANING OF A FORMULA
IN A COMBINATION OF INTUITIONISTIC AND CLASSICAL PROPOSITIONAL
LOGIC
MASANOBU TOYOOKAGRADUATE SCHOOL OF HUMANITIES AND HUMAN SCIENCES, HOKKAIDO UNIVERSITY, JAPANA BSTRACTThis paper studies a proof-theoretic analysis of the meaning of a formula in a combination of
intuitionistic and classical propositional logic, called C + J.1 This logic was provided in [1, 2],
and the analysis uses the sequent calculus G(C + J), which was proposed in [3]. The analysis
given in this paper is based on the one provided by Restall [4], but by employing the method
proposed by Takano [5], a more refined analysis is proposed. This paper has four sections.
Section 1 reviews Restall’s analysis and states two points of improvement. Section 2 introduces
the sequent calculus G(C + J). Section 3 displays the proof-theoretic analysis of the meaningof a formula in C + J, by making use of Takano’s method. Section 4 introduces two positions
called unilateralism and bilateralism and describes an outline of a unilateral analysis for C + J.
1. R ESTALL’ S A NALYSIS
The main idea of Restall [4]’s analysis is to interpret inference rules in a sequent calculus
as normative constraints on assertion and denial, and obtaining a model from those inference
rules. Thus, his analysis is different from proof-theoretic semantics in that a syntactical object
such as an argument or a deduction is not directly used to explain the meaning of a formula.
Since Restall’s analysis technically corresponds a canonical model argument, it provides a viewof a proof of semantic completeness from linguistic acts, such as assertion and denial. Restall
applied his analysis to classical logic, intuitionistic logic, and modal logic S5. Only the case ofclassical logic, where the sequent calculus LK is used, is described here. Since the arguments
about conjunction and disjunction are similar, only the former is mentioned here.
The central notion of his analysis is the notion of a position, defined as below.
Definition 1. [4, Definition 1] A pair (Γ : ∆) of sets of formulas is a position if Γ ⇒ ∆ is notderivable in LK.
In the rest of this paper, (Γ ∪ {A} : ∆ ∪ {B}) is abbreviated as (Γ, A : ∆, B), for any sets Γ, ∆of formulas and formulas A, B. For example, (p : p ∧ q) is a position, while (p : p ∨ q) is not.
Restall regards the antecedents and the succedents of a sequent as asserted and denied formulas
respectively. A position is an expression of a consistent situation with respect to assertion anddenial. The example described above represents the fact that it is consistent to assert p anddeny p ∧ q, while it is inconsistent to assert p and deny p ∨ q. Accordingly, the derivabilityof Γ ⇒ ∆ implies that it is inconsistent to assert all of the formulas in Γ and deny all of theformulas in ∆. In Restall’s analysis, inference rules are interpreted in light of the notion of aposition. Consider the following rule:A, Γ ⇒ ∆A ∧ B, Γ ⇒ ∆.
1

Katsuhiko Sano, the supervisor of the author, gave many helpful comments. This research is partially supported by Grant-in-Aid for JSPS Fellows Grant Number JP22J20341.

77

The following is obtained by the contrapositive of this rule: if (A ∧ B, Γ : ∆) is a position, then(A, Γ : ∆) is also a position. Since a position expresses a consistent situation with respect toassertion and denial, this rule functions as a normative constraint on a linguistic act.
Definition 2. [4, Definition 4] A pair (Γ : ∆) of sets of formulas is a limit position if it satisfies
the following:• For any finite sets Γ′ ⊆ Γ, ∆′ ⊆ ∆ of formulas, the pair (Γ′ : ∆′ ) is a position.
• The union of Γ and ∆ contains all formulas in classical logic.
Fact 3. [4, Fact 4] For any position (Γ : ∆), there is a limit position (Γ∗ : ∆∗ ) such that Γ ⊆ Γ∗and ∆ ⊆ ∆∗ .
Fact 4. [4, Fact 5] For any limit position (Γ : ∆), all of the following hold:(1) A ∧ B ∈ Γ iff both A ∈ Γ and B ∈ Γ hold,(2) A ∧ B ∈ ∆ iff either A ∈ ∆ or B ∈ ∆ hold,(3) ¬A ∈ Γ iff A ∈ ∆,(4) ¬A ∈ ∆ iff A ∈ Γ.
Fact 3 ensures that any position can be extended to some limit position, and Fact 4 establishes
that a formula in Γ and ∆ can be regarded as true and false, respectively. Although the meaningof a formula is explained by a model, since it is not introduced as given but obtained from theadmissibility of inference rules in LK, Restall’s analysis should be regarded as proof-theoretic.
However, two points of improvement exist in Restall’s analysis. Firstly, although Restall’s
analysis obtains the satisfaction relation for a complex formula from the admissibility of corresponding inference rule, an analysis which establishes the equivalence between them is better,
since it describes the tighter relation between them. Secondly, if an analysis without assuming
the admissibility of the rule (Cut) is proposed, such an analysis will be better than Restall’s inthat it uses less assumptions, since Restall’s analysis depends on (Cut), formulated as follows:Γ ⇒ ∆, A A, Γ ⇒ ∆
(Cut)
Γ⇒∆
.
The analysis without depending on (Cut) will be presented in Section 3. In addition to theanalysis assuming less assumptions, it provides a semantic condition corresponding to the admissibility of (Cut), as other rules for connectives. In the rest of this paper, if the side condition
A ∈ Sub(Γ ∪ ∆) is imposed, this restricted rule will be called “analytic cut rule”, denoted by(Cuta ).
2. A C OMBINATION C + J
A syntax of the combination C + J has a countably infinite set of propositional variables
and the following logical connectives: falsum ⊥, conjunction ∧, disjunction ∨, intuitionistic
implication →i , and classical negation ¬c . Intuitionistic negation ¬i and classical implication→c are defined as follows: ¬i A := A →i ⊥, A →c B := ¬c A ∨ B. A logical connective otherthan implications and negations is regarded as common to intuitionistic and classical logic. The
semantics of C + J is provided in [1, 2] and is obtained by adding to ordinary intuitionistic
Kripke semantics (cf. [6]) the satisfaction relation for “¬c ”, described as follows: w |=M ¬c A
iff w ̸|=M A, where M is an intuitionistic Kripke model, and w is a possible world in M .
As a proof theory to analyze the meaning of a formula in C + J, this paper employs thesequent calculus G(C + J), proposed in [3]. This calculus is obtained by adding to the propositional fragment of the classical sequent calculus LK the following rules for “→i ”:
Γ ⇒ ∆, A B, Γ ⇒ ∆A, C1 →i D1 , . . . , Cm →i Dm , p1 , . . . , pn ⇒ B(⇒ →i )(→i ⇒)
C1 →i D1 , . . . , Cm →i Dm , p1 , . . . , pn ⇒ A →i B
A →i B, Γ ⇒ ∆.

78

Although a form of an antecedent is restricted in the right rule for “→i ”, no restriction is needed
for the left rule. The sequent calculus G(C+J) is cut-free and satisfies the subformula property.
3. I MPROVEMENT OF R ESTALL’ S A NALYSIS BY TAKANO ’ S M ETHOD
Since Takano [5]’s method obtains a model from the admissibility of inference rules, its
fundamental idea is similar to Restall’s idea, though the interpretation based on assertion anddenial was not employed in [5]. As in Section 1, the argument about disjunction is omitted.
Stipulation 5. [5, Stipulation 1] The sequent calculus is the calculus having A ⇒ A as anaxiom for any A and having weakening and contraction rules.
By Stipulation 5, a sequent calculus containing only some rules in G(C + J) can be discussed.
It is noted that the existence of (Cut) or (Cut)a is not assumed. In the following, let GL be asequent calculus.
Definition 6. Let A be any formula. We define Sub(A) as the set of all subformulas of A. LetΓ be a finite set of formulas. Then, we define Sub(Γ) as the set of all subformulas of someformulas in Γ. A set Γ of formulas is subformula-closed (sf-closed) if Sub(Γ) ⊆ Γ and ⊥ ∈ Γ.
In the following, a sf-closed finite set Ξ of formulas is considered, while it is not consideredin [5]. However, such a set is considered in [7, 8, 9], and a finite model will be obtained byconsidering it.
Definition 7. [7, Definition 2 (1)] Let Ξ be a sf-closed finite set of formulas and Γ ∪ ∆ ⊆ Ξ. A
sequent Γ ⇒ ∆ is Ξ-derivable in GL if it has a derivation in GL consisting solely of formulasin Ξ.
Instead of the notion of a position, the notion of a Ξ-analytically saturated pair is introduced.
Definition 8. Let Ξ be a sf-closed finite set of formulas. A pair (Γ : ∆) of finite sets of formulasis Ξ-analytically saturated in a sequent calculus GL if it satisfies all of the following:(1) Γ ⇒ ∆ is not Ξ-derivable in GL.
(2) For any formula A ∈ Ξ,
• A ∈ Γ if A, Γ ⇒ ∆ is not Ξ-derivable in GL,• A ∈ ∆ if Γ ⇒ ∆, A is not Ξ-derivable in GL,(3) Sub(Γ ∪ ∆) ⊆ Ξ.
Lemma 9. Let Ξ be a sf-closed finite set of formulas. Consider a set Γ ∪ ∆ of formulas andsuppose that Γ ⇒ ∆ is not derivable in GL and Sub(Γ ∪ ∆) ⊆ Ξ. Then, there exists a
Ξ-analytically saturated pair (Γ∗ : ∆∗ ) such that Γ ⊆ Γ∗ , ∆ ⊆ ∆∗ , and Γ∗ ∪ ∆∗ ⊆ Ξ.
Lemma 9 ensures that any underivable pair of sets of formulas can be extended to some Ξanalytically saturated pair. In the following, the set of all Ξ-analytically saturated pairs isdenoted by W Ξ . From (3) of Definition 8, W Ξ is finite, since Ξ is finite.
Definition 10. For any (Γ : ∆), (Π : Σ) ∈ W Ξ , (Γ : ∆)RΞ (Π : Σ) if the following hold:• For any propositional variable p ∈ Ξ, if p ∈ Γ, then p ∈ Π,
• For any formulas A →i B ∈ Ξ, if A →i B ∈ Γ, then A →i B ∈ Π.
Definition 11. For any propositional variable p ∈ Ξ and any (Γ : ∆) ∈ W Ξ , a valuation V Ξ isdefined as follows:(Γ : ∆) ∈ V Ξ (p) iff p ∈ Γ.
The obtained tuple ⟨W Ξ , RΞ , V Ξ ⟩ is a Kripke model for C + J, provided in [1, 2].
Definition 12. An inference rule is Ξ-admissible in GL if whenever all of the upper sequentsare Ξ-derivable in GL, then the lower sequent is also Ξ-derivable in GL.

79

Theorem 1. Define the following conditions for A, B and (Γ : ∆) ∈ W Ξ :(¬+c ) ¬c A ∈ Γ implies A ∈ ∆,(¬−c ) ¬c A ∈ ∆ implies A ∈ Γ,(∧+ ) A ∧ B ∈ Γ implies A ∈ Γ and B ∈ Γ,(∧− ) A ∧ B ∈ ∆ implies A ∈ ∆ or B ∈ ∆,ΞΞ(→+i ) A →i B ∈ Γ implies for any (Π : Σ) ∈ W s.t. (Γ : ∆)R (Π : Σ), A ∈ Σ or B ∈ Π,ΞΞ(→−i ) A →i B ∈ ∆ implies for some (Π : Σ) ∈ W s.t. (Γ : ∆)R (Π : Σ), A ∈ Π andB ∈ Σ,(⊥) ⊥ ̸∈ Γ,(M axa ) A ∈ Sub(Γ ∪ ∆) implies A ∈ Γ or A ∈ ∆,(M ax) A ∈ Γ or A ∈ ∆.
Then, for any sf-closed finite set Ξ of formulas, all of the following hold:(1) The left rule for “¬c ” is Ξ-admissible in GL for any contexts iff (¬+c ) holds for any(Γ : ∆) ∈ W Ξ ,(2) The right rule for “¬c ” is Ξ-admissible in GL for any contexts iff (¬−c ) holds for any(Γ : ∆) ∈ W Ξ ,(3) The left rule for “∧” is Ξ-admissible in GL for any contexts iff (∧+ ) holds for any(Γ : ∆) ∈ W Ξ ,(4) The right rule for “∧” is Ξ-admissible in GL for any contexts iff (∧− ) holds for any(Γ : ∆) ∈ W Ξ ,(5) The left rule for “→i ” is Ξ-admissible in GL for any contexts iff (→+i ) holds for any(Γ : ∆) ∈ W Ξ ,(6) The right rule for “→i ” is Ξ-admissible in GL for any contexts iff (→−i ) holds for any(Γ : ∆) ∈ W Ξ ,(7) The rule for “⊥” is Ξ-admissible in GL iff (⊥) holds for any (Γ : ∆) ∈ W Ξ ,(8) The rule (Cuta ) is Ξ-admissible in GL for any contexts iff (M axa ) holds for any(Γ : ∆) ∈ W Ξ ,(9) The rule (Cut) is Ξ-admissible in GL for any contexts iff (M ax) holds for any (Γ :∆) ∈ W Ξ .
+−−Theorem 2. Let Ξ be a sf-closed finite set of formulas. Suppose (¬+c ), (¬c ), (∧ ), (∧ ),
−
Ξ
Ξ
(→+
i ), (→i ), and (⊥) are satisfied for W . Then, for any (Γ : ∆) ∈ W and any formulaC ∈ Ξ which has no occurrence of “∨”, the following hold:C ∈ Γ implies (Γ : ∆) |= C and C ∈ ∆ implies (Γ : ∆) ̸|= C.
The analysis in this section overcomes the two points of improvement of Restall’s analysis.
Firstly, in Theorem 1, the equivalence between the admissibility of an inference rule and thecorresponding satisfaction relation is shown. Secondly, this analysis does not depend on (Cut).
Although the absence of (Cut) restricts the obtained satisfaction relations to one direction, as
is seen in Theorem 1, this is sufficient to show Theorem 2. In addition, semantic conditions(M axa ) and (M ax) for the admissibility of (Cuta ) and (Cut) are provided.
4. U NILATERAL A PPROACH
This section explains briefly how the meaning of a formula in C + J can be analyzed basedon unilateralism. Unilateralism is a position considering only one type of linguistic act is(should be regarded as) primitive when the meaning of a formula is considered, while bilateralism considers two types are (should be regarded as) primitive. Bilateralism has been chosen
so far, since both assertion and denial are used to interpret an inference rule and the derivabilityof a sequent. The most direct unilateral way of analyzing the meaning of a formula in C + J,80

employing the method in Section 3, is to regard the antecedents of an underivable pair (Γ : ∆)as formulas which are asserted and the succedents of it as those which are not asserted.
However, since a problem of this interpretation was already pointed out in [10], this section
overviews a different way of providing unilateral analysis, which is realized by constructing the
calculus for C + J where only the antecedents of a sequent are manipulated. In the following,a set ¬c ∆ of formulas is defined for any ∆ as follows: ¬c ∆ = {¬c C | C ∈ ∆} .
Lemma 13. Let GL¬c be a sequent calculus obtained by adding the left and the right rules for“¬c ”. For any set Γ ∪ ∆ of formulas, the following equivalence hold:Γ ⇒ ∆ is derivable in GL¬c iff Γ, ¬c ∆ ⇒ is derivable in GL¬c .
By making use of this lemma, an inference rule in G(C + J) can be transformed into a different
rule, where only antecedents of sequents are manipulated and no formula occurs in succedents.
For example, the left rule for “→i ” in G(C + J) is transformed as follows:¬c A, Γ, ¬c ∆ ⇒ B, Γ¬c ∆ ⇒Γ ⇒ ∆, A B, Γ ⇒ ∆A →i B, Γ ⇒ ∆⇝A →i B, Γ, ¬c ∆ ⇒.
To transform all the rules in G(C + J) in this way provides the calculus, where only the antecedents of a sequent is manipulated. The derivability of Γ ⇒ in the calculus is interpreted asfollows: it is inconsistent to assert all the formulas in Γ. This interpretation does not use thenotion of denial. By employing this calculus, a unilateral analysis is realized by arranging themethod in Section 3. It is also possible to arrange Restall’s analysis in a unilateral manner.
It should be noted that this way of realizing unilateral analysis depends on the existence of theleft and the right rules for classical negation. If classical negation were replaced with intuitionistic one, the transformation described above would be impossible, since a lemma corresponding to Lemma 13 would no longer hold. Thus, although this unilateral analysis is possible for
C + J, it does not follow that the same unilateral analysis is possible for another combination,
since some combination might lack the left and the right rules for classical negation.
R EFERENCES[1] Lloyd Humberstone. Interval semantics for tense logic: Some remarks. Journal of Philosophical Logic,
8:171–196, 1979.
[2] Luis Fariñas del Cerro and Andreas Herzig. Combining classical and intuitionistic logic or: Intuitionistic
implication as a conditional. In Franz Badder and Klaus U Schulz, editors, Frontiers of Combining Systems:
FroCoS 1996, pages 93–102. Springer, March 1996.
[3] Masanobu Toyooka and Katsuhiko Sano. Analytic multi-succedent sequent calculus for combining intuitionistic and classical propositional logic. In Sujata Ghosh and R Ramanujam, editors, ICLA 2021 Proceedings:
9th Indian Conference on Logic and its Applications, pages 128–133. March 2021.
[4] Greg Restall. Truth values and proof theory. Studia Logica, 92:241–264, 2009.
[5] Mitio Takano. A semantical analysis of cut-free calculi for modal logics. Reports on Mathematical Logic,
53:43–65, 2018.
[6] Dirk van Dalen. Logic and Structure. Springer, fifth edition, 2013.
[7] Mitio Takano. A modified subformula property for the modal logics K5 and K5D. Bulletin of the Section of
Logic, 30(2):115–122, 2001.
[8] Katsuhiko Sano and Sakiko Yamasaki. Subformula property and Craig interpolation theorem of sequentcalculi for tense logics. In Nicola Olivetti and Rineke Verbrugge, editors, Short Papers of Advances in Modal
Logic (AiML 2020), pages 97–101. July 2020.
[9] Hiroakira Ono and Katsuhiko Sano. Analytic cut and Mints’ symmetric interpolation method for biintuitionistic tense logic. In David Fernández-Duque, Alessandra Palmigiano, and Sophie Pinchinat, editors,
Advances in Modal Logic, volume 14, pages 601–623. July 2022.
[10] Greg Restall. Multiple conclusions. In Peter Hájek, Luis Valdés-Villanueva, and Dag Westerståhl,
editors, Logic, Methodology and Philosophy of Science Proceedings of the Twelfth International Congress, pages 189–205. King’s College Publications, page reference to the pdf version:
https://consequently.org/writing/multipleconclusions/ edition, 2005.

