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

(1) A police offjcer guards every exit. ⇒ ∃x.offjcer′x ∧ ∀y.exit′y → guard′xy (surface scope) ⇒ ∀y.exit′y → ∃x.offjcer′x ∧ guard′xy (inverse scope) F pred ‘guard’ subj G pred ‘police offjcer’ spec I pred ‘a’

• bj

H pred ‘exit’ spec J pred ‘every’

3/40

Scope ambiguity in English

(1) A police offjcer guards every exit. ⇒ ∃x.offjcer′x ∧ ∀y.exit′y → guard′xy (surface scope) ⇒ ∀y.exit′y → ∃x.offjcer′x ∧ guard′xy (inverse scope) F :              pred ‘guard’ subj G :   pred ‘police offjcer’ spec I :

• pred

‘a’

• 



• bj

H :   pred ‘exit’ spec J :

• pred

‘every’

• 

             

3/40

The Glue account: multiple proofs

a λP.λQ.∃x.Px ∧ Qx : ((spec ↑) ⊸ ↑) ⊸ (((spec ↑) ⊸ %A) ⊸ %A) %A = (gf* ↑) police offjcer offjcer′ : (spec ↑) ⊸ ↑ guards guard′ : (↑ subj) ⊸ ((↑ obj) ⊸ ↑) every λP.λQ.∀y.Py → Qy : ((spec ↑) ⊸ ↑) ⊸ (((spec ↑) ⊸ %B) ⊸ %B) %B = (gf* ↑) exit exit′ : (spec ↑) ⊸ ↑

4/40

The Glue account: multiple proofs

a λP.λQ.∃x.Px ∧ Qx : (G ⊸ I) ⊸ ((G ⊸ F) ⊸ F) %A := F police offjcer offjcer′ : G ⊸ I guards guard′ : G ⊸ (H ⊸ F) every λP.λQ.∀y.Py → Qy : (H ⊸ J) ⊸ ((H ⊸ F) ⊸ F) %B := F exit exit′ : H ⊸ J

4/40

Surface scope interpretation

[G]1 guard′ : G ⊸ (H ⊸ F) H ⊸ F every′ : (H ⊸ J) ⊸ ((H ⊸ F) ⊸ F) exit′ : H ⊸ J (H ⊸ F) ⊸ F F G ⊸ F 1 a′ : (G ⊸ I) ⊸ ((G ⊸ F) ⊸ F)

• ffjcer′ :

G ⊸ I (G ⊸ F) ⊸ F a′offjcer′(λx.every′exit′(guard′x)) : F ≡ ∃x.offjcer′x ∧ ∀y.exit′y → guard′xy : F

5/40

Inverse scope interpretation

[H]2 [G]1 guard′ : G ⊸ (H ⊸ F) H ⊸ F F G ⊸ F 1 a′ : (G ⊸ I) ⊸ ((G ⊸ F) ⊸ F)

• ffjcer′ :

G ⊸ I (G ⊸ F) ⊸ F F H ⊸ F 2 every′ : (H ⊸ J) ⊸ ((H ⊸ F) ⊸ F) exit′ : H ⊸ J (H ⊸ F) ⊸ F every′exit′(λy.a′offjcer′(λx.guard′xy)) : F ≡ ∀y.exit′y → ∃x.offjcer′x ∧ guard′xy : F

6/40

Scope rigidity in other languages

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

7/40

Scope rigidity in English

(4) Hilary gave a student every grade. ⇒ ∃y.student′y ∧ ∀x.grade′x → give′hilary′xy ∀x.grade′x → ∃y.student′y ∧ give′hilary′xy (surface scope only within the double object)

8/40

Not scope ‘islands’

(5) If every student passes, the lecturer will be happy. ⇒ (∀y.student′y → pass′y) → happy′( ι x.lecturer′x) ∀y.student′y → (pass′y → happy′( ι x.lecturer′x))

9/40

Constraining the path

(5) If every student passes, the lecturer will be happy. F :                pred ‘happy’ subj [“the lecturer”] adj                  G :         pred ‘pass’ compform ‘if’ subj H :   pred ‘student’ spec I :

• pred

‘every’

