Upping the ante: the full Lambek Calculus Robert Levine Ohio State - - PowerPoint PPT Presentation

Upping the ante: the full Lambek Calculus

Robert Levine

Ohio State University levine.1@osu.edu

Robert Levine The Lambek Calculus 1 / 24

Coordination

◮ Coordination is a cross-categorial phenomenon

Robert Levine The Lambek Calculus 2 / 24

Coordination

◮ Coordination is a cross-categorial phenomenon

Robert Levine The Lambek Calculus 2 / 24

Coordination

◮ Coordination is a cross-categorial phenomenon

(1)

• a. John [VPfin ate the cake] and [VPfin seemed to like it].

Robert Levine The Lambek Calculus 2 / 24

Coordination

◮ Coordination is a cross-categorial phenomenon

(1)

• a. John [VPfin ate the cake] and [VPfin seemed to like it].
• b. John wanted to [VPbse walk] and [VPbse talk].

Robert Levine The Lambek Calculus 2 / 24

Coordination

◮ Coordination is a cross-categorial phenomenon

(1)

• a. John [VPfin ate the cake] and [VPfin seemed to like it].
• b. John wanted to [VPbse walk] and [VPbse talk].
• c. John [VPfin/NP likes] and [VPfin/NP admires] Mary.

Robert Levine The Lambek Calculus 2 / 24

Coordination

◮ Coordination is a cross-categorial phenomenon

(1)

• a. John [VPfin ate the cake] and [VPfin seemed to like it].
• b. John wanted to [VPbse walk] and [VPbse talk].
• c. John [VPfin/NP likes] and [VPfin/NP admires] Mary.
• d. John talked to the members of the jury [VP\

VP

passionately] and [VP\

VP eloquently]. Robert Levine The Lambek Calculus 2 / 24

Coordination

◮ Coordination is a cross-categorial phenomenon

(1)

• a. John [VPfin ate the cake] and [VPfin seemed to like it].
• b. John wanted to [VPbse walk] and [VPbse talk].
• c. John [VPfin/NP likes] and [VPfin/NP admires] Mary.
• d. John talked to the members of the jury [VP\

VP

passionately] and [VP\

VP eloquently].

• e. [NP John ] and [NP Mary] went to a movie.

Robert Levine The Lambek Calculus 2 / 24

Coordination

◮ Coordination is a cross-categorial phenomenon

(1)

• a. John [VPfin ate the cake] and [VPfin seemed to like it].
• b. John wanted to [VPbse walk] and [VPbse talk].
• c. John [VPfin/NP likes] and [VPfin/NP admires] Mary.
• d. John talked to the members of the jury [VP\

VP

passionately] and [VP\

VP eloquently].

• e. [NP John ] and [NP Mary] went to a movie.
• f. John drove [PP out of the garage ] and [PP onto the

street]

Robert Levine The Lambek Calculus 2 / 24

Coordination

◮ Coordination is a cross-categorial phenomenon

(1)

• a. John [VPfin ate the cake] and [VPfin seemed to like it].
• b. John wanted to [VPbse walk] and [VPbse talk].
• c. John [VPfin/NP likes] and [VPfin/NP admires] Mary.
• d. John talked to the members of the jury [VP\

VP

passionately] and [VP\

VP eloquently].

• e. [NP John ] and [NP Mary] went to a movie.
• f. John drove [PP out of the garage ] and [PP onto the

street]

◮ How should we analyze it?

Robert Levine The Lambek Calculus 2 / 24

Logical conjunction vs. linguistic coordination

◮ In standard logic, conjunction is defined by the rules of ∧

Introduction and ∧ Elimination:

Robert Levine The Lambek Calculus 3 / 24

Logical conjunction vs. linguistic coordination

◮ In standard logic, conjunction is defined by the rules of ∧

Introduction and ∧ Elimination:

Robert Levine The Lambek Calculus 3 / 24

Logical conjunction vs. linguistic coordination

◮ In standard logic, conjunction is defined by the rules of ∧

Introduction and ∧ Elimination: φ, ψ

∧ Intro

φ ∧ ψ φ ∧ ψ

∧ Elim1

φ , φ ∧ ψ

∧ Elim2

ψ

Robert Levine The Lambek Calculus 3 / 24

Logical conjunction vs. linguistic coordination

◮ In standard logic, conjunction is defined by the rules of ∧

Introduction and ∧ Elimination: φ, ψ

∧ Intro

φ ∧ ψ φ ∧ ψ

∧ Elim1

φ , φ ∧ ψ

∧ Elim2

ψ

◮ But logical conjunction is restricted to propositions.

Robert Levine The Lambek Calculus 3 / 24

Logical conjunction vs. linguistic coordination

◮ In standard logic, conjunction is defined by the rules of ∧

Introduction and ∧ Elimination: φ, ψ

∧ Intro

φ ∧ ψ φ ∧ ψ

∧ Elim1

φ , φ ∧ ψ

∧ Elim2

ψ

◮ But logical conjunction is restricted to propositions. ◮ In the case of John walked and talked, what is syntactically

coordinated are the (property-denoting) words walked and talked.

Robert Levine The Lambek Calculus 3 / 24

Logical conjunction vs. linguistic coordination

◮ In standard logic, conjunction is defined by the rules of ∧

Introduction and ∧ Elimination: φ, ψ

∧ Intro

φ ∧ ψ φ ∧ ψ

∧ Elim1

φ , φ ∧ ψ

∧ Elim2

ψ

◮ But logical conjunction is restricted to propositions. ◮ In the case of John walked and talked, what is syntactically

coordinated are the (property-denoting) words walked and talked.

◮ How do we know this?

Robert Levine The Lambek Calculus 3 / 24

Logical conjunction vs. linguistic coordination

◮ In standard logic, conjunction is defined by the rules of ∧

Introduction and ∧ Elimination: φ, ψ

∧ Intro

φ ∧ ψ φ ∧ ψ

∧ Elim1

φ , φ ∧ ψ

∧ Elim2

ψ

◮ But logical conjunction is restricted to propositions. ◮ In the case of John walked and talked, what is syntactically

coordinated are the (property-denoting) words walked and talked.

◮ How do we know this? ◮ The only other possibility is that John walked is coordinated

with talked.

Robert Levine The Lambek Calculus 3 / 24

Logical conjunction vs. linguistic coordination

◮ In standard logic, conjunction is defined by the rules of ∧

Introduction and ∧ Elimination: φ, ψ

∧ Intro

φ ∧ ψ φ ∧ ψ

∧ Elim1

φ , φ ∧ ψ

∧ Elim2

ψ

◮ But logical conjunction is restricted to propositions. ◮ In the case of John walked and talked, what is syntactically

coordinated are the (property-denoting) words walked and talked.

◮ How do we know this? ◮ The only other possibility is that John walked is coordinated

with talked.

◮ Do you think it plausible that these two expressions are

coordinated?

Robert Levine The Lambek Calculus 3 / 24

(2) *Walked and John talked.

Robert Levine The Lambek Calculus 4 / 24

(2) *Walked and John talked.

◮ So we have the two properties coordinated. . .

Robert Levine The Lambek Calculus 4 / 24

(2) *Walked and John talked.

◮ So we have the two properties coordinated. . . ◮ . . . , i.e, λx.walk(x) coordinated with λz.talk(z)

Robert Levine The Lambek Calculus 4 / 24

(2) *Walked and John talked.

◮ So we have the two properties coordinated. . . ◮ . . . , i.e, λx.walk(x) coordinated with λz.talk(z) ◮ . . . but you can’t conjoin properties.

Robert Levine The Lambek Calculus 4 / 24

(2) *Walked and John talked.

◮ So we have the two properties coordinated. . . ◮ . . . , i.e, λx.walk(x) coordinated with λz.talk(z) ◮ . . . but you can’t conjoin properties. ◮ What we want to wind up with is walk(j) ∧ talk(j) . . .

Robert Levine The Lambek Calculus 4 / 24

(2) *Walked and John talked.

◮ So we have the two properties coordinated. . . ◮ . . . , i.e, λx.walk(x) coordinated with λz.talk(z) ◮ . . . but you can’t conjoin properties. ◮ What we want to wind up with is walk(j) ∧ talk(j) . . . ◮ . . . which we could get from λw[walk(w) ∧ talk(w)](j) . . .

Robert Levine The Lambek Calculus 4 / 24

(2) *Walked and John talked.

◮ So we have the two properties coordinated. . . ◮ . . . , i.e, λx.walk(x) coordinated with λz.talk(z) ◮ . . . but you can’t conjoin properties. ◮ What we want to wind up with is walk(j) ∧ talk(j) . . . ◮ . . . which we could get from λw[walk(w) ∧ talk(w)](j) . . . ◮ . . . because walk(w) ∧ talk(w) IS a proposition.

Robert Levine The Lambek Calculus 4 / 24

(2) *Walked and John talked.

