Quantifjcational subordination as anaphora to a function Matthew Gotham University of Oxford 24th Conference on Formal Grammar, University of Latvia 11 August 2019 1/38
Outline Background Outline of the proposal Examples Refset anaphora Telescoping Discussion Comparison with TTS Conclusion 2/38
Outline Background Outline of the proposal Examples Refset anaphora Telescoping Discussion Comparison with TTS Conclusion 2/38
Outline Background Outline of the proposal Examples Refset anaphora Telescoping Discussion Comparison with TTS Conclusion 2/38
Outline Background Outline of the proposal Examples Refset anaphora Telescoping Discussion Comparison with TTS Conclusion 2/38
Background
Quantifjcational subordination (QS) (1) If you give every child a present, some child will open it. (Ranta 1994) (2) Every student bought a book. Most of them read it. (3) Every player chooses a pawn. He puts it on square one. (Groenendijk & Stokhof 1991) Examples like (3) are often called ‘telescoping’. 3/38
Pronouns inaccessible in fjrst-generation dynamic semantics E.g. DRT: If you give every child a present, some child will open it. x child x y present y you give x y z u child z z opens u u=? 4/38 ⇒ ⇒
Pronouns inaccessible in fjrst-generation dynamic semantics Every player chooses a pawn. He puts it on square one. z u x player x y pawn y x chooses y z puts u on square one z=? u=? 5/38 ⇒
Second-generation dynamic semantics h dependency, though. • It’s quite a complex and roudabout way to get to that the necessary dependency between pawns and players. H h y is a pawn chosen by h x . H therefore encodes h H is the set of players, and for every • I.e., h x 6/38 Via a generalization to using sets of assignments: � every x player chooses a y pawn � = � � F , H � | ∃ G : ( ∀ f ∈ F : ∃ g ∈ G : f ≈ x g & ∀ g ∈ G : ∃ f ∈ F : f ≈ x g ) & { g ( x ) | g ∈ G } = � player � & ( ∀ g ∈ G : ∃ h : g ≈ y h & h ( y ) ∈ � pawn � & � h ( x ) , h ( y ) � ∈ � choose � ) � & H = { h | ∃ g ∈ G : g ≈ y h & h ( y ) ∈ � pawn � & � h ( x ) , h ( y ) � ∈ � choose � }
Second-generation dynamic semantics Via a generalization to using sets of assignments: the necessary dependency between pawns and players. • It’s quite a complex and roudabout way to get to that dependency, though. 6/38 � every x player chooses a y pawn � = � � F , H � | ∃ G : ( ∀ f ∈ F : ∃ g ∈ G : f ≈ x g & ∀ g ∈ G : ∃ f ∈ F : f ≈ x g ) & { g ( x ) | g ∈ G } = � player � & ( ∀ g ∈ G : ∃ h : g ≈ y h & h ( y ) ∈ � pawn � & � h ( x ) , h ( y ) � ∈ � choose � ) � & H = { h | ∃ g ∈ G : g ≈ y h & h ( y ) ∈ � pawn � & � h ( x ) , h ( y ) � ∈ � choose � } • I.e., { h ( x ) | h ∈ H } is the set of players, and for every h ∈ H , h ( y ) is a pawn chosen by h ( x ) . H therefore encodes
Second-generation dynamic semantics Via a generalization to using sets of assignments: the necessary dependency between pawns and players. • It’s quite a complex and roudabout way to get to that dependency, though. 6/38 � every x player chooses a y pawn � = � � F , H � | ∃ G : ( ∀ f ∈ F : ∃ g ∈ G : f ≈ x g & ∀ g ∈ G : ∃ f ∈ F : f ≈ x g ) & { g ( x ) | g ∈ G } = � player � & ( ∀ g ∈ G : ∃ h : g ≈ y h & h ( y ) ∈ � pawn � & � h ( x ) , h ( y ) � ∈ � choose � ) � & H = { h | ∃ g ∈ G : g ≈ y h & h ( y ) ∈ � pawn � & � h ( x ) , h ( y ) � ∈ � choose � } • I.e., { h ( x ) | h ∈ H } is the set of players, and for every h ∈ H , h ( y ) is a pawn chosen by h ( x ) . H therefore encodes
1 v 1 u 1 w 1 f w Type-theoretical semantics (TTS) suggests an answer x 1 open e child z z w give you e present y y v e child x u • Propositions-as-types principle f (1) If you give every child a present, some child will open it. B a x A , f a for any a A B —the type of functions with domain A such that, x • B a x b A and A B —the type of ordered pairs a b , where a x • 7/38
1 v 1 u 1 w 1 f w Type-theoretical semantics (TTS) suggests an answer x 1 open e child z z w give you e present y y v e child x u • Propositions-as-types principle f (1) If you give every child a present, some child will open it. B a x A , f a for any a A B —the type of functions with domain A such that, x • B a x b A and A B —the type of ordered pairs a b , where a x • 7/38