81

Logic Operators and Quantifiers in Type-Theory of
Algorithms
Roussanka Loukanova
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria
rloukanova@gmail.com

Abstract. In this paper introduce an extension of Moschovakis Type-Theory of Algorithms
and its reduction calculus, by adding quantifiers to the logic operators. There are two kinds ofterms of formulae, for designating pure and state-dependent propositions and predications. The
logic operators include conjunction, disjunction, and negation. I extend the formal language
by pure and state-dependent quantifiers, for enhancing the standard quantifiers of predicate
logic. I provide extended reduction calculus of the Type-Theory of Acyclic Algorithms, for
reductions of terms to their canonical forms.

1

Introduction to Type-Theory of Acyclic Recursion

The set TypesLλar is defined recursively, in Backus-Naur Form:τ ::= e | t | s | (τ → τ )(Types)

We shall use the following abbreviations:σe ≡ (s → σ), state-dependent objects of type σee ≡ (s → e), state-dependent entitieset ≡ (s → t) state-dependent truth values−(→
τ → σ) ≡ (τ1 → · · · → (τn → σ)) ∈ Types(n ≥ 1)
currying coding, for σ, τi ∈ Types, i = 1, . . . , n

(1a)
(1b)
(1c)
(1d)

Typed Vocabulary of Lλar : For every τ ∈ Types, denumerable sets of typed constants,
Kτ = Constsτ = {cτ0 , . . . , cτk , . . . } and two kinds of infinite, denumerable sets of typed
variables, pure variables, PureVarsτ = PureVτ = {v0τ , v1τ , . . . } and recursion (memory)
variables, RecVarsτ = RecV = {r0τ , r1τ , . . . }.
Definition 1 (Terms). Terms = Terms(Lλar ) = ∪ Termsτ where, for each τ ∈ Types,
Termsτ is the set of the terms of type τ , defined by (2a)–(2g), in a typed style of
Backus-Naur Form (TBNF), and notations for type assignments, A : τ and Aτ :A :≡ cτ : τ | xτ : τ(constants and variables)(2a)| B (σ→τ ) (C σ ) : τ| λ(v σ )(B τ ) : (σ → τ )(application terms)(λ-abstraction terms)

(2b)
(2c)

| Aσ0 0 where { pσ1 1 := Aσ1 1 , . . . , pσnn := Aσnn } : σ0

(recursion terms)

(2d)

82

| ∧(Aτ2 )(Aτ1 ) : τ | ∨ (Aτ2 )(Aτ1 ) : τ | → (Aτ2 )(Aτ1 ) : τ(conjunction / disjunction / implication terms)(2e)| ¬(B τ ) : τ| ∀(v σ )(B τ ) : τ | ∃(v σ )(B τ ) : τ

(2f)
(2g)

(negation terms)(logic quantifier terms)

given that
(1) c ∈ Kτ = Constsτ , x ∈ PureVτ ∪ RecVτ , v ∈ PureVσ(2) In (2d): for i = 1, . . . , n, pi ∈ RecVσi are pairwise different recursion (memory)
variables, such that the sequence of assignments { pσ1 1 := Aσ1 1 , . . . , pσnn := Aσnn } isacyclic by the Acyclicity Constraint (AC)(3) In (2e)–(2g), τ ∈ { t, et } are for state-independent and state-dependent truth values
Acyclicity Constraint (AC) The sequence of assignments (3a) is acyclic iff there
is a function rank : {p1 , . . . , pn } → N, such that:{ pσ1 1 := Aσ1 1 , . . . , pσnn := Aσnn } (n ≥ 0)
if pj occurs freely in Ai , then rank(pi ) > rank(pj )

(3a)
(3b)

Without the AC, (2a)–(2d) define the language Lλr of full recursion, to which I add
terms of the logic operators and quantifiers (2e)–(2g).
We say that a term A is explicit iff the constant where designating the recursion
operator does not occur in it. A is a λ-calculus term iff it is explicit and no recursion
variable occurs in it.
Denotational Semantics of the Logic Operators and Quantifiers The detailed denotational semantics of Lλar is presented in Moschovakis [3] and Loukanova [1]. For any
given semantic structure A having a semantic frame T = { Tσ | σ ∈ Types } of typed
domains Tσ , and the set G of all variable valuations g in T, the denotation function
denA (A) is defined by structural induction on the terms A in (2a)–(2d).
In this paper, I extend denA (A) for terms of the form (2e)–(2g). For instance, thedenotation of the state-dependent existential quantifier is:for τ = et, denA ∃(v σ )(B τ ) (g) : Ts → Tt is a function, such that:
(4a)

 
for every state s ∈ Ts : denA ∃(v σ )(B τ ) (g) (s) = 1 (true in s)(4b)
iff there is a ∈ Tσ , in the semantic domain Tσ , such that: A τ
den B (g{ x := a }) (s) = 1(4c)

The set of the immediate terms consists of all terms of the form (5), for p ∈ RecV,ui , vj , ∈ PureV (i = 1, . . . , n, j = 1, . . . , m, m, n ≥ 0), V ∈ Vars:
T :≡ V | p(v1 ) . . . (vm ) | λ(u1 ) . . . λ(un )p(v1 ) . . . (vm )(5)Every term A that is not immediate is called proper. Intuitively, a term is immediate
if its denotation den(A) is obtained by the values provided via the variable valuationsg ∈ G, without any algorithmic steps of computations.

83

Algorithmic Semantics The algorithmic meaning of a proper A ∈ Terms, i.e., a nonimmediate, algorithmically meaningful term, is designated by alg(A) and is determined by its canonical form cf(A). Informally, for each meaningful term A, the algorithm alg(A) for computing its denotation den(A) consists of computations provided
by the basic parts Ai of cf(A) ≡ A0 where {p1 := A1 , . . . , pn := An }, by their structural rank, by recursive iteration.
For every A ∈ Terms, cf(A) is obtained from A by the reduction calculus of Lλar .

2Reduction Rules of Extended Lλar

I designate the logic operators as specialised, logic constants. By that, I classify the
reduction rules for the terms (2e)–(2f), as special cases of the reduction rule forapplication terms. In this section, I extend the set of the Lλar -reduction rules introducedin [3], by adding the reduction rules (ξ) for the quantifier terms (2g).
Congruence The congruence relation between terms is the closure on renamingbound variables (pure and recursion), without causing variable collisions, and on
reordering of recursion assignments:
If A ≡c B, then A ⇒ B(cong)

Transitivity If A ⇒ B and B ⇒ C, then A ⇒ C(trans)Compositionality Replacement of sub-terms with correspondingly reduced ones respects the term structure by the definition of the term syntax:
If A ⇒ A′ and B ⇒ B ′ , then A(B) ⇒ A′ (B ′ )(comp-ap)
If A ⇒ B, and ξ ∈ { λ, ∃, ∀ }, then ξ(u)(A) ⇒ ξ(u)(B)(comp-lq)
If Ai ⇒ Bi , for i = 0, . . . , n, then(comp-rec)
A0 where { p1 := A1 , . . . , pn := An } ⇒ B0 where { p1 := B1 , . . . , pn := Bn }Head Rule Given that pi ̸= qj and pi does not occur freely in Bj
→
− 
→
−
−
−
A0 where { →p := A } where { →q := B }
→
− −
→
−
−
⇒ A where { →p := A , →q := B }(head)

0

Bekič-Scott Rule Given that pi ̸= qj and qj does not occur freely in Ai
→
−  −
→
−
−
A0 where { p := B0 where { →q := B } , →p := A }
→
− −
→
−
−
⇒ A where { p := B , →q := B , →p := A }
0

(B-S)

0

Recursion-Application Rule Given that pi does not occur freely in B
→
− 
→
−
−
−
(A0 where { →p := A } (B) ⇒ A0 (B) where { →p := A }84(recap)Application Rule Given that B ∈ Terms is proper and b ∈ RecV is fresh
A(B) ⇒ A(b) where { b := B }(ap)

λ and Quantifier Rules Let ξ ∈ { λ, ∃, ∀ }ξ(u) (A0 where { p1 := A1 , . . . , pn := An })⇒ ξ(u) A′0 where { p′1 := λ(u) A′1 , . . . , p′n := λ(u) A′n }(ξ)

given that for every i = 1, . . . , n, p′i ∈ RecV is a fresh recursion (memory) variable,
and A′i is the result of the replacement of all the free occurrences of p1 , . . . , pn in
Ai with p′1 (u), . . . , p′n (u), respectively, i.e.:−−→−A′i ≡ Ai {p1 :≡ p′1 (u), . . . , pn :≡ p′n (u)} ≡ Ai {→p :≡ p′ (u)}(7)

Theorem 1 (Canonical Form Theorem). See [1]–[3]. For every term A ∈ Terms,
there is a term cf(A), called the canonical form of A, such that:cf(A) ≡ A0 where { p1 := A1 , . . . , pn := An }(n ≥ 0)
(1) A ⇒ cf(A)(2) for every B, if A ⇒ B and B is irreducible, then B ≡c cf(A), i.e., cf(A) is theunique, up to congruence, term to which A can be reduced(3) FreeV(cf(A)) = FreeV(A); Consts(cf(A)) = Consts(A)Proof. It is long and outlined in Moschovakis [3]. A version is given in Loukanova [1].
⊔⊓Coordinated Predication: a class of sentences with coordinated VPs render [Φj ]np [ΘL and ΨH ] [Ww ]np vp −−−→ λxj λyw L(xj )(yw ) ∧ H(xj )(yw ) (w) (j) (8){z}|algorithmic pattern with memory parameters L, H, w, j

Specific Instantiations of Parametric Algorithms, e.g., (8), by (9c)–(9d):render[John]j loves and honors [his]j wife. −−−→ A
(9a)



A ≡ λxj λyw loves(yw )(xj ) ∧ honors(yw )(xj ) (wife(xj )) (john) (9b)



(9c)
⇒ · · · ⇒ cf(A) ≡ λxj λyw L′′ (xj )(yw ) ∧ H ′′ (xj )(yw ) (w′ (xj )) (j){z}|algorithmic pattern with memory parameters L′′ , H ′′ , w′ , jwhere { L′′ := λxj λyw loves(yw )(xj ),H ′′ := λxj λyw honors(yw )(xj ),w′ := λxj wife(xj ), j := john }|{z}instantiations of memory L′′ , H ′′ , w′ , j

85

(9d)

Logic Quantifiers and Reductions with Quantifier Rules: Explicit logic terms, e.g.,(10a), can be reduced to recursive canonical forms, (10c). Assume that cube, large 0 ∈Consts(ee→ et) . and large ∈ Consts((ee→ et)→(ee→ et)) is a modifier.
renderSome cube is large −−−→ B ≡ ∃x(cube(x) ∧ large 0 (x))(10a)⇒ ∃x((c ∧ l) where { c := cube(x), l := large 0 (x) }) 2x(ap) to ∧; (comp-lq) (10b)⇒ ∃x(c′ (x) ∧ l′ (x)) where { c′ := λ(x)(cube(x)), l′ := λ(x)(large 0 (x)) } (ξ) (10c)
|
|
{z
}
{z
}
instantiations of memory c′ , l′B0 algorithmic pattern

Proposition 1. The Lλar -terms C ≈ cf(C) in (11a)–(11d), similarly to many otherLλar -terms, are not algorithmically equivalent to any explicit term. Thus, Lλar is a strict,proper extension of Montagovian semantics.

render(11a)
Some cube is large −−−→ C ≡ ∃x c′ (x) ∧ large(c′ )(x) where { c′ := cube }
{z
}
|
E0
 ′


′
⇒ ∃x c (x) ∧ l where { l := large(c )(x) } where { c′ := cube }
(11b)
|
{z
}
E1

by (ap) to ∧ of E0 ; (comp-lq); (comp-rec) ⇒ ∃x c (x) ∧ l (x) where { l := λ(x) large(c′ )(x) } where { c′ := cube }
(11c)
{z
}
|
E2


′
′
⇒ ∃x c (x) ∧ l (x) where { c′ := cube, l′ := λ(x) large(c′ )(x) } ≡ cf(C) (11d)
{z
}
{z
}
|
|


′

′



C0 an algorithmic pattern′

instantiations of memory c′ , l′by (head), from (11c); (cong)
Motivation for the Type Theory Lλar : Provides Algorithmic Patterns. Memory variables in Lλar -terms can be instantiated by the corresponding canonical forms, depending on the specific areas of applications and domain specific texts, e.g., as in (10c)and (11d); and (8) as in (9d).
Motivation for Lλar with Logical Operators and Standard Quantifiers: Reasoning
and Semantic Inferences. Canonical forms can be used for reasoning and inferencesof semantic information by automatic provers and prove assistants.
In the full paper, I shall extend the formal language of γ ∗ -reduction calculus, seeLoukanova [2], by the logic operators and standard quantifiers.

References
1. Loukanova, R.: Gamma-Reduction in Type Theory of Acyclic Recursion. Fundamenta Informaticae
170(4), 367–411 (2019). URL https://doi.org/10.3233/FI-2019-1867
2. Loukanova, R.: Gamma-Star Canonical Forms in the Type-Theory of Acyclic Algorithms. In: J. van den
Herik, A.P. Rocha (eds.) Agents and Artificial Intelligence. ICAART 2018, Lecture Notes in Computer
Science, vol. 11352, pp. 383–407. Springer International Publishing, Cham (2019). URL https://doi.
org/10.1007/978-3-030-05453-3_18
3. Moschovakis, Y.N.: A Logical Calculus of Meaning and Synonymy. Linguistics and Philosophy 29(1),27–89 (2006). URL https://doi.org/10.1007/s10988-005-6920-7

86Deriving formal semantic representations
from dependency structures— extended abstract —Philippe de Groote
LORIA, UMR 7503, Université de Lorraine, CNRS, Inria, 54000 Nancy, France

1

introduction

Dependency grammars provide an interesting alternative to phrase structure grammars.
They derive from a long linguistic tradition [3], and are gaining more and more interestin the computational linguistic international community [6]. One of their advantages isthat dependency parsing appears to be more robust than constituency parsing. Indeed,
while parsing an agrammatical sentence with a phrase structure grammar usually leads toa failure, parsing it with a dependency grammar can result in an incomplete dependency
structure that nevertheless carries some semantic information.
Dependency grammars, however, do not seem suitable for a formal semantic treatment, in the tradition of Montague [5]. Formal semantics [2], being compositional, relies
heavily on the notion of constituent, a notion that does not appear explicitly within dependency structures.
A possible remedy to this this situation is to normalize the dependency structures inorder to recover an implicite notion of constituent (see [7], for instance). This approach,however, is not robust in the sense that it does not allow for the interpretation of partial
dependency structures. The goal of this paper is to remedy this problem by laying the
grounds for a new formal theory of dependency semantics, in the spirit of Montague.

2

The basic concept

Consider the following simple sentence:(1)

Michael smiles

Parsing it with a phrase structure grammar would yield a constituency parse tree akin to
the following one1 :1

All the parse trees and the constituency structures occurring in this abstract have been obtained using the Stanfordparser

87

S
NP

VP

NNP VBZ
Michael smiles
Following Montague [4], one may take advantage of the above parse tree in order toderive the truth-conditional meaning of sentence (1). To this end, let us write S, NP, and
VP for the semantic interpretations of the syntactic categories S, NP, and VP. As usual,
one posits:
S=t
VP = e → t
NP = VP → S = (e → t) → t
Then, one assigns the following lexical semantic recipes to Michael and smiles:
MICHAEL = λp. p m : VP → S
SMILE = λx. smile x : VP

This allows one to compute the the truth-conditional semantics of sentence (1) by reducing
the corresponding λ-term:(2)

MICHAEL SMILE →→β smile mNow, we want to apply the same kind of technique to dependency grammars, i.e, to derive
the truth-conditional semantics of sentence (1) form the following dependency structure:
nsubj

NNP
Michael

VBZ
smiles

The solution we develop in this paper is based on a simple idea, which consists in assigning a semantic role to the dependency relations, and computing the desired semantics
from a term akin to nsubj (SMILE, MICHAEL) (or simply, nsubj SMILE MICHAEL, using
the λ-calculus notation). For our current example, it suffices to let nsubj = λvn. n v.
However, as we will see, this interpretation is too simple.

3

The coherence requirement

In pursuing the basic idea we sketched in the previous section, we soon run into an obstacle. There is indeed no canonical way of representing a dependency structure as a term.
Consider, for instance, the following sentence together with its dependency structure:

88

(3)

Michael praises Samuel
nsubj

NNP
Michaelobj

VBZ
praises

NNP
SamuelThere is in fact two ways of encoding the above dependency structure as a term:
(4)

a.
b.

nsubj (obj PRAISE SAMUEL) MICHAEL
obj (nsubj PRAISE MICHAEL) SAMUEL

One could try to circumvent this difficulty by preferring one of these representations to
the other, but such a choice would be arbitrary. Moreover, the resulting solution would not
be robust in the sense that it would not allow for the interpretation of partial dependency
structures. Consequently, we require the coherence condition that both terms (4-a) and(4-b) must yield the same semantic interpretation.
In order to satisfy this coherence condition, we adopt a Neo-Davisdsonian event semantics, and interpet a sentence as a set of sets of events, as suggested by Champolion [1].
This gives rise to the following semantic interpretation:
GS = (v → t) → t
MICHAEL = λp. p m : NP
SAMUEL = λp. p s : NP
PRAISES = λp. ∃e. (praise e) ∧ (p e) : GS

nsubj = λvn. λp. n (λx. v (λe. (agent e x) ∧ (p e))) : GS → NP → GS
obj = λvn. λp. n (λx. v (λe. (theme e x) ∧ (p e))) : GS → NP → GS

4Interpreting the noun phrases

Montagovian semantics assigns to the (common) nouns the semantic category N = e → t.
It assigns to the adnominal modifiers, such as the adjectives, the category ADJ = N → N,
and to the determiners, the category DET = N → NP. This approach is not directlytransferable to the case of dependency structures. Consider indeed the folowing nounphrase and its associated dependency structure:(5)

a red car
det
amod

DT
a

JJ
red

89

NN
car

As a consequence of the coherence condition, the following expressions must be assigned
the same semantic type:
(6)

a.
b.
c.
d.
e.