◮ So we have the two properties coordinated. . . ◮ . . . , i.e, λx.walk(x) coordinated with λz.talk(z) ◮ . . . but you can’t conjoin properties. ◮ What we want to wind up with is walk(j) ∧ talk(j) . . . ◮ . . . which we could get from λw[walk(w) ∧ talk(w)](j) . . . ◮ . . . because walk(w) ∧ talk(w) IS a proposition. ◮ If we could get the semantics of and to combine λx.walk(x)

and λz.talk(z) to give us λw.walk(w) ∧ talk(w), we would have what we needed.

Robert Levine The Lambek Calculus 4 / 24

Cross-categorial analysis of coordination

◮ Can we do that? How can we get

λx.walk(x) λz.talk(z) ⇒ λw.walk(w) ∧ talk(w)

Robert Levine The Lambek Calculus 5 / 24

Cross-categorial analysis of coordination

◮ Can we do that? How can we get

λx.walk(x) λz.talk(z) ⇒ λw.walk(w) ∧ talk(w)

◮ (where is the name of the ‘mystery operation’ required) Robert Levine The Lambek Calculus 5 / 24

Cross-categorial analysis of coordination

◮ Can we do that? How can we get

λx.walk(x) λz.talk(z) ⇒ λw.walk(w) ∧ talk(w)

◮ (where is the name of the ‘mystery operation’ required) Robert Levine The Lambek Calculus 5 / 24

Cross-categorial analysis of coordination

◮ Can we do that? How can we get

λx.walk(x) λz.talk(z) ⇒ λw.walk(w) ∧ talk(w)

◮ (where is the name of the ‘mystery operation’ required)

Generalized conjunction P ⊓ Q =def P ∧ Q if P and Q are of type t λh.P(h) ⊓ Q(h)

• therwise

Robert Levine The Lambek Calculus 5 / 24

Cross-categorial analysis of coordination

◮ Can we do that? How can we get

λx.walk(x) λz.talk(z) ⇒ λw.walk(w) ∧ talk(w)

◮ (where is the name of the ‘mystery operation’ required)

Generalized conjunction P ⊓ Q =def P ∧ Q if P and Q are of type t λh.P(h) ⊓ Q(h)

• therwise

Examples: (3) walk ⊓ talk = λx.walk(x)

• P

⊓ λz.talk(z)

• Q

Robert Levine The Lambek Calculus 5 / 24

Cross-categorial analysis of coordination

◮ Can we do that? How can we get

λx.walk(x) λz.talk(z) ⇒ λw.walk(w) ∧ talk(w)

◮ (where is the name of the ‘mystery operation’ required)

Generalized conjunction P ⊓ Q =def P ∧ Q if P and Q are of type t λh.P(h) ⊓ Q(h)

• therwise

Examples: (3) walk ⊓ talk = λx.walk(x)

• P

⊓ λz.talk(z)

• Q

= λw[λx[walk(x)](w) ⊓ λz[talk(z)](w)]

Robert Levine The Lambek Calculus 5 / 24

Cross-categorial analysis of coordination

◮ Can we do that? How can we get

λx.walk(x) λz.talk(z) ⇒ λw.walk(w) ∧ talk(w)

◮ (where is the name of the ‘mystery operation’ required)

Generalized conjunction P ⊓ Q =def P ∧ Q if P and Q are of type t λh.P(h) ⊓ Q(h)

• therwise

Examples: (3) walk ⊓ talk = λx.walk(x)

• P

⊓ λz.talk(z)

• Q

= λw[λx[walk(x)](w) ⊓ λz[talk(z)](w)] = λw[walk(w) ∧ talk(w)]

Robert Levine The Lambek Calculus 5 / 24

Cross-categorial analysis of coordination

◮ Can we do that? How can we get

λx.walk(x) λz.talk(z) ⇒ λw.walk(w) ∧ talk(w)

◮ (where is the name of the ‘mystery operation’ required)

Generalized conjunction P ⊓ Q =def P ∧ Q if P and Q are of type t λh.P(h) ⊓ Q(h)

• therwise

Examples: (3) walk ⊓ talk = λx.walk(x)

• P

⊓ λz.talk(z)

• Q

= λw[λx[walk(x)](w) ⊓ λz[talk(z)](w)] = λw[walk(w) ∧ talk(w)]

Robert Levine The Lambek Calculus 5 / 24

◮

passionatelye,t,e,t ⊓ eloquentlye,t,e,t = λRλx.passionately(R)(x) ⊓ λSλy.eloquently(S)(y) Robert Levine The Lambek Calculus 6 / 24

◮

passionatelye,t,e,t ⊓ eloquentlye,t,e,t = λRλx.passionately(R)(x) ⊓ λSλy.eloquently(S)(y) Robert Levine The Lambek Calculus 6 / 24

◮

passionatelye,t,e,t ⊓ eloquentlye,t,e,t = λRλx.passionately(R)(x) ⊓ λSλy.eloquently(S)(y) = λT.λR[λx.passionately(R)(x)](T ) ⊓ λS[λy.eloquently(S)(y)](T ) Robert Levine The Lambek Calculus 6 / 24

◮

passionatelye,t,e,t ⊓ eloquentlye,t,e,t = λRλx.passionately(R)(x) ⊓ λSλy.eloquently(S)(y) = λT.λR[λx.passionately(R)(x)](T ) ⊓ λS[λy.eloquently(S)(y)](T ) = λT λx.passionately(T )(x) ⊓ λy.eloquently(T )(y) Robert Levine The Lambek Calculus 6 / 24

◮

passionatelye,t,e,t ⊓ eloquentlye,t,e,t = λRλx.passionately(R)(x) ⊓ λSλy.eloquently(S)(y) = λT.λR[λx.passionately(R)(x)](T ) ⊓ λS[λy.eloquently(S)(y)](T ) = λT λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)

◮

Then talked to the members of the jury passionately and eloquently will apply λT [λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)] to the semantics of the VP talked to the members of the jury, talked(mj). . . Robert Levine The Lambek Calculus 6 / 24

◮

passionatelye,t,e,t ⊓ eloquentlye,t,e,t = λRλx.passionately(R)(x) ⊓ λSλy.eloquently(S)(y) = λT.λR[λx.passionately(R)(x)](T ) ⊓ λS[λy.eloquently(S)(y)](T ) = λT λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)

◮

Then talked to the members of the jury passionately and eloquently will apply λT [λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)] to the semantics of the VP talked to the members of the jury, talked(mj). . .

◮

. . . and we wind up with Robert Levine The Lambek Calculus 6 / 24

◮

passionatelye,t,e,t ⊓ eloquentlye,t,e,t = λRλx.passionately(R)(x) ⊓ λSλy.eloquently(S)(y) = λT.λR[λx.passionately(R)(x)](T ) ⊓ λS[λy.eloquently(S)(y)](T ) = λT λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)

◮

Then talked to the members of the jury passionately and eloquently will apply λT [λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)] to the semantics of the VP talked to the members of the jury, talked(mj). . .

◮

. . . and we wind up with

◮

λT [λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)](talked(mj)) Robert Levine The Lambek Calculus 6 / 24

◮

passionatelye,t,e,t ⊓ eloquentlye,t,e,t = λRλx.passionately(R)(x) ⊓ λSλy.eloquently(S)(y) = λT.λR[λx.passionately(R)(x)](T ) ⊓ λS[λy.eloquently(S)(y)](T ) = λT λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)

◮

Then talked to the members of the jury passionately and eloquently will apply λT [λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)] to the semantics of the VP talked to the members of the jury, talked(mj). . .

◮

. . . and we wind up with

◮

λT [λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)](talked(mj)) Robert Levine The Lambek Calculus 6 / 24

◮

passionatelye,t,e,t ⊓ eloquentlye,t,e,t = λRλx.passionately(R)(x) ⊓ λSλy.eloquently(S)(y) = λT.λR[λx.passionately(R)(x)](T ) ⊓ λS[λy.eloquently(S)(y)](T ) = λT λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)

◮

Then talked to the members of the jury passionately and eloquently will apply λT [λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)] to the semantics of the VP talked to the members of the jury, talked(mj). . .

◮

. . . and we wind up with

◮

λT [λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)](talked(mj)) = λx.passionately(talked(mj))(x) ⊓ λy.eloquently(talked(mj))(y) Robert Levine The Lambek Calculus 6 / 24

◮

passionatelye,t,e,t ⊓ eloquentlye,t,e,t = λRλx.passionately(R)(x) ⊓ λSλy.eloquently(S)(y) = λT.λR[λx.passionately(R)(x)](T ) ⊓ λS[λy.eloquently(S)(y)](T ) = λT λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)

◮

Then talked to the members of the jury passionately and eloquently will apply λT [λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)] to the semantics of the VP talked to the members of the jury, talked(mj). . .