• 

                                         every λP.λQ.∀y.Py → Qy : ((spec ↑) ⊸ ↑) ⊸ (((spec ↑) ⊸ %B) ⊸ %B) %B = (path ↑)

10/40

Constraining the path

(5) If every student passes, the lecturer will be happy. F :                pred ‘happy’ subj [“the lecturer”] adj                  G :         pred ‘pass’ compform ‘if’ subj H :   pred ‘student’ spec I :

• pred

‘every’

• 

                                         every λP.λQ.∀y.Py → Qy : (H ⊸ I) ⊸ ((H ⊸ %B) ⊸ %B) %B = (path ↑) (where path is such that %B can be G but not F)

10/40

Not an available strategy here

(2) Ein Polizist bewacht jeden Ausgang. F :            pred ‘guard’ topic G : [“Ein Polizist”] subj

• bj

H :   pred ‘exit’ spec J :

• pred

‘every’

• 

            jeden λP.λQ.∀y.Py → Qy : : ((spec ↑) ⊸ ↑) ⊸ (((spec ↑) ⊸ %B) ⊸ %B) %B = (path ↑)

11/40

Not an available strategy here

(2) Ein Polizist bewacht jeden Ausgang. F :            pred ‘guard’ topic G : [“Ein Polizist”] subj

• bj

H :   pred ‘exit’ spec J :

• pred

‘every’

• 

            jeden λP.λQ.∀y.Py → Qy : : (H ⊸ I) ⊸ ((H ⊸ %B) ⊸ %B) %B = (path ↑) We have %B := F for both the surface scope and the inverse scope interpretation.

11/40

A previous proposal

Node orderings

Crouch & van Genabith (1999) propose to analzye scope rigidity like this: bewacht V guard′ : (↑ subj) ⊸ ((↑ obj) ⊸ ↑) (↑ subj) = (↑ topic) ⇒ (↑ subj) ≻ (↑ obj)

• 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

• ccurs strictly lower down than every

instance of .

12/40

Node orderings

Crouch & van Genabith (1999) propose to analzye scope rigidity like this: bewacht V guard′ : (↑ subj) ⊸ ((↑ obj) ⊸ ↑) (↑ subj) = (↑ topic) ⇒ (↑ subj) ≻ (↑ obj)

• 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

• ccurs strictly lower down than every

instance of .

12/40

Node orderings

Crouch & van Genabith (1999) propose to analzye scope rigidity like this: bewacht V guard′ : (↑ subj) ⊸ ((↑ obj) ⊸ ↑) (↑ subj) = (↑ topic) ⇒ (↑ subj) ≻ (↑ obj)

• The last line is a node ordering: a constraint on linear

logic proofs.

• Roughly, α ≻ β means that in every licit linear logic proof,

no instance of β occurs strictly lower down than every instance of α.

12/40

Node orderings in action

(2) Ein Polizist bewacht jeden Ausgang. F :        pred ‘guard’ topic G : [“Ein Polizist”] subj

• bj

H : [“jeden Ausgang”]        bewacht V guard′ : (↑ subj) ⊸ ((↑ obj) ⊸ ↑) (↑ subj) = (↑ topic) ⇒ (↑ subj) ≻ (↑ obj)

13/40

Node orderings in action

(2) Ein Polizist bewacht jeden Ausgang. F :        pred ‘guard’ topic G : [“Ein Polizist”] subj

• bj

H : [“jeden Ausgang”]        bewacht V guard′ : G ⊸ (H ⊸ F) G = G ⇒ G ≻ H

13/40

G ≻ H

[G]1 G ⊸ (H ⊸ F) H ⊸ F jeden Ausgang ⇓ (H ⊸ F) ⊸ F F G ⊸ F 1 ein Polizit ⇓ (G ⊸ F) ⊸ F F Surface scope

• [H]2

[G]1 G ⊸ (H ⊸ F) H ⊸ F F G ⊸ F 1 ein Polizist ⇓ (G ⊸ F) ⊸ F F H ⊸ F 2 jeden Ausgang ⇓ (H ⊸ F) ⊸ F F Inverse scope ×

14/40

What is a proof?

