Constraining scope ambiguity in LFG+Glue Matthew Gotham University - PowerPoint PPT Presentation

Constraining scope ambiguity in LFG+Glue Matthew Gotham University of Oxford 24th International LFG Conference, Australian National University 8–10 July 2019 1/40

Outline Scope (non-)ambiguity in LFG+Glue Background Scope rigidity–what this talk is about A previous proposal Node orderings Problems with the node ordering approach My proposal Using a counter Re-enabling scope fmexibility Refmections 2/40

Outline Scope (non-)ambiguity in LFG+Glue Background Scope rigidity–what this talk is about A previous proposal Node orderings Problems with the node ordering approach My proposal Using a counter Re-enabling scope fmexibility Refmections 2/40

Outline Scope (non-)ambiguity in LFG+Glue Background Scope rigidity–what this talk is about A previous proposal Node orderings Problems with the node ordering approach My proposal Using a counter Re-enabling scope fmexibility Refmections 2/40

Outline Scope (non-)ambiguity in LFG+Glue Background Scope rigidity–what this talk is about A previous proposal Node orderings Problems with the node ordering approach My proposal Using a counter Re-enabling scope fmexibility Refmections 2/40

Scope (non-)ambiguity in LFG+Glue

Scope ambiguity in English spec ‘every’ pred J spec ‘exit’ pred H obj ‘a’ pred I ‘police offjcer’ (1) pred G subj ‘guard’ pred F (inverse scope) (surface scope) A police offjcer guards every exit. 3/40 ⇒ ∃ x . offjcer ′ x ∧ ∀ y . exit ′ y → guard ′ xy ⇒ ∀ y . exit ′ y → ∃ x . offjcer ′ x ∧ guard ′ xy

Scope ambiguity in English pred ‘every’ pred spec ‘exit’ pred obj ‘a’ (1) spec ‘police offjcer’ pred subj ‘guard’ pred 3/40 A police offjcer guards every exit. (surface scope) (inverse scope) ⇒ ∃ x . offjcer ′ x ∧ ∀ y . exit ′ y → guard ′ xy ⇒ ∀ y . exit ′ y → ∃ x . offjcer ′ x ∧ guard ′ xy           G :    � �    I : F :             H :   � �    J : 

4/40 The Glue account: multiple proofs a � λ P .λ Q . ∃ x . Px ∧ Qx : (( spec ↑ ) ⊸ ↑ ) ⊸ ((( spec ↑ ) ⊸ % A ) ⊸ % A ) % A = ( gf* ↑ ) police offjcer � offjcer ′ : ( spec ↑ ) ⊸ ↑ guards � guard ′ : ( ↑ subj ) ⊸ (( ↑ obj ) ⊸ ↑ ) every � λ P .λ Q . ∀ y . Py → Qy : (( spec ↑ ) ⊸ ↑ ) ⊸ ((( spec ↑ ) ⊸ % B ) ⊸ % B ) % B = ( gf* ↑ ) exit � exit ′ : ( spec ↑ ) ⊸ ↑

4/40 The Glue account: multiple proofs a � λ P .λ Q . ∃ x . Px ∧ Qx : ( G ⊸ I ) ⊸ (( G ⊸ F ) ⊸ F ) % A := F police offjcer � offjcer ′ : G ⊸ I guards � guard ′ : G ⊸ ( H ⊸ F ) every � λ P .λ Q . ∀ y . Py → Qy : ( H ⊸ J ) ⊸ (( H ⊸ F ) ⊸ F ) % B := F exit � exit ′ : H ⊸ J

Surface scope interpretation F 5/40 every ′ : guard ′ : exit ′ : ( H ⊸ J ) ⊸ a ′ : (( H ⊸ F ) ⊸ F ) H ⊸ J [ G ] 1 G ⊸ ( H ⊸ F ) offjcer ′ : ( G ⊸ I ) ⊸ ( H ⊸ F ) ⊸ F H ⊸ F (( G ⊸ F ) ⊸ F ) G ⊸ I G ⊸ F 1 ( G ⊸ F ) ⊸ F a ′ offjcer ′ ( λ x . every ′ exit ′ ( guard ′ x )) : F ≡ ∃ x . offjcer ′ x ∧ ∀ y . exit ′ y → guard ′ xy : F

Inverse scope interpretation F F 6/40 guard ′ : a ′ : [ G ] 1 G ⊸ ( H ⊸ F ) offjcer ′ : ( G ⊸ I ) ⊸ [ H ] 2 H ⊸ F every ′ : (( G ⊸ F ) ⊸ F ) G ⊸ I exit ′ : ( H ⊸ J ) ⊸ ( G ⊸ F ) ⊸ F G ⊸ F 1 (( H ⊸ F ) ⊸ F ) H ⊸ J H ⊸ F 2 ( H ⊸ F ) ⊸ F every ′ exit ′ ( λ y . a ′ offjcer ′ ( λ x . guard ′ xy )) : F ≡ ∀ y . exit ′ y → ∃ x . offjcer ′ x ∧ guard ′ xy : F