◮

. . . and we wind up with

◮

λT [λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)](talked(mj)) = λx.passionately(talked(mj))(x) ⊓ λy.eloquently(talked(mj))(y) = λv[λx[passionately(talked(mj))(x)](v) ⊓ λy[eloquently(talked(mj))(y)](v)] Robert Levine The Lambek Calculus 6 / 24

◮

passionatelye,t,e,t ⊓ eloquentlye,t,e,t = λRλx.passionately(R)(x) ⊓ λSλy.eloquently(S)(y) = λT.λR[λx.passionately(R)(x)](T ) ⊓ λS[λy.eloquently(S)(y)](T ) = λT λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)

◮

Then talked to the members of the jury passionately and eloquently will apply λT [λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)] to the semantics of the VP talked to the members of the jury, talked(mj). . .

◮

. . . and we wind up with

◮

λT [λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)](talked(mj)) = λx.passionately(talked(mj))(x) ⊓ λy.eloquently(talked(mj))(y) = λv[λx[passionately(talked(mj))(x)](v) ⊓ λy[eloquently(talked(mj))(y)](v)] = λv.passionately(talked(mj))(v) ∧ eloquently(talked(mj))(v) Robert Levine The Lambek Calculus 6 / 24

◮

passionatelye,t,e,t ⊓ eloquentlye,t,e,t = λRλx.passionately(R)(x) ⊓ λSλy.eloquently(S)(y) = λT.λR[λx.passionately(R)(x)](T ) ⊓ λS[λy.eloquently(S)(y)](T ) = λT λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)

◮

Then talked to the members of the jury passionately and eloquently will apply λT [λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)] to the semantics of the VP talked to the members of the jury, talked(mj). . .

◮

. . . and we wind up with

◮

λT [λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)](talked(mj)) = λx.passionately(talked(mj))(x) ⊓ λy.eloquently(talked(mj))(y) = λv[λx[passionately(talked(mj))(x)](v) ⊓ λy[eloquently(talked(mj))(y)](v)] = λv.passionately(talked(mj))(v) ∧ eloquently(talked(mj))(v)

◮

The VP talked to the members of the jury passionately and eloquently can now apply to John Robert Levine The Lambek Calculus 6 / 24

◮

passionatelye,t,e,t ⊓ eloquentlye,t,e,t = λRλx.passionately(R)(x) ⊓ λSλy.eloquently(S)(y) = λT.λR[λx.passionately(R)(x)](T ) ⊓ λS[λy.eloquently(S)(y)](T ) = λT λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)

◮

Then talked to the members of the jury passionately and eloquently will apply λT [λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)] to the semantics of the VP talked to the members of the jury, talked(mj). . .

◮

. . . and we wind up with

◮

λT [λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)](talked(mj)) = λx.passionately(talked(mj))(x) ⊓ λy.eloquently(talked(mj))(y) = λv[λx[passionately(talked(mj))(x)](v) ⊓ λy[eloquently(talked(mj))(y)](v)] = λv.passionately(talked(mj))(v) ∧ eloquently(talked(mj))(v)

◮

The VP talked to the members of the jury passionately and eloquently can now apply to John

◮

to give Robert Levine The Lambek Calculus 6 / 24

◮

passionatelye,t,e,t ⊓ eloquentlye,t,e,t = λRλx.passionately(R)(x) ⊓ λSλy.eloquently(S)(y) = λT.λR[λx.passionately(R)(x)](T ) ⊓ λS[λy.eloquently(S)(y)](T ) = λT λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)

◮

Then talked to the members of the jury passionately and eloquently will apply λT [λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)] to the semantics of the VP talked to the members of the jury, talked(mj). . .

◮

. . . and we wind up with

◮

λT [λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)](talked(mj)) = λx.passionately(talked(mj))(x) ⊓ λy.eloquently(talked(mj))(y) = λv[λx[passionately(talked(mj))(x)](v) ⊓ λy[eloquently(talked(mj))(y)](v)] = λv.passionately(talked(mj))(v) ∧ eloquently(talked(mj))(v)

◮

The VP talked to the members of the jury passionately and eloquently can now apply to John

◮

to give

◮

passionately(talked(mj))(j) ∧ eloquently(talked(mj))(j) Robert Levine The Lambek Calculus 6 / 24

◮

passionatelye,t,e,t ⊓ eloquentlye,t,e,t = λRλx.passionately(R)(x) ⊓ λSλy.eloquently(S)(y) = λT.λR[λx.passionately(R)(x)](T ) ⊓ λS[λy.eloquently(S)(y)](T ) = λT λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)

◮

Then talked to the members of the jury passionately and eloquently will apply λT [λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)] to the semantics of the VP talked to the members of the jury, talked(mj). . .

◮

. . . and we wind up with

◮

λT [λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)](talked(mj)) = λx.passionately(talked(mj))(x) ⊓ λy.eloquently(talked(mj))(y) = λv[λx[passionately(talked(mj))(x)](v) ⊓ λy[eloquently(talked(mj))(y)](v)] = λv.passionately(talked(mj))(v) ∧ eloquently(talked(mj))(v)

◮

The VP talked to the members of the jury passionately and eloquently can now apply to John

◮

to give

◮

passionately(talked(mj))(j) ∧ eloquently(talked(mj))(j)

◮

How about auxiliaries? Robert Levine The Lambek Calculus 6 / 24

◮

passionatelye,t,e,t ⊓ eloquentlye,t,e,t = λRλx.passionately(R)(x) ⊓ λSλy.eloquently(S)(y) = λT.λR[λx.passionately(R)(x)](T ) ⊓ λS[λy.eloquently(S)(y)](T ) = λT λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)

◮

Then talked to the members of the jury passionately and eloquently will apply λT [λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)] to the semantics of the VP talked to the members of the jury, talked(mj). . .

◮

. . . and we wind up with

◮

λT [λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)](talked(mj)) = λx.passionately(talked(mj))(x) ⊓ λy.eloquently(talked(mj))(y) = λv[λx[passionately(talked(mj))(x)](v) ⊓ λy[eloquently(talked(mj))(y)](v)] = λv.passionately(talked(mj))(v) ∧ eloquently(talked(mj))(v)

◮

The VP talked to the members of the jury passionately and eloquently can now apply to John

◮

to give

◮

passionately(talked(mj))(j) ∧ eloquently(talked(mj))(j)

◮

How about auxiliaries? Robert Levine The Lambek Calculus 6 / 24

◮

passionatelye,t,e,t ⊓ eloquentlye,t,e,t = λRλx.passionately(R)(x) ⊓ λSλy.eloquently(S)(y) = λT.λR[λx.passionately(R)(x)](T ) ⊓ λS[λy.eloquently(S)(y)](T ) = λT λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)

◮

Then talked to the members of the jury passionately and eloquently will apply λT [λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)] to the semantics of the VP talked to the members of the jury, talked(mj). . .

◮

. . . and we wind up with

◮

λT [λx.passionately(T )(x) ⊓ λy.eloquently(T )(y)](talked(mj)) = λx.passionately(talked(mj))(x) ⊓ λy.eloquently(talked(mj))(y) = λv[λx[passionately(talked(mj))(x)](v) ⊓ λy[eloquently(talked(mj))(y)](v)] = λv.passionately(talked(mj))(v) ∧ eloquently(talked(mj))(v)

◮

The VP talked to the members of the jury passionately and eloquently can now apply to John

◮

to give

◮

passionately(talked(mj))(j) ∧ eloquently(talked(mj))(j)

◮

How about auxiliaries? (4) Mary can and should apply for the job. Robert Levine The Lambek Calculus 6 / 24

Syntax and semantics of coordination

(5) and; λPλQ.Q ⊓ P; (X\X)/X

john; j; NP walked; talk; VP[fin] and; λPλQ.Q ⊓ P; (X\X)/X talked; talk; VP[fin]

/E

and • talked; λQ.Q ⊓ talk; VP

fin\VP fin /E

walked • and • talked; walk ⊓ talk; VP

fin

............................................. walked • and • talked; λy.walk(y) ⊓ talk(y); VP

fin

\E

john • walked • and • talked; λy[walk(y) ∧ talk(y)](j); Sfin ................................................................. john • walked • and • talked; walk(r) ∧ talk(j); Sfin

Robert Levine The Lambek Calculus 7 / 24

But wait, there’s more: nonconstituent coordination

Robert Levine The Lambek Calculus 8 / 24

But wait, there’s more: nonconstituent coordination

Right node raising

Robert Levine The Lambek Calculus 8 / 24

But wait, there’s more: nonconstituent coordination

Right node raising (6) [John gave the pair of pliers], and [Mary offered the hammer drill], to Anne.

Robert Levine The Lambek Calculus 8 / 24

But wait, there’s more: nonconstituent coordination

Right node raising (6) [John gave the pair of pliers], and [Mary offered the hammer drill], to Anne. Dependent cluster coordination

Robert Levine The Lambek Calculus 8 / 24

But wait, there’s more: nonconstituent coordination

Right node raising (6) [John gave the pair of pliers], and [Mary offered the hammer drill], to Anne. Dependent cluster coordination (7) Bill gave [Ann the book] and [Sue the pair of pliers].

