Type Theory as a Formal Basis of Natural Language Semantics

An invited talk at Yanaka Lab on May 21, 2025.

Transcript

1. Type Theory as a Formal Basis of Natural Language Semantics Daiki Matsuoka (Yanaka Lab, D1) Invited Talk (May 21, 2025)
2. Introduction One key aspect of natural language is that humans can convey information with it. sound ? meaning More abstractly: the human cognitive mechanism can translate between sounds and mental representations for meaning. 1 / 83
3. Why “Formal”? Some empirical facts about meaning can be well captured by formal logic. Classic example: inference (1) a. No dog ran. ∀𝑥.(dog(𝑥) ⊃ ¬run(𝑥)) b. Every puppy is a dog. ∀𝑥.(puppy(𝑥) ⊃ dog(𝑥)) c. ⟹ No puppy ran. ∀𝑥.(puppy(𝑥) ⊃ ¬run(𝑥)) Formalsemanticsisabranchoftheoreticallinguisticsthatuses a logical system as a framework of semantic representation. 2 / 83
4. Empirical Domain: Anaphora This talk focuses on anaphora. Anaphora is a phenomenon where the interpretation of a pronoun depends on another expression, called its antecedent. (2) Alex𝑖 loves her𝑖 mother. (I’ll use subscripts to indicate anaphora) In particular, we investigate cases where the antecedent is a quantifier. (3) [Every girl]𝑖 loves her𝑖 mother. (i.e., the girl A loves A’s mother, B loves B’s mother ...) Here, “her” does not refer to any particular girl. 3 / 83
5. Empirical Domain: Anaphora [contd.] The anaphoric interpretation of (3) can be easily expressed with a logical formula. (3) [Every girl]𝑖 loves her𝑖 mother. ⇝ ∀𝑥.(girl(𝑥) ⊃ love(𝑥, mother(𝑥))) This phenomenon is called bound variable anaphora. • The bound variable interpretation seems to be adequately captured by logical formulas. • However, natural language shows some differences from the standard first-order logic ... 4 / 83
6. Puzzle (i): Scope Problem Pronominal binding is sometimes “looser” than binding in the logical language in terms of variable scope. (4) a. Kim greeted [a girl]. ⇝ ∃𝑥.(girl(𝑥) ∧ greet(k, 𝑥)) b. Kim greeted [a girl]𝑖 , and she𝑖 smiled. ⇝ ∃𝑥.(girl(𝑥) ∧ greet(k, 𝑥)) ∧ smile( ??? ) Since the scope of ∃𝑥 is closed on the left of ∧, we cannot express the anaphoric dependency between a girl and she. 5 / 83
7. Puzzle (i): Scope Problem [contd.] Although one might think of “extending” the effect of a girl to the second sentence, things are not so simple ... Quantifiers in certain environments are inaccessible from outside (Groenendijk & Stokhof, 1991).1 (5) a. *Kim greeted [{every/no} girl]𝑖 , and she𝑖 smiled. b. If Kim buys [a book]𝑖 , she reads it𝑖 through. *It𝑖 is long. Q. How can we formally account for inter-sentential variable binding? 1Here, I use the asterisk * to indicate the unacceptability of the interpretation indicated by the subscripts, not the sentence itself. 6 / 83
8. Puzzle (ii): Crossover Problem Certain syntactic configurations disallow logically possible bound variable interpretations. It has been observed that quantifiers cannot bind pronouns from a structurally lower position. (6) *Her𝑖 mother loves [every girl]𝑖 . ∀𝑥.(girl(𝑥) ⊃ love(mother(𝑥), 𝑥)) This is called the crossover effect (Postal, 1971; Safir, 2017). 7 / 83
9. Puzzle (ii): Crossover Problem [contd.] N.B. in principle, the subject can be in the scope of the object quantifier. (7) Alex loves every girl. ⇝ ∀𝑥.(girl(𝑥) ⊃ love(a, 𝑥)) Q. How can we constrain the bound variable interpretation in line with the crossover effect? 8 / 83
10. Outline of This Talk Thistalkdescribeshowtypetheorycanbeusedasaframework of semantic representation. • Concretely, I will introduce a theory called Dependent Type Semantics (DTS) (Bekki, 2014; Bekki & Mineshima, 2017). • DTS provides a type-theoretical account of bound variable interpretations. • If time permits, we will see how the DTS-based analysis is extended to a phenomenon called presupposition. 9 / 83