Node orderings are defjned over derivations A derivation is a tree-like structure of sequents […] Rep- resent derivations D as triples S, >S, $ where S is the set of points in the tree, >S is a transitive , asymmetric

• rdering over them, and $ is a function mapping the

points onto their corresponding sequents. (Crouch & van Genabith 1999: 131) But (natural deduction) derivations are representations of proofs, not the proofs themselves. Gentzen calculus, labelled and unlabelled natural de- ductions, proof nets, categorical calculus, etc. are all of repute, all have their respective advantages and disad- vantages, and are all notations for the same theory. (Corbalán & Morrill 2016: fn. 4), emphasis mine

15/40

What is a proof?

Node orderings are defjned over derivations A derivation is a tree-like structure of sequents […] Rep- resent derivations D as triples S, >S, $ where S is the set of points in the tree, >S is a transitive , asymmetric

• rdering over them, and $ is a function mapping the

points onto their corresponding sequents. (Crouch & van Genabith 1999: 131) But (natural deduction) derivations are representations of proofs, not the proofs themselves. Gentzen calculus, labelled and unlabelled natural de- ductions, proof nets, categorical calculus, etc. are all of repute, all have their respective advantages and disad- vantages, and are all notations for the same theory. (Corbalán & Morrill 2016: fn. 4), emphasis mine15/40

Sequent calculus

G ⊢ G H ⊸ F ⊢ H ⊸ F G, G ⊸ (H ⊸ F) ⊢ H ⊸ F ⊸L F ⊢ F G, G ⊸ (H ⊸ F), (H ⊸ F) ⊸ F ⊢ F ⊸L G ⊸ (H ⊸ F), (H ⊸ F) ⊸ F ⊢ G ⊸ F ⊸R F ⊢ F (G ⊸ F) ⊸ F, G ⊸ (H ⊸ F), (H ⊸ F) ⊸ F ⊢ F ⊸L Surface scope G ⊢ G H ⊢ H F ⊢ F H, H ⊸ F ⊢ F ⊸L G, H, G ⊸ (H ⊸ F) ⊢ F ⊸L H, G ⊸ (H ⊸ F) ⊢ G ⊸ F ⊸R F ⊢ F H, (G ⊸ F) ⊸ F, G ⊸ (H ⊸ F) ⊢ F ⊸L (G ⊸ F) ⊸ F, G ⊸ (H ⊸ F) ⊢ H ⊸ F ⊸R F ⊢ F (G ⊸ F) ⊸ F, G ⊸ (H ⊸ F), (H ⊸ F) ⊸ F ⊢ F ⊸L Inverse scope

16/40

Proof nets I

(Moot 2002: Chapter 5)

F+ ⊸− F− ⊸+ F+ H− ⊸− ⊸− F− H+ G+ ⊸− F− ⊸+ F+ G− Surface scope F+ ⊸− F− ⊸+ F+ H− ⊸− ⊸− F− H+ G+ ⊸− F− ⊸+ F+ G− Inverse scope

17/40

Proof nets II

(Adapted from Andrews 2010)

F+ F− ⊸− ⊸+ F+ F− ⊸− ⊸+ F+ F− ⊸− ⊸− G+ H+ H− G− F+ F− ⊸− ⊸+ F+ F− ⊸− ⊸+ F+ F− ⊸− ⊸− G+ H+ G− H−

Surface scope Inverse scope

18/40

• The point is not that an equivalent notion of node
• rdering couldn’t be defjned for these other proof formats.

(In face, I’ve actually done this in adapting the defjnition that Crouch & van Genabith (1999) give for a slightly difgerent proof format.)

• The point is that if we have properly linguistic constraint
• n the form of derivations, we’re not doing logic any more.

Rather than make such nonlogical restrictions on our proof theory, I turn to an alternative approach (Carpenter 1998: 203)

19/40

• The point is not that an equivalent notion of node
• rdering couldn’t be defjned for these other proof formats.

(In face, I’ve actually done this in adapting the defjnition that Crouch & van Genabith (1999) give for a slightly difgerent proof format.)

• The point is that if we have properly linguistic constraint
• n the form of derivations, we’re not doing logic any more.

