Modal Subordination in Type Theoretic Dynamic Logic (2016)

Abstract: Classical theories of discourse semantics, such as Discourse Representation Theory (DRT), Dynamic Predicate Logic (DPL), predict that an indefinite noun phrase cannot serve as antecedent for an anaphor if the noun phrase is, but the anaphor is not, in the scope of a modal expression. However, this prediction meets with counterexamples. The phenomenon modal subordination is one of them. In general, modal subordination is concerned with more than two modalities, where the modality in subsequent sentences is interpreted in a context ‘subordinate’ to the one created by the first modal expression. In other words, subsequent sentences are interpreted as being conditional on the scenario introduced in the first sentence. One consequence is that the anaphoric potential of indefinites may extend beyond the standard limits of accessibility constraints. This paper aims to give a formal interpretation on modal subordination. The theoretical backbone of the current work is Type Theoretic Dynamic Logic (TTDL), which is a Montagovian account of discourse semantics. Different from other dynamic theories, TTDL was built on classical mathematical and logical tools, such as λ-calculus and Church’s theory of types. Hence it is completely compositional and does not suffer from the destructive assignment problem. We will review the basic set-up of TTDL and then present Kratzer’s theory on natural language modality. After that, by integrating the notion of conversation background, in particular, the modal base usage, we offer an extension of TTDL (called Modal-TTDL, or M-TTDL in short) which properly deals with anaphora across modality. The formal relation between Modal-TTDL and TTDL will be discussed as well. We uncover the difficulty of specific sense distinctions by investigating distributional bias and reducing the sparsity of existing small-scale corpora used in prior work. We build a semantically enriched model for modal sense classification by designing novel features related to lexical, proposition-level and discourse-level semantic factors. Besides improved classification performance, closer examination of interpretable feature sets unveils relevant semantic and contextual factors in modal sense classification. Finally, we investigate genre effects on modal sense distribution and how they affect classification performance. Our investigations uncover the difficulty of specific sense distinctions and how they are affected by training set size and distributional bias. Our large-scale experiments confirm that semantically enriched models outperform models built on shallow feature sets. Cross-genre experiments shed light on differences in sense distributions across genres and confirm that semantically enriched models have high generalization capacity, especially in unstable distributional settings.

## type theoretic dynamic logic around a decade ago, de groote proposed a new dynamic framework type theoretic dynamic logic (ttdl) @xcite . this framework aims to study the semantics of sentences and discourses in a uniform and classical way. in order to achieve dynamics, ttdl integrates the notion of left and right contexts into mg: given a sentence, its left context denotes the discourse that precedes it, namely what has already been processed; its right context is the continuation @xcite , denoting the discourse that follows it, namely what will be processed in future. a sentence is interpreted with respect to both its left and right contexts, and its semantics is abstracted over the two contexts. technically, ttdl sticks to the tradition of mg. it only makes use of standard mathematical and logical tools, such as λ-calculus and theory of types. logical notions such as free and bound variables, quantifier scopes, are as usual. and the only operations involved are standard αconversions and β-reductions. this property enables it to inherit all nice properties of well-established mathematics and logics. in what follows, we present the formal details of ttdl. akin to other systems based on the simply typed λ-calculus, the syntax of ttdl can be defined in terms of the notion of higher order signature (de groote, 2001), which is a triplet consisting of a finite set of atomic types, a finite set of constant symbols, and a function that assigns each constant a type. definition 2.1. the signature of ttdl, in notation σ t t dl , is defined august 2016 as follows: σ t t @xmath0, o, γ}, { , ∧, ¬, ∃, :: , sel, nil}, logical constants such as (tautology), ∧ (conjunction), ¬ (negation) and ∃ (existential quantifier) are exactly the same as in first order logic (fol). however, besides the two atomic types in church's simple type theory @xcite : ι denoting the type of individuals and o denoting the type of propositions, there is a third one in ttdl: γ denoting the type of left contexts. the right context, which is interpreted as the continuation of the sentence, is a function from left contexts to truth values. so its type is γ → o. for instance, assume e is a left-context variable, then the empty right context can be defined in the compact term stop as follows: stop λe. in order to solve pronominal anaphora in ttdl, the left context is modeled as a list of individuals. this explains the type of the empty context "nil" in σ t t dl . in addition to that, two operators are introduced in definition 2.1. the first is the list constructor "::". its function is to add new individuals (of type ι) into existing contexts (of type γ). hence the type of "::" is ι → γ → γ. the second is the choice operator "sel", it takes a left context (of type γ) as