11. Roadmap Propositions-as-Types Dependent Types Underspecification Presupposition Conclusion and Future Outlook 10 / 83
12. Propositions-as-Types
13. Type Theory A type is a formal object that classifies terms. type term nat 0, 1, 2, ... nat → nat 𝜆𝑥.𝑥 + 1 nat → nat → nat 𝜆𝑦.𝜆𝑥.𝑥 + 𝑦 The relationship between terms and types are stated in terms of typing judgements, written Γ ⊢ 𝑀 ∶ 𝐴. typing context 𝑥 ∶ nat, 𝑓 ∶ nat → nat ⊢ the term 𝑓𝑥 has type nat 𝑓𝑥 ∶ nat 11 / 83
14. Typing Rules Typing judgments are derived with typing rules. For each type, we have ... • Introduction rule: how to construct a term of the type • Elimination rule: how to use a term of the type Function type 𝐴 → 𝐵 Γ, 𝑥 ∶ 𝐴 ⊢ 𝑡 ∶ 𝐵 (→𝐼) Γ ⊢ 𝜆𝑥.𝑡 ∶ 𝐴 → 𝐵 Γ ⊢ 𝑓 ∶ 𝐴 → 𝐵 Γ ⊢ 𝑎 ∶ 𝐴 (→𝐸) Γ ⊢ 𝑓𝑎 ∶ 𝐵 12 / 83
15. Typing Rules [contd.] Product type 𝐴 × 𝐵 Γ ⊢ 𝑎 ∶ 𝐴 Γ ⊢ 𝑏 ∶ 𝐵 (×𝐼) Γ ⊢ ⟨𝑎, 𝑏⟩ ∶ 𝐴 × 𝐵 Γ ⊢ 𝑡 ∶ 𝐴 × 𝐵 (×𝐸 1 ) Γ ⊢ 𝜋1 𝑡 ∶ 𝐴 Γ ⊢ 𝑡 ∶ 𝐴 × 𝐵 (×𝐸 2 ) Γ ⊢ 𝜋2 𝑡 ∶ 𝐵 We also have structural rules, which specify how to manipulate the typing context. (var) (𝑥 ∶ 𝐴 ∈ Γ) Γ ⊢ 𝑥 ∶ 𝐴 13 / 83
16. Correspondence with Proof Theory Typing rules have some parallels with inference rules in proof theory (the Curry-Howard correspondence (Howard, 1980)). Example: modus ponens Γ ⊢ 𝑓 ∶ 𝐴 → 𝐵 Γ ⊢ 𝑎 ∶ 𝐴 (→𝐸) Γ ⊢ 𝑓𝑎 ∶ 𝐵 𝐴 ⊃ 𝐵 𝐴 (⊃𝐸) 𝐵 Example: conjunction introduction Γ ⊢ 𝑎 ∶ 𝐴 Γ ⊢ 𝑏 ∶ 𝐵 (×𝐼) Γ ⊢ ⟨𝑎, 𝑏⟩ ∶ 𝐴 × 𝐵 𝐴 𝐵 (∧𝐼) 𝐴 ∧ 𝐵 14 / 83
17. Correspondence with Proof Theory [contd.] A bit deeper intuition … (called the Brouwer-Heyting-Kolmogorov interpretation) A proof of 𝐴 ⊃ 𝐵 is, essentially, a method (≃ function) that turns any given proof of 𝐴 into a proof of 𝐵. 𝐴 𝐵 • 𝑎 •𝑏1 •𝑏2 𝑓1 𝑓2 15 / 83
18. Correspondence with Proof Theory [contd.] Likewise, a proof of 𝐴 ∧ 𝐵 can be viewed as a pair of a proof of 𝐴 and a proof of 𝐵. 𝐴 𝐵 • 𝑎 •𝑏1 •𝑏2 ⟨𝑎, 𝑏1 ⟩ ⟨𝑎, 𝑏2 ⟩ 16 / 83
19. Propositions-as-Types Thus, we can identify a proposition with a type to which its proofs belong (see Dyberempty citation for a review). The principle of propositions-as-types A proposition and its proofs correspond to a type and its terms. More concretely, Γ ⊢ 𝑀 ∶ 𝐴 can be interpreted in two ways: in this typing context with these hypotheses Γ ⊢ the term 𝑀 has type 𝐴 the proof 𝑀 verifies the proposition 𝐴 𝑀 ∶ 𝐴 17 / 83
20. Propositions-as-Types: Example Exercise: derive the typing judgment corresponding to the following proof of 𝐴 ∧ 𝐵 ⊃ 𝐵 ∧ 𝐴. [𝐴 ∧ 𝐵]1 𝐵 [𝐴 ∧ 𝐵]1 𝐴 𝐵 ∧ 𝐴 𝐴 ∧ 𝐵 ⊃ 𝐵 ∧ 𝐴 Answer: (var) 𝑥 ∶ 𝐴 × 𝐵 ⊢ 𝑥 ∶ 𝐴 × 𝐵 (×𝐸 2 ) 𝑥 ∶ 𝐴 × 𝐵 ⊢ 𝜋2 𝑥 ∶ 𝐵 (var) 𝑥 ∶ 𝐴 × 𝐵 ⊢ 𝑥 ∶ 𝐴 × 𝐵 (×𝐸 1 ) 𝑥 ∶ 𝐴 × 𝐵 ⊢ 𝜋1 𝑥 ∶ 𝐴 (×𝐼) 𝑥 ∶ 𝐴 × 𝐵 ⊢ ⟨𝜋2 𝑥, 𝜋1 𝑥⟩ ∶ 𝐵 × 𝐴 (→𝐼) ⊢ 𝜆𝑥.⟨𝜋2 𝑥, 𝜋1 𝑥⟩ ∶ 𝐴 × 𝐵 → 𝐵 × 𝐴 18 / 83
21. Dependent Types
22. Dependent Type A dependent type is a type that depends on values (Martin-Löf, 1984). Example: the type of lists with a specified length 𝑛 • [ ] ∶ Vec nat (0) • [0], [1], ... ∶ Vec nat (1) • [0; 1], [1; 2], ... ∶ Vec nat (2) Dependent types enriches the type theory by allowing finer classification of types. 19 / 83
23. Dependent Type [contd.] With dependent types, we can generalize functions and pairs. Example: the function that returns a vector filled with zeros. • zeros(0) = [ ] ∶ Vec nat (0) • zeros(1) = [0] ∶ Vec nat (1) • zeros(2) = [0; 0] ∶ Vec nat (2) The type of zeros is (𝑛 ∶ nat) → Vec nat (𝑛). Here, the codomain of the function varies with the argument. This type of dependency will be important later ... 20 / 83
24. Dependent Type Theory and Predicate Logic Dependent types can be used to simulate predicate logic. • “𝑥 is a girl” ⇝ girl(𝑥) • “𝑥 greeted 𝑦” ⇝ greet(𝑥, 𝑦) Here, precidates are functions from entities into types. • ⊢ girl ∶ e → type • ⊢ greet ∶ e → e → type 21 / 83