Robert Levine The Lambek Calculus 8 / 24

But wait, there’s more: nonconstituent coordination

Right node raising (6) [John gave the pair of pliers], and [Mary offered the hammer drill], to Anne. Dependent cluster coordination (7) Bill gave [Ann the book] and [Sue the pair of pliers].

◮ In RNR, what is coordinated?

Robert Levine The Lambek Calculus 8 / 24

But wait, there’s more: nonconstituent coordination

Right node raising (6) [John gave the pair of pliers], and [Mary offered the hammer drill], to Anne. Dependent cluster coordination (7) Bill gave [Ann the book] and [Sue the pair of pliers].

◮ In RNR, what is coordinated? ◮ We need John gave the pair of pliers to combine semantically

with to Anne to yield something with the same meaning as John gave the pair of pliers to Anne.

Robert Levine The Lambek Calculus 8 / 24

But wait, there’s more: nonconstituent coordination

Right node raising (6) [John gave the pair of pliers], and [Mary offered the hammer drill], to Anne. Dependent cluster coordination (7) Bill gave [Ann the book] and [Sue the pair of pliers].

◮ In RNR, what is coordinated? ◮ We need John gave the pair of pliers to combine semantically

with to Anne to yield something with the same meaning as John gave the pair of pliers to Anne.

◮ to Anne is an PP Robert Levine The Lambek Calculus 8 / 24

But wait, there’s more: nonconstituent coordination

Right node raising (6) [John gave the pair of pliers], and [Mary offered the hammer drill], to Anne. Dependent cluster coordination (7) Bill gave [Ann the book] and [Sue the pair of pliers].

◮ In RNR, what is coordinated? ◮ We need John gave the pair of pliers to combine semantically

with to Anne to yield something with the same meaning as John gave the pair of pliers to Anne.

◮ to Anne is an PP ◮ Bill gave the pair of pliers to Anne is an S. Robert Levine The Lambek Calculus 8 / 24

But wait, there’s more: nonconstituent coordination

Right node raising (6) [John gave the pair of pliers], and [Mary offered the hammer drill], to Anne. Dependent cluster coordination (7) Bill gave [Ann the book] and [Sue the pair of pliers].

◮ In RNR, what is coordinated? ◮ We need John gave the pair of pliers to combine semantically

with to Anne to yield something with the same meaning as John gave the pair of pliers to Anne.

◮ to Anne is an PP ◮ Bill gave the pair of pliers to Anne is an S. ◮ So what is the category we have to posit for Bill gave the pair

• f pliers?

Robert Levine The Lambek Calculus 8 / 24

◮ Examples such as (8) show that the ‘raised’ material must in

some sense count as a constituent of both conjuncts:

Robert Levine The Lambek Calculus 9 / 24

◮ Examples such as (8) show that the ‘raised’ material must in

some sense count as a constituent of both conjuncts:

Robert Levine The Lambek Calculus 9 / 24

◮ Examples such as (8) show that the ‘raised’ material must in

some sense count as a constituent of both conjuncts:

Robert Levine The Lambek Calculus 9 / 24

◮ Examples such as (8) show that the ‘raised’ material must in

some sense count as a constituent of both conjuncts: (8)

• a. I put the teapot, and John carefully placed the

vase, on the shelf.

Robert Levine The Lambek Calculus 9 / 24

◮ Examples such as (8) show that the ‘raised’ material must in

some sense count as a constituent of both conjuncts: (8)

• a. I put the teapot, and John carefully placed the

vase, on the shelf.

Robert Levine The Lambek Calculus 9 / 24

◮ Examples such as (8) show that the ‘raised’ material must in

some sense count as a constituent of both conjuncts: (8)

• a. I put the teapot, and John carefully placed the

vase, on the shelf.

• b. John carefully placed the vase, and I put the

teapot, on the shelf.

Robert Levine The Lambek Calculus 9 / 24

◮ Examples such as (8) show that the ‘raised’ material must in

some sense count as a constituent of both conjuncts: (8)

• a. I put the teapot, and John carefully placed the

vase, on the shelf.

• b. John carefully placed the vase, and I put the

teapot, on the shelf.

Robert Levine The Lambek Calculus 9 / 24

◮ Examples such as (8) show that the ‘raised’ material must in

some sense count as a constituent of both conjuncts: (8)

• a. I put the teapot, and John carefully placed the

vase, on the shelf.

• b. John carefully placed the vase, and I put the

teapot, on the shelf.

◮ What do we know about put?

Robert Levine The Lambek Calculus 9 / 24

(9)

• a. The Nobel Committee awarded–and the King of Sweden

presented—the Physics prize to Peter Higgs.

• b. John wanted to wager—and Mary insisted on betting–a

ton of money with the bookies on the outcome of the game.

Robert Levine The Lambek Calculus 10 / 24

(9)

• a. The Nobel Committee awarded–and the King of Sweden

presented—the Physics prize to Peter Higgs.

• b. John wanted to wager—and Mary insisted on betting–a

ton of money with the bookies on the outcome of the game.

◮ In (9a), we have a RNRed string corresponding to an NP and

a PP; in b, to a sequence NP PP

with PP

• n.

Robert Levine The Lambek Calculus 10 / 24

(9)

• a. The Nobel Committee awarded–and the King of Sweden

presented—the Physics prize to Peter Higgs.

• b. John wanted to wager—and Mary insisted on betting–a

ton of money with the bookies on the outcome of the game.

◮ In (9a), we have a RNRed string corresponding to an NP and

a PP; in b, to a sequence NP PP

with PP

• n.

◮ How are we going to get all of these examples to go through?

Robert Levine The Lambek Calculus 10 / 24

What needs doing?

◮ We continue to assume that coordination applies to constituents. . .

Robert Levine The Lambek Calculus 11 / 24

What needs doing?

◮ We continue to assume that coordination applies to constituents. . . ◮ by which, as always, we mean a string of words to which a syntactic type

is assigned

Robert Levine The Lambek Calculus 11 / 24

What needs doing?

◮ We continue to assume that coordination applies to constituents. . . ◮ by which, as always, we mean a string of words to which a syntactic type

is assigned

◮ either via the lexicon Robert Levine The Lambek Calculus 11 / 24

What needs doing?

◮ We continue to assume that coordination applies to constituents. . . ◮ by which, as always, we mean a string of words to which a syntactic type

is assigned

◮ either via the lexicon ◮ or by our proof theory. Robert Levine The Lambek Calculus 11 / 24

What needs doing?

◮ We continue to assume that coordination applies to constituents. . . ◮ by which, as always, we mean a string of words to which a syntactic type

is assigned

◮ either via the lexicon ◮ or by our proof theory.

◮ Consider again (10):

Robert Levine The Lambek Calculus 11 / 24

What needs doing?

◮ We continue to assume that coordination applies to constituents. . . ◮ by which, as always, we mean a string of words to which a syntactic type

is assigned

◮ either via the lexicon ◮ or by our proof theory.

◮ Consider again (10):

Robert Levine The Lambek Calculus 11 / 24

What needs doing?

◮ We continue to assume that coordination applies to constituents. . . ◮ by which, as always, we mean a string of words to which a syntactic type

is assigned

◮ either via the lexicon ◮ or by our proof theory.

◮ Consider again (10):

(10) [John gave the pair of pliers], and [Mary offered the hammer drill], to Anne.

Robert Levine The Lambek Calculus 11 / 24

What needs doing?

◮ We continue to assume that coordination applies to constituents. . . ◮ by which, as always, we mean a string of words to which a syntactic type

is assigned

◮ either via the lexicon ◮ or by our proof theory.

◮ Consider again (10):

(10) [John gave the pair of pliers], and [Mary offered the hammer drill], to Anne.

◮ We are conjoining John gave the pair of pliers and Mary offered the

hammar drill. Let’s examine a bit more closely what the problem with these is.

Robert Levine The Lambek Calculus 11 / 24

What needs doing?

◮ We continue to assume that coordination applies to constituents. . . ◮ by which, as always, we mean a string of words to which a syntactic type

is assigned

◮ either via the lexicon ◮ or by our proof theory.

◮ Consider again (10):

(10) [John gave the pair of pliers], and [Mary offered the hammer drill], to Anne.

◮ We are conjoining John gave the pair of pliers and Mary offered the

hammar drill. Let’s examine a bit more closely what the problem with these is.

Robert Levine The Lambek Calculus 11 / 24

What needs doing?

◮ We continue to assume that coordination applies to constituents. . . ◮ by which, as always, we mean a string of words to which a syntactic type

is assigned

◮ either via the lexicon ◮ or by our proof theory.

◮ Consider again (10):

(10) [John gave the pair of pliers], and [Mary offered the hammer drill], to Anne.