Rather than make such nonlogical restrictions on our proof theory, I turn to an alternative approach (Carpenter 1998: 203)

19/40

• The point is not that an equivalent notion of node
• rdering couldn’t be defjned for these other proof formats.

(In face, I’ve actually done this in adapting the defjnition that Crouch & van Genabith (1999) give for a slightly difgerent proof format.)

• The point is that if we have properly linguistic constraint
• n the form of derivations, we’re not doing logic any more.

Rather than make such nonlogical restrictions on our proof theory, I turn to an alternative approach (Carpenter 1998: 203)

19/40

My proposal

The name of the game

• Assign linear logic formula to lexical items such that all

and only the desired interpretations have a corresponding proof.

• I.e., not fjltering out proofs by non-logical means.

20/40

The name of the game

• Assign linear logic formula to lexical items such that all

and only the desired interpretations have a corresponding proof.

• I.e., not fjltering out proofs by non-logical means.

20/40

In a bit more detail

Expand the fragment of linear logic used such that

• f-structure nodes are linear logic predicates (not

formulae),

• the arguments to those predicates ‘keep track’ of the
• rder of application of quantifjers, and
• set things up so that only by applying quantifjers in the

desired order can a valid proof be constructed.

21/40

In a bit more detail

Expand the fragment of linear logic used such that

• f-structure nodes are linear logic predicates (not

formulae),

• the arguments to those predicates ‘keep track’ of the
• rder of application of quantifjers, and
• set things up so that only by applying quantifjers in the

desired order can a valid proof be constructed.

21/40

In a bit more detail

Expand the fragment of linear logic used such that

• f-structure nodes are linear logic predicates (not

formulae),

• the arguments to those predicates ‘keep track’ of the
• rder of application of quantifjers,

and

• set things up so that only by applying quantifjers in the

desired order can a valid proof be constructed.

21/40

In a bit more detail

Expand the fragment of linear logic used such that

• f-structure nodes are linear logic predicates (not

formulae),

• the arguments to those predicates ‘keep track’ of the
• rder of application of quantifjers, and
• set things up so that only by applying quantifjers in the

desired order can a valid proof be constructed.

21/40

Provenance

The approach is inspired by work in Abstract Categorial Grammar (Pogodalla & Pompigne 2012, Kanazawa 2015). A crude characterisation would be that glue semantics is like categorial grammar and its semantics, but with-

• ut the categorial grammar.

(Crouch & van Genabith 2000: 91)

22/40

Provenance

The approach is inspired by work in Abstract Categorial Grammar (Pogodalla & Pompigne 2012, Kanazawa 2015). A crude characterisation would be that glue semantics is like categorial grammar and its semantics, but with-

• ut the categorial grammar.

(Crouch & van Genabith 2000: 91)

22/40

Linear logic fragment

Given a set P of predicates (f-structure nodes) and a set V of variables, the fragment of linear logic used is: n V 0 s n (terms) P n V (formulae) (where s is the successor function)

23/40

Linear logic fragment

Given a set P of predicates (f-structure nodes) and a set V of variables, the fragment of linear logic used is: n ::= V | 0 | s n (terms) φ, ψ ::= P n | φ ⊸ ψ | ∀V.φ (formulae) (where s is the successor function)

23/40

Linear logic fragment

Given a set P of predicates (f-structure nodes) and a set V of variables, the fragment of linear logic used is: n ::= V | 0 | s n (terms) φ, ψ ::= P n | φ ⊸ ψ | ∀V.φ (formulae) (where s is the successor function)

23/40

Our German example

bewacht guard′ : ∀i.∀j.(↑ subj) i ⊸ ((↑ obj) j ⊸ ↑j) det det′ : ∀i.[(spec ↑) 0 ⊸ ↑0] ⊸ ([(spec ↑)(s i) ⊸ %A (s i)] ⊸ %A i) %A = (gf* ↑) bewacht guard i j Gi Hj Fj ein Polizist P x offjcer x Px i G si F si Fi jeden Ausgang Q y exit y Qy i H si F si Fi A F

