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A Model-Theoretic Reconstruction of Type-Theoretic Semantics for Anaphora Matthew Gotham University of Oslo 22 nd Conference on Formal Grammar, 22 July 2017 Matthew Gotham (Oslo) Semantics for Anaphora Formal Grammar 2017 1 / 26

What this talk is about A framework for the semantics of anaphora and accessibility constraints. Inspired by analyses in type-theoretical approaches to semantics using dependent types, reconstructed in (more or less) simple type theory. We’ll look at a couple of examples of cross-sentential binding and a ‘donkey sentence’, and see how the the system blocks inaccessible antecedents. There are more examples (negation, proportional quantifiers, weak and strong readings) in the paper. Matthew Gotham (Oslo) Semantics for Anaphora Formal Grammar 2017 2 / 26

‘Model-Theoretic’? What I mean is that meanings will be given as expressions of a logical language, which are taken to be dispensable in favour of their interpretations in a model (as in Montague 1973), which is where the ‘real’ semantics is. Expressions of the language of type theory are not understood this way in TTS—see Luo 2014 and Ranta 1994: §2.27. However, I don’t want to lean to heavily on this point from now on. Matthew Gotham (Oslo) Semantics for Anaphora Formal Grammar 2017 3 / 26

Pronouns bound outside of scope (1) A donkey brays. Giles feeds it. ∃ x ( donkey ( x ) ∧ bray ( x )) ∧ feed ( giles , ?) (2) Every farmer who owns a donkey feeds it. ∀ y . ( farmer ( y ) ∧ ∃ x . donkey ( x ) ∧ own ( y , x )) → feed ( y , ?) Various options pursued: ? := x , change the model theory to extend the scope of ∃ x ? is a description, possibly indexed to situations ? is a constant manipulated by functions ...etc. Matthew Gotham (Oslo) Semantics for Anaphora Formal Grammar 2017 4 / 26

In Type-Theoretic Semantics (1) A donkey brays. Giles feeds it. � �� � Σ y : (Σ x : DONKEY )( BRAY ( x )) FEED ( giles , π 1 y ) (Sundholm 1986, Ranta 1994) � �� � λ c . Σ w : (Σ u : (Σ x : e )( DONKEY ( x )))( BRAY ( π 1 u )) FEED ( giles , π 1 π 1 w ) (Bekki 2014) Matthew Gotham (Oslo) Semantics for Anaphora Formal Grammar 2017 5 / 26

Witnesses dependent pairs (1) A donkey brays. Giles feeds it. � �� � Σ y : (Σ x : DONKEY )( BRAY ( x )) FEED ( giles , π 1 y ) The type of ordered pairs �� a , b � , c � such that: a is a donkey, and b is a proof that a brays, and c is a proof that Giles feeds a . Matthew Gotham (Oslo) Semantics for Anaphora Formal Grammar 2017 6 / 26

In Type-Theoretic Semantics (2) Every farmer who owns a donkey feeds it. � �� � Π z : (Σ x : FARMER )((Σ y : DONKEY )( OWN ( x , y ))) FEED ( π 1 z , π 1 π 2 z ) (Sundholm 1986, Ranta 1994) � λ c . Π u :(Σ x : e ) � ( FARMER ( x ) × (Σ v : (Σ y : e )( DONKEY ( y )))( OWN ( x , π 1 v ))) � � FEED ( π 1 u , π 1 π 1 π 2 π 2 u ) (Bekki 2014) Matthew Gotham (Oslo) Semantics for Anaphora Formal Grammar 2017 7 / 26

Witnesses dependent functions (2) Every farmer who owns a donkey feeds it. � �� � Π z : (Σ x : FARMER )((Σ y : DONKEY )( OWN ( x , y ))) FEED ( π 1 z , π 1 π 2 z ) The type of functions f such that: the domain of f is the set of ordered pairs � a , � b , c �� such that: a is a farmer, and b is a donkey, and c is a proof that a owns b , and f maps every � a , � b , c �� in its domain to a proof that a feeds b . Matthew Gotham (Oslo) Semantics for Anaphora Formal Grammar 2017 8 / 26