◮ We are conjoining John gave the pair of pliers and Mary offered the

hammar drill. Let’s examine a bit more closely what the problem with these is. gave; give; VP/PP/NP the • pair • of • pliers; pl; NP gave • the • pair • of • pliers;

• ffer(pl); VP/PP

john; j; NP FAIL

Robert Levine The Lambek Calculus 11 / 24

◮ There’s no way forward here.

Robert Levine The Lambek Calculus 12 / 24

◮ There’s no way forward here. ◮ But there HAS to be, no?

Robert Levine The Lambek Calculus 12 / 24

◮ There’s no way forward here. ◮ But there HAS to be, no? ◮ What we want isn’t all that unreasonable:

Robert Levine The Lambek Calculus 12 / 24

◮ There’s no way forward here. ◮ But there HAS to be, no? ◮ What we want isn’t all that unreasonable:

◮ We know that John gave the pair of pliers is really a sentence lacking a

PP on the right, i.e.

Robert Levine The Lambek Calculus 12 / 24

◮ There’s no way forward here. ◮ But there HAS to be, no? ◮ What we want isn’t all that unreasonable:

◮ We know that John gave the pair of pliers is really a sentence lacking a

PP on the right, i.e.

◮ a sign of type S/PP, Robert Levine The Lambek Calculus 12 / 24

◮ There’s no way forward here. ◮ But there HAS to be, no? ◮ What we want isn’t all that unreasonable:

◮ We know that John gave the pair of pliers is really a sentence lacking a

PP on the right, i.e.

◮ a sign of type S/PP, ◮ and we’ve already got a VP/PP, Robert Levine The Lambek Calculus 12 / 24

◮ There’s no way forward here. ◮ But there HAS to be, no? ◮ What we want isn’t all that unreasonable:

◮ We know that John gave the pair of pliers is really a sentence lacking a

PP on the right, i.e.

◮ a sign of type S/PP, ◮ and we’ve already got a VP/PP, ◮ so if we could just temporarily forget about the PP and let the VP

combine with John,

Robert Levine The Lambek Calculus 12 / 24

◮ There’s no way forward here. ◮ But there HAS to be, no? ◮ What we want isn’t all that unreasonable:

◮ We know that John gave the pair of pliers is really a sentence lacking a

PP on the right, i.e.

◮ a sign of type S/PP, ◮ and we’ve already got a VP/PP, ◮ so if we could just temporarily forget about the PP and let the VP

combine with John,

◮ and then ‘remember’ the missing PP that we briefly ignored, Robert Levine The Lambek Calculus 12 / 24

◮ There’s no way forward here. ◮ But there HAS to be, no? ◮ What we want isn’t all that unreasonable:

◮ We know that John gave the pair of pliers is really a sentence lacking a

PP on the right, i.e.

◮ a sign of type S/PP, ◮ and we’ve already got a VP/PP, ◮ so if we could just temporarily forget about the PP and let the VP

combine with John,

◮ and then ‘remember’ the missing PP that we briefly ignored, ◮ we’d have S/PP just as we want. Robert Levine The Lambek Calculus 12 / 24

◮ There’s no way forward here. ◮ But there HAS to be, no? ◮ What we want isn’t all that unreasonable:

◮ We know that John gave the pair of pliers is really a sentence lacking a

PP on the right, i.e.

◮ a sign of type S/PP, ◮ and we’ve already got a VP/PP, ◮ so if we could just temporarily forget about the PP and let the VP

combine with John,

◮ and then ‘remember’ the missing PP that we briefly ignored, ◮ we’d have S/PP just as we want. ◮ You can think of it as taking out a ‘bridge loan’ to put together the type

we need,

Robert Levine The Lambek Calculus 12 / 24

◮ There’s no way forward here. ◮ But there HAS to be, no? ◮ What we want isn’t all that unreasonable:

◮ We know that John gave the pair of pliers is really a sentence lacking a

PP on the right, i.e.

◮ a sign of type S/PP, ◮ and we’ve already got a VP/PP, ◮ so if we could just temporarily forget about the PP and let the VP

combine with John,

◮ and then ‘remember’ the missing PP that we briefly ignored, ◮ we’d have S/PP just as we want. ◮ You can think of it as taking out a ‘bridge loan’ to put together the type

we need,

◮ and then paying it back. Robert Levine The Lambek Calculus 12 / 24

◮ There’s no way forward here. ◮ But there HAS to be, no? ◮ What we want isn’t all that unreasonable:

◮ We know that John gave the pair of pliers is really a sentence lacking a

PP on the right, i.e.

◮ a sign of type S/PP, ◮ and we’ve already got a VP/PP, ◮ so if we could just temporarily forget about the PP and let the VP

combine with John,

◮ and then ‘remember’ the missing PP that we briefly ignored, ◮ we’d have S/PP just as we want. ◮ You can think of it as taking out a ‘bridge loan’ to put together the type

we need,

◮ and then paying it back.

◮ But we can’t do that,

Robert Levine The Lambek Calculus 12 / 24

◮ There’s no way forward here. ◮ But there HAS to be, no? ◮ What we want isn’t all that unreasonable:

◮ We know that John gave the pair of pliers is really a sentence lacking a

PP on the right, i.e.

◮ a sign of type S/PP, ◮ and we’ve already got a VP/PP, ◮ so if we could just temporarily forget about the PP and let the VP

combine with John,

◮ and then ‘remember’ the missing PP that we briefly ignored, ◮ we’d have S/PP just as we want. ◮ You can think of it as taking out a ‘bridge loan’ to put together the type

we need,

◮ and then paying it back.

◮ But we can’t do that, ◮ because our proof theory gives us no way to do the

ignoring-and-remembering that we need to do.

Robert Levine The Lambek Calculus 12 / 24

◮ There’s no way forward here. ◮ But there HAS to be, no? ◮ What we want isn’t all that unreasonable:

◮ We know that John gave the pair of pliers is really a sentence lacking a

PP on the right, i.e.

◮ a sign of type S/PP, ◮ and we’ve already got a VP/PP, ◮ so if we could just temporarily forget about the PP and let the VP

combine with John,

◮ and then ‘remember’ the missing PP that we briefly ignored, ◮ we’d have S/PP just as we want. ◮ You can think of it as taking out a ‘bridge loan’ to put together the type

we need,

◮ and then paying it back.

◮ But we can’t do that, ◮ because our proof theory gives us no way to do the

ignoring-and-remembering that we need to do.

◮ But it SHOULD, if standard logics are any guide.

Robert Levine The Lambek Calculus 12 / 24

Implication introduction in logic.

◮ Let’s look at classical logic again for a clue as to why:

Robert Levine The Lambek Calculus 13 / 24

Implication introduction in logic.

◮ Let’s look at classical logic again for a clue as to why:

Robert Levine The Lambek Calculus 13 / 24

Implication introduction in logic.

◮ Let’s look at classical logic again for a clue as to why:

(11) φ ⊃ ψ φ

⊃ Elim

ψ

Robert Levine The Lambek Calculus 13 / 24

Implication introduction in logic.

◮ Let’s look at classical logic again for a clue as to why:

(11) φ ⊃ ψ φ

⊃ Elim

ψ

◮ We’ve talked about this rule in connection with the /, \

connectives at the very beginning.

Robert Levine The Lambek Calculus 13 / 24

Implication introduction in logic.

◮ Let’s look at classical logic again for a clue as to why:

(11) φ ⊃ ψ φ

⊃ Elim

ψ

◮ We’ve talked about this rule in connection with the /, \

connectives at the very beginning.

◮ But in logic, rules for connectives come in pairs.

Robert Levine The Lambek Calculus 13 / 24

Implication introduction in logic.

◮ Let’s look at classical logic again for a clue as to why:

(11) φ ⊃ ψ φ

⊃ Elim

ψ

◮ We’ve talked about this rule in connection with the /, \

connectives at the very beginning.

◮ But in logic, rules for connectives come in pairs. ◮ The ⊃ Elimination rule tells us that

Robert Levine The Lambek Calculus 13 / 24

Implication introduction in logic.

◮ Let’s look at classical logic again for a clue as to why:

(11) φ ⊃ ψ φ

⊃ Elim

ψ

◮ We’ve talked about this rule in connection with the /, \

connectives at the very beginning.

◮ But in logic, rules for connectives come in pairs. ◮ The ⊃ Elimination rule tells us that

◮ if in some logical context the antecedent of an implication is

available,

Robert Levine The Lambek Calculus 13 / 24

Implication introduction in logic.

◮ Let’s look at classical logic again for a clue as to why:

(11) φ ⊃ ψ φ

⊃ Elim

ψ

◮ We’ve talked about this rule in connection with the /, \

connectives at the very beginning.

◮ But in logic, rules for connectives come in pairs. ◮ The ⊃ Elimination rule tells us that

◮ if in some logical context the antecedent of an implication is

available,

◮ then the consequent of that implication can be legally inferred. Robert Levine The Lambek Calculus 13 / 24

Implication introduction in logic.

◮ Let’s look at classical logic again for a clue as to why:

(11) φ ⊃ ψ φ

⊃ Elim

ψ

◮ We’ve talked about this rule in connection with the /, \

connectives at the very beginning.