25. Remark: Well-Formedness of Types type is the (higher-order) type of all the well-formed types. Well-formed types are derived through typing rules. Example: girl(𝑥) (con) 𝑥 ∶ e ⊢ girl ∶ e → type (var) 𝑥 ∶ e ⊢ 𝑥 ∶ e (→𝐸) 𝑥 ∶ e ⊢ girl(𝑥) ∶ type N.B. unlike in simply typed lambda calculus, we cannot define the set of well-formed types in advance. 22 / 83
26. Remark: Well-Formedness of Types [contd.] What does well-formedness mean linguistically? A well-formed type corresponds to a proposition that makes sense in the current context. Example: if 𝑥 ∶ e is not in Γ, then Γ ⊬ girl(𝑥) ∶ type Analogously, if there is nothing in the context referred to by 𝑥, girl(𝑥) does not make sense (neither true nor false). Thus, this notion is related to the notion of felicity of an utterance (i.e., whether the utterance is pragmatically appropriate). 23 / 83
27. Quantifiers What about quantifiers? • Π-type (𝑥 ∶ 𝐴) → 𝐵 • The dependent version of the function type 𝐴 → 𝐵. • It corresponds to ∀𝑥 ∈ 𝐴.𝐵. • Σ-type: (𝑥 ∶ 𝐴) × 𝐵 • The dependent version of the product type 𝐴 × 𝐵. • It corresponds to ∃𝑥 ∈ 𝐴.𝐵. 24 / 83
28. Π-type: Correspondence with ∀ A proof of ∀𝑥 ∈ 𝐴.𝐵 can be regarded as a function that turns any given element 𝑎 of 𝐴 into a proof of 𝐵[𝑥 ∶= 𝑎]. 𝐴 𝐵[𝑥 ∶= 𝑎1 ] 𝐵[𝑥 ∶= 𝑎2 ] ⋮ • 𝑎1 • 𝑎2 •𝑏1 1 •𝑏1 2 •𝑏2 1 •𝑏2 2 𝑓1 𝑓2 𝑓1 𝑓2 25 / 83
29. Π-type: Typing Rules Π-type (𝑥 ∶ 𝐴) → 𝐵 Γ, 𝑥 ∶ 𝐴 ⊢ 𝑡 ∶ 𝐵 (Π𝐼) Γ ⊢ 𝜆𝑥.𝑡 ∶ (𝑥 ∶ 𝐴) → 𝐵 Γ ⊢ 𝑓 ∶ (𝑥 ∶ 𝐴) → 𝐵 Γ ⊢ 𝑎 ∶ 𝐴 (Π𝐸) Γ ⊢ 𝑓𝑎 ∶ 𝐵[𝑥 ∶= 𝑎] Note: if 𝑥 is not free in 𝐵, the Π-type is equivalent to 𝐴 → 𝐵. 26 / 83
30. Π-type: Typing Rules [contd.] In addition to the intro/elim rules, we have the formation rule, specifying the well-formedness condition. Γ ⊢ 𝐴 ∶ type Γ, 𝑥 ∶ 𝐴 ⊢ 𝐵 ∶ type (Π𝐹) Γ ⊢ (𝑥 ∶ 𝐴) → 𝐵 ∶ type Crucially, the well-formedness of 𝐵 can depend on 𝑥 ∶ 𝐴 for the whole Π-type to be well-formed. 27 / 83
31. Σ-type: Correspondence with ∃ A proof of ∃𝑥 ∈ 𝐴.𝐵 can be regarded as a pair of an element 𝑎 of 𝐴 and a proof of 𝐵[𝑥 ∶= 𝑎]. 𝐴 𝐵[𝑥 ∶= 𝑎1 ] 𝐵[𝑥 ∶= 𝑎2 ] ⋮ • 𝑎1 • 𝑎2 •𝑏1 1 •𝑏1 2 •𝑏2 1 •𝑏2 2 ⟨𝑎1 , 𝑏1 1 ⟩ ⟨𝑎1 , 𝑏1 2 ⟩ ⟨𝑎2 , 𝑏2 1 ⟩ ⟨𝑎2 , 𝑏2 2 ⟩ 28 / 83
32. Σ-type: Typing Rules Σ-type (𝑥 ∶ 𝐴) × 𝐵 Γ ⊢ 𝑎 ∶ 𝐴 Γ ⊢ 𝑏 ∶ 𝐵[𝑥 ∶= 𝑎] (Σ𝐼) Γ ⊢ ⟨𝑎, 𝑏⟩ ∶ (𝑥 ∶ 𝐴) × 𝐵 Γ ⊢ 𝑡 ∶ (𝑥 ∶ 𝐴) × 𝐵 (Σ𝐸 1 ) Γ ⊢ 𝜋1 𝑡 ∶ 𝐴 Γ ⊢ 𝑡 ∶ (𝑥 ∶ 𝐴) × 𝐵 (Σ𝐸 2 ) Γ ⊢ 𝜋2 𝑡 ∶ 𝐵[𝑥 ∶= 𝜋1 𝑡] Γ ⊢ 𝐴 ∶ type Γ, 𝑥 ∶ 𝐴 ⊢ 𝐵 ∶ type (Σ𝐹) Γ ⊢ (𝑥 ∶ 𝐴) × 𝐵 ∶ type Note: if 𝑥 is not free in 𝐵, the Σ-type is equivalent to 𝐴 × 𝐵. 29 / 83
33. Some Examples (8) Kim greeted every girl. ⇝ (𝑥 ∶ e) → ((𝑢 ∶ girl(𝑥)) → greet(k, 𝑥)) cf. ∀𝑥.(girl(𝑥) → greet(k, 𝑥)) (9) Kim greeted a girl. ⇝ (𝑥 ∶ e) × ((𝑢 ∶ girl(𝑥)) × greet(k, 𝑥)) cf. ∃𝑥.(girl(𝑥) ∧ greet(k, 𝑥)) 30 / 83
34. Π-type and Σ-type: Example Exercise: derive the typing judgment corresponding to the proof of ((∃𝑥 ∈ 𝐴.𝐵) ⊃ 𝐶) ⊃ ∀𝑥 ∈ 𝐴.(𝐵 ⊃ 𝐶) Ans: let Γ be 𝑓 ∶ (𝑢 ∶ (𝑥 ∶ 𝐴) × 𝐵) → 𝐶, 𝑥 ∶ 𝐴, 𝑢 ∶ 𝐵. (var) Γ ⊢ 𝑓 ∶ (𝑢 ∶ (𝑥 ∶ 𝐴) × 𝐵) → 𝐶 (var) Γ ⊢ 𝑥 ∶ 𝐴 (var) Γ ⊢ 𝑢 ∶ 𝐵 (Σ𝐼) Γ ⊢ ⟨𝑥, 𝑢⟩ ∶ (𝑥 ∶ 𝐴) × 𝐵 (Π𝐸) Γ ⊢ 𝑓⟨𝑥, 𝑢⟩ ∶ 𝐶 (Π𝐼) 𝑓 ∶ ⋯ , 𝑥 ∶ 𝐴 ⊢ 𝜆𝑢.𝑓⟨𝑥, 𝑢⟩ ∶ (𝑢 ∶ 𝐵) → 𝐶 (Π𝐼) 𝑓 ∶ ⋯ ⊢ 𝜆𝑥.𝜆𝑢.𝑓⟨𝑥, 𝑢⟩ ∶ (𝑥 ∶ 𝐴) → (𝑢 ∶ 𝐵) → 𝐶 (Π𝐼) ⊢ 𝜆𝑓.𝜆𝑥.𝜆𝑢.𝑓⟨𝑥, 𝑢⟩ ∶ (𝑓 ∶ (𝑢 ∶ (𝑥 ∶ 𝐴) × 𝐵) → 𝐶) → (𝑥 ∶ 𝐴) → (𝑢 ∶ 𝐵) → 𝐶 31 / 83
35. Remark: Box Notation For readability, we use a box-like notation for the Σ-type. (9) Kim greeted a girl. ⇝ ⎡ ⎢ ⎢ ⎣ 𝑥 ∶ e [ 𝑢 ∶ girl(𝑥) greet(k, 𝑥) ] ⎤ ⎥ ⎥ ⎦ 32 / 83
36. Representing Inter-Sentential Anaphora Now we are ready to represent inter-sentential bound variable interpretations with dependent type theory (Ranta, 1995; Sundholm, 1986). (10) Kim greeted [a girl]𝑖 , and she𝑖 smiled. Since the two sentences are combined with conjunction, the whole representation should be a Σ-type. ⎡ ⎢ ⎢ ⎢ ⎣ 𝑣 ∶ ⎡ ⎢ ⎢ ⎣ 𝑥 ∶ e [ 𝑢 ∶ girl(𝑥) greet(k, 𝑥) ] ⎤ ⎥ ⎥ ⎦ smile( ? ) ⎤ ⎥ ⎥ ⎥ ⎦ 33 / 83