1 v 1 u 1 w 1 f w Type-theoretical semantics (TTS) suggests an answer v 1 open e child z z w give you e present y y e child x • Propositions-as-types principle x u f (1) If you give every child a present, some child will open it. B a x A , f a for any a A B —the type of functions with domain A such that, x • 7/38 • (Σ x : A ) B —the type of ordered pairs � a , b � , where a : A and b : B [ a / x ]
1 v 1 u 1 w 1 f w Type-theoretical semantics (TTS) suggests an answer e present y 1 open e child z z w give you y • Propositions-as-types principle v e child x x u f (1) If you give every child a present, some child will open it. 7/38 • (Σ x : A ) B —the type of ordered pairs � a , b � , where a : A and b : B [ a / x ] • (Π x : A ) B —the type of functions with domain A such that, for any a : A , f ( a ) : B [ a / x ]
Type-theoretical semantics (TTS) suggests an answer • Propositions-as-types principle (1) If you give every child a present, some child will open it. 7/38 • (Σ x : A ) B —the type of ordered pairs � a , b � , where a : A and b : B [ a / x ] • (Π x : A ) B —the type of functions with domain A such that, for any a : A , f ( a ) : B [ a / x ] � � � Π f : Π u : (Σ x : e ) child ( x ) � (Σ v : (Σ y : e ) present ( y )) give ( you ′ , π 1 ( v ) , π 1 ( u )) (Σ w : (Σ z : e ) child ( z )) open ( π 1 ( w ) , π 1 ( π 1 ( f ( w ))))
Using the function every child to a present you give him/her. the function is overtly present in the second sentence. • BUT it is actually crucial that an appropriate argument to makes this kind of dependency possible. • The fact that the fjrst sentence expresses a function open f z . a child z and a proof that you • some child will open it a function f mapping • you give every child a present 8/38 � � � Π f : Π u : (Σ x : e ) child ( x ) � (Σ v : (Σ y : e ) present ( y )) give ( you ′ , π 1 ( v ) , π 1 ( u )) (Σ w : (Σ z : e ) child ( z )) open ( π 1 ( w ) , π 1 ( π 1 ( f ( w ))))
Using the function every child to a present you give him/her. the function is overtly present in the second sentence. • BUT it is actually crucial that an appropriate argument to makes this kind of dependency possible. • The fact that the fjrst sentence expresses a function 8/38 � � � Π f : Π u : (Σ x : e ) child ( x ) � (Σ v : (Σ y : e ) present ( y )) give ( you ′ , π 1 ( v ) , π 1 ( u )) (Σ w : (Σ z : e ) child ( z )) open ( π 1 ( w ) , π 1 ( π 1 ( f ( w )))) • you give every child a present � a function f mapping • some child will open it � a child z and a proof that you open f ( z ) .
Using the function every child to a present you give him/her. the function is overtly present in the second sentence. • BUT it is actually crucial that an appropriate argument to makes this kind of dependency possible. • The fact that the fjrst sentence expresses a function 8/38 � � � Π f : Π u : (Σ x : e ) child ( x ) � (Σ v : (Σ y : e ) present ( y )) give ( you ′ , π 1 ( v ) , π 1 ( u )) (Σ w : (Σ z : e ) child ( z )) open ( π 1 ( w ) , π 1 ( π 1 ( f ( w )))) • you give every child a present � a function f mapping • some child will open it � a child z and a proof that you open f ( z ) .
Using the function every child to a present you give him/her. the function is overtly present in the second sentence. • BUT it is actually crucial that an appropriate argument to makes this kind of dependency possible. • The fact that the fjrst sentence expresses a function 8/38 � � � Π f : Π u : (Σ x : e ) child ( x ) � (Σ v : (Σ y : e ) present ( y )) give ( you ′ , π 1 ( v ) , π 1 ( u )) (Σ w : (Σ z : e ) child ( z )) open ( π 1 ( w ) , π 1 ( π 1 ( f ( w )))) • you give every child a present � a function f mapping • some child will open it � a child z and a proof that you open f ( z ) .
Limitations (3) Every player chooses a pawn. He puts it on square one. Ranta (1994: 73): the only way to interpret the text […] is by treating the pronoun ‘he’ as an abbreviation of ‘every player’ Obviously, this ‘abbreviation’ strategy is unsatisfactory. (2) Every student bought a book. Most of them read it. No mechanism for plural anaphora (yet). 9/38
Limitations (3) Every player chooses a pawn. He puts it on square one. Ranta (1994: 73): the only way to interpret the text […] is by treating the pronoun ‘he’ as an abbreviation of ‘every player’ Obviously, this ‘abbreviation’ strategy is unsatisfactory. (2) Every student bought a book. Most of them read it. No mechanism for plural anaphora (yet). 9/38
Outline of the proposal
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