24/40

Our German example

bewacht guard′ : ∀i.∀j.(↑ subj) i ⊸ ((↑ obj) j ⊸ ↑j) det det′ : ∀i.[(spec ↑) 0 ⊸ ↑0] ⊸ ([(spec ↑)(s i) ⊸ %A (s i)] ⊸ %A i) %A = (gf* ↑) ⇓ bewacht guard′ : ∀i.∀j.Gi ⊸ (Hj ⊸ Fj) ein Polizist λP.∃x.offjcer′x ∧ Px : ∀i.(G(si) ⊸ F(si)) ⊸ Fi jeden Ausgang λQ.∀y.exit′y → Qy : ∀i.(H(si) ⊸ F(si)) ⊸ Fi %A := F

24/40

How it works

• There’s a ‘counter’.
• Applying a quantifjer reduces the counter by
• ne:

spec

s i

A s i A i

• So if Q1 immediately outscopes Q2, then you have to set

the counter for Q1 to one lower than for Q2.

• So to get the inverse scope reading, you’d have to set the

counter for the subject position one higher than for the

• bject position.
• But the lexical entry for the verb guarantees that if you do

that, no proof can be constructed: subj i

• bj j

j

25/40

How it works

• There’s a ‘counter’.
• Applying a quantifjer reduces the counter by
• ne:

[(spec ↑)(s i) ⊸ %A (s i)] ⊸ %A i

• So if Q1 immediately outscopes Q2, then you have to set

the counter for Q1 to one lower than for Q2.

• So to get the inverse scope reading, you’d have to set the

counter for the subject position one higher than for the

• bject position.
• But the lexical entry for the verb guarantees that if you do

that, no proof can be constructed: subj i

• bj j

j

25/40

How it works

• There’s a ‘counter’.
• Applying a quantifjer reduces the counter by
• ne:

[(spec ↑)(s i) ⊸ %A (s i)] ⊸ %A i

• So if Q1 immediately outscopes Q2, then you have to set

the counter for Q1 to one lower than for Q2.

• So to get the inverse scope reading, you’d have to set the

counter for the subject position one higher than for the

• bject position.
• But the lexical entry for the verb guarantees that if you do

that, no proof can be constructed: subj i

• bj j

j

25/40

How it works

• There’s a ‘counter’.
• Applying a quantifjer reduces the counter by
• ne:

[(spec ↑)(s i) ⊸ %A (s i)] ⊸ %A i

• So if Q1 immediately outscopes Q2, then you have to set

the counter for Q1 to one lower than for Q2.

• So to get the inverse scope reading, you’d have to set the

counter for the subject position one higher than for the

• bject position.
• But the lexical entry for the verb guarantees that if you do

that, no proof can be constructed: subj i

• bj j

j

25/40

How it works

• There’s a ‘counter’.
• Applying a quantifjer reduces the counter by
• ne:

[(spec ↑)(s i) ⊸ %A (s i)] ⊸ %A i

• So if Q1 immediately outscopes Q2, then you have to set

the counter for Q1 to one lower than for Q2.

• So to get the inverse scope reading, you’d have to set the

counter for the subject position one higher than for the

• bject position.
• But the lexical entry for the verb guarantees that if you do

that, no proof can be constructed: (↑ subj) i ⊸ ((↑ obj) j ⊸ ↑ j)

25/40

The inverse scope reading is underivable

ein Polizist ⇓ (G2 ⊸ F2) ⊸ F1 [H1]2 [G2]1 guard′ : ∀i.∀j.Gi ⊸ (Hj ⊸ Fj) G2 ⊸ (H1 ⊸ F1) ∀E × 2 H1 ⊸ F1 F1 G2 ⊸ F1 1 ∗

back

26/40

The surface scope reading is derivable

Ein Polizist ⇓ (G1 ⊸ F1) ⊸ F0 jeden Ausgang ⇓ (H2 ⊸ F2) ⊸ F1 [G1]1 guard′ : ∀i.∀j.Gi ⊸ (Hj ⊸ Fj) G1 ⊸ (H2 ⊸ F2) ∀E × 2 H2 ⊸ F2 F1 G1 ⊸ F1 1 F0

27/40

The relevance of topicalization

(2) Ein Polizist bewacht jeden Ausgang. (6) Jeden Ausgang bewacht ein Polizist. (2) (6)        pred ‘guard’ topic [“Ein Polizist”] subj

• bj

[“jeden Ausgang”]               pred ‘guard’ topic [“jeden Ausgang”] subj [“Ein Polizist”]

• bj

       (6), unlike (2), has both the surface scope and inverse scope readings.