◮ But in logic, rules for connectives come in pairs. ◮ The ⊃ Elimination rule tells us that

◮ if in some logical context the antecedent of an implication is

available,

◮ then the consequent of that implication can be legally inferred.

◮ But the ‘flip side’ of this rule is that,

Robert Levine The Lambek Calculus 13 / 24

Implication introduction in logic.

◮ Let’s look at classical logic again for a clue as to why:

(11) φ ⊃ ψ φ

⊃ Elim

ψ

◮ We’ve talked about this rule in connection with the /, \

connectives at the very beginning.

◮ But in logic, rules for connectives come in pairs. ◮ The ⊃ Elimination rule tells us that

◮ if in some logical context the antecedent of an implication is

available,

◮ then the consequent of that implication can be legally inferred.

◮ But the ‘flip side’ of this rule is that,

◮ in a logical context where assuming a certain premise φ

guarantees the truth of some formula ψ,

Robert Levine The Lambek Calculus 13 / 24

Implication introduction in logic.

◮ Let’s look at classical logic again for a clue as to why:

(11) φ ⊃ ψ φ

⊃ Elim

ψ

◮ We’ve talked about this rule in connection with the /, \

connectives at the very beginning.

◮ But in logic, rules for connectives come in pairs. ◮ The ⊃ Elimination rule tells us that

◮ if in some logical context the antecedent of an implication is

available,

◮ then the consequent of that implication can be legally inferred.

◮ But the ‘flip side’ of this rule is that,

◮ in a logical context where assuming a certain premise φ

guarantees the truth of some formula ψ,

◮ you know that in that context, if φ were true, then ψ would be

true.

Robert Levine The Lambek Calculus 13 / 24

Implication rules of propositional logic φ ⊃ ψ φ

⊃ Elim

ψ φ . . . ψ

⊃ Intro

φ ⊃ ψ

Robert Levine The Lambek Calculus 14 / 24

Implication rules of propositional logic φ ⊃ ψ φ

⊃ Elim

ψ φ . . . ψ

⊃ Intro

φ ⊃ ψ

◮ We have some eggs. Let’s pretend that we have some ham.

Then we can prove that we can have ham and eggs.

Robert Levine The Lambek Calculus 14 / 24

Implication rules of propositional logic φ ⊃ ψ φ

⊃ Elim

ψ φ . . . ψ

⊃ Intro

φ ⊃ ψ

◮ We have some eggs. Let’s pretend that we have some ham.

Then we can prove that we can have ham and eggs.

◮ It follows that in the context where we have just some eggs,

we know that IF we actually had some ham, THEN we could have ham and eggs.

Robert Levine The Lambek Calculus 14 / 24

Hypothetical reasoning

◮ In the same way, given a sign of type VP/PP, pretend that we have some

sign of type PP; then, by / Elim (think ‘⊃ Elim’), we would have VP.

Robert Levine The Lambek Calculus 15 / 24

Hypothetical reasoning

◮ In the same way, given a sign of type VP/PP, pretend that we have some

sign of type PP; then, by / Elim (think ‘⊃ Elim’), we would have VP.

◮ If we combine this VP with an NP, we get an S.

Robert Levine The Lambek Calculus 15 / 24

Hypothetical reasoning

◮ In the same way, given a sign of type VP/PP, pretend that we have some

sign of type PP; then, by / Elim (think ‘⊃ Elim’), we would have VP.

◮ If we combine this VP with an NP, we get an S. ◮ But since we don’t actually have a PP, what we have to conclude is that

in the context where we have VP/PP and NP, we are allowed to deduce that if we did have a PP, we’d have an S,

Robert Levine The Lambek Calculus 15 / 24

Hypothetical reasoning

◮ In the same way, given a sign of type VP/PP, pretend that we have some

sign of type PP; then, by / Elim (think ‘⊃ Elim’), we would have VP.

◮ If we combine this VP with an NP, we get an S. ◮ But since we don’t actually have a PP, what we have to conclude is that

in the context where we have VP/PP and NP, we are allowed to deduce that if we did have a PP, we’d have an S,

◮ so that VP/PP and NP should be enough to allow us to deduce a sign of

type S/PP: something that would have been an S if we’d been able to supply it with a PP.

Robert Levine The Lambek Calculus 15 / 24

Hypothetical reasoning

◮ In the same way, given a sign of type VP/PP, pretend that we have some

sign of type PP; then, by / Elim (think ‘⊃ Elim’), we would have VP.

◮ If we combine this VP with an NP, we get an S. ◮ But since we don’t actually have a PP, what we have to conclude is that

in the context where we have VP/PP and NP, we are allowed to deduce that if we did have a PP, we’d have an S,

◮ so that VP/PP and NP should be enough to allow us to deduce a sign of

type S/PP: something that would have been an S if we’d been able to supply it with a PP.

◮ The rule licensing this reasoning on the basis of temporarily assumed

linguistic signs which are subsequently ‘withdrawn’, allowing us to infer an implication, is the ‘dual’ of slash elimination,

Robert Levine The Lambek Calculus 15 / 24

Hypothetical reasoning

◮ In the same way, given a sign of type VP/PP, pretend that we have some

sign of type PP; then, by / Elim (think ‘⊃ Elim’), we would have VP.

◮ If we combine this VP with an NP, we get an S. ◮ But since we don’t actually have a PP, what we have to conclude is that

in the context where we have VP/PP and NP, we are allowed to deduce that if we did have a PP, we’d have an S,

◮ so that VP/PP and NP should be enough to allow us to deduce a sign of

type S/PP: something that would have been an S if we’d been able to supply it with a PP.

◮ The rule licensing this reasoning on the basis of temporarily assumed

linguistic signs which are subsequently ‘withdrawn’, allowing us to infer an implication, is the ‘dual’ of slash elimination,

◮ just as implication introduction in logic is the dual of implication

elimination.

Robert Levine The Lambek Calculus 15 / 24

The Lambek Calculus

Forward Slash Introduction . . . . . . . . . . . . [ϕ; x; A]n . . . . . . . . . . . . . . . . . . b • ϕ; F ; B

/In

b; λx.F ; B/A Forward Slash Elimination a; F ; A/B b; G; B

/E

a • b; F (G); A Backward Slash Introduction . . . . . . . . . . . . [ϕ; x; A]n . . . . . . . . . . . . . . . . . . ϕ • b; F ; B

\In

b; λx.F ; A\B Backward Slash Elimination b; G; B a; F ; B\A

\E

b • a; F (G); A Robert Levine The Lambek Calculus 16 / 24

The Lambek Calculus

Forward Slash Introduction . . . . . . . . . . . . [ϕ; x; A]n . . . . . . . . . . . . . . . . . . b • ϕ; F ; B

/In

b; λx.F ; B/A Forward Slash Elimination a; F ; A/B b; G; B

/E

a • b; F (G); A Backward Slash Introduction . . . . . . . . . . . . [ϕ; x; A]n . . . . . . . . . . . . . . . . . . ϕ • b; F ; B

\In

b; λx.F ; A\B Backward Slash Elimination b; G; B a; F ; B\A

\E

b • a; F (G); A

◮ If a set of resources plus one additional assumption gives you a result,

then you can infer from that set of resources something which would give you that result if you restored that assumption.

Robert Levine The Lambek Calculus 16 / 24

. . . . . . . . . . . . [ϕ; x; A]n . . . . . . . . . . . . . . . . . . b • ϕ; F ; B

/In

b; λx.F ; B/A

Robert Levine The Lambek Calculus 17 / 24

. . . . . . . . . . . . [ϕ; x; A]n . . . . . . . . . . . . . . . . . . b • ϕ; F ; B

/In

b; λx.F ; B/A ◮ If some arbitrary sign with phonology ϕ (whatever it is), semantics x (whatever that is) and type A lets you deduce a sign with ϕ on its right edge, a meaning F and a type B,

Robert Levine The Lambek Calculus 17 / 24

. . . . . . . . . . . . [ϕ; x; A]n . . . . . . . . . . . . . . . . . . b • ϕ; F ; B

/In

b; λx.F ; B/A ◮ If some arbitrary sign with phonology ϕ (whatever it is), semantics x (whatever that is) and type A lets you deduce a sign with ϕ on its right edge, a meaning F and a type B, ◮ then you know that what you have WITHOUT that arbitrary sign is something

Robert Levine The Lambek Calculus 17 / 24

. . . . . . . . . . . . [ϕ; x; A]n . . . . . . . . . . . . . . . . . . b • ϕ; F ; B

/In

b; λx.F ; B/A ◮ If some arbitrary sign with phonology ϕ (whatever it is), semantics x (whatever that is) and type A lets you deduce a sign with ϕ on its right edge, a meaning F and a type B, ◮ then you know that what you have WITHOUT that arbitrary sign is something

◮ whose phonology would be B’s except that A’s prososdy is missing (so, in

effect, (b+ϕ)−ϕ = b);

