course 5: montague's approach to semantics · 2016. 1. 11. · montague semantics the...

Course 5: Montague’s approach to semantics

27. June 2014NASSLLI 6

On Montague Grammar

Barbara Partee (Encylopedia of Language and Linguistics)

Before Montague, semanticists focused on the explication ofambiguity, anomaly, and “semantic relatedness”; data were oftensubjective and controversial. The introduction of truth-conditions

and entailment relations as core data profoundly affected theadequacy criteria for semantics, and led to a great expansion of

semantic research.

Montague Semantics

The principles of Montague semantics are:

I an algebraic interpretation of compositionality

I higher-order typed intentional logic

I truth-conditional semantics.

(Dowty Wall Peters)

A truth-conditional theory of semantics is one which adheres to thefollowing dictum: To know the meaning of a (declarative) sentenceis to know what the world would have to be like for the sentence to

be true

Montague Semantics

(Dowty Wall Peters)

be true

Montague Semantics

(Dowty Wall Peters)

be true

Montague Semantics

(Dowty Wall Peters)

be true

Languages of λ-terms and λ-CFG

A λ-CFG is a tuple (Σ1,Σ2,S ,L) where:

I Σ1 is a second order signature (i.e. a multi-sorted freealgebra),

I Σ2 is a string signature (an alphabet of non-terminal)

I S is an atomic type of Σ1

I L is a homomorphism (the lexicon):I given α an atomic type of Σ1, L(α) is in L(Σ2), andL(α→ β) = L(α) → H(β)

I L(S) = o → oI given c a constant of Σ1 of type α, H(c) is a closed term of

ΛαΣ1

The language defined by such a grammar is:

{s | ∃M ∈ ΛSΣ1.s =βη L(M)}

Languages of λ-terms and λ-CFG

A λ-CFG of λ-terms is a tuple (Σ1,Σ2,S ,L) where:

I Σ1 is a second order signature (i.e. a multi-sorted freealgebra),

I Σ2 is a signature (an alphabet of non-terminal)

I S is an atomic type of Σ1

I L is a homomorphism (the lexicon):I given α an atomic type of Σ1, L(α) is in L(Σ2), andL(α→ β) = L(α) → H(β)

IL(S) = o → o

I given c a constant of Σ1 of type α, H(c) is a closed term ofΛα

The language defined by such a grammar is:

{s | ∃M ∈ ΛSΣ1.s =βη L(M)}

λ-Context-Free Grammars

A λ-CFG (Σ1,Σ2,S ,L) of λ-terms

ΛSΣ1

ΛL(S)Σ2

Montague Grammar as a pair of λ-CFG

(de Groote 2001 and Muskens 2001)

Σstruct

A toy example

sentence : NP → VP → Sverbp : V → NP → VPloves : VMary : NPJohn : NP

Lsyn(S) = Lsyn(NP) = Lsyn(VP)= Lsyn(V ) = string

Lsyn(sentence) = λxy .x + yLsyn(verbp) = λxy .x + yLsyn(loves) = lovesLsyn(Mary) = MaryLsyn(John) = John

Lsem(S) = t, Lsem(NP) = e,Lsem(VP) = e → t, Lsem(V ) = e →e → t

Lsem(sentence) = λS V .V SLsem(verbp) = λV O S .V S OLsem(loves) = λxy .love x yLsem(Mary) = mLsem(John) = j

sentence

John verbp

loves Mary

sentence

John verbp

loves Mary

A toy example

sentence

John verbp

loves Mary

sentence

John verbp

loves Mary

A toy example

sentence

John verbp

loves Mary

sentence

John verbp

loves Mary

A toy example

sentence

John verbp

loves Mary

sentence

John verbp

loves Mary

A toy example

λxy .x + y

John λxy .x + y

loves Mary

sentence

John verbp

loves Mary

A toy example

λxy .x + y

John loves Mary

loves Mary

sentence

John verbp

loves Mary

A toy example

John loves Mary

loves Mary

sentence

John verbp

loves Mary

A toy example

John loves Mary

loves Mary

λS V .V S

j λV O S .V S O

λxy .love x y m

A toy example

John loves Mary

loves Mary

λS V .V S

j λS .love S m

λxy .love x y m

A toy example

John loves Mary

loves Mary

love j m

j λS .love S m

λxy .love x y m

Yet another example

sentence : NP → VP → Sverbp : V → NP → VPnounp : Det → N → NPloves : VMary : NPJohn : NPevery : Deta : Detman : Nwoman : N