28/40

Conditional meaning constructors

It seems that we want something like this: bewacht V (↑ pred) = ‘guard’ (↑ subj) = (↑ topic) ⇒ guard′ : ∀i.∀j.(↑ subj) i ⊸ ((↑ obj) j ⊸ ↑ (f i j)) (↑ subj) = (↑ topic) ⇒ guard′ : ∀i.∀j.∀k.(↑ subj) i ⊸ ((↑ obj) j ⊸ ↑ k) But this is an abuse of notation, since meaning constructors aren’t defjning equations.

29/40

A possible implementation

bewacht V (↑ pred) = ‘guard’ guard′ : ∀i.∀j.(↑ subj) i ⊸ ((↑ obj) j ⊸ ↑ (f i j)) (@reset) where reset := (↑ subj) = (↑ topic) λp.p : ∀i.∀j.↑i ⊸ ↑j

30/40

• If the subject is the topic, calling reset will cause failure.

So, the scope is frozen.

• If the subject is not the topic, then reset may or may not

be called. If it is, then both scope ordering are possible since the counter can be changed.

31/40

• If the subject is the topic, calling reset will cause failure.

So, the scope is frozen.

• If the subject is not the topic, then reset may or may not

be called. If it is, then both scope ordering are possible since the counter can be changed.

31/40

Deriving the inverse scope reading with reset

Remember this derivation? With reset it can be completed. jeden Ausgang H1 F1 F0 ein Polizist G2 F2 F1 H1 2 G2 1 guard . . . . F1 i j Fi Fj F1 F2

E

2 F2 G2 F2 1 F1 H1 F1 2 F0

32/40

Deriving the inverse scope reading with reset

Remember this derivation? With reset it can be completed. jeden Ausgang ⇓ (H1 ⊸ F1) ⊸ F0 ein Polizist ⇓ (G2 ⊸ F2) ⊸ F1 [H1]2, [G2]1, guard′ . . . . F1 ∀i.∀j.Fi ⊸ Fj F1 ⊸ F2 ∀E × 2 F2 G2 ⊸ F2 1 F1 H1 ⊸ F1 2 F0

32/40

The English double object construction

(7) Most teachers gave a student every grade. most a every a most every every most a most every a a every most every a most (Bruening 2001) The only way for the secondary object not to take narrowest scope is for both objects to scope over the subject (in surface

• rder).

gave give i j k subj i

• bj j
• bj

k

fijk

where f is the function such that fijk i if j k i k otherwise

33/40

The English double object construction

(7) Most teachers gave a student every grade. most a every a most every every most a most every a a every most every a most (Bruening 2001) The only way for the secondary object not to take narrowest scope is for both objects to scope over the subject (in surface

• rder).

gave give i j k subj i

• bj j
• bj

k

fijk

where f is the function such that fijk i if j k i k otherwise

33/40

The English double object construction

(7) Most teachers gave a student every grade. most a every a most every every most a most every a a every most every a most (Bruening 2001) The only way for the secondary object not to take narrowest scope is for both objects to scope over the subject (in surface

• rder).

gave give′ : ∀i.∀j.∀k.(↑ subj) i ⊸ ((↑ obj) j ⊸ ((↑ objθ) k ⊸ ↑(fijk))) where f is the function such that fijk =

• i if j < k < i

k otherwise

33/40

Refmections

Features on this account

Scope rigidity because

• quantifjers are not modifjers on the linear logic side, and
• verb forms can specify which argument takes narrowest

scope. This has been stated as particular to verb lexical entries, but of course we’d want to generalize to every transitive/ditransitive verb in the language.

34/40

Features on this account

Scope rigidity because

• quantifjers are not modifjers on the linear logic side

, and

• verb forms can specify which argument takes narrowest

scope. This has been stated as particular to verb lexical entries, but of course we’d want to generalize to every transitive/ditransitive verb in the language.