CAR

amod CAR RED
det CAR A
det (amod CAR RED) A
amod (det CAR A) RED




, we
More generally, if dep is the semantic recipe associated to a dependency edge dep must have:dep : α → β → αwhere α is the semantic type assigned to the source of the edge, and β, the semantic typeassigned to its target.
A way of satisfying the above requirement is to parametrize the type assigned to thehead of a dependency relation with the types assigned to all its possible dependents. In
the case of the expressions listed in (6), this type is then the following one:
GNP = DET → ADJ → NP

Accordingly, the semantic recipes associated to the lexical items and dependency relations
are as follows:A = λpq. ∃x. (p x) ∧ (q x) : DET
RED = λnx. (n x) ∧ (red x) : ADJ
CAR = λda. d (a car) : GNP

amod = λna. λdb. n d (λz. b (a z)) : GNP → ADJ → GNP
det = λnd. λea. n d a : GNP → DET → GNP

5

Revisiting the subject and object dependencies

The typing principle we posited in the previous section must be propagated throughout thegrammar. Therefore, the type assigned to nsubj and obj should no longer be GS → NP →GS but GS → GNP → GS . Similarly, the semantic type assigned to a proper name should
be GNP rather than NP. let us illustrate this with a last example. Consider the following
sentence:(7)

Michael drives a red car
obj
det
nsubj

NNP
Michaelamod

VBZ
drives

DT
a

90

JJ
red

NN
car

This example can be handled using the following semantic recipes, which implement the
principles we have discussed in this abstract:
A = SOME = λpq. ∃x. (p x) ∧ (q x) : DET
RED = λnx. (n x) ∧ (red x) : ADJ
CAR = λda. d (a car) : GNP
MICHEL = λdap. p m : GNP
DRIVE = λp. ∃e. (drive e) ∧ (p e) : GSamod = λna. λdb. n d (λz. b (a z)) : GNP → ADJ → GNP
det = λnd. λea. n d a : GNP → DET → GNP
nsubj = λvn. λp. n SOME (λx. x) (λx. v (λe. (agent e x) ∧ (p e))) : GS → NP → GS
obj = λvn. λp. n SOME (λx. x) (λx. v (λe. (theme e x) ∧ (p e))) : GS → NP → GS
The reader may then check that the four possible expressions that encode the above dependency structure yield all the same semantic interpretation2 of sentence (7), namely:
λf. ∃x. (car x) ∧ (red x) ∧ (∃e. (drive e) ∧ (agent e m) ∧ (theme e x) ∧ (f e))

6

Conclusions

We have discussed and elaborated some principles that provide the basis for a formal theory of dependency semantics. The resulting system satisfies several interesting properties
that the format of this extended abstract does not allow us to illustrate further. In particular,
the toy semantic grammar that supports our last example allows incomplete dependency
structures to be assigned semantic interpretations, showing therefore some robustness. It
also provides an effective treatment of scope ambiguities.

References
1. L. Champollion. The interaction of compositional semantics and event semantics. Linguistic & Philosophy,
38(1):31–66, 2015.
2. I. Heim and A. Kratzer. Semantics in Generative Grammar. Blackwell Publishing, 1998.
3. Igor Mel’čuk. Dependency Syntax: Theory and Practice. State University of New York Press, 1988.
4. R. Montague. The proper treatment of quantification in ordinary english. In J. Hintikka, J. Moravcsik, and P. Suppes,
editors, Approaches to natural language: proceedings of the 1970 Stanford workshop on Grammar and Semantics,
Dordrecht, 1973. Reidel. Reprinted: [5, pages 247–270].
5. R. Montague. Formal Philosophy: selected papers of Richard Montague, edited and with an introduction by Richmond Thomason. Yale University Press, 1974.
6. J. Nivre, M.-C. de Marneffe, F. Ginter, Y. Goldberg, J. Hajič, C. D. Manning, R. McDonald, S. Petrov, S. Pyysalo,
N. Silveira, R. Tsarfaty, and D. Zeman. Universal Dependencies v1: A multilingual treebank collection. In Proceedings of the Tenth International Conference on Language Resources and Evaluation (LREC’16), pages 1659–1666,
Portorož, Slovenia, May 2016. European Language Resources Association (ELRA).
7. S Reddy, O. Täckström, M. Collins, T. Kwiatkowski, D. Das, M. Steedman, and M. Lapata. Transforming dependency structures to logical forms for semantic parsing. Transactions of the Association for Computational
Linguistics, 4:127–140, 2016.
2

up to conjunction commutativity

91

Modal Reasoning and Theorizing
in Quantified Modal Logic
Kohei Kishida, University of Illinois
Logicians have developed various forms of quantified modal logic and its
semantics. In many of these logics, certain metaphysical facts and principles,
and typically those involving singular names and other linguistic features, areendorsed as theorems or necessary truths. For instance, Hesperus is necessarily identical with Phosphorus, or all true identities are necessarily true.
This talk concerns how logic and semantics can represent a cognizer’s reasoning about these facts and principles. On the one hand, many philosophers
maintain that metaphysical facts of this type can be necessary a posteriori.
Most of the semantics they propose, on the other hand, have these facts(or metaphysical principles underlying them) built into the mathematicalmakeup. Therefore, while these semantics deem those facts necessarily true,
they cannot account for their aposteriority. I will take an approach based on
intensional logic that treats all terms, predicates, and quantifiers as uniformly
intensional. Augmenting it with epistemic logic, I will demonstrate how thislogic can represent metaphysical facts and principles as a posteriori facts
and substantial principles that a cognizer can learn and use in their modal
reasoning and theorizing. In a certain sense, this talk attempts to draw a
new line of distinction between logic and theory of modal metaphysics.

92Factivity Alternation Types and Compensatory Prosodic Focus Marking
Chungmin Lee clee@snu.ac.kr
Seoul National University
1. Introduction
This talk explores different cross-linguistic factivity alternation types to see how different
languages mobilize different means of representation for factivity alternation in
epistemic/doxastic attitudes. Altaic languages including Korean show factivity alternation of‘know’ (Japanese alone has no alternation of ‘know’) and other cognitive attitude verbs suchas ‘remember,’ ‘recognize,’ ‘understand,’ etc. by distinct complement endings (Lee 1978, Lee2019). Ch(inese) is the least alternating language, with mostly strictly factive epistemic attitude
verbs, including the representative one 知道[zhidao] ‘know,’ which does not reveal any
alternation signs (‘recognize,’ ‘understand’ are also factives with no alternation). The verb 记
得 [jide] ‘remember,’ in contrast, is an alternation verb with no surface complement structure
differentiation (Lee 2021). Then, how are the two semantic alternants of the verb ‘remember’with no complementizers detected? I argue that it is by means of compensatory prosodic focus
marking, by which the embedding verb part is focused in contrast with the factively
presupposed unfocused complement part. This verbal focus marking occurs in compensation
with the unrealized covert ‘fact’ head noun in apposition with its complement content. Suchcompensation occurs also occurs when alternation-marking complement case endings delete
in Korean (Lee 2020, cf. Jeong 2021). E(nglish) is a moderately alternating language, with nonfactive alternants occurring in some non-veridical contexts such as negative, interrogative,
before, conditional, and a denial context such as but --- not, (in which contexts non-factive
complement marking may occur in Altaic as well). The interrogative whether complementizer
with P or not P is typically presuppositional, conceptually co-related to know, as in remember.
If lexically or conceptually negated as in moru- (K(orean) ‘not know’) and forget, theircomplement presupposition is blocked. Preferential attitudes such as hope and fear can takewhether cross-linguistically, which even engender expletive negation in K (and J(apanese)).
The counterfactual attitude ‘imagine whether’ is endorsed only by amodal/futuristic/conditional complement cross-linguistically (Lee 2022, Liefke 2022). Overall,complements determine factivity.
2. Complement Structures for Factivity Alternation
In Altaic languages with SOV, center-embedded factively presupposed complements havenominal/DP endings (with ACC markers), whereas non-factive complements have REPORT C
regularly (in Uyghur, and Manchurian as well, Lee (2019)).
(1) a. Bat [Mia –giin yav-san-iig] mede-j baina
Factive (Mongolian)B M-GEN leave-NPST-ACC know ST‘Bat knows that Mia left.’ [[-j baina: ST = result state]]b. Ken [Mia-g yav-san gej] mede-j baina
Non-factive (Mongolian)K M-ACC leave-PST REPORT know ST‘Ken non-factively knows [believes] that Mia left.’(2) a. Da [Sue-nun git–tiğin-i]
bil-iyor
Factive (Turkish)
D S-GEN leave-pstN-ACC know-PRES.3Sg‘Da knows that Sue left.’ (Sue’s having left)b. Da [Sue git-ti-diye] bil-iyor D S-NOM leave-PST-REPORT know-PRES-3rdSg‘Da non-factively knows that Sue left’
Non-factive (Turkish)(3) a. Mia-nun [Hia-ka ttena-n kes-ul]
al-ko iss-ta Factive (Korean)

93

M-TOP H-NOM leave-ADNpst ProFactN-ACC ‘know’-ST-DEC‘Mia knows that Hia left.’ [ADNpst: ADNominal with past; ST=-ko iss-: result state]
b. ?Mia-nun [Hia-ka ttena-ess-ta-ko] al-ko iss-ta
Non-factive (Korean)
M-TOP H-NOM leave-PST-DEC-C ‘know’-ST-DEC (Lit.)‘Mia non-factively knows [i.e., believes with evidence] Hia left.’(C=REPORTative complementizer; not in the reportative evidential meaning)
In Chinese, non-Altaic, 知道[zhidao]‘know,’ ‘understand,’ and ‘recognize’ are only factive but
记得[jide]’remember’ is a factivity alternation verb, as follows:(4) a. Ta jide/bu jide [jintian shi faxinri]. (the complement is presupposed) Factive (Chinese)
he remember/not remember today is payday‘He remembers/does not remember today’s payday.’ [Yuan 2020 argues jide is only
factive with no alternation]b. Ta jide [jintian shi faxinri] Qishi jintian bu shi faxinri.
Non-factive (Chinese)
he remember today is payday in fact today not is payday‘He (falsely) remembers today’s payday. In fact, today is not payday.’ (non-factive)
The factive and non-factive complements [jintian shi faxinri] ‘today is payday’ have the same
complement clause form with bare C. 60 native speakers didn’t mention intonation in theirsurvey responses but most responded that they feel the presence of (ting-)shuo ‘(hear-)say’ infront of the complement. It functions as a grammaticalized C (confirmed by Jim Huang, via email, 2022). This ‘say’ C for non-factive complements is common with Altaic and covertly
English. I asked several native speakers whether they feel intonation differences between thefactive vs. non-factive jide sentences and they said ‘yes.’ So, I conducted the intonation
experiment (with PRAAT), the result of which is as follows, as expected. The factively
presupposed complement has no focus and the main V is focused on Fig. 1 below.

Figure 1 The main V jide is focal and high.

Figure 1 The complement part jintian is focal and high.

94

In contrast, the complement element jintian ‘today’ is focused on Fig.2 above. The above
contrast is due to the compensatory focus marking (CFM), I argue, in compensation with thecovert conceptual, structural representation of the FACT DP with its appositional,
presuppositional proposition. The focus marking arises when there occurs rivalry between
factive vs. non-factive alternation in particular. The main attitude verb is foregrounded with its
presupposition backgrounded, non-presupposed elements foregrounded. This happens crosslinguistically. Let’s consider the often-cited English example (Beaver 2010):(5) a. If the T.A. DISCOVERS that your work is plagiarized, I will be forced to notify the dean.
b. If the T.A. discovers that your work is PLAGIARIZED, I will be forced to notify the dean.
In case a conditional operator precedes an epistemic attitude verb, presupposition projection
depends, I argue, on the location of the focus: If the epistemic verb DISCOVERS itself isfocused as in (5a), this factive alternant gets its non-focal complement presupposition triggered.
In contrast, if a part PLAGIARIZED of the complement is foreground focused as in (5b), thenthe complement content is not presupposed. The main verb focus in (5a) is in compensation
with the covert FACT complement. In (5b), in a conditional clause, the complement element
PLAGIARIZED is foreground focused, and it receives a non-factive reading. In this case,
Altaic complements get the overt REPORT C. CFM occurs if alternation rivalry arises.
In Korean, another ProFact noun cwul gets factivity alternating grammatical markings of
ACC -ul for the factive complement of ‘know’ vs. DIRECTIONAL -lo for the non-factive
complement of ‘know’. In conversational Korean, however, these distinguishing case markers
are easily deleted and the bare cwul produces factivity ambiguity. Here, the CFM intervenes:
if the ProFact N occurs with no ACC, then the higher embedding ‘know’ is focused to get its
factive reading (6a) but if the same N with no ProFact function because of the oblique
DIRECTIONAL postposition/case -lo, then a part or the whole of the complement is focusedto receive its non-factive reading (6b). If the main V remains and the complement is deleted,as “na al-a” in K or “I know” in E, it is factive, not non-factive (impossible for ‘believe’).
(6) a. Ken-un [Mia-ka ttena-n cwul]-ø AL-A. b. Ken-un [MIA-ka ttena-n cwul]-ø al-a.
K -TOP M-NOM leave-C ProFact N know K-TOP M -NOM leave-C N
know
Examine the English example of non-factive know in the context of before, which is ananti-veridical operator, not an alternating context, below in (7a) and compare it with otherfactivity alternation languages such as K (Altaic), and factive only Ch (non-Altaic) and J. The
only interpretation of (7a) in E, is its non-factive reading because of the anti-veridical
operator before. The non-factive knew has a covert SAY REPORTative C associated with‘that,’ so the main V cannot be focused.
(7) a. Medieval Koreans knew that Ch characters were the best before Hangul was invented.
b. cwungsei hankwukin-tul-un Hangul-i changcey-toyki cen-ey-nun [Hanca-ka choyko-i
M
K-PL-TOP H -NOM invent become before-TOP H -NOM best-be-n
kes uroal-ass-ta
A(dn)C kes DIR
know-PST-DEC <by Non-F DIR complement ending>c. Zai Hanwen faming yiqian, zhongshiji de Hanguoren renwei/yiwei/*zhidao Hanzi shi.
at Korean letters invent before, Middle Age of Koreans thought/believed/#knew C
characters are the best<by different verb in Ch (and J), belief doxastic only>
Another common non-factive context in English is the consecutive denial but --- not. Observe.
(8) Ken remembers that a hippo sang, but (in fact) no hippo sang.
In this context again, K (and other Altaic as well) uses the non-factive complement ending
REPORT C (or DIRectional PP).
The polar interrogative whether complementizer with P or not P is typically presuppositional,

95

conceptually correlated to know, as in remember (similarly, in Egré’s Vp ⇒ p veridicality indeclarative and White’s whether-licensing Generalization V (‘a predicate is responsive iff it isveridical’). Several epistemic verbs have the same ⇒ relation (Ziever 2022) like ‘x remembersthat p’ ⇒‘x knows that p’ and the same ⇒ relation applies to the whether complement iff thecomplement is presuppositional. Preferential attitude predicates such as ‘hope’ and ‘fear’ take,I claim, the non-presuppositional whether complement with cautious polarity consideration andeven engender expletive negation in Korean and Japanese. The negation anh-, logically not
required, occurs with modal(-futurity) in (9) in K. (Similarly in J).
(9) Mia-ka ipen-ey-nun[caki-ka iki-ci anh-ul-kka]
himangha-n-ta
M -NOM this time-at-TOP self-NOM win-NEG-MOD-Q hope -PRES-DEC‘Mia hopes whether she may ((notexpl)) win this time.’
The rare case of counterfactual attitude predicate imagine whether will also be examined crosslinguistically (K, Ch, and Italian), to see why modal/conditional complement only is allowed,
whereas the ProFact N kes(-ul) K/koto(-o) J is odd as a counterfactual attitude predicate.
3. Fact vs. REPORT C Semantics for Semantic Universal(s)In line with Kiparsky and Kiparsky 1971, Kratzer 2006, 2015, and Schueler 2016, I analyze
a factively presupposed complement as a complement with the FACT DP head noun (or at least
with thatF in English) and the ProFact N DP with kes in K/or koto in J. This ProFact N isequivalent to FACT, referring to the embedded clause content (like a pronoun) via internal
perception and exemplification. Its adnominal C –(nu)n (‘high pitch’ in J) connects the
propositional clause content in apposition to the ProFact N. In contrast, the (overt or covert)
REPORT C, derived from ‘say,’ functions as C, searching for a higher non-factive attitude
predicate. If it happens to hit higher epistemic verbs, they become non-factive doxastic verbs.
The FACT/ProFact DP composes with [[knowF]] and other epistemic verbs that entail[[knowF]]
suchas[[recognizeF]](⇒[[knowF]]),[[be-awareF]](⇒[[knowF]]),[[rememberF]](⇒[[knowF]]), [[understand F]] (⇒[[knowF]]), [[discoverF]](⇒[[knowF]]),[[realizeF]](⇒[[knowF]]), [[see (that)]](⇒[[knowF]]), etc.
The FACT/ProFact DP also composes with those higher embedding lexically negative, antiveridical verbs that entail [[∼knowF]] such as [[moru-]](‘not know,’ K), [[forgetF]], [[ic-]](‘forget,’ K), [[bu-jide]](‘not-remember,’ Ch)/忘记 [wàngjì] ‘forget,’ Ch), etc. (Lee 2019,
Jeong 2021 for K). In this group, a wide scope negation is blocked and the factivity readingalone remains.
The verb [[forgetF]], is factive and its complement is presupposed, whereas the implicative
verb [[forgetIMPL]] cannot have a presuppositional component; its construction is limited to theinfinitive and involves volitional modality (as in action V). This implicative meaning is crosslinguistcally attested: K Ken-un mwun-ul camku-l kes-ul ic-ess-ta ‘Ken forgot to lock the door’;
J wasureru ‘(forgetF’ or ‘forgetIMPL’ tense); Ch jide ‘forgetIMPL’ modal yao).
[[wonder]] has its ingredient of [[knowF]]) but because it is embedded by a bouletic modal[[want]] as in J shiri-tai ‘know-want’, implying ‘not-know,’ it takes the interrogative C whether.
Karttunen’s (1977) thesis that verbs of communications (such as tell, indicate, inform,
disclose, etc.) when used with the complementizer whether give rise to “truth conveying”
operators meets some pragmatic objections (Tsohatzidis 1997).
4. Concluding Remarks
By dividing between FACT DP presuppositional and REPORT C CP non-preuppositional
complement clauses, we can come up with factive vs. non-factive attitude predicates (with96

factivity alternation) by composition. Alternation (conceptual and structural), with no surface
distinction, is compensated for by prosodic focus-marking intonation, if rivalry occurs,
triggering or not triggering presupposition, along the positions in speech.
This way, we can approach the goal of attaining the semantic universal(s) of factivity vs.
non-factivity alternation.