Robert Levine The Lambek Calculus 17 / 24

. . . . . . . . . . . . [ϕ; x; A]n . . . . . . . . . . . . . . . . . . b • ϕ; F ; B

/In

b; λx.F ; B/A ◮ If some arbitrary sign with phonology ϕ (whatever it is), semantics x (whatever that is) and type A lets you deduce a sign with ϕ on its right edge, a meaning F and a type B, ◮ then you know that what you have WITHOUT that arbitrary sign is something

◮ whose phonology would be B’s except that A’s prososdy is missing (so, in

effect, (b+ϕ)−ϕ = b); ◮ whose semantics is something that would be

Robert Levine The Lambek Calculus 17 / 24

. . . . . . . . . . . . [ϕ; x; A]n . . . . . . . . . . . . . . . . . . b • ϕ; F ; B

/In

b; λx.F ; B/A ◮ If some arbitrary sign with phonology ϕ (whatever it is), semantics x (whatever that is) and type A lets you deduce a sign with ϕ on its right edge, a meaning F and a type B, ◮ then you know that what you have WITHOUT that arbitrary sign is something

◮ whose phonology would be B’s except that A’s prososdy is missing (so, in

effect, (b+ϕ)−ϕ = b); ◮ whose semantics is something that would be ◮ F IF it HAD A’s semantics to ‘get its hands on’

Robert Levine The Lambek Calculus 17 / 24

. . . . . . . . . . . . [ϕ; x; A]n . . . . . . . . . . . . . . . . . . b • ϕ; F ; B

/In

b; λx.F ; B/A ◮ If some arbitrary sign with phonology ϕ (whatever it is), semantics x (whatever that is) and type A lets you deduce a sign with ϕ on its right edge, a meaning F and a type B, ◮ then you know that what you have WITHOUT that arbitrary sign is something

◮ whose phonology would be B’s except that A’s prososdy is missing (so, in

effect, (b+ϕ)−ϕ = b); ◮ whose semantics is something that would be ◮ F IF it HAD A’s semantics to ‘get its hands on’ ◮ (i.e., it would be a function which takes something of the type assigned to x and returns F , aka λx.F )

Robert Levine The Lambek Calculus 17 / 24

. . . . . . . . . . . . [ϕ; x; A]n . . . . . . . . . . . . . . . . . . b • ϕ; F ; B

/In

b; λx.F ; B/A ◮ If some arbitrary sign with phonology ϕ (whatever it is), semantics x (whatever that is) and type A lets you deduce a sign with ϕ on its right edge, a meaning F and a type B, ◮ then you know that what you have WITHOUT that arbitrary sign is something

◮ whose phonology would be B’s except that A’s prososdy is missing (so, in

effect, (b+ϕ)−ϕ = b); ◮ whose semantics is something that would be ◮ F IF it HAD A’s semantics to ‘get its hands on’ ◮ (i.e., it would be a function which takes something of the type assigned to x and returns F , aka λx.F ) ◮ and whose type would be B if it could actually combine with something of type A (that is, B/A).

Robert Levine The Lambek Calculus 17 / 24

. . . . . . . . . . . . [ϕ; x; A]n . . . . . . . . . . . . . . . . . . b • ϕ; F ; B

/In

b; λx.F ; B/A ◮ If some arbitrary sign with phonology ϕ (whatever it is), semantics x (whatever that is) and type A lets you deduce a sign with ϕ on its right edge, a meaning F and a type B, ◮ then you know that what you have WITHOUT that arbitrary sign is something

◮ whose phonology would be B’s except that A’s prososdy is missing (so, in

effect, (b+ϕ)−ϕ = b); ◮ whose semantics is something that would be ◮ F IF it HAD A’s semantics to ‘get its hands on’ ◮ (i.e., it would be a function which takes something of the type assigned to x and returns F , aka λx.F ) ◮ and whose type would be B if it could actually combine with something of type A (that is, B/A). ◮ and likewise for \ Intro.

Robert Levine The Lambek Calculus 17 / 24

Deriving RNR

gave; give; VP/PP/NP the • pair • of • pliers; pl; NP

/E

gave • the • pair • of • pliers; gave(pl); VP/PP [ϕ1; u; PP]1

/E

gave • the • pair • of • pliers • ϕ1; gave(pl)(u); VP john; j; NP

\E

john • gave • the • pair • of • pliers • ϕ1; gave(pl)(u)(j); S

Robert Levine The Lambek Calculus 18 / 24

Deriving RNR

gave; give; VP/PP/NP the • pair • of • pliers; pl; NP

/E

gave • the • pair • of • pliers; gave(pl); VP/PP [ϕ1; u; PP]1

/E

gave • the • pair • of • pliers • ϕ1; gave(pl)(u); VP john; j; NP

\E

john • gave • the • pair • of • pliers • ϕ1; gave(pl)(u)(j); S

/I1

john • gave • the • pair • of • pliers; λu.gave(pl)(u)(j); S/PP

Robert Levine The Lambek Calculus 18 / 24

Deriving RNR

gave; give; VP/PP/NP the • pair • of • pliers; pl; NP

/E

gave • the • pair • of • pliers; gave(pl); VP/PP [ϕ1; u; PP]1

/E

gave • the • pair • of • pliers • ϕ1; gave(pl)(u); VP john; j; NP

\E

john • gave • the • pair • of • pliers • ϕ1; gave(pl)(u)(j); S

/I1

john • gave • the • pair • of • pliers; λu.gave(pl)(u)(j); S/PP

◮ Now what??

Robert Levine The Lambek Calculus 18 / 24

RNR via hypothetical proof

and; ⊓; (X\X)/X . . . . . . mary • offered • the • hammer • drill; λw.offer(hd)(w)(m); S/PP and • mary • offered • the • hammer • drill; ⊓ offer(hd)(w)(m); (S/PP)\(S/PP) john • gave • the • pair • of • pliers; λu.gave(pl)(u)(j); S/PP john • gave • the • pair • of • pliers • and • mary • offered • the • hammer • drill; λu.gave(pl)(u)(j) ⊓ λw.offer(hd)(w)(m); S/PP

Robert Levine The Lambek Calculus 19 / 24

RNR via hypothetical proof

and; ⊓; (X\X)/X . . . . . . mary • offered • the • hammer • drill; λw.offer(hd)(w)(m); S/PP and • mary • offered • the • hammer • drill; ⊓ offer(hd)(w)(m); (S/PP)\(S/PP) john • gave • the • pair • of • pliers; λu.gave(pl)(u)(j); S/PP john • gave • the • pair • of • pliers • and • mary • offered • the • hammer • drill; λu.gave(pl)(u)(j) ⊓ λw.offer(hd)(w)(m); S/PP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . john • gave • the • pair • of • pliers • and • mary • offered • the • hammer • drill; λx.gave(pl)(x)(j) ∧ offer(hd)(x)(m); S/PP

Robert Levine The Lambek Calculus 19 / 24

RNR via hypothetical proof

and; ⊓; (X\X)/X . . . . . . mary • offered • the • hammer • drill; λw.offer(hd)(w)(m); S/PP and • mary • offered • the • hammer • drill; ⊓ offer(hd)(w)(m); (S/PP)\(S/PP) john • gave • the • pair • of • pliers; λu.gave(pl)(u)(j); S/PP john • gave • the • pair • of • pliers • and • mary • offered • the • hammer • drill; λu.gave(pl)(u)(j) ⊓ λw.offer(hd)(w)(m); S/PP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . john • gave • the • pair • of • pliers • and • mary • offered • the • hammer • drill; λx.gave(pl)(x)(j) ∧ offer(hd)(x)(m); S/PP to • anne; a; PP john • gave • the • pair • of • pliers • and • mary • offered • the • hammer • drill • to • anne; gave(pl)(a)(j) ∧ offer(hd)(a)(m); S

Robert Levine The Lambek Calculus 19 / 24

◮ The critical proof step is

john • gave • the • pair • of • pliers • ϕ1; gave(pl)(u)(j); S

/I1

john • gave • the • pair • of • pliers; λu.gave(pl)(u)(j); S/PP

Robert Levine The Lambek Calculus 20 / 24

◮ The critical proof step is

john • gave • the • pair • of • pliers • ϕ1; gave(pl)(u)(j); S

/I1

john • gave • the • pair • of • pliers; λu.gave(pl)(u)(j); S/PP Prosody: john • gave • the • pair • of • pliers

• b
• ϕ1

Robert Levine The Lambek Calculus 20 / 24

◮ The critical proof step is

john • gave • the • pair • of • pliers • ϕ1; gave(pl)(u)(j); S

/I1

john • gave • the • pair • of • pliers; λu.gave(pl)(u)(j); S/PP Prosody: john • gave • the • pair • of • pliers

• b
• ϕ1

Robert Levine The Lambek Calculus 20 / 24

◮ The critical proof step is

john • gave • the • pair • of • pliers • ϕ1; gave(pl)(u)(j); S

/I1

john • gave • the • pair • of • pliers; λu.gave(pl)(u)(j); S/PP Semantics: gave(pl)(u)(j)

• F

Robert Levine The Lambek Calculus 20 / 24

◮ The critical proof step is

john • gave • the • pair • of • pliers • ϕ1; gave(pl)(u)(j); S

/I1

john • gave • the • pair • of • pliers; λu.gave(pl)(u)(j); S/PP Semantics: λu. gave(pl)(u)(j)

• F

Robert Levine The Lambek Calculus 20 / 24

◮ The critical proof step is

john • gave • the • pair • of • pliers • ϕ1; gave(pl)(u)(j); S

/I1

john • gave • the • pair • of • pliers; λu.gave(pl)(u)(j)); S/PP Syntax: S

• B

Robert Levine The Lambek Calculus 20 / 24

◮ The critical proof step is

john • gave • the • pair • of • pliers • ϕ1; gave(pl)(u)(j); S

/I1

john • gave • the • pair • of • pliers; λu.gave(pl)(u)(j); S/PP Syntax: S/PP

B/A

Robert Levine The Lambek Calculus 20 / 24

More examples: dependent cluster coordination (DCC)

Robert Levine The Lambek Calculus 21 / 24

More examples: dependent cluster coordination (DCC)

(12) Bill gave [Anne the book] and [Sue the pair of pliers].