37. Representing Inter-Sentential Anaphora [contd.] We can use 𝜋1 to represent the anaphoric interpretation! ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 𝑣 ∶ ⎡ ⎢ ⎢ ⎣ 𝑥 ∶ e [ 𝑢 ∶ girl(𝑥) greet(𝑥) ] ⎤ ⎥ ⎥ ⎦ smile( 𝜋1 𝑣 ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ • 𝑣 ranges over the proofs of Kim greeted a girl, which should have the form ⟨𝑥, ⟨𝑝1 , 𝑝2 ⟩⟩. • The crucial point is that we view conjunction as an existential quantification of the proofs of the first conjunct. 34 / 83
38. Representing Inter-Sentential Anaphora [contd.] The same strategy does not apply to the universal quantifier. (11) *Kim greeted [every girl]𝑖 , and she𝑖 smiled. ⇝ [ 𝑣 ∶ (𝑥 ∶ e) → (𝑢 ∶ girl(𝑥)) → greet(𝑥) smile( ? ) ] • The typing rules of the Π-type do not allow us to access 𝑥 ∶ 𝐴 via 𝑣 ∶ (𝑥 ∶ 𝐴) → 𝐵. • Thus, we can correctly predict the bound variable interpretation in (11). (we will give a more formal account later ...) 35 / 83
39. Representing Inter-Sentential Anaphora [contd.] Another case: (12) a. If Kim recommends [a book]𝑖 , Alex reads it𝑖 . ⇝ ⎛ ⎜ ⎜ ⎜ ⎝ 𝑣 ∶ ⎡ ⎢ ⎢ ⎣ 𝑥 ∶ e [ 𝑢 ∶ book recommend(k, 𝑥) ] ⎤ ⎥ ⎥ ⎦ ⎞ ⎟ ⎟ ⎟ ⎠ → read(a, 𝜋1 𝑣 ) b. *It𝑖 is interesting. (It is impossible to refer to 𝑥 ∶ e from outside.) 36 / 83
40. Interim Summary So far, we have seen … • The principle of propositions-as-types allows us to use types as semantic representations. • Dependent types can be used to simulate predicate logic. • The inter-sentential anaphoric interpretation can be represented via quantification over the proofs of propositions. ⎡ ⎢ ⎢ ⎣ 𝑣 ∶ [ 𝑥 ∶ e ⋯ ] 𝑃(𝜋1 𝑣) ⎤ ⎥ ⎥ ⎦ 37 / 83
41. Interim Summary [contd.] However, it remains to see how to systematically derive those representations ... • For example, how did the term 𝜋1 𝑣 come about? • We cannot simply stipulate that she is translated into 𝜋1 𝑣, since its specific content is context-dependent. Q. How can we determine the representation for pronouns based on the contextual information? 38 / 83
42. Underspecification
43. Underspecified Type One solution: assume an intermediate step where the meaning of the pronoun is underspecified. DTS extends dependent type theory with the underspecified type (𝑥 @ 𝐴) × 𝐵 (@-type, for short) (Bekki, 2023) (13) She smiled. ⇝ (𝑥 @ e) × smile(𝑥) 𝑥 @ 𝐴 serves as a placeholder for a concrete term of type 𝐴. 39 / 83
44. Underspecified Type: Formation Rule The @-type is characterized by the following formation rule. Γ ⊢ 𝐴 ∶ type Γ ⊢ 𝑀 ∶ 𝐴 Γ ⊢ 𝐵[𝑥 ∶= 𝑀] ∶ type (@𝐹) Γ ⊢ (𝑥 @ 𝐴) × 𝐵 ∶ type Intuitively, (@𝐹) specifies the felicity condition of sentences with pronouns. • Recall: well-formedness ≃ felicity • For (𝑥 @ 𝐴) × 𝐵 to be well-formed, the context Γ should provide a term 𝑀 of type 𝐴 (≃ the antecedent). 40 / 83
45. Underspecified Type: Example (14) [Context: Kim greeted a student.] She smiled. Assuming that the typing context contains 𝑣 ∶ (𝑥 ∶ e) × (⋯), we can derive the following. (con) Γ ⊢ e ∶ type (var) Γ ⊢ 𝑣 ∶ [ 𝑥 ∶ e [⋯] ] (Σ𝐸1 ) Γ ⊢ 𝜋1 𝑣 ∶ e ⋮ Γ ⊢ smile(𝜋1 𝑣) ∶ type (@𝐹) Γ ⊢ (𝑥 @ e) × smile(𝑥) ∶ type 41 / 83
46. Eliminating @-Types Semantic representation with @-types are only intermediate: we need to derive the final representation by replacing them. ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 𝑣 ∶ ⎡ ⎢ ⎣ 𝑥 ∶ e [ 𝑢 ∶ girl(𝑥) greet(k, 𝑥) ] ⎤ ⎥ ⎦ [ 𝑦 @ e smile(𝑦) ] ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ??? = = = = = = ⇒ ⎡ ⎢ ⎢ ⎢ ⎣ 𝑣 ∶ ⎡ ⎢ ⎣ 𝑥 ∶ e [ 𝑢 ∶ girl(𝑥) greet(k, 𝑥) ] ⎤ ⎥ ⎦ smile(𝜋1 𝑢) ⎤ ⎥ ⎥ ⎥ ⎦ We formalize the rewriting process as part of type checking. 42 / 83
47. Type Checking Type checking is a computation that checks if Γ ⊢ 𝑀 ∶ 𝐴 holds. • Here, we are interested in judgments of the form Γ ⊢ 𝐴 ∶ type, where 𝐴 may contain @-types. • If Γ ⊢ 𝐴 ∶ type is derivable, then we predict that an utterance of 𝐴 is felicitous in Γ. Example: (Σ𝐹) (i) Γ ⊢ 𝐴 ∶ type (ii) Γ, 𝑥 ∶ 𝐴 ⊢ 𝐵 ∶ type (Σ𝐹) Γ ⊢ (𝑥 ∶ 𝐴) × 𝐵 ∶ type We first derive (i), and then (ii). 43 / 83