Yet another example

Lsyn(S) = Lsyn(NP) = Lsyn(VP)= Lsyn(V ) = Lsyn(Det) = Lsyn(N) =string

Lsyn(sentence) = λxy .x + yLsyn(verbp) = λxy .x + yLsyn(nounp) = λxy .x + yLsyn(loves) = lovesLsyn(Mary) = MaryLsyn(John) = JohnLsyn(every) = everyLsyn(a) = aLsyn(woman) = womanLsyn(man) = man

Lsem(S) = t, Lsem(NP) = e,Lsem(VP) = e → t, Lsem(V ) = e →e → t, Lsem(Det) =

(e → t) → t

,Lsem(N) =

e → t

Lsem(sentence) = λS V .V SLsem(verbp) = λV O S .V S OLsyn(nounp) =

Lsem(loves) = λxy .love x yLsem(Mary) = mLsem(John) = jLsem(a) =

λP.∃(λx .P x)

Lsem(every) =

λP.∀(λx .P x)

Lsem(woman) =

λx .woman x

Lsem(man) =

λx .man x

Yet another example

Lsem(S) = t, Lsem(NP) = e,Lsem(VP) = e → t, Lsem(V ) = e →e → t, Lsem(Det) = (e → t) → t,Lsem(N) =

e → t

λP.∃(λx .P x)

Lsem(every) =

λP.∀(λx .P x)

Lsem(woman) =

λx .woman x

Lsem(man) =

λx .man x

Yet another example

Lsem(S) = t, Lsem(NP) = e,Lsem(VP) = e → t, Lsem(V ) = e →e → t, Lsem(Det) = (e → t) → t,Lsem(N) = e → t

λP.∃(λx .P x)

Lsem(every) =

λP.∀(λx .P x)

Lsem(woman) =

λx .woman x

Lsem(man) =

λx .man x

Yet another example

λP.∃(λx .P x)

Lsem(every) =

λP.∀(λx .P x)

Lsem(woman) =

λx .woman x

Lsem(man) =

λx .man x

Yet another example

λP.∃(λx .P x)

Lsem(every) =

λP.∀(λx .P x)

Lsem(woman) = λx .woman xLsem(man) =

λx .man x

Yet another example

λP.∃(λx .P x)

Lsem(every) =

λP.∀(λx .P x)

Lsem(woman) = λx .woman xLsem(man) = λx .man x

Yet another example

Lsem(loves) = λxy .love x yLsem(Mary) = mLsem(John) = jLsem(a) = λP.∃(λx .P x)Lsem(every) =

λP.∀(λx .P x)

Lsem(woman) = λx .woman xLsem(man) = λx .man x

Yet another example

Lsem(loves) = λxy .love x yLsem(Mary) = mLsem(John) = jLsem(a) = λP.∃(λx .P x)Lsem(every) = λP.∀(λx .P x)Lsem(woman) = λx .woman xLsem(man) = λx .man x

Yet another example

Lsem(sentence) = λS V .V SLsem(verbp) = λV O S .V S OLsyn(nounp) = ??Lsem(loves) = λxy .love x yLsem(Mary) = mLsem(John) = jLsem(a) = λP.∃(λx .P x)Lsem(every) = λP.∀(λx .P x)Lsem(woman) = λx .woman xLsem(man) = λx .man x

The semantic problem of nounp

nounp : Det → N → NPLsem(Det) = (e → t) → tLsem(N) = e → tLsem(NP) = e

λD(e→t)→tNe→t .

We need to change Lsem(V ) and Lsem(VP).

We had:

Lsem(V ) = e → e → t

Lsem(VP) = e → t

nounp : Det → N → NPLsem(Det) = (e → t) → (e → t) → tLsem(N) = e → tLsem(NP) = (e → t) → t

λD(e→t)→(e→t)→tNe→t .