REFERENCES
Lee, Chungmin. 1978. Analysis of the Word al-ta ‘know’ [in Korean] ('알다'라는 말의분석), 마음(Mind), 1, 60-65, 1978.
Lee, Chungmin.2019. Factivity Alternation of Attitude ‘know’ in Korean, Mongolian,
Uyghur, Manchu, Azeri, etc. and Content Clausal Nominals. Journal of Cognitive
Science 20(4), 449-504.
Lee, Chungmin. 2020. Content Nominals with and without Factive Presupposition.
LENLS 17 Proceed’s. Nov. Tokyo.
Jeong, Sunwoo 2021. Melodies of Truth: Deriving Factive Inferences from Alternatives. At
the MECORE Workshop.
Lee, Chungmin. 2021. Factivity in Chinese: Viewed from the Altaic Alternation Type. At
LENLS 18 (Proceed’s).
Lee, Chungmin. 2022. How Factivity Alternates for ‘remember’/’imagine’ (?) as well. At theWorkshop, ESSLLI (Galway, Ireland): The Semantics of Imagination.
Liefke, Kritina 2022. Meaning-Driven Combinatorial Restrictions and ‘imagine whether’. At
the ESSLLI Workshop: The Semantics of Imagination.
Egré, P. 2008. Question-embedding and Factivity. Ms. https://jeannicod.ccsd.cnrs.fr/ijn
00226386/ (2008).
Karttunen, L. 1977. Syntax and Semantics of Questions. Linguistics and Philosophy, 1, 3–44.
Kiparsky Paul and Carol Kiparsky 1970. Fact. In Progress in Linguistics. Mouton.
Kratzer, A. 2006. Decomposing Attitude Verbs. Ms., University of Masachusetts at Amherst.
Kratzer, A. 2015. Modal Domains. Handout.
Schueler, D. 2016. FACTIVITY AND COMPLEMENT-TYPES. Studia Linguistics 70.3. 207335.
Tsohatzidis, S. L. 1997. More Telling Examples: A reply to Holton. Journal of Pragmatics, 28,
625–638.
White, A.S. 2019. Nothing’s Wrong with Believing (or Hoping) whether. Under review at
Semantics and Pragmatics.
Zuber, Richard 2022. Properties of Propositional Operators. Linguistics and Philosophy.
(Internet)

97

The Discourse Function of Aspect in Japanesea

Kei Yoshimoto

Joseph TaboltbZhen Zhouc

Hiromi Kajib

Tamami Shimadab

1 Introduction
Aspect in Japanese, particularly the presence of the te iru form and its absence (i.e.,
the non-te iru form), has mainly been studied on a single sentence basis. In order to get amore exact picture of the aspectual system of the language, this paper sheds light on how it
works in discourse, specifically considering whether reference points (hereafter RP’s) should
be adopted for this purpose. As a theoretical framework, we rely on Segmented Discourse
Representation Theory (SDRT; Asher and Lascarides 2003), which formalizes the discourse
coherence.

2 Previous studies
Kamp and Reyle (1993; hereafter K&R), after Hinrichs (1986), apply event semantics
to the elucidation of tense and aspect in English by specifying relationships of an event(represented as e) or a state (s) to a temporal location (t; the denotation of temporaladverbial phrases, which is hypothesized to be present even without them) and the utterance
time (n). The temporal relationships are denoted, e.g., by ‘<’ for the earlier/later relation,by ‘⊆’ for inclusion, and by ‘◦’ for overlapping. They use an RP (Reichenbach 1947) torelate the temporal meanings of sentences. An event (e) establishes a new RP, which occurs
after the old one, while a state (s) does not add a new RP, instead including the previousone.
Lascarides and Asher (1993; L&A) cover more complicated temporal relationships between eventualities than K&R by introducing, following Hobbs (1985), discourse relations(hereafter DR’s), which capture a rhetorical role of an utterance by investigating how it
relates to others. They solve inconsistencies between rules by giving priority to the most
specific of the applicable defeasible rules in the nonmonotonic logic. They do away withRP’s, directly establishing relationships between eventuality pairs by reference to a DR oran Aktionsart.
Asher and Lascarides (2003; A&L) give accounts of discourse understanding, similarly to
L&A. The logical form of a discourse is constructed eﬃciently through the glue logic, which
has only restricted access to various information sources. A&L don’t oﬀer a systematic
treatment of tense and aspect, but do analyze some temporal relations as directly derived
from DR’s, as in L&A.

3 Interpretation of tense and aspect and an RP
We propose to adopt RP’s for the interpretation of tense and aspect, in contrast to L&A
and A&L, for the following reasons.
First, in a te iru sentence with a perfective meaning, a ni wa-marked temporal adverbial
often indicates a limit by which the event is accomplished. Such a temporal adverbial clearly
introduces an RP.
(1) Kinoo
wa, go-ji
ni wa kitaku-shi
teita.
yesterday tpc five o’clock limit come home-lv ptcl perf pst‘I was home by five o’clock yesterday.’a

Tohoku University (professor emeritus)
Meikai UniversitycRitsumeikan Asia Pacific Universityb98Likewise, a temporal adverbial or a noun phrase often introduces a discourse referent, which
provides a basis for the temporal interpretation of the following part of the text, typicallyin the beginning of a story or embedded by a predicate for belief or speech.
On the other hand, for the lexical description of temporal adverbials with contextdependent meanings (e.g., sono go ‘after that’ and zenjitsu ‘the previous day’), an RP isneeded for the basis of interpretation.
Furthermore, a te iru sentence can occur alone without neighboring sentences, whichleads to hypothesizing an implicit RP supported by the extralinguistic context.
Lastly, some of te iru sentences with the experiential usage perform the role of anevent in the discourse, but are depicted from the perspective of a certain temporal point(see Section 5). This kind of sentence can be accounted for only with the concept of RP.
These facts motivate us to establish RP’s as discourse referents. In doing so, we haveto integrate information not only from predicate meanings, but also from those of temporaladverbials, in distinction from K&R. Moreover, a task still remains for us to find a way toapply RP’s to real texts with complicated text structures.

4 Continuous usage of te iru form
The usage of the te iru form has been traditionally classified into four types: the continuous, resultative, experiential, and collocations with verbs which are virtually adjectives.
This section mainly discusses the continuous and later mentions the resultative. In bothusages, RP’s are brought forth by events and aﬀect the interpretation of states. Let us see
how the tense and aspect in text (2) are interpreted.
(2)

i

A
kun wa, tadachini
dooshiokyuugoo-shi te,name title tpc immediately like-minded people acc rally-lv
ptcl
ii
iii
sutoraiki ohakat ta. Zengakkyuu nooosawagi ni natta.
strikeacc plan pst the whole class adnm confusion cpl become pst
iv
Watashi wa, kyoohu notame ni wanawana hurue teita.
Itpc fearadnm cause for quivering tremble ptcl cont pst‘A immediately rallied students of like mind and planned a strike. The whole class
was thrown into confusion. I was trembling all over for fear.’

Sentence (2.i) is analyzed as the Segmented Discourse Representation Structure (SDRS)
tagged with π1 in Figure 1. Only the relevant part of the meaning is shown.
Sentence (2.ii) is parsed likewise as π2 . If two SDRS’s α and β stand in a certain DR
in some constituent λ, and additionally it is from world knowledge assumed that events ofthe sort described by α lead to events of the sort described as β (i.e., if α occasions β),then the DR Narration is inferred by default, as stipulated by (3) (A&L). We constrain
both the eventualities to be nonstative. me outputs the main eventuality of the relevant
constituent, i.e., the eventuality introduced by the main verb in its head (A&L).
(3) (?(α, β.λ)∧ occasion(α, β)∧ event(me(α))∧ event(me(β))) > N arration(α, β, λ)Defeasible Modus Ponens produces from this the inference that the DR Narration between
π1 and π2 . This and the following rule establish a relationship between the RP’s of π1 andπ2 .
(4) ϕN arration(α,β,λ) ⇒ natural sequence(rp(α), rp(β))
r (eference-)p(oint) is a function which outputs the RP of the SDRS it takes. Here is atentative definition, which still needs prescriptions for other cases of temporal adverbials:

99

s4 t4 x
t4 < nπ4 ∶ s4 ◦ t4
s4 ◦ srp(π4 )s4 : x PROG(tremble)

e1 t1 x y
t1 < nπ1 ∶ e ⊆ t
11
e1 : x rally y
Figure 1: SDRS for (2.i)

Figure 2: SDRS for (2.iv), underspecified(5) If limit(temporal adv(α)), then rp(α) = temporal adv(α);
Otherwise, if event(me(α)), then rp(α) = me(α).
If the DRS is a sentence with a temporal adverbial which limits the accomplishment of anevent, the value of the function is the meaning of the temporal adverbial. Otherwise, if therelevant SDRS represents an event, its value is identified with the main event in it. In thiscase, r1 = rp(π1 ) = e1 and r2 = rp(π2 ) = e2 .
When natural sequence holds between two RP’s, they stand in an earlier/later relationship separated by an interval of some length which is possibly null and is compliant withthe world knowledge (this is glossed as r1 ⪯ r2 ). A&L take an entirely diﬀerent approach:they constrain the relationship between the poststate of the first event and the prestate of
the second to overlap. However, we don’t follow them, owing to the diﬃculty in defining
pre/poststates.
DR’s are classified into three groups: subordination, in which one proposition adds descriptions to the other (e.g., Elaboration), coordination, i.e., text developmemt as successionof events (e.g., Narration), and the other group (e.g., Parallel ).
The semantics of sentence (2.iii) is given in exactly the same manner as for (2.i) and(2.ii), with (2.ii) and (2.iii) related by Narration. Furthermore, r2 ⪯ r3 .
Next, the analysis of sentence (2.iv) produces the underspecified SDRS in Figure 2. The
function s(ister-)r (eference-)p(oint) supplies the RP of the SDRS existing, not necessarily
last, in the context which is combined by a DR with the SDRS it takes. The second andthird from last lines in Figure 2 are by default shared by all the stative predicates, including
the continuous usage of the te iru form of verb.
Following K&R’s analysis of the progressive, we assume the SDRS condition for thecontinuous predicate. s4 stands for the state of the speaker’s being trembling.
π4 is connected to π3 by the DR Background (i.e., Background(π3 , π4 )). Because of its
behavior in terms of anaphoric reference and tense interpretation, it is treated as a kind ofsubordination, avoiding complicated manipulations to include it in coordinating relationsas practiced by A&L.
The rule (6) applies to two SDRS’s for which a subordinating relation holds. At thestage of having analyzed sentence (2.iv), the SDRS π4 remains underspecified with the valueof srp(π4 ) being undetermined. (6) equates it with the RP of π3 (rp(π3 ) = srp(π4 ) ◦ s4 ).
Furthermore, rp(π3 ) = r3 . Note that the relevant value is fully specified in Figure 3,
the result of resolution embedded within the SDRS for the whole text. The relationshipbetween the state and the RP is specified with overlapping (◦), which is a relation withwider coverage. Since r3 = e3 is not an interval, r3 ⊆ s4 is inferred from s4 ◦ r3 .
(6) ϕSubordination(α,β) ⇒ srp(β) = rp(α)
This works even when the subordinate proposition β occurs before α.
The whole SDRS for text (2) explicitly represents the relationships between RP’s as r1⪯ r2 ⪯ r3 , additionally with the specification s4 ◦ r3 in the embedded SDRS π4 (see Figure100estxyt= ne⊆ tπ∶ e ss ◦ r0e: x write ys4 t4 xt4 < nπ4 ∶ s4 ◦ t4
s4 ◦ r3
s4 : x PROG(tremble)
Figure 3: SDRS for (2.iv), specified

Figure 4: SDR for (5.ii)

3). Thus the addition of RP’s as discourse referents, which L&A and A&L lack, represents
the temporal development of texts in an intuitively comprehensible manner.
Our formalization has captured a parallelism between tense/aspect and reference by
NP’s (Partee 1973) in that an event introduces into the context a new RP, which provides abasis for understanding states including those denoted by te iru clauses. Specifically, boththe stative aspect and anaphora remain underspecified when a single sentence has been
analyzed, needing information from the context. On the other hand, a diﬀerence has been
made clear. Reference by anaphora is essentially ambiguous since its antecedent needs to bechosen from among multiple candidates. By contrast, the RP is exceptionlessly identified
with that of the constituent with which the stative sentence is connected by a DR, without
a need for ambiguity resolution, once the connection has been made.
The resultative usage is analyzed similarly, except that the main predicate is RESULT
which outputs the result state caused by the event.

5 Experiential usage of te iru form
The so-called experiential usage of the te iru form indicates that an eventuality that
occurred in the past exerts an indirect influence until a certain time, which is usually theutterance time. It includes cases in which none of its result but only its record remains, asin (7).
(7)iKagemichi wa, Ooshuu Shirakawa de kuma ga kuma no
ko o
name
tpc place
loc bear nom bear adnm child acc
ii
tsure
te
tooru no omikake ta. Sokode nikki ni ‘Kodomo ni
accompany ptcl pass nml acc see
pst thendiary loc childdat
hitome mi setai’ toshirushi teiru.
a look see caus desid quot write
ptcl expr‘At Shirakawa in Ooshuu, Kagemichi saw a bear walking on the street with its cubs.
He wrote in his diary that he wanted to let his son have a look at them.’

This type of te iru-sentence like (7.ii) has a noticeable feature of being combined by a DR
Narration with another clause. Thus we need to treat it as an event, while assigning it
other pieces of information typical with the te iru form, including assignment of an RP.
Figure 4 is an SDRS for this sentence, which is analyzed as an event and therefore canbe combined with Narration. Simultaneously, a state s is specified in such a way that it
starts when e ends (only the temporal relation is shown) and overlaps with an RP r0 . All
these are introduced by the lexical description for this usage of te iru. r0 is identified withan RP at the global level, the utterance time in this case. Note that an RP is indispensable
to clarify the distinction from sentences without te iru marking, which is neutral in termsof a point of view.

101π1 : r 1

.56 RP’s in complex text structures
Elaboration
The studies on interpretation of tense and as?
pect in texts (K&R, L&A, and A&L) have so far′
π : r1been limited to cases with relatively simple text

H

H

HH
structures. It is yet to be seen if the RP defined

-H
π
:
π3 :
above can be applied as it is to real texts or its es2
Narration
r
⊆
r
r
⊆ r1
2
1
3
sential revisions are needed. In this section, we investigate how we should modify it for one of typicaltexts with complexity, where a constituent which Figure 5: Discourse relations in (8)elaborates a preceding sentence is a complex one made up of sentences with a Narration
relation.
(8) π1 : Taroo wa tanoshii nichiyoobi osugoshi ta.
name tpc happy Sunday
acc spend pst‘Taroo spent a happy Sunday.’
π2 : Asa
wa kooen osampo-shi ta.
morning tpc park path walk-lv pst‘In the morning, he walked in the park.’
π3 : Sorekara, gaaruhurendo to eiga omi ni
it ta.
thengirlfriend
with movie acc see goal go pst‘Then he went to see a movie with his girlfriend.’
In this text, sentences π2 and π3 , which elaborate π1 , share the time of spending a happy
Sunday with the first sentence. However, they don’t inherit its RP as it is; rather, it is split
into two parts, one of which precedes the other. To cope with such cases, we need to revise′
the rule (4 ) so that the two consecutive sentences (π2 and π3 in (8)) may each share only
a part of the RP of the constituent they elaborate (π1 in (8)). See Figure 5.
′(4 ) ϕN arration(α,β,λ) ⇒ natural sequence(rp(α), rp(β)) ∧ rp(α) ⊆ rp(λ)∧ rp(β) ⊆ rp(λ)
The RP is by default an instant, but it is compatible with being a temporal interval.

7 Conclusion
Evidenced by temporal adverbials, lexical items, and te iru’s requirement for RP’s, wehave situated RP’s as discourse referents and proposed a framework based on SDRT to
explain the behavior of te iru-marked and non-te iru-marked sentences. This has madeclearer and more formal the intuitive observation that tense and aspect are like anaphora.
We have also proposed a way to apply the concept of RP to real complex texts.

References
Asher, Nicholas and Alex Lascaride. (2003) Logics of Conversation. Cambridge University Press.
Hinrichs, Erhard. (1986) Temporal Anaphora in Discourses of English. Linguistics and Philosophy
9, 63-82.
Hobbs, Jerry R. (1985) On the Coherence and Structure of Discourse. Report No. CSLI-85-37,
Center for the Study of Language and Information, Stanford University.
Kamp, Hans and Uwe Reyle. (1993) From Discourse to Logic. Kluwer Academic Publishers.
Lascarides, Alex and Nicholas Asher. (1993) Temporal Interpretation, Discourse Relations andCommonsense Entailment. Linguistics and Philosophy 16, 437-493.
Partee, Barbara. (1973) Some Structural Analogies between Tenses and Pronouns in English. The
Jounal of Philosophy 70, 601-609.