48. Type Checking [contd.] Idea: we type check and eliminate the @-type at the same time. (i) Γ ⊢ 𝐴 ∶ type (ii) Γ ⊢ 𝑀 ∶ 𝐴 (iii) Γ ⊢ 𝐵[𝑥 ∶= 𝑀] ∶ type (@𝐹) Γ ⊢ (𝑥 @ 𝐴) × 𝐵 ∶ type We derive (i), (ii), and (iii), and then return (iii) as the result. We can visualize the process as shown below. Γ ⊢ (𝑥 @ 𝐴) × 𝐵 ∶ type Γ ⊢ 𝐵[𝑥 ∶= 𝑀] ∶ type type checking 𝑥 @ 𝐴 𝑥 ∶= 𝑀 proof search Γ ⊢ ∶ 𝐴 44 / 83
49. Type Checking [contd.] Remark: the type checking process is now non-deterministic, since there can be more than one potential antecedent. Example: Γ ≡ 𝑥 ∶ e, 𝑦 ∶ e and 𝐴 ≡ (𝑧 @ e) × 𝑃𝑧 Γ ⊢ (𝑧 @ e) × 𝑃𝑧 ∶ type Γ ⊢ 𝑃𝑥 ∶ type Γ ⊢ 𝑃𝑦 ∶ type 𝑧 @ e 𝑧 ∶= 𝑦 𝑧 ∶= 𝑥 Γ ⊢ ∶ e 45 / 83
50. Type Checking: Definition (Semi)formally, we define the function TC that maps a tuple of Γ, 𝑀, 𝐴 to a set of derivable judgments. The @-type involves the proof search Prv. TC(Γ ⊢ (𝑥 @ 𝐴) × 𝐵 ∶ type) = ⎧ { { ⎨ { { ⎩ Γ ⊢ 𝐵′ ∶ type ∣ ∣ ∣ ∣ ∣ Γ ⊢ 𝐴′ ∶ type ∈ TC(Γ ⊢ 𝐴 ∶ type) Γ ⊢ 𝑀 ∶ 𝐴′ ∈ Prv(Γ, 𝐴′) Γ ⊢ 𝐵′ ∶ type ∈ TC(Γ ⊢ 𝐵[𝑥 ∶= 𝑀] ∶ type) ⎫ } } ⎬ } } ⎭ Remark: Prv is undecidable, so we need to assume that it is computed via some heuristics. 46 / 83
51. Type Checking: Definition [contd.] Other rules propagate the non-deterministic results. Example: (Σ𝐹) TC(Γ ⊢ (𝑥 ∶ 𝐴) × 𝐵 ∶ type) = ⎧ { ⎨ { ⎩ Γ ⊢ (𝑥 ∶ 𝐴′) × 𝐵′ ∶ type ∣ Γ ⊢ 𝐴′ ∶ type ∈ TC(Γ ⊢ 𝐴 ∶ type) Γ, 𝑥 ∶ 𝐴′ ⊢ 𝐵′ ∶ type ∈ TC(Γ, 𝑥 ∶ 𝐴′ ⊢ 𝐵 ∶ type) ⎫ } ⎬ } ⎭ Intuition: eliminate the @-types in 𝐴 and 𝐵 (which then result in 𝐴′ and 𝐵′) and “reconstruct” them into a Σ-type. 47 / 83
52. Deriving the Anaphoric Interpretation Let us apply the @-elimination mechanism to the previous example. (15) Kim greeted a girl, and she smiled. ⇝ ⎡ ⎢ ⎢ ⎣ 𝑣 ∶ 𝐴𝑘 [ 𝑦 @ e smile(𝑦) ] ⎤ ⎥ ⎥ ⎦ ⎛ ⎜ ⎜ ⎜ ⎝ 𝐴𝑘 def = ⎡ ⎢ ⎢ ⎣ 𝑥 ∶ e [ 𝑢 ∶ girl(𝑥) greet(k, 𝑥) ] ⎤ ⎥ ⎥ ⎦ ⎞ ⎟ ⎟ ⎟ ⎠ Assuming there is no previous context, we compute the following. TC ⎛ ⎜ ⎜ ⎜ ⎝ ⊢ ⎡ ⎢ ⎢ ⎣ 𝑣 ∶ 𝐴𝑘 [ 𝑦 @ e smile(𝑦) ] ⎤ ⎥ ⎥ ⎦ ∶ type ⎞ ⎟ ⎟ ⎟ ⎠ 48 / 83
53. Deriving the Anaphoric Interpretation [contd.] (i) Check the 1st conjunct: TC( ⊢ 𝐴𝑘 ∶ type) • trivial: { ⊢ 𝐴𝑘 ∶ type} (ii) Check the 2nd conjunct: TC(𝑣 ∶ 𝐴𝑘 ⊢ (𝑦 @ e) × smile(𝑦) ∶ type) (i) TC(𝑣 ∶ 𝐴𝑘 ⊢ e ∶ type) (trivial) (ii) Proof search: Prv(𝑣 ∶ 𝐴𝑘 , e) = {𝑣 ∶ 𝐴𝑘 ⊢ 𝜋1 𝑣 ∶ e} (iii) TC(𝑣 ∶ 𝐴𝑘 ⊢ smile(𝜋1 𝑣) ∶ type) (trivial) As a result, we obtain { ⊢ [ 𝑣 ∶ 𝐴𝑘 smile(𝜋1 𝑣) ] ∶ type}. 49 / 83
54. Deriving the Anaphoric Interpretation [contd.] Crucially, we can use the local variable 𝑣 ∶ 𝐴𝑘 in the proof search (as the well-formedness of 𝐵 can depend on 𝑥 ∶ 𝐴 in (𝑥 ∶ 𝐴) × 𝐵). ⊢ ⎡ ⎢ ⎢ ⎣ 𝑣 ∶ 𝐴𝑘 [ 𝑦 @ e smile(𝑦) ] ⎤ ⎥ ⎥ ⎦ ∶ type ⊢ [ 𝑣 ∶ 𝐴𝑘 smile(𝜋1 𝑣) ] ∶ type type checking 𝑦 @ e 𝑦 ∶= 𝜋1 𝑣 𝑣 ∶ 𝐴𝑘 ⊢ ∶ e We will refer to a typing context with local variables (here, 𝑣 ∶ 𝐴𝑘 ) as a local typing context. 50 / 83
55. Constraints on Anaphoric Interpretations N.B. if the order of the conjuncts is the opposite, the anaphoric interpretation is disallowed. (16) *Kim greeted her𝑖 , and [a girl]𝑖 smiled. This asymmetry is correctly predicted since 𝐵 in (𝑥 ∶ 𝐴) × 𝐵 is not in the local typing context when 𝐴 is type checked. ⎡ ⎢ ⎢ ⎢ ⎣ 𝑣 ∶ [ 𝑦 @ e greet(k, 𝑦) ] [ 𝑥 ∶ e [⋯] ] ⎤ ⎥ ⎥ ⎥ ⎦ 51 / 83
56. Constraints on Anaphoric Interpretations [contd.] We can account for the unavailability of inter-sentential anaphora as failure of the proof search. (11) *Kim greeted [every girl]𝑖 , and she𝑖 smiled. 𝑣 ∶ (𝑥 ∶ e) → (𝑢 ∶ girl(𝑥)) → greet(k, 𝑥) ⊢ ??? ∶ e In summary ... DTS identifies the availability of anaphora with the proof con- structability in the local typing context. 52 / 83