Scope rigidity in other languages (2) (surface scope only) (Chinese) exit chukou. every meige guards kanshou police offjcer jingcha One-CL Yi-ming (3) (German) exit Ausgang. every jeden guards bewacht police offjcer Polizist A Ein 7/40 ⇒ ∃ x . offjcer ′ x ∧ ∀ y . exit ′ y → guard ′ xy � ∀ y . exit ′ y → ∃ x . offjcer ′ x ∧ guard ′ xy

Scope rigidity in English (4) Hilary gave a student every grade. (surface scope only within the double object) 8/40 ⇒ ∃ y . student ′ y ∧ ∀ x . grade ′ x → give ′ hilary ′ xy � ∀ x . grade ′ x → ∃ y . student ′ y ∧ give ′ hilary ′ xy

Not scope ‘islands’ (5) If every student passes, the lecturer will be happy. 9/40 ⇒ ( ∀ y . student ′ y → pass ′ y ) → happy ′ ( x . lecturer ′ x ) ι � ∀ y . student ′ y → ( pass ′ y → happy ′ ( x . lecturer ′ x )) ι

Constraining the path ‘if’ subj [“the lecturer”] adj pred ‘pass’ compform subj pred (5) If every student passes, the lecturer will be happy. pred ‘student’ spec pred ‘every’ ‘happy’ 10/40                         F :                       G :                 H :      � �         I :          every � λ P .λ Q . ∀ y . Py → Qy : (( spec ↑ ) ⊸ ↑ ) ⊸ ((( spec ↑ ) ⊸ % B ) ⊸ % B ) % B = ( path ↑ )

Constraining the path subj [“the lecturer”] adj pred ‘pass’ compform ‘if’ (5) If every student passes, the lecturer will be happy. ‘happy’ pred ‘student’ spec pred ‘every’ (where path is such that %B can be G but not F ) subj 10/40 pred                         F :                       G :                 H :      � �         I :          every � λ P .λ Q . ∀ y . Py → Qy : ( H ⊸ I ) ⊸ (( H ⊸ % B ) ⊸ % B ) % B = ( path ↑ )

Not an available strategy here pred ‘every’ pred spec (2) Ein Polizist bewacht jeden Ausgang. pred obj subj topic ‘guard’ ‘exit’ 11/40     G : [“Ein Polizist”]       F :            H :   � �    J : jeden � λ P .λ Q . ∀ y . Py → Qy : : (( spec ↑ ) ⊸ ↑ ) ⊸ ((( spec ↑ ) ⊸ % B ) ⊸ % B ) % B = ( path ↑ )

Not an available strategy here pred scope interpretation. ‘every’ pred (2) Ein Polizist bewacht jeden Ausgang. ‘exit’ pred obj subj topic ‘guard’ spec 11/40     G : [“Ein Polizist”]       F :            H :   � �    J : jeden � λ P .λ Q . ∀ y . Py → Qy : : ( H ⊸ I ) ⊸ (( H ⊸ % B ) ⊸ % B ) % B = ( path ↑ ) We have % B := F for both the surface scope and the inverse

A previous proposal

Node orderings Crouch & van Genabith (1999) propose to analzye scope rigidity like this: bewacht V • The last line is a node ordering : a constraint on linear logic proofs. • Roughly, means that in every licit linear logic proof, no instance of occurs strictly lower down than every instance of . 12/40 guard ′ : ( ↑ subj ) ⊸ (( ↑ obj ) ⊸ ↑ ) ( ↑ subj ) = ( ↑ topic ) ⇒ ( ↑ subj ) ≻ ( ↑ obj )

Node orderings Crouch & van Genabith (1999) propose to analzye scope rigidity like this: bewacht V • The last line is a node ordering : a constraint on linear logic proofs. • Roughly, means that in every licit linear logic proof, no instance of occurs strictly lower down than every instance of . 12/40 guard ′ : ( ↑ subj ) ⊸ (( ↑ obj ) ⊸ ↑ ) ( ↑ subj ) = ( ↑ topic ) ⇒ ( ↑ subj ) ≻ ( ↑ obj )

Node orderings Crouch & van Genabith (1999) propose to analzye scope rigidity like this: bewacht V • The last line is a node ordering : a constraint on linear logic proofs. 12/40 guard ′ : ( ↑ subj ) ⊸ (( ↑ obj ) ⊸ ↑ ) ( ↑ subj ) = ( ↑ topic ) ⇒ ( ↑ subj ) ≻ ( ↑ obj ) • Roughly, α ≻ β means that in every licit linear logic proof, no instance of β occurs strictly lower down than every instance of α .

Node orderings in action pred V bewacht obj (2) Ein Polizist bewacht jeden Ausgang. topic ‘guard’ subj 13/40     G : [“Ein Polizist”]   F :         H : [“jeden Ausgang”] guard ′ : ( ↑ subj ) ⊸ (( ↑ obj ) ⊸ ↑ ) ( ↑ subj ) = ( ↑ topic ) ⇒ ( ↑ subj ) ≻ ( ↑ obj )

Node orderings in action pred V bewacht obj (2) Ein Polizist bewacht jeden Ausgang. topic ‘guard’ subj 13/40     G : [“Ein Polizist”]   F :         H : [“jeden Ausgang”] guard ′ : G ⊸ ( H ⊸ F ) G = G ⇒ G ≻ H