102

[Abstract]
Left-branching tree in CCG with D combinator∗
Masaya Taniguchi1 and Satoshi Tojo1
1Japan Advanced Institute of Science and Technology{taniguchi,tojo}@jaist.ac.jp

The essence of incremental parsing is to construct a partial syntactic structure stepwise from the head of a sentence. This parsing algorithm is especially
preferable when we analyze the process for us to listen to/ read a sentence in thetemporal order, and is beneficial to reveal an intermediate state of natural language
understanding and to parse a long natural language sentence.
Some may argue that we possess a short term memory and can look ahead multiple words at a time prior to compose a tree structure, and thus, our parsing process
may not be incremental. However, even though we can pool a few number of words
as a chunk, such chunks must be processed incrementally since these chunks could
hardly be pooled again unless there is a secondary short-term memory. In addition, Chomskian syntactic theory may generate an arbitrarily long sentence thoughgenerative rules are finite, and thus, we cannot predefine a ceiling of memory size.
Therefore, we consider a word as the minimal chunk for the base case, and implement the word-by-word process. For example, the human chops Sentence (1)into chunks: ‘High above the city,’ ‘on a tall column,’ ‘stood a statue,’ and ‘of theHappy Prince.’ For every chunk processed by the reader, they could understandthe meaning of the sentence incrementally.
(1)

High above the city, on a tall column, stood a statue of the Happy Prince. ¹

In categorial grammar (CG) and its variants, there are incremental parsing algorithms [1, 2, 3], but not all grammatical sentences could be parsed by thesealgorithms. It is empirically known that all the grammatical sentences is parsed∗ This work was supported by the JSPS KAKENHI Grant Number 22H00597 and and the Grant-

in-Aid for JSPS Fellows Number 21J15207.
¹Oscar Wilde, The Happy Prince and Other Tales, 1888.

103

incrementally by CG with combinator rules. As the previous researches were software simulations of the incremental parsing, we could not obtain that those algorithms work on a sentence of any length. Thus far, we have empirically shown thatany grammatical sentences could be incrementally parsed [4], but no proofs to thisis known to us.
A naïve structure generated of incremental parsing is a left branchingSStree, which has binary nodes only on
the left branches. We obtain the increNP
VP
VP∗
NP
mental parsing by our transformationfrom a given arbitrary tree to the leftVBP NP NP VBP
branching tree.
First, we define CG and its variants.
CG is proof system 𝐴 consists of three
I love you I
love youaxiom schemata with a cut rule, where
the lower letters are arbitrary any cate- Figure 1: Non-incremental parsing (left)gory of CG and the capital Greek let- and Incremental parsing (right)
ters are sequences of categories.
𝑥 ` 𝑥 Id

𝑥/𝑦, 𝑦 ` 𝑥

A>𝑥, 𝑥\𝑦 ` 𝑦A<Γ ` 𝑥 Σ, 𝑥, Δ ` 𝑦CutΣ, Γ, Δ ` 𝑦

For example, (I) “I love you” is parsed as 𝑛𝑝, (𝑛𝑝\𝑠)/𝑛𝑝, 𝑛𝑝 ` 𝑠 with thefollowing derivation.
A>A<(𝑛𝑝\𝑠)/𝑛𝑝, 𝑛𝑝 ` 𝑛𝑝\𝑠
𝑛𝑝, 𝑛𝑝\𝑠 ` 𝑠Cut𝑛𝑝, (𝑛𝑝\𝑠)/𝑛𝑝, 𝑛𝑝 ` 𝑠
According to the proof net’s convention, we use a graph. The axiom schemata
A>, A<are represented by the graphs in Figure 2, and the cut rule is denoted as theconnection of the two graphs. Then, the graph of the above example is represented
𝑠
𝑠

𝑛𝑝\𝑠
A<

A<
𝑛𝑝

𝑛𝑝\𝑠

t
Cu

A>

(𝑛𝑝\𝑠)/𝑛𝑝

𝑛𝑝

A>

𝑛𝑝
(𝑛𝑝\𝑠)/𝑛𝑝

Figure 2: Graph of 𝐴 < and 𝐴 >𝑛𝑝

Figure 3: Graph of (I)104as the graph in Figure 3. For the sake of convenience, we employ a new notation ofgraphs; the graph in Figure 2 is A< [𝑛𝑝, 𝑛𝑝\𝑠] 𝑠 and A> [(𝑛𝑝\𝑠)/𝑛𝑝, 𝑛𝑝] 𝑛𝑝\𝑠 . We
use the binary operator as adjoining tree by Cut. Then, the graph in Figure 3 isthe following expression.
A> [(𝑛𝑝\𝑠)/𝑛𝑝, 𝑛𝑝] 𝑛𝑝\𝑠

A< [𝑛𝑝, 𝑛𝑝\𝑠] 𝑠 = A> [𝑛𝑝, A< [(𝑛𝑝\𝑠)/𝑛𝑝, 𝑛𝑝] 𝑛𝑝\𝑠 ] 𝑠Generally, only the following eight patterns are possible for each triplet ofnodes, where capital Roman letters are expressions with the root category denotedas the subscript. Especially, we call the patterns (7) and (8) the backward long reference. In these patterns, the third category includes a hidden category for the firstcategory, e.g., The category 𝑧 in the pattern (7) is appeared in the left-most/ rightmost nodes as a part of functional categories. For example, post-posited adverbs‘only’, in linguistics, is corresponding to this reference, which are grammaticalrules but are not preferable in the conversation.
(1)
(2)
(3)
(4)

A> [A> [𝑋 (𝑥/𝑧)/𝑦 , 𝑌𝑦 ], 𝑍 𝑧 ] 𝑥
A> [A< [𝑋𝑥 , 𝑌𝑥\(𝑦/𝑧) ], 𝑍 𝑧 ] 𝑦
A< [A> [𝑋𝑥/𝑦 , 𝑌𝑦 ], 𝑍𝑥\𝑧 ] 𝑧
A< [A< [𝑋𝑥 , 𝑌𝑥\𝑦 ], 𝑍 𝑦\𝑧 ] 𝑧

(5)
(6)
(7)
(8)

A> [𝑋𝑥/𝑦 , A> [𝑌𝑦/𝑧 , 𝑍 𝑧 ]] 𝑥
A< [𝑋𝑥 , A> [𝑌(𝑥\𝑧)/𝑦 , 𝑍 𝑦 ]] 𝑧
A> [𝑋𝑥/𝑧 , A< [𝑌𝑦 , 𝑍 𝑦\𝑧 ]] 𝑥
A< [𝑋𝑥 , A< [𝑌𝑧 , 𝑍 𝑧\(𝑥\𝑦) ]] 𝑦

In the system 𝐴, (I) is not parsed incrementally because there is a binary nodeon the right branch in Figure 3. Hence, we employ a new proof system 𝐴𝐵𝑇 𝐷 ∗ ,which parses (I) without binary nodes on the right branch. Note that the systemallows the cut rule only on the left-most category. Therefore, it is exactly corresponding to the left-branching tree that has not binary nodes on the right branches.
Hereafter, we use the binary operator ~ corresponding to Cut∗ .
A>A<
B>𝑥/𝑦, 𝑦 `∗ 𝑥𝑥, 𝑥\𝑦 `∗ 𝑦𝑥/𝑦, 𝑦/𝑧 `∗ 𝑥/𝑧∗
𝑥, Δ `∗ 𝑦T< Γ ` 𝑥D>Cut∗𝑥 `∗ 𝑦/(𝑥\𝑦)𝑥/𝑦 `∗ (𝑥/𝑧)/(𝑦/𝑧)Γ, Δ `∗ 𝑦𝑥 `∗ 𝑥Id

In this paper, we show the main theorem; For a certain parsing tree generated by
CG, we can generate the equivalent parsing tree without binary nodes on the rightbranch by the combinator rules 𝐵, 𝐷, 𝑇, where the condition is that there are no
the backward long references (7) and (8) on the above eight patterns.
Theorem. If Γ ` 𝑎 is derived by a tree without the backward long reference, thenΓ `∗ 𝑎, which is derived by a left-branching tree.
Proof. We use 1 and 2 to denote applicable single/ binary axiomata. To declarethe correspondence, we attach the subscript to these symbols if they are needed.
We prove this theorem by the mathematical induction with respect to the number105

of leaves. If the tree has less than three leaves, it is obviously the left-branchingtree. We define the transformation ⊲ of the tree. The following is the inductive
definition, that is, we need to process the transformation on the upside to process
ones on the downside. Let the capital Roman letters be parsing trees of CG.
𝑥⊲𝑥 I

2 [𝑥, 𝑦] ⊲ 2 [𝑥, 𝑦] B

21 [𝑋, 𝑌 ] ⊲ 𝐴𝑎
22 [𝑎, 𝑍] ⊲ 𝐵E
22 [21 [𝑋, 𝑌 ], 𝑍] ⊲ 𝐴 ~ 𝐵

1 [𝑋] ⊲ 𝐴 ~ 1 [𝑥] U
𝑋⊲𝐴

B> [𝑋, T< [𝑌 ]] ⊲ 𝐴𝑎
A> [𝑎, 𝑍] ⊲ 𝐵
1
A> [𝑋, A> [𝑌 , 𝑍]] ⊲ 𝐴 ~ 𝐵

2 [𝑋, 𝑌 ] ⊲ 𝐴𝑎
B> [𝑎, T< [𝑍]] ⊲ 𝐵
1L
B> [ 2 [𝑋, 𝑌 ], T< [𝑍]] ⊲ 𝐴 ~ 𝐵

D> [𝑋] ⊲ 𝐴𝑎
A> [𝑎, A> [𝑌 , 𝑍]] ⊲ 𝐵
1R
B> [𝑋, T< [A> [𝑌 , 𝑍]]] ⊲ 𝐴 ~ 𝐵

T< [𝑋] ⊲ 𝐴𝑎
A> [𝑎, A> [𝑌 , 𝑍]] ⊲ 𝐵
5
A< [𝑋, A> [𝑌 , 𝑍]] ⊲ 𝐴 ~ 𝐵

Here, we show the following two properties of the transformation ⊲.
(i) The transformation works for any tree without the backward long reference.
(ii) The transformation terminate and we obtain the left-branching tree.
First, we prove (i) with respect to the outermost form of the tree.
• If the tree is just a category, then it is transformed by I.
• If the tree has only two leaves, then it is transformed by B.
• If the outermost form of the tree is the pattern (1), (2), (3), and (4), then it istransformed by E. Following that, as 21 [𝑋, 𝑌 ] and 22 [𝑎, 𝑍] are the trees ofless than 𝑛 leaves in CG, it can be transformed by ⊲.
• Note that category 𝑎 in all the transformations appears at the top-most in atree. This means that the category 𝑎 remains unchanged by the transformation because there are no more applicable transformation rules to 𝑎.
• If the outermost form of the tree is the unary rule, then it is transformed by
U. As the tree 𝑋 is also the tree in CG, it can be transformed to 𝐴 inductively.
• If the outermost form of the tree is the pattern (5), then it is transformed by
1. There are no backward long reference in the tree by the assumption.
– For A> [𝑎, 𝑍], there are two cases: 𝑍 is a leaf or A> [. . .]. If it is aleaf, then it is transformed by B. Otherwise, as it is a tree of less than𝑛 leaves in CG, it is inductively transformed by ⊲.
– For B> [𝑋, T< [𝑌 ]], it is not a tree in CG because of B>and T<. Thus,
we must define the transformation for each situation. If the outermost
form of 𝑋 is 2 [. . .], then it is transformed by 1L. If the outermost formof 𝑌 is 2 [. . .], then it is transformed by 1R.
• If the outermost form of the tree is B> [2 [𝑋, 𝑌 ], T< [𝑍]], then it is transformed by 1L. 2 [𝑋, 𝑌 ] is inductively transformed by ⊲ because it is the treeof less than 𝑛 leaves in CG. B> [𝑎, T< [𝑍]] is transformed by 1R.

106• If the outermost form of the tree is B > [𝑋, T < [A > [𝑌 , 𝑍]]], then it istransformed by 1R. D > [𝑋] is transformed by U. A > [𝑎, A > [𝑌 , 𝑍]] istransformed inductively because the tree has less than 𝑛 leaves.
• If the outermost form of the tree is the pattern (6), it is reduced into thepattern (5) by the transformation rule 5. T< [𝑋] is transformed by U. A>[𝑎, A> [𝑌 , 𝑍]] is transformed by (5).
• There are no other outermost form because of the assumption of the theorem.
Next, we prove (ii) for the transformation ⊲. For each transformation, we adjoin
the transformed subtree by ~ to adjoin the left-branching trees without any rightbranching. Thus, the transformation generates left-branching tree only. Moreover,For each transformation step, the depth of tree and number of leaves of subtree isdecreasing. Since the number of leaves and the depth of tree is finite, the transformation should terminate in finite steps. If we have Γ ` 𝑎 holding the assumption,then there is a tree 𝑋, and we obtain a certain 𝑌 by 𝑋 ⊲𝑌 . 𝑌 is a graph representationof the proof in 𝐴𝐵𝑇 𝐷 ∗ . Thus, we obtain the proof of Γ `∗ 𝑎.

We have shown the efficient transformation from a given syntactic tree to a leftbranching one regarding the pattern (1) – (6), though we excluded those sentenceswith the backward long reference. The naïve implementation [4] of incremental
parser produces the backtracking many times. As a consequence, the complexityof incremental parser to obtain the left branching tree is exponential, though a nonincremental parser, in general, O (𝑛 𝑘 ). However, we could obtain a transforming
algorithm with O (𝑛), and thus, the parsing complexity to obtain the left-branchingtree is the same as O (𝑛 𝑘 ) because O (𝑛 𝑘 ) + O (𝑛) = O (𝑛 𝑘 ).
We have not dealt with the backward long reference. Empirically, we see thetree for such a reference by the Q combinator rules. However, the left branchingtree is drastically changed and several variations from the original trees. Hence,
we remain the transformation for these patterns as future work.

References[1] Vera Demberg. Incremental derivations in ccg. In Proceedings of the 11th International
Workshop on Tree Adjoining Grammars and Related Formalisms, 2012.
[2] Ahmed Hefny, Hany Hassan, and Mohamed Bahgat. Incremental combinatory categorial
grammar and its derivations. In International Conference on Intelligent Text Processing andComputational Linguistics, 2011.
[3] Miloš Stanojević and Mark Steedman. Max-margin incremental ccg parsing. In Proceedingsof the 58th Annual Meeting of the Association for Computational Linguistics, 2020.
[4] Masaya Taniguchi and Satoshi Tojo. Incremental derivations with q combinator in ccg. In
Proceedings of Logic and Engineering of Natural Language Semantics 18, 2021. This is notin the postproceedings.

107

To be Canceled, or Not to be Canceled: Reconsidering the Caused Possession in the
Dative Alternation Experimentally
Junya Fukuta1 and Koji Shimamura2
1

1

Chuo University, 2 Kanazawa Gakuin University/Kobe City University of Foreign Studies
Reviewing Literature on the Dative Alternation

The alternation between the prepositional dative construction (PDC) in (1a) and the double
object construction (DOC) in (1b) (i.e. the dative alternation) has a long history of research in
the generative tenet, and there should be at least two headings of the pertinent research: (i) they
are derived from the same structure, setting aside the issue concerning which of (1a) or (1b) is
the base, or (ii) they are not derivationally intertwined, each endowed with different syntactic
makeups, i.e. what Harley (2002) terms the “alternative projection” view.
(1)

a.
b.

John gave a ring to Mary. (PDC)
John gave Mary a ring. (DOC)

Those who advocate the alternative projection approach sometimes observe that DOC, not
PDC, is accompanied by the entailment that the goal has got the theme because of the agent’s
giving/passing/sending/throwing etc. event. However, the facts are, as pointed out by Rappaport Hovav and Levin (2008) among others, not so simple, and they observe that verbs likegive, lend, loan and sell (GIVE-type verbs) do not allow the cancellation of caused possession,irrespective of the DOC or PDC frame whereas other verbs like send are compatible with sucha cancelled interpretation:(2)a. #My aunt {gave/lent/loaned} my brother some money for new skis, but he never got
it.
b. #My aunt {gave/lent/loaned} some money to my brother for new skis, but he nevergot it.

(3)

a. #My brother sold Caroline his old car, but she never owned it.
b. #My brother sold his old car to Caroline, but she never owned it.

(4)

a.
b.

Lewis sent Sam a bicycle, but it never arrived.
Lewis sent a bicycle to Sam, but it never arrived.
2008: 146-147)(Rappaport Hovav and Levin

This state of affairs then rejects a uniform analysis of DOC with the caused possession lexicalized as HAVE in the DOC structure (Beck and Johnson 2004, Harley 2002 inter alia). If HAVE
lexicalizes the caused possession, we need it even in the PDC frame, and it should be absent in(4) in both frames.
Also, Basilico (2008) observes that inanimate subjects force the DOC frame to be accompanied by the caused possession construal, even for non-GIVE-type verbs:

108

(5)

Those heavy April showers brought us some nice May flowers (#but we didn’t get niceflowers this May). (Basilico 2008, 760)

Proposing any specific structures building on the set of data collected from only a handful of
speakers may jeopardize the reliability of the analysis, so we conducted a survey regarding thecancelled possession, finding out that even give and other GIVE-type verbs are possible with it(if the subject is animate).
2

Method and Result

In an attempt to empirically test the above point in an experiment, the research hypothesis is asfollows. The dative alternation presupposes that both DOC and PDC sentences containing thesame verb are acceptable. On this premise, when a sentence containing cancellation follows
them, only DOC becomes unacceptable, which proves the existence of HAVE in the syntactic
structure only in DOC. On the other hand, if only GIVE-type verbs are affected by cancellationand no difference is found between DO and PO, then the above-discussed explanation by Rappaport Hovav and Levin (2008) is supported. We confirmed through Experiment 1 with 100
American native speakers of English whether both DOC and PDC stimuli used in this experiment are acceptable as Figure 1 shows, and through Experiment 2 with another 100 American
native speakers of English how the acceptability changes when a cancellation follows the sentence used in Experiment 1. Bayes factor (BF10 ) and Bayesian 95% credible intervals were used
to assess whether the participants tended to judge each stimulus as grammatical or ungrammatical. The results (Figure 2) showed that acceptability declined when stimuli accompany thecancellation of the caused possession as an overall tendency, but DOC and PDC showed almost the same trends as Rappaport Hovav and Levin (2008) have shown. Interestingly, theacceptability of give and loan declined when followed by cancellation, but it did not decline tothe point of being judged completely ungrammatical, contra what Rappaport Hovav and Levinobserve, and we found that show, sell, lend were judged as ungrammatical when followed bycancellation, although there was not much difference between DOC and PDC.
Figure 2: With Cancellation

Figure 1: Without Cancellation

We also conducted two more judgment experiments: Experiment 3 tested the acceptabilityof the inanimate subject of DOC and PDC sentences, and Experiment 4 investigated the changeof these sentences when cancellation was added to these sentences. One hundred American
native speakers of English were recruited for each experiment and engaged in the acceptability judgment task. The data obtained analyzed in the same manner as Experiment 1 and 2.
The results showed that DOC showed a significantly higher acceptance rate than PDC without
cancellation, while neither condition became completely unacceptable. When cancellation is109

added, however, both showed significantly lower acceptance rates and there is not significant
difference between DOC and PDC as the following figure shows:
Figure 3: Inanimate Subject with/without Cancellation

Factor 1: level 1 = without cancellation, level 2 = with cancellation
Factor 2: 1 = DOC, 2 = PDC
What is interesting about Figure 3 is that inanimate subjects are possible even for the PDC
frame, and this state of affairs is contrary to what Lechner (2006) and Pesetsky (1995) point
out, whereas supporting Basilico’s (2008) observation that the caused possession is obligatory
for the inanimate subject DOC.
3

The Syntax of the Dative Alternation

The results of our survey indicate that postulating HAVE for even GIVE-type verbs is not wellmotivated, and importantly, they show that the obligation of the caused possession interpretation does not concern the dative alternation itself but more intricate factors such as the meaningof individual verbs and the animacy of their subject. We thus do not assume HAVE, and following Pesetsky (1995), we argue that both DOC and PDC involve PP, and that only the former
has a silent preposition, viz. Pesetsky’s G. Under his analysis, it incorporates to V, so we have:
(6)

a.

b.