57. Accounting for the Crossover Effect We are now in a position to predict the crossover effect. (17) a. [Every girl]𝑖 loves her𝑖 mother. ⇝ (𝑥 ∶ e) → (𝑢 ∶ girl(𝑥)) → [ 𝑦 @ e love(𝑥, mother(𝑦)) ] b. *Her𝑖 mother loves [every girl]𝑖 . ⇝ [ 𝑦 @ e (𝑥 ∶ e) → (𝑢 ∶ girl(𝑥)) → love(mother(𝑦), 𝑥) ] 𝑥 ∶ e is a local variable for 𝑦 @ e in (17a), but not in (17b). In short, DTS predicts the crossover effect when the the quantifier is type checked later than the @-type. 53 / 83
58. Accounting for the Crossover Effect: Remark Note that DTS treats pronouns as scope-taking expressions. English often allows inverse scope, where the scope relationship does not follow the structural hierarchy (cf. Ruys and Winter (2011)). (18) A drone survails every building. ⇝ ∀𝑦.(bldg(𝑦) ⊃ ∃𝑥.(drone(𝑥) ∧ survail(𝑥, 𝑦))) Hence, we need some constraints to prevent pronouns to be subject to inverse scope (see Matsuoka et al. (to appear) for details). (17b) *Her𝑖 mother loves [every girl]𝑖 . NG: (𝑥 ∶ e) → (𝑢 ∶ girl(𝑥)) → [ 𝑦 @ e love(𝑥, mother(𝑦)) ] 54 / 83
59. Advanced Case: Quantificational Subordination As a showcase, let us look at another interesting case. Basically, an existential quantifier in the scope of a universal quantifier cannot be accessed from outside. (19) *Every student wrote [a paper]𝑖 , and it𝑖 was well written. However, when the pronoun is also in the scope of a quantifier rainging over the same domain, anaphora is allowed. (20) Every student wrote [a paper]𝑖 , and every advanced student submitted it𝑖 to ICLR. This is called quantificational subordination (van den Berg, 1996). 55 / 83
60. Advanced Case: Quantificational Subordination [contd.] In DTS, quantificational subordination can be accounted for as proof construction with a function (Tanaka, 2021). ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 𝑓 ∶ (𝑥 ∶ e) → (𝑢 ∶ student(𝑥)) → ⎡ ⎢ ⎣ 𝑦 ∶ e [ 𝑣 ∶ paper(𝑦) write(𝑥, 𝑦) ] ⎤ ⎥ ⎦ (𝑥 ∶ e) → (𝑢′ ∶ [ 𝑢 ∶ student(𝑥) advanced(𝑥) ]) → [ 𝑧 @ e submit(𝑥, 𝑧) ] ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ The result of proof search: 𝑓 ∶ (𝑥 ∶ e) → ⋯ , 𝑥 ∶ e, 𝑢′ ∶ [ 𝑢 ∶ student(𝑥) ⋯ ] ⊢ 𝜋1 ((𝑓𝑥)(𝜋1 𝑢′)) ∶ e 56 / 83
61. Interim Summary So far, we have seen … • DTS uses the @-type for an underspecified representation of pronouns. • @-types are eliminated through type checking. ∘ The concrete term for an @-type is obtained via proof search. • The availability of an anaphoric interpretation is identified with the proof constructability in the local typing context. ∘ The crossover effect can be derived with this assumption. 57 / 83
62. Interim Summary [contd.] We have only considered the type e for the 𝐴 in (𝑥 @ 𝐴) × 𝐵. Can it be otherwise? Yes, and it leads to an explanation of another phenomenon, called presupposition, which we turn to next. 58 / 83
63. Presupposition
64. Presupposition Certain classes of expressions require a proposition to hold in the preceding context (Stalnaker, 1974; Tonhauser et al., 2013) (In what follows, ⟹ indicates the implication relation in a broad sense) (21) a. Alex saw Kim’s dog. ⟹ Kim has a dog. b. Kim knows that Alex danced. ⟹ Alex danced. This type of content is referred to as a presupposition. An expression associated with a presupposition is called its trigger. 59 / 83
65. Property: Felicity Constraint A presupposition is relevant to the felicity of an utterance, not the truth of the content. For example, with a false presupposition, the whole sentence is (typically) neither true nor false (Strawson, 1950; von Fintel, 2004). (22) #The king of France is wise. (# indicates pragmatic oddity.) 60 / 83
66. Property: Projection Presuppositions are basically not subject to entailment-canceling operators. This behavior is called projection (Heim, 1983; Karttunen, 1973). (23) a. Alex did not see Kim’s dog. b. Alex might see Kim’s dog. c. If Alex see Kim’s dog, she pets it. ⟹ Kim has a dog. In contrast, a presupposition can be canceled by some other content in the same sentence. This effect is called filtering. (24) If Alex has a dog, she takes her dog for a walk. ⇏ Alex has a dog. 61 / 83