We had:

Lsem(VP) = e → t

λD(e→t)→(e→t)→tNe→t .D N

We had:

Lsem(VP) = e → t

We had:

Lsem(VP) = e → t

We need to change Lsem(V ) and Lsem(VP).We had:

Lsem(VP) = e → t

Lsem(V ) = Lsem(NP) → Lsem(NP) → t

Lsem(VP) = Lsem(NP) → t

Lsem(V ) = ((e → t) → t) → ((e → t) → t) → t

Lsem(VP) = ((e → t) → t) → t

Updating the rest of the semantics interpretation

Lsem(S) = t, Lsem(NP) = e, Lsem(VP) = np → t, Lsem(V ) = np → np → t,Lsem(Det) = (e → t) → (e → t)t, Lsem(N) = e → t, with np = (e → t) → t

sentence : NP → VP → S Lsem(sentence) = λS V .V Sverbp : V → NP → VP Lsem(verbp) = λV O S .V S Onounp : Det → N → NP Lsem(nounp) = λD N.D Nloves : V Lsem(loves) = λxy .love x yMary : NP Lsem(Mary) = mJohn : NP Lsem(John) = jevery : Det Lsem(every) = λP.∀(λx .P x)a : Det Lsem(a) = λP.∃(λx .P x)man : N Lsem(man) = λx .man xwoman : N Lsem(woman) = λx .woman x

sentence : NP → VP → S Lsem(sentence) = λS V .V Sverbp : V → NP → VP Lsem(verbp) = λV O S .V S Onounp : Det → N → NP Lsem(nounp) = λD N.D Nloves : V Lsem(loves) = λOS .S(λx .O(λy .love x y))Mary : NP Lsem(Mary) = mJohn : NP Lsem(John) = jevery : Det Lsem(every) = λP.∀(λx .P x)a : Det Lsem(a) = λP.∃(λx .P x)man : N Lsem(man) = λx .man xwoman : N Lsem(woman) = λx .woman x

sentence : NP → VP → S Lsem(sentence) = λS V .V Sverbp : V → NP → VP Lsem(verbp) = λV O S .V S Onounp : Det → N → NP Lsem(nounp) = λD N.D Nloves : V Lsem(loves) = λOS .S(λx .O(λy .love x y))Mary : NP Lsem(Mary) = λP.P mJohn : NP Lsem(John) = jevery : Det Lsem(every) = λP.∀(λx .P x)a : Det Lsem(a) = λP.∃(λx .P x)man : N Lsem(man) = λx .man xwoman : N Lsem(woman) = λx .woman x

sentence : NP → VP → S Lsem(sentence) = λS V .V Sverbp : V → NP → VP Lsem(verbp) = λV O S .V S Onounp : Det → N → NP Lsem(nounp) = λD N.D Nloves : V Lsem(loves) = λOS .S(λx .O(λy .love x y))Mary : NP Lsem(Mary) = λP.P mJohn : NP Lsem(John) = λP.P jevery : Det Lsem(every) = λP.∀(λx .P x)a : Det Lsem(a) = λP.∃(λx .P x)man : N Lsem(man) = λx .man xwoman : N Lsem(woman) = λx .woman x

sentence : NP → VP → S Lsem(sentence) = λS V .V Sverbp : V → NP → VP Lsem(verbp) = λV O S .V S Onounp : Det → N → NP Lsem(nounp) = λD N.D Nloves : V Lsem(loves) = λOS .S(λx .O(λy .love x y))Mary : NP Lsem(Mary) = λP.P mJohn : NP Lsem(John) = λP.P jevery : Det Lsem(every) = λP Q.∀(λx .P x ⇒ Q x)a : Det Lsem(a) = λP.∃(λx .P x)man : N Lsem(man) = λx .man xwoman : N Lsem(woman) = λx .woman x

sentence : NP → VP → S Lsem(sentence) = λS V .V Sverbp : V → NP → VP Lsem(verbp) = λV O S .V S Onounp : Det → N → NP Lsem(nounp) = λD N.D Nloves : V Lsem(loves) = λOS .S(λx .O(λy .love x y))Mary : NP Lsem(Mary) = λP.P mJohn : NP Lsem(John) = λP.P jevery : Det Lsem(every) = λP Q.∀(λx .P x ⇒ Q x)a : Det Lsem(a) = λP Q.∃(λx .P x ∧ Q x)man : N Lsem(man) = λx .man xwoman : N Lsem(woman) = λx .woman x