argument and yields back an individual (of type ι). hence its type is γ → ι. in standard truth-conditional semantics, a sentence expresses a proposition, which is of type o. while in ttdl, a sentence will be interpreted with respect to both its left and right contexts, which are of type γ and γ → o, respectively. if we use s to denote the syntactic category of sentences, t t dl to denote the semantic interpretation under ttdl, then: discourses, which also express propositions, are interpreted in the same way as single sentences. so, letting d be the syntactic category of discourses, we have: in order to contrast with o, which is the type of (standard/static) propositions, we call γ → (γ → o) → o the type of dynamic proposi-tions. hereinafter, we will use ω as an abbreviation for γ → (γ → o) → o, namely: ω γ → (γ → o) → o (2.4) after presenting the typing information in ttdl, let us proceed to the logics of the framework. let a and b be variables denoting dynamic propositions, e and e be variables denoting left contexts, φ be a variable denoting the right context, then the dynamic conjunction ∧ d t t dl in ttdl, which conjoins two dynamic propositions, is defined as follows foot_2 : in order to negate dynamic propositions, ttdl defines the dynamic negation operator ¬ d t t dl as follows: ## modal subordination under ttdl in this section, we will integrate epistemic modality within the continuationbased dynamic framework ttdl as introduced earlier in section 2, we will call the new framework modal ttdl (m-ttdl). as explained in section 3.2, a conversational background is a function from possible worlds to sets of propositions, which are the ones that are necessarily true at the given world. they serve as the common ground information, or premise assumption, for subsequent modally subordinated utterances. hence our strategy for achieving m-ttdl is to enrich the context of ttdl with the notion of conversational background @xcite , in particular the modal base. in the following, we will first present the formal framework, including the particular signature for m-ttdl, and the typing information and the way in which (modal) proposition, left context, right context, etc., are respectively interpreted; then we will define some preliminary functions that facilitate later presentation; after that, we propose the formal framework, including the syntax and semantics; finally, the lexical entries, together with the treatments of some puzzling examples will be provided. ## conclusion and future work anaphora interpretation is a critical piece of machinery in natural language interpretation. the aim of this paper was to study the semantics of one specific type of anaphora: inter-sentential pronominal anaphora from the discourse perspective. more specifically, this paper was concerned with the phenomenon of modal subordination, and the con-straints on interpretation of anaphors and modal sentences. to summarize, the solution we proposed involves extending the dynamic framework ttdl. another system in the simply typed λcalculus family, namely m-ttdl, has been developed. we have briefly sketched the relation between m-ttdl and ttdl: the former has at least the same empirical coverage as the latter (the formal proof is not presented due to the length of the paper). furthermore, m-ttdl is able to account for both the felicitous and infelicitous anaphors under various number of modality-involved discourses. ttdl and m-ttdl have advantage over other dynamic theories because they stick to the traditional montagovian style: the only operations involved are standard α-conversions and β-reductions. hence the principle of compositionality is retained without adding any new concept to classical logic. however, the empirical data addressed here is only a small part in the semantics of anaphora and modality. on the theoretical side, a potential continuation of the current research is to broaden the coverage of our framework. for instance, in the following set of examples, the lifespan of a discourse referent is longer than m-ttdl would expect: in addition, the accessibility of discourse referents seems to be exceptional when modality and negation are involved at the same time: ttdl. similar examples include: (19) a. i didn't submit a paper i . they wouldn't have published it i . (kibble, 1994) b. john didn't buy a mystery novel i . he would be reading it i by now. (krifka, 2001) c. mary didn't buy a microwave i . she would never use it i . besides the above theoretical continuations, the current framework face a number of practical challenges as well. it is well-known that defining propositions as functions from possible worlds to truth values (this is what most modal systems do to integrate the notion of possible world) creates hyperintensional problems. since m-ttdl is configured exactly in this fashion, it inevitably suffers from the same problems. also, from the implementational point of view, possible worlds are costly and have been shown to present serious complexity problems even for the propositional case @xcite . consequently, it might not be an easy task to put the m-ttdl framework into practice in the short term, particularly in those real-time anaphora resolution systems. however, we are still full of optimism because a fragment of m-ttdl has already been successfully implemented in the abstract categorial grammar toolkit (de groote, 2001), programmed in the ocaml language. this can be considered as a first step towards a real application in language technologies, computational issues such as complexity will definitely be a direction for further research. possible world, is of type o i .