67. Presupposition in DTS DTS adopts the view that presupposition triggers are anaphoric to information in the context (van der Sandt, 1992). Namely, DTS represents presuppositions with an @-types (Bekki & Satoh, 2015; Tanaka et al., 2017). (25) Alex saw Kim’s dog. ⇝ ⎡ ⎢ ⎢ ⎣ 𝑣 @ ⎡ ⎢ ⎣ 𝑥 ∶ e [ 𝑢 ∶ dog(𝑥) have(k, 𝑥) ] ⎤ ⎥ ⎦ see(a, 𝜋1 𝑣) ⎤ ⎥ ⎥ ⎦ For the semantic representation of (25) to be well-formed, there needs to be a proof of Kim has a dog. (i.e., it is presupposed.) 62 / 83
68. Presupposition in DTS [contd.] Projection can be predicted because the well-formedness condition of the @-type is inherited to a more complex type. Example: negation (cf. (23a)) Γ ⊢ (𝑥 @ 𝐴) × 𝐵 ∶ type ⋯ (Π𝐹) Γ ⊢ (𝑢 ∶ (𝑥 @ 𝐴) × 𝐵) → ⊥ ∶ type To complete the type checking of the whole Π-type, we anyway need to perform proof search for 𝐴. 63 / 83
69. Presupposition in DTS [contd.] The filtering effect can be viewed as a result of proof search. (24) If Alex has a dog, she takes her dog for a walk. ⇝ ⎛ ⎜ ⎜ ⎜ ⎝ 𝑣′ ∶ ⎡ ⎢ ⎢ ⎣ 𝑥 ∶ e [ 𝑢 ∶ dog(𝑥) have(a, 𝑥) ] ⎤ ⎥ ⎥ ⎦ ⎞ ⎟ ⎟ ⎟ ⎠ → ⎡ ⎢ ⎢ ⎢ ⎣ 𝑣 @ ⎡ ⎢ ⎢ ⎣ 𝑥 ∶ e [ 𝑢 ∶ dog(𝑥) have(a, 𝑥) ] ⎤ ⎥ ⎥ ⎦ takeWalk(a, 𝜋 1 𝑣) ⎤ ⎥ ⎥ ⎥ ⎦ The proof search here returns 𝑣′ for 𝑣 @ [⋯]. Hence, the preceding context does not have to entail Kim has a dog. 64 / 83
70. Accommodation Even when a presupposition is not in the context, the hearer can usually understand the utterance. (26) [Context: the hearer does not know much about the speaker’s personal information] My sister is coming to lunch tomorrow. In such cases, the hearer updates their prior belief to include the presupposed content. This process is called accommodation (Lewis, 1979; von Fintel, 2008). 65 / 83
71. Accommodation [contd.] Then, how can we model accommodation in DTS? Previous studies assume the following operation (in line with other frameworks (e.g., Beaver (2001) and van der Sandt (1992))). Accommodation If proof search Γ ⊢ ∶ 𝐴 fails during the type checking of 𝐴, add 𝑥 ∶ 𝐴 to the typing context and rerun the type checking. However, this type of “add-and-rerun” strategy faces a problem ... 66 / 83
72. Generalized Crossover A presupposition can provide an antecedent for a pronoun. (27) Alex didn’t know that Kim wrote [a paper]𝑖 , and reviewed it𝑖 . This kind of dependency is not always possible ... (28) *Its𝑖 reviewer did not know that Kim wrote [a paper]𝑖 . Crucially, there is an asymmetry parallel to the (weak) crossover cases. This effect is called generalized crossover (Elliott & Sudo, 2021). 67 / 83
73. Generalized Crossover [contd.] If we add the presupposed content and rerun the type checking, it allows its to access a paper in (28)! • Initial state: Γ ⊢ (𝑦 @ e) × ((𝑣 @ 𝐴𝑘 ) × ⋯) ∶ type • Accommodation for 𝑣: Γ ↦ Γ, 𝑣 ∶ 𝐴𝑘 • Rerun: Γ, 𝑣 ∶ 𝐴𝑘 ⊢ (𝑦 @ e) × ⋯ ∶ type Problem: the scope relation between the @-types is ignored. • The same applies to other theories of presupposition as well. • Furthermore, no previous account of crossover can account for why quantifiers and presuppositions show similar behavior. 68 / 83
74. Dynamic Accommodation Solution: we perform accommodation during type checking. Dynamic Accommodation (Matsuoka et al., to appear) If the proof search fails in computing TC(Γ ⊢ (𝑥 @ 𝐴) × 𝐵), then we can replace the result of this type checking with that of TC(Γ, 𝑥 ∶ 𝐴 ⊢ 𝐵 ∶ type). 69 / 83
75. Dynamic Accommodation [contd.] (29) Alex did not know that Kim wrote a paper. ⇝ ¬ [ 𝑢 @ 𝐴𝑘 know(a, 𝐴𝑘 ) ] ⎛ ⎜ ⎜ ⎜ ⎝ 𝐴𝑘 def = ⎡ ⎢ ⎣ 𝑥 ∶ e [ 𝑢 ∶ paper(𝑥) write(k, 𝑥) ] ⎤ ⎥ ⎦ ⎞ ⎟ ⎟ ⎟ ⎠ In the type checking, 𝑢 ∶ 𝐴𝑘 is added to the typing context. ⊢ ¬ [ 𝑢 @ 𝐴𝑘 know(a, 𝐴𝑘 ) ] ∶ type 𝑢 ∶ 𝐴𝑘 ⊢ ¬know(a, 𝐴𝑘 ) ∶ type type checking 𝑢 @ 𝐴𝑘 (accomm.) ⊢ ∶ 𝐴𝑘 70 / 83