34/40

Features on this account

Scope rigidity because

• quantifjers are not modifjers on the linear logic side, and
• verb forms can specify which argument takes narrowest

scope. This has been stated as particular to verb lexical entries, but of course we’d want to generalize to every transitive/ditransitive verb in the language.

34/40

Features on this account

Scope rigidity because

• quantifjers are not modifjers on the linear logic side, and
• verb forms can specify which argument takes narrowest

scope. This has been stated as particular to verb lexical entries, but of course we’d want to generalize to every transitive/ditransitive verb in the language.

34/40

Possible alternatives

• Provide f-/s-structure with more internal structure (cf.

Andrews (2018) on the relative scope of adjectives).

• Read linear logic formulae ofg c-structure instead.

I can’t seen either of these options being popular.

35/40

Possible alternatives

• Provide f-/s-structure with more internal structure (cf.

Andrews (2018) on the relative scope of adjectives).

• Read linear logic formulae ofg c-structure instead.

I can’t seen either of these options being popular.

35/40

Possible alternatives

• Provide f-/s-structure with more internal structure (cf.

Andrews (2018) on the relative scope of adjectives).

• Read linear logic formulae ofg c-structure instead.

I can’t seen either of these options being popular.

35/40

Possible alternatives

• Provide f-/s-structure with more internal structure (cf.

Andrews (2018) on the relative scope of adjectives).

• Read linear logic formulae ofg c-structure instead.

I can’t seen either of these options being popular.

35/40

Thanks!

This research is funded by the

36/40

References

Andrews, Avery D. 2010. Propositional glue and the projection architecture of LFG. Linguistics and Philosophy 33. 141–170. https://doi.org/10.1007/s10988-010-9079-9. Andrews, Avery D. 2018. Sets, heads, and spreading in LFG. Journal of Language Modelling 6(1). 131–174. https://doi.org/10.15398/jlm.v6i1.175. Bruening, Benjamin. 2001. QR obeys superiority: frozen scope and ACD. Linguistic Inquiry 32(2). 233–273. Carpenter, Bob. 1998. Type-logical semantics. Cambridge, MA: MIT Press.

37/40

Corbalán, María Inés & Glyn Morrill. 2016. Overtly anaphoric control in type logical grammar. In Annie Foret, Glyn Morrill, Reinhard Muskens, Rainer Osswald & Sylvain Pogodalla (eds.), Formal grammar: FG 2015, FG 2016 (Lecture Notes in Computer Science 9804), 183–199. Berlin, Heidelberg: Springer. https://doi.org/10.1007/978-3-662-53042-9_11. Crouch, Richard & Josef van Genabith. 1999. Context change, underspecifjcation and the structure of Glue language

• derivations. In Mary Dalrymple (ed.), Semantics and syntax in

Lexical Functional Grammar: The resource logic approach, 117–189. Cambridge, MA: MIT Press.

38/40

Crouch, Richard & Josef van Genabith. 2000. Linear logic for

• linguists. ESSLLI 2000 course notes. Archived 2006-10-19 in

the Internet Archive at http: //web.archive.org/web/20061019004949/http: //www2.parc.com/istl/members/crouch/esslli00_ notes.pdf. Kanazawa, Makoto. 2015. Syntactic features for regular constraints and an approximation of directional slashes in abstract categorial grammars. In Yusuke Kubota & Robert Levine (eds.), Empirical advances in categorial grammar: Proceedings of the ESSLLI 2015 workshop (CG 2015), 34–70. http://www.u.tsukuba.ac.jp/~kubota. yusuke.fn/cg2015-proceedings.pdf. Last accessed 2019-02-12.

39/40

Moot, Richard. 2002. Proof nets for linguistic analysis. University of Utrecht dissertation. Pogodalla, Sylvain & Florent Pompigne. 2012. Controlling extraction in abstract categorial grammars. In Philippe de Groote & Mark-Jan Nederhof (eds.), Formal grammar: FG 2010, FG 2011 (Lecture Notes in Computer Science 7395), 162–177. Berlin, Heidelberg: Springer. https://doi.org/10.1007/978-3-642-32024-8_11.

40/40