The idea behind this paper (is very simple) Formalize those glosses in higher-order logic (Jacobs & Melham (1993) have shown how). Work backwards to the lexical entries we need to derive them compositionally. N.B.: Limited polymorphism required. Event(ualitie)s play the role of proof objects. Discourse-level existential closure plays the role of the non-empty type condition. Matthew Gotham (Oslo) Semantics for Anaphora Formal Grammar 2017 9 / 26

Types Base types 1 unit e entities v eventualities t booleans Binary type constructors functional types � × product types ( � and × associate to the right, and × binds more tightly than � ) Matthew Gotham (Oslo) Semantics for Anaphora Formal Grammar 2017 10 / 26

Terms ∗ : 1 unit f α � β ( a α ) : β application λ v α ( φ β ) : α � β abstraction ( a α , b β ) : α × β pairing [ c α × β ] 0 : α lef projection [ c α × β ] 1 : β right projection Matthew Gotham (Oslo) Semantics for Anaphora Formal Grammar 2017 11 / 26

Example lexical entries and ✱ ; �→ λ p α � β � t .λ q α × β � γ � t .λ i α .λ a β × γ . p ( i )([ a ] 0 ) ∧ q ( i , [ a ] 0 )([ a ] 1 ) . �→ λ p 1 � α � t . ∃ z α . p ( ∗ )( z ) a �→ λ P e × α � β � t .λ V .λ i β .λ a ( e × α ) × γ . P ([ a ] 0 )( i ) ∧ V ([[ a ] 0 ] 0 )( i , [ a ] 0 )([ a ] 1 ) where V : e � β × e × α � γ � t donkey �→ λ a e × 1 .λ i α . donkey ([ a ] 0 ) brays �→ λ x e .λ i α .λ e v . bray ( x , e ) Giles �→ λ P e � α × e � β � t .λ i α .λ a e × β . P ([ a ] 0 )( i , [ a 0 ])([ a ] 1 ) ∧ [ a ] 0 = giles owns �→ λ D ( e � α � v � t ) � β � γ � t .λ x e . D λ y e .λ a α .λ e v . own ( x , y , e ) � � it �→ λ V α � β � γ � t .λ i β . V ( g β � α ( i ))( i ) where g stands for an arbitrarily-chosen free variable Matthew Gotham (Oslo) Semantics for Anaphora Formal Grammar 2017 12 / 26

Instantiated lexical entries a �→ λ P e × 1 � 1 � t .λ V e � 1 × e × 1 � v � t .λ i 1 .λ a ( e × 1 ) × v . P ([ a ] 0 )( i ) ∧ V ([[ a ] 0 ] 0 )( i , [ a ] 0 )([ a ] 1 ) donkey �→ λ a e × 1 .λ i 1 . donkey ([ a ] 0 ) brays �→ λ x e .λ i 1 × e × 1 .λ e v . bray ( x , e ) lef context for the whole sentence NP witness, part of the lef context for the VP VP witness a donkey brays �→ λ i 1 .λ a ( e × 1 ) × v . donkey ([[ a ] 0 ] 0 ) ∧ bray ([[ a ] 0 ] 0 , [ a ] 1 ) Matthew Gotham (Oslo) Semantics for Anaphora Formal Grammar 2017 13 / 26

Instantiated lexical entries Giles �→ λ P e � ( 1 × ( e × 1 ) × v ) × e � v � t .λ i 1 × ( e × 1 ) × v .λ a e × v . P ([ a ] 0 )( i , [ a ] 0 )([ a ] 1 ) ∧ [ a ] 0 = giles λ y e .λ a ( 1 × ( e × 1 ) × v ) × e .λ e v . own ( x , y , e ) owns �→ λ D .λ x e . D � � where D : ( e � ( 1 × ( e × 1 ) × v ) × e � v � t ) � ( 1 × ( e × 1 ) × v ) × e � v � t it �→ λ V e � ( 1 × ( e × 1 ) × v ) × e � v � t .λ i ( 1 × ( e × 1 ) × v ) × e . V ( g ( 1 × ( e × 1 ) × v ) × e � e ( i ))( i ) lef context NP witness VP witness Giles owns it �→ λ i 1 × ( e × 1 ) × v .λ a e × v . own ([ a ] 0 , g ( 1 × ( e × 1 ) × v ) × e � e ( i , [ a ] 0 ) , [ a ] 1 ) ∧ [ a ] 0 = giles Matthew Gotham (Oslo) Semantics for Anaphora Formal Grammar 2017 14 / 26