Double Object Construction
VP

Prepositional
struction

Dative

Con-

VP
V

PP
V

give/throw/tell-G1

DPP

give/throw/tell
Goal

PP

0

P

DP

t1

Theme

P0

DP
Theme

P

DP

to

Goal

Note that in both frames there is not direct selectional relation between the verb and the twoarguments. This is supported by (7), which shows that omitting a goal argument (in an out-of
blue context) leads to ungrammaticality.
(7)

a. *John gave a book.

110b. ?*John sent a letter.

(Takami 2003: 219)

Then, immediate questions should be concerned with how the complex facts exhibited by theabove results can be captured as well as what motivates the dative alternation. Regarding theseissues, we however do not ask syntax for the answers; instead, we try to incorporate ideas put
forth by cognitive grammar to our analysis.
4

Explaining The Dative Alternation – Where Cognitive Linguistics and Generative
Grammar Meet

We contend that the dative alternation should be understood in terms of cognitive and pragmatic factors such as salience, attention shift, and information structure. Langacker (2008)
argues that when a verb profiles three participants (agent, mover, receiver), the agent is focusedas the trajector because it is the most prominent, followed by the object adjacent to the verb
as the first landmark. Thus, to express the event of transfer, English speakers have the optionof either PDC where the mover is focused as the (first) landmark, or DOC where the recipient is focused instead. The length of the noun phrase, (in)definitness, (in)animacy, and otherdiscourse-related variables are also related to these factors and affect choice of these constructions (Bresnan et al. 2007). The focus of the event is then determined by factors such as how
the language speaker conceptualizes the content of consciousness and what kind of linguistic
expression the context requires. We also argue that whether a given verb tends to (or must) be
accompanied by the caused possession construal depends on to what extent such an interpretation is conventionalized for the verb and/or what kind of situation a given language speaker isfacing.
Concerning the inanimate subject in the dative alternation, Ikegami (1981) proposes anfigure-ground alternation between action and result in the construal of the meaning expressedby the sentence, and argued that inanimate subjects are more acceptable when action by an
inanimate subject is not in the foreground (Ikegami 1981). As Goldberg (1995) also showsthat volitionality is diminished in the inanimate subject construction, this could increase theacceptability of DOC over PDC for inanimate subjects. In addition, since inanimate subjects
draw interpreters’ attention on result rather than action, inanimate subjects are likely to be lesscompatible with cancellation of the result than when animate subjects are positioned. This
could reduce the acceptability of the sentences, regardless of the types of construction.
The above kinds of cognitive/pragmatic processes are done after constructing a structure
in narrow syntax, namely after Spell-Out. Therefore, it can be considered that they consult
the conventionalized/encyclopedic knowledge of each language user. This does not necessarily
contradict what the current generative approach assumes. Under the framework of Distributed
Morphology (DM), word formation is done in narrow syntax, and all the interpretational processes are rendered after Spell-Out, and DM also assumes the encyclopedic knowledge of thelanguage user, and in this connection, Lohndal (2014) claims that the argument structure, a caseof non-word level interpretation, may also consult speakers’ conventionalized/encyclopedic
knowledge. However, what kinds of determinants are significant in the relevant processes are,if not all of them, yet to be known. If syntax is only for recursion, cognitive linguistics andgenerative grammar can help each other move forward to a better understanding of human language, especially what will actually happen after narrow syntax – the language faculty in the
broad sense (Hauser et al. 2002).

111

References
Basilico, David. 2008. Particle verbs and benefactive double objects in English: high and low
attachments. Natural Language and Linguistic Theory 26:731–773.
Beck, Sigrid, and Kyle Johnson. 2004. Double objects again. Linguistic Inquiry 35:97–124.
Bresnan, Joan, Anna Cueni, Tatiana Nikitina, and R Harald Baayen. 2007. Predicting the dative
alternation. In Cognitive foundations of interpretation, 69–94. KNAW.
Goldberg, Adele E. 1995. Constructions: A construction grammar approach to argument
structure. Chicago, IL: University of Chicago Press.
Harley, Heidi. 2002. Possession and the double object construction. Yearbook of Linguistic
Variation 2:29–68.
Hauser, Marc D., Noam Chomsky, and W. Tecumseh Fitch. 2002. The faculty of language:
What is it, who has it, and how did it evolve? Science 298:1569–1579.
Ikegami, Yoshihiko. 1981. Suru to naru no gengogaku: Gengo to bunka no taiporozi eno shiro[Linguistics of doing and becoming: An essay towards a typology of language and culture].
Tokyo: Taishukan.
Langacker, Ronald W. 2008. Cognitive grammar: A basic introduction. Oxford: Oxford
University Press.
Lechner, Winfried. 2006. Double object constructions (II). Hanout # 6 for Syntax II in Universität Stuttgart.
Lohndal, Terje. 2014. Phrase structure and argument structure: A case study of thesyntax-semantics interface. Oxford: Oxford University Press.
Pesetsky, David. 1995. Zero syntax. Cambridge, MA: The MIT Press.
Rappaport Hovav, Malka, and Beth Levin. 2008. The English dative alternation: The case forverb sensitivity. Journal of Linguisitcs 44:129–167.
Takami, Ken-ichi. 2003. A semantic constraint on the benefactive double object construction.
English Linguistics 20:197–224.

112

Events and Relative Clauses
Oleg Kiselyov and Haruki Watanabe
Tohoku University, Japanoleg@okmij.org

Abstract. This work is the continuation of the development of polynomial event semantics (adialect of Neo-Davidsonian event semantics), using the FraCaS textual entailment corpus as awhetstone. This time we grapple with various, often complicated, relative clauses.
Relative clauses have hardly been analyzed before in event semantics. Although simple cases are
straightforward, challenges arise when a clause contains quantification, coordination or negation.
We deal with such complications in the present paper, focusing on entailments.

This work is the continuation of [3–5] on polynomial event semantics and textual
entailment.
Deciding entailments ‘by pure logic’, without resorting to meaning postulates is oneof the most attractive features of event semantics. However, beyond the classical “Brutus stabbed Caesar violently”, one quickly runs into problems. One is quantification,
described and dealt with in [3, 5]; another is negation [4]. There are also relative clauses,
which are rarely considered in event semantics. In fact, the recent survey [7] and theextensive study [1] give, among the multitude of examples, not a single analysis of asentence with a relative clause.
A relative clause appears already in the very first problem in the FraCaS textual
inference problem set [2, 6]:(1)

There was an Italian who became the world’s greatest tenor.

Such simple case was analyzed in [5]. But even a slightly more complicated problem 018
below requires quite a non-trivial entailment reasoning involving the relative clause.
(2)
(3)
(4)
(5)

Every European has the right to live in Europe.
Every European is a person.
Every person who has the right to live in Europe
can travel freely within Europe.
Every European can travel freely within Europe.

As in all FraCaS problems, the goal is to determine the entailment of the last sentence(in our case, (5)) from the others. We must stress that FraCaS collects not only positive
examples of expected entailments, but also negative examples where entailment does
not hold – and also “yes and no” cases where entailment comes through only on somereadings. Our goal is hence not only to derive entailments where expected, but also
to explain why entailment does not hold in negative examples, as well as to reproduce
several readings where present.
FraCaS has quite a few problems similar to the above, with copula relative clauses(problems 005, 006, 028) and quantifiers like ‘most’ (problem 074). Object relative
clauses also appear (e.g., problems 133 and 344). There are further complications, with113

quantified or coordinated relative clauses:
(6)
(7)
(8)

There was one auditor who signed all the reports.
There is a car that John and Bill own.
There is a representative that Smith wrote to every week.

The following NPs, although not appearing in FraCaS, are also common:
(9)
(10)
(11)

two students who skipped three classes
every student who skipped no classes
a student who didn’t skip all classes

The present paper gives analysis of all such sentences and NPs, focusing on entailments. Due to the lack of space in this extended abstract, many details are elided.

1

Background

First a brief reminder of polynomial event semantics. It deals with events, notated e,such as ‘having become the world’s greatest tenor’ or ‘being Italian’ or ‘having the rightto live in Europe’ (we denote the latter set of events as RtlE). It should be clear that wetake events in broad sense (as detailed in [7]): associated not only with actions but also
states. Besides events, there are also individuals, notated i, such as john, and relations
between events and individuals (written as rel′ ) such as
subj′ = {(e, i) | ag(e) = i}ob1′ = {(e, i) | th(e) = i}where ag and th are thematic functions (for agents and themes of events, resp.).
Remark 1. The terms ‘agent’ and ‘theme’ are used purely formally. After all, event
semantics is widely praised for avoiding meaning postulates and deriving entailmentsfrom the structure alone. Likewise, the focus of FraCaS is textual entailment without
relying on world knowledge. We, too, concentrate on the structure: Just as verbs havearguments, events – records in a world database – have attributes. We use ‘agent’ and‘theme’ to refer to the attributes. What actions or attributes really mean and what istheir connection with agency, etc., is out of scope.
If rel′ is a relation of events to individuals, rel′ / i = {e | (e, i) ∈ rel′ } is the set of
events related to i. In particular,
subj′ / john = {e | (e, john) ∈ subj′ } = {e | ag(e) = john}
is the set of events whose agent is john. The semantics of a simple sentence such as“John has the right to live in Europe” is given compositionally as(12)subj′ / john ∩ RtlE

Subject, predicate, complements all denote event sets, and the whole sentence is theirintersection. In particular, our sentence denotes – or, witnessed by – events of having
the right to live in Europe whose agent is John. The denotation is hence an event set –114

or the formula representing it, as (12), which one may think of as a query of the recordof events in the world. A sentence is true in that world just in case the result of thequery is a non-empty event set.
The denotation of the subject is also determined compositionally, by applying subj′
to the denotation of NP, in our case, john.
We often deal not with individuals but with sets of individuals such as Student
or European, which are the denotations of common nouns. Determiners pick which individuals from this set to consider. Correspondingly, they call for generalization: theintroduction of (internal) choice ⊔ (for narrow-scope existentials and indefinites), external choice ⊕ (wide-scope ones) and grouping ⊗. Thus john ⊔ bill is a choice between
John and Bill, whereas john ⊗ bill is a group of John and Bill: both have to be involved,
not necessarily in the same action however. Likewise, event sets are generalized to polyconcepts, such as d1 ⊗ d2 for two disjoint event sets d1 and d2 , which specifies that an
event from d1 and an event from d2 must have transpired. We define for convenience
Ec = ⊔j∈c j

Ic = ⊕j∈c j

Ac = ⊗j∈c j

The meaning of “All Europeans/Every European” is then A European and “A European” (narrow scope) is E European. Thus (2) has as its denotation(13)(subj′ / AEuropean) ⊓ RtlE

(14)

= A(subj′ / European) ⊓ RtlE
O
(subj′ / i ∩ RtlE)=i∈European

where ⊓ is the generalization of set intersection ∩ to polyconcepts; subj′ / is likewise
generalized to apply to sets of individuals and poly-individuals – as homomorphism.
The distribution laws detailed in [5] lead to (14), which asserts there is a group of
non-empty events of having right to live in Europe, and each European is an agent ofsome event in that group.

2

Subject Relative Clauses

The problem is then determining the meaning of “who has the right to live in Europe.”
If RtlE is the set of events of having the right to live in Europe, then who has that rightis the agent of these events. Thus the denotation of our subject relative clause, to be′notated as subj / RtlE, is the set of individuals(15)′

subj / RtlE = {ag(e) | e ∈ RtlE}

Then (4) has as its denotation


′
′
(16)
subj / A Person ∩ (subj / RtlE) ⊓ CtfE
where CtfE is the set of events of having the possibility to freely travel within Europe.
This analysis is more or less what was described in [5], but recast now in simpler terms.
It takes us quite far: many more FraCaS problems can be analyzed similarly.
However, relative clauses with quantifiers, coordination or negation present a problem. Again we need to generalize. Remembering the definition of the subj′ relation, we
115may re-write (15) as′

subj / RtlE = {ag(e) | e ∈ RtlE}= {i | e ∈ RtlE , (i, e) ∈ subj′ } = {i ∈ dom(subj′ ) | subj′ / i ∩ RtlE ̸= ∅}One may notice that subj′ / i ∩ RtlE is exactly the meaning of “i has the right to live inEurope”. Thus “who has the right to live in Europe” is the set of those i in the domainof the subj′ who make the sentence true (in the world of the discourse). We may thus
define the denotation of a subject relative clause as(17)′

subj / d = {i ∈ dom(subj′ ) | subj′ / i ∩ d ̸= ∅}where d is the event set that is the denotation of the rest of the clause (without “who”).
This is already helpful: recall (2) and its denotation (14), which says that subj′ / i ∩ RtlE′
is non-empty for all i ∈ European. Then (17) immediately gives European ⊂ subj / RtlE;
in words: the set of who has the right to live in Europe includes all Europeans. Likewise,′(3) gives European ⊂ Person, leading to European ⊂ (Person ∩ subj / RtlE). Then, by
monotonicity of A, (16) entails subj′ / (AEuropean) ⊓ CtfE, which is the denotation of(5). This is the solution to FraCaS 018.
Definition (17), unlike (15), now easily generalizes to the case when the denotationof the rest of the clause d is not an event set but a polyconcept with choice or grouping:(18)′

subj / d = {i ∈ dom(subj′ ) | subj′ / i ⊓ d ̸= ⊥}where ⊥ is the null polyconcept. The generalization lets us analyze quantified andcoordinated relative clauses such as (6)-(8).
′′Let us first notice that subj is sort of an inverse of subj′ : we have subj / subj′ / i = {i}′for any individual i. Further, subj / subj′ / ES = S for any sets of individuals S (and′the same for I). However, unlike subj′ , subj is not a homomorphism. In particular,′
subj / subj′ / AS is always the empty set. More interestingly, since the external choice ⊕′distributes over ⊓ (and any other operation for that matter), we have subj / (d1 ⊕ d2 ) =′′subj / d1 ∪ subj / d2 and furtherMsubj′ / IS ⊓ d =subj′ / i ⊓ di∈Swhich eventually leads to′′S ∩ subj / d = subj / (subj′ / IS ⊓ d)(19)
or, in concrete terms,′′

Person ∩ subj / RtlE = subj / (subj′ / IPerson ⊓ RtlE)
The formula in the parentheses is the denotation of “A person has the right to live in′Europe.”. The role of subj then is collecting such persons into a set.
Hence (4) will have as its denotation′

A subj′ / subj / (subj′ / IPerson ⊓ RtlE) ⊓ CtfE
116

which may be regarded as a database join, of “A person has the right to live in Europe.”with “can travel freely within Europe” on agent. Such database join may be illustrated
by a (bit contrived) paraphrase: “Some people have the right to live in Europe. Every
one of them can travel freely within Europe.” Generalizing, a more complicated “This
sport trains muscles you never thought existed.” can hence be paraphrased as “Younever thought some muscles existed. This sport trains them.”More interesting is the application to quantified relative clauses, such as (6), whose
denotation is′
subj′ / E Auditor ∩ subj / (Sign ⊓ ob1′ / AReport) ⊓ E Be
where EBe is an existence event (see [5]). This denotation is equivalent1 to′

Auditor ∩ subj / (Sign ⊓ ob1′ / AReport)
which, by (19), is′

subj / (subj′ / IAuditor ⊓ Sign ⊓ ob1′ / AReport)
and, in turn, is equivalent to subj′ / IAuditor ⊓ Sign ⊓ ob1′ / AReport. In other words, (6)is equivalent to, or mutually entails, “One particular auditor/the same auditor signed
all the reports” – which is what FraCaS problem 196 is all about.
For object relative clauses such as (7), we use the thematic function th in place ofag, and ob1′ instead of subj′ . For example, we obtain that (7) is equivalent to “John andBill own the same car”. For problem 308, we obtain (8) is equivalent to “Smith wrote
to a representative every week.” on the wide-scope reading of the indefinite – with noentailment for the narrow-scope reading.
Negation calls for one more generalization of (18):′subj / d = {i ∈ dom(subj′ ) | justified(subj′ / i ⊓ d)}where justified(d) means that either d is marked as negative (as a counter-example)
and is ⊥, or not marked as negative and is not ⊥ (see [4, 5]). We obtain that “student′
who didn’t skip all classes” is the set Student ∩ subj / (¬ Skip ⊓ ob1′ / AClass), and it is
the complement of the set of students who skipped all classes. Likewise, “student who
skipped no classes” is the complement of the set of students who skipped a class.
[1] Lucas Champollion. The interaction of compositional semantics and event semantics. Linguistics andPhilosophy, 38(1):31–66, 2015.
[2] Robin Cooper, Dick Crouch, Jan van Eijck, Chris Fox, Josef van Genabith, Jaspars Jan, Hans Kamp,
David Milward, Manfred Pinkal, Massimo Poesio, Steve Pulman, Ted Briscoe, Holger Maier, and Karsten
Konrad. Using the framework. Deliverable D16, FraCaS Project, 1996.
[3] Oleg Kiselyov. Polynomial event semantics - Non-Montagovian proper treatment of quantifiers. In JSAIisAI Workshops, volume 11717 of Lecture Notes in Computer Science, pages 313–324. Springer, 2019.
[4] Oleg Kiselyov. Polynomial event semantics: Negation. In JSAI-isAI Workshops: New Frontiers in Artificial
Intelligence, volume 12758 of Lecture Notes in Computer Science, pages 82–95. Springer, 2021.
[5] Oleg Kiselyov and Haruki Watanabe. QNP textual entailment with polynomial event semantics.
[6] Bill MacCartney. The FRACAS textual inference problem set. https://nlp.stanford.edu/~wcmac/
downloads/fracas.xml.
[7] Claudia Maienborn. Event semantics, chapter 8, pages 232–266. Semantics - Theories. De Gruyter Mouton,
2019.
1

We call two polyconcepts/event sets to be equivalent just in case when one is not-⊥/not-empty, so is theother – that is, when they entail each other.