Robert Levine The Lambek Calculus 21 / 24

More examples: dependent cluster coordination (DCC)

(12) Bill gave [Anne the book] and [Sue the pair of pliers].

◮ What is coordinated?

Robert Levine The Lambek Calculus 21 / 24

More examples: dependent cluster coordination (DCC)

(12) Bill gave [Anne the book] and [Sue the pair of pliers].

◮ What is coordinated? ◮ How can we do that?

Robert Levine The Lambek Calculus 21 / 24

More examples: dependent cluster coordination (DCC)

(12) Bill gave [Anne the book] and [Sue the pair of pliers].

◮ What is coordinated? ◮ How can we do that? ◮ Reason it out:

Robert Levine The Lambek Calculus 21 / 24

More examples: dependent cluster coordination (DCC)

(12) Bill gave [Anne the book] and [Sue the pair of pliers].

◮ What is coordinated? ◮ How can we do that? ◮ Reason it out:

◮ Anne the book can’t be a constituent by virtue of Anne

combining with the book in a functor/argument relationship.

Robert Levine The Lambek Calculus 21 / 24

More examples: dependent cluster coordination (DCC)

(12) Bill gave [Anne the book] and [Sue the pair of pliers].

◮ What is coordinated? ◮ How can we do that? ◮ Reason it out:

◮ Anne the book can’t be a constituent by virtue of Anne

combining with the book in a functor/argument relationship.

◮ But it must be a constituent somehow. Robert Levine The Lambek Calculus 21 / 24

More examples: dependent cluster coordination (DCC)

(12) Bill gave [Anne the book] and [Sue the pair of pliers].

◮ What is coordinated? ◮ How can we do that? ◮ Reason it out:

◮ Anne the book can’t be a constituent by virtue of Anne

combining with the book in a functor/argument relationship.

◮ But it must be a constituent somehow. ◮ What other way can it get to be a constituent? Robert Levine The Lambek Calculus 21 / 24

More examples: dependent cluster coordination (DCC)

(12) Bill gave [Anne the book] and [Sue the pair of pliers].

◮ What is coordinated? ◮ How can we do that? ◮ Reason it out:

◮ Anne the book can’t be a constituent by virtue of Anne

combining with the book in a functor/argument relationship.

◮ But it must be a constituent somehow. ◮ What other way can it get to be a constituent? ◮ HINT: how did we get assign a syntactic type each of the

coordinated sequences in RNR??

Robert Levine The Lambek Calculus 21 / 24

More examples: dependent cluster coordination (DCC)

(12) Bill gave [Anne the book] and [Sue the pair of pliers].

◮ What is coordinated? ◮ How can we do that? ◮ Reason it out:

◮ Anne the book can’t be a constituent by virtue of Anne

combining with the book in a functor/argument relationship.

◮ But it must be a constituent somehow. ◮ What other way can it get to be a constituent? ◮ HINT: how did we get assign a syntactic type each of the

coordinated sequences in RNR??

◮ In the case of RNR, we were able to prove that each of the

conjuncts was S/PP: an S missing a PP.

Robert Levine The Lambek Calculus 21 / 24

More examples: dependent cluster coordination (DCC)

(12) Bill gave [Anne the book] and [Sue the pair of pliers].

◮ What is coordinated? ◮ How can we do that? ◮ Reason it out:

◮ Anne the book can’t be a constituent by virtue of Anne

combining with the book in a functor/argument relationship.

◮ But it must be a constituent somehow. ◮ What other way can it get to be a constituent? ◮ HINT: how did we get assign a syntactic type each of the

coordinated sequences in RNR??

◮ In the case of RNR, we were able to prove that each of the

conjuncts was S/PP: an S missing a PP.

◮ Can we do something similar in the case of (12)?

Robert Levine The Lambek Calculus 21 / 24

Proofs for DCC

◮ The task here is twofold:

Robert Levine The Lambek Calculus 22 / 24

Proofs for DCC

◮ The task here is twofold:

◮ first, DECIDE what category Anne the book should be, and Robert Levine The Lambek Calculus 22 / 24

Proofs for DCC

◮ The task here is twofold:

◮ first, DECIDE what category Anne the book should be, and ◮ second, PROVE that syntactic type for that string. Robert Levine The Lambek Calculus 22 / 24

Proofs for DCC

◮ The task here is twofold:

◮ first, DECIDE what category Anne the book should be, and ◮ second, PROVE that syntactic type for that string.

◮ What type could Anne the book be? Think about RNR!

Robert Levine The Lambek Calculus 22 / 24

Proofs for DCC

◮ The task here is twofold:

◮ first, DECIDE what category Anne the book should be, and ◮ second, PROVE that syntactic type for that string.

◮ What type could Anne the book be? Think about RNR! ◮ It’s a sentence minus WHAT? What do we want it to combine with

to form a sentence?

Robert Levine The Lambek Calculus 22 / 24

Proofs for DCC

◮ The task here is twofold:

◮ first, DECIDE what category Anne the book should be, and ◮ second, PROVE that syntactic type for that string.

◮ What type could Anne the book be? Think about RNR! ◮ It’s a sentence minus WHAT? What do we want it to combine with

to form a sentence?

◮ If Anne the book picks up first a ditransitive verb, and then an NP,

we’ll get an S, right?

Robert Levine The Lambek Calculus 22 / 24

Proofs for DCC

◮ The task here is twofold:

◮ first, DECIDE what category Anne the book should be, and ◮ second, PROVE that syntactic type for that string.

◮ What type could Anne the book be? Think about RNR! ◮ It’s a sentence minus WHAT? What do we want it to combine with

to form a sentence?

◮ If Anne the book picks up first a ditransitive verb, and then an NP,

we’ll get an S, right?

◮ So: (VP/NP/NP)

• ditransitive

\ (NP\S)

NP to S

Robert Levine The Lambek Calculus 22 / 24

Proofs for DCC

◮ The task here is twofold:

◮ first, DECIDE what category Anne the book should be, and ◮ second, PROVE that syntactic type for that string.

◮ What type could Anne the book be? Think about RNR! ◮ It’s a sentence minus WHAT? What do we want it to combine with

to form a sentence?

◮ If Anne the book picks up first a ditransitive verb, and then an NP,

we’ll get an S, right?

◮ So: (VP/NP/NP)

• ditransitive

\ (NP\S)

NP to S

◮ Can we prove a sign anne • the • book; G ; (VP/NP/NP)\(NP\S)?

Robert Levine The Lambek Calculus 22 / 24

ϕ1; P; VP/NP/NP anne; a; NP

/E

ϕ1 • anne; P(a); VP/NP the • book; the-book; NP

/E

ϕ1 • anne • the • book; P(a)(the-book); VP

Robert Levine The Lambek Calculus 23 / 24

ϕ1; P; VP/NP/NP anne; a; NP

/E

ϕ1 • anne; P(a); VP/NP the • book; the-book; NP

/E

ϕ1 • anne • the • book; P(a)(the-book); VP

/I1

anne • the • book; λP.P(a)(the-book); (VP/NP/NP)\VP

Robert Levine The Lambek Calculus 23 / 24

ϕ1; P; VP/NP/NP anne; a; NP

/E

ϕ1 • anne; P(a); VP/NP the • book; the-book; NP

/E

ϕ1 • anne • the • book; P(a)(the-book); VP

/I1

anne • the • book; λP.P(a)(the-book); (VP/NP/NP)\VP We similarly obtain sue • the • pair • of • pliers; λQ.Q(s)(pl); (VP/NP/NP)\VP

Robert Levine The Lambek Calculus 23 / 24

. . . Now what??

Robert Levine The Lambek Calculus 24 / 24