Instantiated lexical entries ; �→ λ p 1 � ( e × 1 ) × v � t .λ q 1 × ( e × 1 ) × v � e × v � t .λ i 1 .λ a (( e × 1 ) × v ) × e × v . p ( i )([ a ] 0 ) ∧ q ( i , [ a ] 0 )([ a ] 1 ) . �→ λ p 1 � (( e × 1 ) × v ) × e × v � t . ∃ z (( e × 1 ) × v ) × e × v . p ( ∗ )( z ) lef context first sentence witness, part of the lef context for the second sentence second sentence witness A donkey brays; Giles owns it. �→ ∃ z (( e × 1 ) × v ) × e × v . � � donkey ([[[ z ] 0 ] 0 ] 0 ) ∧ bray ([[[ z ] 0 ] 0 ] 0 , [[ z ] 0 ] 1 ) [[ z ] 1 ] 0 , g ( 1 × ( e × 1 ) × v ) × e � e (( ∗ , [ z ] 0 ) , [[ z ] 1 ] 0 ) , [[ z ] 1 ] 1 � � � ∧ own � ∧ [[ z ] 1 ] 0 = giles Matthew Gotham (Oslo) Semantics for Anaphora Formal Grammar 2017 15 / 26

Resolution of the free variable g ( 1 × ( e × 1 ) × v ) × e � e Natural resolution: a function that selects an element of (an element of...) a tuple (of tuples...) For any types α, β and γ : λ b α . b is a natural resolution function (NRF). λ b α × β . [ b ] 0 is an NRF. λ b α × β . [ b ] 1 is an NRF. For any terms F : β � γ and G : α � β , λ b α . F ( G ( b )) is an NRF if F and G are NRFs. In this case, the resolution that we want gives us g := λ b ( 1 × ( e × 1 ) × v ) × e . [[[[ b ] 0 ] 1 ] 0 ] 0 Matthew Gotham (Oslo) Semantics for Anaphora Formal Grammar 2017 16 / 26

With the pronoun resolution g := λ b ( 1 × ( e × 1 ) × v ) × e . [[[[ b ] 0 ] 1 ] 0 ] 0 ⇒ β ∃ z (( e × 1 ) × v ) × e × v . � � donkey ([[[ z ] 0 ] 0 ] 0 ) ∧ bray ([[[ z ] 0 ] 0 ] 0 , [[ z ] 0 ] 1 ) � � � ∧ own [[ z ] 1 ] 0 , ([[[[(( ∗ , [ z ] 0 ) , [[ z ] 1 ] 0 )] 0 ] 1 ] 0 ] 0 ) , [[ z ] 1 ] 1 � ∧ [[ z ] 1 ] 0 = giles ⇒ β ∃ z (( e × 1 ) × v ) × e × v . � � donkey ([[[ z ] 0 ] 0 ] 0 ) ∧ bray ([[[ z ] 0 ] 0 ] 0 , [[ z ] 0 ] 1 ) � � � � ∧ [[ z ] 1 ] 0 , [[[ z ] 0 ] 0 ] 0 , [[ z ] 1 ] 1 ∧ [[ z ] 1 ] 0 = giles own ∃ x e . ∃ e v . ∃ y e . ∃ d v . � � � � ≡ donkey ( x ) ∧ bray ( x , e ) ∧ own ( y , x , d ) ∧ y = giles Matthew Gotham (Oslo) Semantics for Anaphora Formal Grammar 2017 17 / 26