117

Truthmaker Semantics for Degreeism of VaguenessS HIMPEI E NDO
Hitotsubashi University
e-mail: endoshimpeiendo@gmail.com
Philosophers have been long discussed vagueness and its related paradox — thesorites — for many reasons, mostly linguistic, sometimes metaphysical, and more (cf.
[5]). When philosophers talk about vagueness, they often end up talking about semantics. In fact, many solutions to the paradox suggest revising semantics: supervaluationism renovates semantics with a new formal concept of supervalues, degreeism
suggests many-valued logic and its coresponding semantics, and epistemicism suggests keeping classical logic and semantics fixed, but ascribes vagueness to an epistemic issue. In the market of semantic builders, truthmaker is a rising star with its
expressive power powerful enough for many including hyperintensionality ([3]).
Still, few have employed truthmakers for vagueness. An exceptional case [7] suggests an argument appealing to truthmaker gaps but only for his particular version of
epistemicism. But might there be other applications of truthmaker semantics in the
study of vagueness?
The goal of this paper is to offer an affirmative answer to this question, by designing a truthmaker semantics for another position on vagueness. Among several
positions, this paper works on a popular one: degreeism (degree theory). True to its
name, degreeism revises the semantic concept of a truth value from binary one (truth 1
and false 0 and nothing else) to a many-valued one (often infinite). However, importing truthmakers into degreeism is not straightforward. While truthmakers are about
quality and use mereological part-whole relations in their formalization, degreeism
is based on a more quantitative idea, namely a segment of real numbers [0, 1] ⊆ R.
But how can we translate mereological structures of truthmaking into degreeists’ real
numbers and the other way around?
One option is to import measure theory (cf. [2], [1]). A measure is, very roughly
put, a mathematical generalization of geometrical measures such as distance, length,area, and volume. This formal notion of “measuring” the size of a given set is applied
to many things such as physical mass and, most importantly for degreeism, probabilityof events. Given that degreeism is often associated with probability theory as theyboth feature a fragment of real numbers [0, 1] as a central part of their formalization,
the match already seems apt. We see an evaluation function µ which assigns a truth
value for a given truthmaker as a measure function, which satisfies standard axiomsof measure theory.
A glance at the definitions shows how naturally these concepts fit degreeism. For
one thing, degreeism stipulates that the measure of the null set is zeroµ(0)/ = 0.
This seems to correspond to our intuitive idea that if a proposition ϕ has no truthmaker at all (in other words, nothing in a world supports ϕ) its truth value should be118

zero. Also, another important definition of (countable) additivity confirms our idea ofthe relationship between truthmakers and truth values — the more truthmakers (e.g.
evidence) a proposition has, the more likely it is that the proposition is true.
Still, measure theory is built upon a set-theoretical setting (i.e. upon families ofsets), which is different from the mereological structures of truthmakers. So someformal work is needed in order to offer a mereological version of a measure functionµT M . In this paper, I offer a truthmaker semantics for degreeism of the following form:
M = ⟨S, ⊑, µT M ⟩,where S is a non-empty set of states (truthmakers), ⊑ is a partial order on S expressing mereological relation for part-wholehoodness, and µT M is a degreeism evaluation
function from propositions to [0, 1] ⊆ R, assigning a real number from 0 to 1 to a given
propositional letter. The first two suggestions follow from the standard formalism of
truthmaker semantics. The last one is original. In particular, I design the following
two properties for µT M so as to behave as a measure. First, we need something corresponding to the null set. In the original measure theory, we have 0/ as an obvious andnatural example. But truthmakers do not have apparent counterparts of the empty set.
Second, we need a truthmaker version of additivity. I provide several operations suchas ⊓ (s ⊓ t is the overlapping part of s and t) and difference \ of truthmakers for thispurpose.
Having introduced truthmaker semantics for degreeism, this paper discusses thebenefits of this semantics to further support how truthmakers are useful in discussionsof vagueness, at least for degreeism. The resulting semantics resolves two formal issues for degreeism. One is about triviality [6]. Some may want to characterize vague
predicates (from non-vague ones) by the formal concept of continuity. For instance,
we may want to characterize vague terms by whether their evaluation function from(a subset of) N (e.g. the number of hairs) to truth values [0, 1]. Unfortunately, thisdoes not work because the domain (the number of hairs, with the most natural topology) is discrete, hence any function from N is trivially continuous. In my suggested
framework, such a worry disappears. The domain is not the natural numbers but a
set of truthmakers, whose topology is not necessarily discrete. The other is called the
problem of “penumbral connection” [4]. This problem is about how to calculate thetruth values of two vague clauses connected by logical connectives. For instance, what
happens if two indefinite clauses (i.e. borderline cases) are connected with a conjunction, say, “This ball is purple and this ball is red”? The truth value of this sentence
should be zero i.e. definitely false because one ball cannot have different colors at thesame time. But typical degreeists say it is indefinite, i.e. somewhere between [0, 1].
Truthmakers provides an easy way out, though. It is clear that a truthmaker for being
red and another truthmaker for being purple are simply incompatible.

119References[1] Marek Capiński and P. E. Kopp. Measure, integral, and probability. Springer
undergraduate mathematics series. London ; New York: Springer, 1999. ISBN:
978-3-540-76260-7.
[2] Joseph L. Doob. Measure theory. Graduate texts in mathematics 143. New York:
Springer-Verlag, 1994. ISBN: 978-0-387-94055-7 978-3-540-94055-5.
[3] Kit Fine. “Truthmaker Semantics”. In: A Companion to the Philosophy of Language. Ed. by Bob Hale, Crispin Wright, and Alexander Miller. 2nd ed. John
Wiley & Sons Ltd., 2017, pp. 556–577. DOI: 10.1002/9781118972090.ch22.
[4] Kit Fine. Vagueness: a global approach. en. New York, NY, United States ofAmerica: Oxford University Press, 2020. ISBN: 978-0-19-751498-6 978-0-19751497-9.
[5] R Keefe and P Smith. Vagueness: A reader. The MIT Press, 1997. (Visited on
12/14/2015).
[6] Nicholas J.J. Smith. Vagueness and Degrees of Truth. Oxford University Press,
2008. ISBN: 978-0-19-171643-0. DOI: 10.1093/acprof:oso/9780199233007.
001.0001.
[7] Roy Sorensen. Vagueness and Contradiction. Oxford University Press, 2001.

120

KK is Wrong Because We Say So
Simon Goldstein and John Hawthorne
1 Against KK
The KK principle says that if you know something, then you know that you know it.
The KK principle plays a central role in theories of linguistic communication. Since at least
Stalnaker 1978, many have interpreted assertion against a background of shared
information, the common ground. The common ground is standardly understood in terms of
iterated attitudes: what the group of interlocutors commonly accept, believe, or know. As
Greco 2014 and others have argued, these common attitudes rely on some version of the
KK principle. In order for a group to know that everyone knows that everyone knows that …
everyone knows p, it is necessary that each member of the group knows that they know
everything that they know. Otherwise, the failure of iterated knowledge (or acceptance, or
belief) for individual group members will generate failures of common knowledge (oracceptance, or belief) for the group.
Unfortunately, we think KK is wrong. Moreover, KK is wrong because it is incompatible with
commonplace linguistic practice. In particular, we’ll argue that KK is wrong because of thelinguistic practices of those who deny KK.
Doubting Dudley says that he knows he has hands, but he doesn’t know that he knows …
that he knows that he has hands. In logician speak, he says that he doesn’t ‘100-know’ that
he has hands. (You 1-know something if you know it. You n-know something if you know thatyou n-1 know it.)
Dudley only says things that he believes. And he never believes things that he knows arefalse. Since he says he doesn’t 100-know that he has hands, he believes he doesn’t
100-know he has hands. So he doesn’t know that he 100-knows that he has hands. But if KK
is true, and if he knows he has hands, then it follows that he does know that he 100-knowsthat he has hands. So KK is wrong.1(It may help to state the argument using the tools of epistemic logic. Let ‘Ap’, ‘Bp’, and ‘Kp’
represent the claims that Dudley asserts, believes, and knows p. Here is the argument:
A~100Kp |= B~100Kp. B~100Kp |= ~K100Kp. So Ap & A~100Kp implies Kp & ~100Kp,
which contradicts KK.)
We don’t even need to say that KK is wrong. KK is wrong if we don’t say that it’s right.
Agnostic Agnes is asked whether she 100-knows she has hands, and she refrains fromanswering, because she is unsure. Agnes doesn’t refrain from answering questions when
she knows the answer. Since she refrains from answering the question of whether she

1

Dudley may say that KK fails without saying exactly where it fails. For example, Dudley may say the
conjunction he doesn’t 100-know he has hands and he doesn’t 100-know he has feet, and either heknows he has hands or he knows he has feet. He says the conjunction, so he believes the
conjunction, so he doesn’t know the conjunction is false. But if KK is true, and if he does know he has
hands, then he does know that the conjunction is false.

121100-knows she has hands, she does not know that she 100-knows she has hands. So KK iswrong.
Here is an analogy. Agnes doesn’t know exactly what love is. Is loving someone a matter ofwanting their life to go well? Or do you also have to enjoy spending time with them? Agnes
has a friend, Boris, who she doesn’t like spending time with, but who she wishes the best for.
Since Agnes doesn’t know exactly what love is, Agnes doesn’t know whether she loves
Boris. In this respect, knowledge is like love. Agnes doesn’t know exactly what knowledge is.
Since she doesn’t know exactly what knowledge is, she doesn’t know whether she
100-knows that she has hands. Knowledge is one relation among others. If you don’t knowwhether a condition is required for something to be related to you in a certain way, and aparticular thing obviously doesn’t satisfy that condition, then you won’t know whether thething is related to you in that way. KK denies this. So KK is wrong.
The defender of KK has one way out. Deny that Dudley and Agnes know they have hands! If
they don’t know they have hands, then they don’t know that they 100-know that they havehands. The KK defender can say that no one who disagrees with them knows anything. This
response is consistent; but it is tasteless and offensive.
2 Fragmentation
Our argument had two premises: Dudley only says what he believes, and Dudley neverbelieves what he knows is false.2 Defenders of KK may try to reject either premise. We’ll
start with the second premise. Perhaps people can believe something in one sense or ‘guise’
while knowing it in another sense. On this proposal, Dudley believes under one guise that hedoes not 100-know that he has hands. He knows under another guise that he does
100-know that he has hands.
The details of this response depend on what guises are. One theory is that guises are
fragments. People believe and know things relative to different fragments. Each fragment is
activated in different situations. Each fragment is logically closed and internally consistent,
and each fragment satisfies KK (Greco 2015b).
For example, perhaps Dudley and Agnes have a classroom fragment and an outside
fragment. In the classroom, they have beliefs that guide their philosophical musings; in theoutside world, they have other beliefs that guide their ongoing relations with the external
world. The beliefs of one fragment can contradict the beliefs of another; but each fragment isinternally consistent.
The fragmenter wants to claim that people only deny KK relative to their classroom fragment.
But when they are in the outside world, they do not deny KK. Sadly, the fragmenter cannot
claim this. The problem is that the defender of KK says that each fragment satisfies KK.

2More carefully: we assume that there is a class of agents who deny KK in the way that Dudley does.
Our premises are that people in this class only say what they believe, and never believe what theyknow to be false.

122Tim thinks he doesn’t 100-know he has hands, because he thinks he doesn’t 100-know any
empirical truth about the world. The fragmenter interprets Tim as believing this only relative
to the classroom fragment. But here’s the problem. Presumably Tim knows some empirical
truth about the world relative to the classroom fragment. As Tim shouts at you about KK, hepoints to his hands as an example of something he doesn’t 100-know, and he tells you that
he does have hands. But if KK holds in the classroom fragment, and if Tim knows he has
hands in the classroom fragment, then it immediately follows that he knows in the classroom
fragment that he 100-knows he has hands. The KK fragmenter is forced to say that when
Tim denies that he 100-knows any empirical truth about the world, he can only do so relative
to a fragment that knows zero empirical truths about the world.
Imagine a tougher opponent to KK, who denies that they 100-know anything at all, including
even the claim that they believe they fail to 100-know anything. Suppose that their classroom
fragment knows some claim p. It follows that their classroom fragment knows they 100-know
p. So what explains why this tougher person is in the classroom denying that they 100-know
p? The KK fragmenter is forced to say that the tougher opponent knows nothing at all
relative to the fragment that denies KK. Again, tasteless and offensive, but now withfragments.
3. Diagonalization
We have explored the prospects of denying the principle that when you believe p, you do notknow that p is false. Now consider our other starting premise: that if someone utters thesentence I don’t 100-know I have hands, then they believe that they don’t 100-know thatthey have hands. One way to deny this premise appeals to the theory of diagonalization(Stalnaker 1978). According to this theory, our beliefs about the non-linguistic world aresystematically conflated with beliefs about language. Dudley believes that he has hands.
Dudley also believes the sentence I have hands is true. Philosophical confusion results frommistaking one kind of belief for the other.
For example, the diagonalizing KK-er will claim that Dudley knows he has hands, and knowsthat he 100-knows he has hands. Crucially, however, Dudley does not know that thesentence I 100-know I have hands is true. Dudley is in the grip of a false theory about theword know. He thinks that know expresses a relation that does not freely iterate. So Dudleydoes not know that I 100-know I have hands expresses a truth.
Dudley utters the sentence I don’t 100-know I have hands. He utters this sentence because
he thinks the sentence is true, and Dudley tries his best to say what is true. But he does not
actually believe he does not 100-know he has hands.3
We are unimpressed. Return to the case of love. Agnes is unsure of what love is, and so she
is unsure of whether she loves Boris. Here, the diagonalizer could intervene, arguing that
Agnes does know that she loves Boris; Agnes is merely ignorant of the fact that thesentence Agnes loves Boris is true. This proposal strikes us as veering towards madness.
3

Note that the defender of diagonalization must in general deny that people know the disquotation
principle that p iff ‘p’ is true. Otherwise, Dudley could infer from the fact that he knows he hands thatthe sentence ‘he knows he has hands’ is true. In place of this principle, the defender of
diagonalization should say that Dudley merely knows that the sentence ‘p iff ‘p’ is true’ is true.

123

Does all ignorance about the nature of worldly relations turn out to be ignorance of
language? At any rate, we feel strong pressure to analogize Agnes’s attitudes towards loveand knowledge. If she is ignorant about who she loves, then she can also be ignorant about
what she knows.
The diagonalization defense of KK risks undercutting the explanatory power of KK. Consider
assertion. Cohen and Comesaña 2013, Greco 2014, and Das and Salow 2018 argue that KK
explains why it is strange to say the ‘dubious’ assertion I have hands but I don’t know that I
know I have hands. First, assertion is governed by a knowledge norm: I should only say
what I know (Williamson 2000). Second, KK predicts that I can’t know a dubious assertion: if
I know it, then (assuming, plausibly, that knowledge distributes over conjunctions), I know I
have hands; so by KK I know that I know I have hands; but then it isn’t true that I don’t knowthat I know I have hands.
Now return to the diagonalization defender. They seek to explain why the dubious assertion I
have hands but I don’t know that I know I have hands is strange to say. But Dudley will say asentence from his own language if he believes he knows the sentence is true, regardless ofwhether he believes the proposition it expresses. The theory under discussion successfully
predicts that Dudley knows that he does not know that: he has hands and doesn’t know that
he knows he has hands. But the problem is that the theory allows Dudley to believe that heknows that the sentence I have hands but I don’t know that I know I have hands is true. Inthat case, he may well assert it.
Bibliography
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Cohen, Stewart & Comesaña, Juan (2013). Williamson on Gettier Cases and Epistemic
Logic. Inquiry: An Interdisciplinary Journal of Philosophy 56 (1):15-29.
Das, Nilanjan & Salow, Bernhard (2018). Transparency and the KK Principle. Noûs 52(1):3-23.
Dorr, Cian (2014). Transparency and the Context-Sensitivity of Attitude Reports. In ManuelGarcía-Carpintero & Genoveva Martí (eds.), Empty Representations: Reference and
Non-Existence. Oxford University Press. pp. 25-66.
Goldman, Alvin (2004). Pathways to Knowledge: Private and Public. Oxford University Press
USA.
Greco, Daniel (2014). Could KK Be OK? Journal of Philosophy 111 (4):169-197.
Greco, Daniel (2015a). Iteration Principles in Epistemology I: Arguments For. Philosophy
Compass 10 (11):754-764.
Greco, Daniel (2015b). Iteration and Fragmentation. Philosophy and PhenomenologicalResearch 91 (3):656-673.
Lederman, Harvey (2021). Fine-grained semantics for attitude reports. Semantics andPragmatics 14 (1).
Percus, Orin and Uli Sauerland (2003). On the LFs of attitude reports. In Proceedings of theConference “SuB7 – Sinn und Bedeutung”, ed. M. Weisgerber, 228–242. Arbeitspapier Nr.
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Salmon, Nathan. (1986). Frege’s puzzle. Ridgeview.

124Soames, Scott (1987). Direct reference, propositional attitudes, and semantic content.
Philosophical Topics 15 (1):47-87.
Stalnaker, Robert (1978). Assertion. Syntax and Semantics (New York Academic Press)
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Willard-Kyle, Christopher (2020). Being in a Position to Know is the Norm of Assertion.
Pacific Philosophical Quarterly 101 (2):328-352.
Williamson, Timothy (2000). Knowledge and its Limits. Oxford University Press.