76. Dynamic Accommodation [contd.] The accommodated content can be accessed from a subsequent proof search. (27) Alex didn’t know that Kim wrote [a paper]𝑖 , and reviewed it𝑖 . ⊢ ⎡ ⎢ ⎢ ⎢ ⎣ 𝑤 ∶ ¬ [ 𝑣 @ 𝐴𝑘 know(a, 𝐴𝑘 ) ] [ 𝑦 @+ e review(a, 𝑦) ] ⎤ ⎥ ⎥ ⎥ ⎦ ∶ type 𝑣 ∶ 𝐴𝑘 ⊢ [ 𝑤 ∶ ¬know(a, 𝐴𝑘 ) review(a, 𝜋1 𝑣) ] ∶ type type checking 𝑣 @ 𝐴𝑘 (accomm.) ⊢ ∶ 𝐴𝑘 𝑦 @+ e 𝑦 ∶= 𝜋1 𝑣 𝑣 ∶ 𝐴𝑘 , 𝑤 ∶ ¬know(a, 𝐴𝑘 ) ⊢ ∶ e 71 / 83
77. Accounting for the Generalized Crossover Effect What happens with the crossover case? (28) *Its𝑖 reviewer did not know that Kim wrote [a paper]𝑖 . ⇝ ⎡ ⎢ ⎣ 𝑦 @ e [ 𝑣 @ 𝐴𝑘 ¬know(reviewer(𝑦), 𝐴𝑘 ) ] ⎤ ⎥ ⎦ Since 𝑣 @ 𝐴𝑘 is added to the context after 𝑦 @ e is type checked, there can be no anaphoric dependencies! 72 / 83
78. Interim Summary • DTS handles presuppositions with the @-type. ∘ Presupposition triggers are anaphoric to a proof of the presupposed content. • With dynamic accommodation, we can account for generalized crossover. pronoun ⋯ quantifier ⇝ 𝑦 @ 𝐵 ⋯ 𝑥 ∶ 𝐴 pronoun ⋯ presup. trigger ⇝ 𝑦 @ 𝐵 ⋯ 𝑥 @ 𝐴 ∘ The crucial point is that, due to propositions-as-types, pesuppositions can have scope relation with other elements. 73 / 83
79. Remark: Not-at-Issueness (Dynamic) accommodation is related to yet another important property of presuppositions ... A presupposition is not at-issue, in the sense that it is not the central part of an utterance (Simons et al., 2010). For instance, a presupposition cannot be normally targeted by a direct response (Tonhauser, 2012). (30) a. A: Alex saw Kim’s dog. b. B: No, that’s not true. 74 / 83
80. Remark: At-Issueness [contd.] Due to accommodation, type checking can extend the typing context (as well as rewrite the semantic representation). • Type checking is a process that determines the utterance content. • The content added to the context duruing this process does not count as the “central message” of the utterance (Matsuoka et al., 2024). Γ ⊢ 𝐴 ∶ type Γ, not-at-issue Δ ⊢ at-issue 𝐴′ ∶ type Γ, Δ, 𝑥 ∶ 𝐴′ (i) type checking (ii) addition 75 / 83
81. Conclusion and Future Outlook
82. Summary • Propositions-as-types • A proposition can be identified with a type. • Dependent types can capture inter-sentential anaphora. • Underspecified type • Anaphora can be analzed as proof construction in the local typing context. • This reflects some constraints on anaphora (e.g., crossover). • Presupposition-as-anaphora • Presupposition can be analized as anaphora to a proof. • Dynamic accommodation predicts the generalized crossover. 76 / 83
83. Current Issue (i): Negation and Disjunction Sometimes, pronouns can be bound by a quantifier in a negative environment (Krahmer & Muskens, 1995). (31) a. It is not true that Kim does not have [a car]𝑖 . She keeps it𝑖 in the garage. b. Either there is [no bathroom]𝑖 in the house or it𝑖 is in a funny place. These two cases can be accounted for if we assume double negation elimination ¬¬𝐴 ⊃ 𝐴. (Note: 𝐴 ∨ 𝐵 ⊢ ¬𝐴 ⊃ 𝐵.) But this makes the logic classical, which leads to ¬∀𝑥.𝐴 ⊃ ∃𝑥.¬𝐴. Then we cannot block some invalid anaphoric dependencies ... (32) [Not every girl]𝑖 came. *She𝑖 was sick. 77 / 83
84. Current Issue (ii): Gradience in Acceptability It has been observed that crossover may lead to various degree of unacceptability, depending on both constructions and individuals. (adapted from Ross et al. (2023)) It would be necessary to integrate some gradable constraints on the scope of @-types, but how? 78 / 83
85. Thank you for listening! For any follow-up questions, please contact me at: daiki.matsuoka@is.s.u-tokyo.ac.jp
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