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Three Rich-Lexicon Theories of Slurs: A Comparison
Dan Zeman
University of Warsaw

Many authors writing on slurs think that they are lexically rich, at least in the sense that their literalor lexical meaning comprises both a descriptive, truth-conditional dimension and an
expressive/evaluative one (e.g., Potts 2007; Richard 2008; Predelli 2013; Camp 2013; Jeshion 2013).
Appeal to an additional meaning dimension helps accounting for the derogatory character of slurs.
However, more fine-grained theories of slurs have been proposed, and they all draw on frameworks
and tools from lexical semantics and the theory of concepts. Such theories give a more precise
description of the lexical entries of slurs and impose a certain structure on them.
My main aim in this paper is to compare three such fine-grained rich-lexicon theories bothvis-à-vis their theoretical commitments and in their ability to explain the whole range of the data they
are meant to apply to. As widely known, slurs have a panoply of uses. While the most common istheir straightforward derogatory use, there are several types of non-derogatory uses of slurs as well.
(For examples and contributions to the literature on this type of use of slurs, see Cepollaro & Zeman(2020).) Re-appropriated (the n-word; “queer”), corrective (“Institutions that treat Chinese as c***are morally depraved.” Hom, 2008: 423), referential (certain forms of the n-word, according toAnderson 2018), metaphorical (“Woman is the n*** of the world.”, John Lennon & Yoko Ono) anddidactical uses all are of this kind. In addition, certain ethnic slurs (e.g., “țigan” in Romanian) havewhat can be called identificatory uses – when a slur is used by members of the target group to nonderogatorily identify as an ethnic group (Zeman 2020).
The view I prefer is a rich-lexicon theory that takes the lexical meaning of slurs to becomposed of an extension and a conceptual structure (on the model of Del Pinal’s 2018 framework).
Among the meaning dimensions of the conceptual structure are the usual ones found in nouns(information about perceptual information about the objects referred to, about what they are made ofor their parts, about how they came to being, about the purpose of their creation, about their typicalfunction etc.), but also, because the targets of slurs belong to a certain group, information about theirprovenance, history, social standing etc. Crucially, such lexical entries comprise an evaluative
dimension, which is responsible for the derogatory character of slurs. While the lexical entry of a slur

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comprises the entire conceptual structure with all the meaning dimensions, in particular uses of slurs
different dimensions get foregrounded/backgrounded. Thus, when a slur is used derogatorily, theevaluative dimension is foregrounded and the others backgrounded. When the same slur is used in areferential or identificatory manner, the evaluative dimension is backgrounded, while others areforegrounded. (In re-appropriated uses, the evaluative dimension is still foregrounded, but the valenceof the evaluation is switched.) According to this view, then, slurs are polysemous, and their variation
in meaning across contexts is construed as polysemy resolution, which is done via the mechanismsof foregrounding and backgrounding mentioned.
At least a couple of similar views can be found in the literature. For Croom 2011, 2013, slurs
encode rich conceptual structures that have as their meaning dimensions properties, both positive andnegative, that the prototypical members of the target groups are taken to possess. While there is no
essential property that all members of the target group need to share for a slur to apply, different
members share some of the properties associated with them. These properties are ranked; this ranking,however, is context-sensitive and helps in communicating with slurs: a speaker selects among theproperties in question those which are suitable for a given communicative situation. When a slur isused derogatorily, the negative properties prototypical members of the target group are taken to haveare selected. When a slur is used in a re-appropriated manner, the positive properties prototypicalmembers of the target group are taken to have are selected. (Croom doesn’t consider referential or
identificatory uses of slurs, but one can extrapolate from the above how they would be treated: noneof the positive or the negative properties from the list are selected, only neutral properties like mere
membership of the relevant group.)
One worry one might have about Croom’s view is that it is too unrestricted: the list of
properties associated with a slur is open, and thus can end up being quite substantive. This has thepotential to create trouble when it comes to storing and computing slurs. A view that postulates fewer,
better regimented, meaning dimensions might fare better on this score. Second, Croom claims thatthe selection of properties in a given context is based on closeness to the members of the target group
and are supposed to serve only communicative purposes. Yet, there are cases in which the use of aslur is restricted by the linguistic material of the sentences in which is appears. For example, one
cannot use the n-word in a re-appropriated manner in a sentence stating the results of a census, even
if the speaker is very close to the members of the target group.
Neufeld 2019, 2022 proposes an “essentialist” theory of slurs, according to which they areterms with null extension. The novelty of the view consists in the claim that part of the meaning127dimensions of a slur is an “essence” that is given a causal-explanatory role for the negative traits thatmembers of the target group are taken to possess. That is, what explains and causes the members of
the target group to have the negative traits perceived is precisely this “essence” (“blackness”,“queerness” etc.); but since no such real essences exist, the terms fail to refer. The view differs from
Croom’s in taking the meaning dimensions to be related in a precise way (the “essences” cause thenegative properties).
Neufeld doesn’t consider re-appropriated, referential or identificatory sues of slurs, and it’s
not straightforward how these are to be accounted for in her view. In fact, there might be a possible
problem with extending the view to such uses. Since the “essences” that “cause” the negative
properties ascribed to the targets don’t exist, slurs have null extension. However, slurs don’t seem tohave null extensions when used in the ways mentioned above (sentences in which slurs are used inthose ways strike us/their users as true). What happens with the “essence” in such cases? It would be
weird to claim that it simply disappears. A view that postulates some selection mechanisms (likeforegrounding and backgrounding) has an easier answer to this issue.
Further, the postulation of an “essence” in the lexical entry of slurs might also lead toproblems. Neufeld cites a lot of psychological evidence according to which essentialization is a corecharacteristic of humans’ conceptual ability. Yet, I wonder whether her view allows for users of slursthat are not committed to, or even explicitly reject, any “essences”. Thus, imagine a social
constructivist who thinks that race is not something people “essentially” possess, but determined by
the milieu in which one grows up. They might hold that the traits associated with a given race are not
immutable, but that, as a matter of statistics, the traits that most members of a certain racial grouppossess are negative. Now, as it happens, our social constructivist is also a racist. When our racist
social constructivist uses a slur, they seem to be employing it without appeal to any “essence”. Suchracist social constructivists seem conceivable/possible, but it is not clear whether Neufeld can makeroom for them in her theory. This is not a problem for the other two rich-lexicon theories considered.

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References:
Anderson, L. 2018. “Calling, Addressing, and Appropriation.” In Bad Words. Philosophical
Perspectives on Slurs, edited by D. Sosa, 6–26. Oxford: Oxford University Press.
Camp, E. 2013. “Slurring Perspectives.” Analytic Philosophy 54 (3): 330–349.
Cepollaro, B., & Zeman, D. (eds.). 2020. “Non-derogatory Uses of Slurs.” Grazer PhilosophischeStudien 97 (1): 1–202.
Croom, A. 2011. “Slurs.” Language Sciences 33 (3): 343–358.
Croom, A. 2013. “How to do Things with Slurs: Studies in the way of Derogatory Words.” Language& Communication 33: 177–204.
Del Pinal, G. 2018. “Meaning, Modulation, and Context: A Multidimensional Semantics for TruthConditional Pragmatics.” Linguistics and Philosophy 41 (2): 165–207.
Hom, C. 2008. “The Semantics of Racial Epithets.” Journal of Philosophy 105 (8): 416–440.
Jeshion, R. 2013. “Expressivim and the Offensiveness of Slurs.” Philosophical Perspectives 27: 231–
259.
Potts, C. 2007. “The Expressive Dimension.” Theoretical Linguistics 33 (2): 165–198.
Predelli, S. 2013. Meaning Without Truth. Oxford: Oxford University Press.
Richard, M. 2008. When Truth Gives Out. Oxford: Oxford University Press.
Neufeld, E. 2019. “An Essentialist Theory of the Meaning of Slurs.” Philosophers’ Imprint 19 (35):
1–29.
Neufeld, E. 2022. “Psychological Essentialism and the Structure of Concepts.” Philosophy Compass
17(5), e12823.
Zeman, D. 2022. “A Rich-lexicon Theory of Slurs and Their Uses.” Inquiry 65 (7), 942-966.

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Contrafactives and learnability*
David Strohmaier1 and Simon Wimmer2
1: Department of Computer Science and Technology, University of Cambridge
2: Department of Philosophy and Political Science, Technical University Dortmund

Holton (2017) has drawn attention to a novel semantic universal, according to which (at least
almost) no natural language features contrafactive attitude verbs. Contrafactives are the mirror
image of factive attitude verbs like know, remember, see, and regret: although both factives
and contrafactives entail a belief, contrafactives differ from factives in presupposing the falsity,
as opposed to the truth, of that belief. So, whilst both Dan knows that it is raining and Dan
contrafactives that it is raining entail that Dan believes that it is raining, the former also presupposes that it is raining, whereas the latter presupposes that it is not. Although some candidate
contrafactives have been discussed (see Holton, 2017, pp.245-9, 262–4), no clear counterexample to the universal has been found: for instance, Anvari, Maldonado, and Soria Ruiz (2019)’s
creerse is built by adjoining the reflexive pronoun to the non-factive verb creer ‘believe’, thefalsity inference of Hsiao (2017)’s Taiwanese Southern Min verb liah-tsun is too easy to cancel,
and Glass (2022)’s Mandarin belief verb yĭwéi carries a post-, rather than presupposition, thatthe reported belief must not be added to the common ground.
The no contrafactives universal raises an important question: why do natural languages universally feature factive verbs like know (Goddard, 2010; Haspelmath and Tadmor, 2009), but(at least almost) universally lack contrafactives? We develop a novel explanation of at least
part of this asymmetry. Drawing on recent discussions of other semantic universals, such as theveridical uniformity universal for responsive verbs (Steinert-Threlkeld, 2019), the conservativity, monotonicity, and quantity universals for determiners (Steinert-Threlkeld and Szymanik,2019), and the convexity universal for color terms (Steinert-Threlkeld and Szymanik, 2020), we
explore the hypothesis that the asymmetry between contrafactives and factives arises at least inpart because the meaning of a contrafactive is harder to learn than that of a factive. We tested
this hypothesis by conducting a computational experiment using an artificial neural network.
As we explain below, the results of the experiment support our hypothesis.
Our hypothesis is inspired by the intuitive idea that languages have words for meanings
that are easier to learn and use compositional methods to express meanings that are harder toacquire (Steinert-Threlkeld and Szymanik, 2019, p.4). This intuitive idea can be applied to theasymmetry between factives and contrafactives. Since a factive presupposes the truth of its
that-complement, a factive attitude ascription represents the way the subject of the ascription
takes the world to be and the way the world is, according to the ascriber, as converging. By
contrast, since a contrafactive presupposes the falsity of its that-complement, a contrafactive
attitude ascription represents the way the subject of the ascription takes the world to be and theway the world is, according to the ascriber, as diverging. A factive attitude ascription thus only
requires ascribing (part of) one’s own take on how things are to another person. By contrast, acontrafactive attitude ascription requires ascribing a take on how things are to another personthat is inconsistent with one’s own. This is a more complex achievement that we expect to beharder to learn (cf. Phillips and Norby, 2021). Given that this difference between factive andcontrafactive attitude ascription appears to be due entirely due to the meaning of factives andcontrafactives, this difference leads us to expect that the semantics of contrafactives is harderto acquire than that of factives.
* This paper reports on research supported by Cambridge University Press and Assessment, University ofCambridge. We thank the NVIDIA Corporation for the donation of the Titan X Pascal GPU used in this research.
Simon Wimmer’s work on this paper was supported by a postdoc stipend of the Fritz Thyssen Foundation.

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To test our expectation, we conducted a computational experiment using an artificial neuralnetwork. This network was trained to predict the truth value of factive, non-factive or contrafactive attitude ascriptions, given a representation of a small world and a representation of the small
world as the attitude holder takes it to be (which may or may not be accurate). The network’s
prediction was expressed in a probability that the target ascription is true. The artificial language
in which the target attitude ascriptions were formulated and which the neural network learned
can be interpreted as a fragment that describes propositions about the relative locations of two
objects to each other plus the attitude taken towards these propositions. To encode this artificial
language and the representations, we used a Transformer encoder. Transformers, which are
based upon the so-called attention mechanism that allows contextualised processing of word
information, are the foundation of current state-of-the-art results in natural language processing(Vaswani et al., 2017; Devlin et al., 2019; Rogers, Kovaleva, and Rumshisky, 2020).
Generally speaking, the results of our experiments show the Transformer-encoder to perform better on factive verbs than contrafactive verbs. While the performance on non-factive
verbs was even worse, we consider this to be at least in part the result of the architecture being
not well-suited for these verbs. The architecture always processes both a mind and a world
representation, even though the latter only contributes noise in the case of non-factive verbs.
We assume that the human brain incorporates a better input-gating mechanism than our model,
which would increase the performance on non-factive verbs. In the following, we report theresults for the comparison between factive and contrafactive verbs.
We evaluated 51 hyperparameter settings in an initial search, 18 of which performed below
60% accuracy and 27 exceeded 90% accuracy. In all except 4 of the 51 settings, the accuracy
was higher for the factive than the contrafactive verb (but not significantly so). We take this tosuggest that the difference is due to the neural architecture rather than the hyperparameters.
The setting which performed best in the hyperparameter search, i.e. the one with the highest overall accuracy, was then applied to a hold-out test set. The results on this test set once
again showed higher performance for the factive than the contrafactive verbs. The differencein accuracy was small, because the model was trained on a large sample of data.
The differences for the mean absolute error are larger. In particular, the error is significantly
larger for contrafactive verbs than factive ones (p < 0.01), see figure 1. The training for thefactive verb also proceeded faster than for the contrafactive verb, see figure 2, providing further
support for a difference in how hard it is to learn the meaning of a contrafactive as opposed tofactive. To provide some numbers for intuition, after 100000 training examples the average lossof the factive predicate is 0.39, while the one for the contrafactive predicate is 0.54.
Post-experimental analysis suggests that the presence or absence of potentially confusing
objects in the world and mind representation was responsible for much of the remaining errors.
That is, the network was paying excessive attention to whether objects named in the artificial
language sentences were present or not, ignoring whether the described relationship between
the objects held. Given this, we plan to carry out a follow-up experiment in which the artificial
language which the neural network learns is simpler, being roughly interpretable as a fragmentthat simply describes a primitive, non-decomposable proposition and the attitude taken towardthat proposition.
Our computational experiment improves on similar ones conducted by Steinert-Threlkeld(2019) and Steinert-Threlkeld and Szymanik (2019) in a number of ways. First, we reportresults from a larger range of hyperparameters (e.g. training epochs, learning rate, etc.). By
exploring the range of models which bear out our expectation that contrafactives are harder
to learn than factives, we provide a better sense of the robustness of our experimental results.
Second, while the cited research used feed-forward neural networks and LSTMs, we switched131to the more advanced Transformer-architecture. Recent results suggest that despite not being
originally designed for cognitive plausibility, Transformer-based networks nonetheless show
greater convergence with human processing than other approaches (e.g. Caucheteux and King,2022). Given this, the results of our computational experiment likely reflect learnability forhuman language learners more closely than previous work.
However, the results from our experiment likely capture only one source of a difference inlearnability between factives and contrafactives. Our results do not reflect pragmatic and syntactic features, since the input sentences do not differ in this regard. But these features likely
contribute to the difference in learnability of factives and contrafactives. To illustrate, on the
pragmatic syntactic bootstrapping model of how infants acquire attitude verb meanings (Hacquard and Lidz, 2022), the meaning of non-factive think is partly inferred from the parallel
between the use of I think P as an indirect assertion and the primary use of P as an assertion,and the meaning of factive know is partly inferred from the parallel between the use of Do you
know Q? as an indirect question and the primary use of Q? as a question. Yet unlike in the caseof factive and non-factive verbs, no such parallels would hold for a contrafactive: use of a contrafactive attitude ascription would not match the primary use of its complement. We leave it
to future work to explore how this pragmatic syntactic difference between factive, non-factive,
and contrafactive attitude ascriptions and the differences suggested by our computational experiment combine to explain the difference in frequency of factive, non-factive, and contrafactive
verbs in natural languages.
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Gricean Pragmatics According to Himself
Nayuta Miki (Osaka University)
Grice presents the theory of conversational implicature based on the cooperative principle
and conversational maxims in his 1975 paper. Typically, Gricean pragmatics refers to a range ofpragmatic theories that account for various cases of conversational implicature by relying on various
conversational maxims. However, it is unclear what Grice himself would think a pragmatic theory of
conversational implicature should be beyond what he stated in his 1975 article.
The following four points must be noted to understand Grice’s perspective on conversational
implicature: (1) he classifies conversational implicature as falling under the category of speaker
meaning (or “utterer’s meaning” in his original term), (2) the concept of speaker meaning is understoodin terms of the concept of speaker’s intention, (3) he believes that the concept of intention is analyzable
in terms of the concepts of will and belief, (4) his perspective on psychological states such as will andbelief is based on his unique axiomatic philosophy of mind.
As Grice explained in his papers from 1968 and 1969, his analysis of speaker meaning is
supposed to provide a basis for his theory of implicature. He attempts to provide the individuallynecessary and jointly sufficient conditions for one’s meaning something in terms of their intentions: aspeaker, S, (indicatively) means that p by uttering x if and only if S utters x intending (1) an audience,
A, to believe that p, (2) A to realize that S intends (1), and (3) A’s realizing that S intends (1) is a partof their reason to believe that p. Thus, what one means by their utterance, and therefore what they
conversationally implicate, is essentially a matter of what they intend.
Grice argues that intention is a complex psychological state composed of two simpler states:
will and belief. Thus, what one intends is essentially a matter of what they will and believe. It meansthat what a speaker conversationally implicates is supposed to be determined by their wills and beliefs.
According to Grice, psychological states such as will and belief are understood as something thatprovides an “explanatory bridge” between the situation an agent is put in and what they do in thatsituation. According to Grice, to explain an agent’s action is to provide a deductive inference with thepremise that they are in such and such a situation and with the consequence that they do the action inquestion. In order to make such an inference, we must fill in the gaps between the premise and theconsequence. Thus, we must rely on certain axioms. Psychological states, according to Grice, appear
in such axioms.
With this overview of Grice’s philosophical system, it is that Gricean pragmatics as Gricehimself envisions would be a search for our psychological axioms that eenable our mutual
understanding in everyday communication.

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