Semantic interpretation at work

sentence

every man

loves nounp

a woman

Lsem(nounpawoman) =

Lsem(nounp everyman) =λQ.∀(λu.man u ⇒ Q u)

Lsem(verbp loves (nounpawoman)) =

Lsem(sentence(nounp everyman)(verbp loves (nounpawoman))) =

sentence

every man

loves nounp

a woman

Lsem(nounpawoman) =

sentence

every man

loves nounp

a woman

Lsem(nounpawoman) =λP Q.∃(λz.P z ∧ Q z)(λx .woman x)

sentence

every man

loves nounp

a woman

Lsem(nounpawoman) =λP Q.∃(λz.P z ∧ Q z)(λx .woman x)

sentence

every man

loves nounp

a woman

Lsem(nounpawoman) =λQ.∃(λz.(λx .woman x) z ∧ Q z)

sentence

every man

loves nounp

a woman

Lsem(nounpawoman) =λQ.∃(λz.(λx .woman x) z ∧ Q z)

sentence

every man

loves nounp

a woman

Lsem(nounpawoman) =λQ.∃(λz.woman z ∧ Q z)

sentence

every man

loves nounp

a woman

sentence

every man

loves nounp

a woman

sentence

every man

loves nounp

a woman

Lsem(verbp loves (nounpawoman)) =λO S .S(λx .O(λy .love x y))(λP.∃(λz.woman z ∧ P z))

sentence

every man

loves nounp

a woman

Lsem(verbp loves (nounpawoman)) =λO S .S(λx .O(λy .love x y))(λP.∃(λz.woman z ∧ P z))

sentence

every man

loves nounp

a woman

Lsem(verbp loves (nounpawoman)) =λS .S(λx .(λP.∃(λz.(λz.woman z ∧ P z)))(λy .love x y))

sentence

every man

loves nounp

a woman

Lsem(verbp loves (nounpawoman)) =λS .S(λx .(λP.∃(λz.(λz.woman z ∧ P z)))(λy .love x y))

sentence

every man

loves nounp

a woman

Lsem(verbp loves (nounpawoman)) =λS .S(λx .∃(λz.woman z ∧ (λy .love x y) z))

sentence

every man

loves nounp

a woman

Lsem(verbp loves (nounpawoman)) =λS .S(λx .∃(λz.woman z ∧ (λy .love x y) z))

sentence

every man

loves nounp

a woman

Lsem(verbp loves (nounpawoman)) =λS .S(λx .∃(λz.woman z ∧ love x z))

sentence

every man

loves nounp

a woman

sentence

every man

loves nounp

a woman

Lsem(sentence(nounp everyman)(verbp loves (nounpawoman))) =(λS.S(λx .∃(λz.woman z ∧ love x z)))(λQ.∀(λu.man u ⇒ Q u))

sentence

every man

loves nounp

a woman

Lsem(sentence(nounp everyman)(verbp loves (nounpawoman))) =(λS.S(λx .∃(λz.woman z ∧ love x z)))(λQ.∀(λu.man u ⇒ Q u))

sentence

every man

loves nounp

a woman

Lsem(sentence(nounp everyman)(verbp loves (nounpawoman))) =(λQ.∀(λu.man u ⇒ Q u))(λx .(∃(λz.woman z ∧ love x z)))

sentence

every man

loves nounp

a woman

Lsem(sentence(nounp everyman)(verbp loves (nounpawoman))) =(λQ.∀(λu.man u ⇒ Q u))(λx .(∃(λz.woman z ∧ love x z)))

sentence

every man

loves nounp

a woman

Lsem(sentence(nounp everyman)(verbp loves (nounpawoman))) =∀(λu.man u ⇒ (λx .∃(λz.woman z ∧ love x z)) u)

sentence

every man

loves nounp

a woman

Lsem(sentence(nounp everyman)(verbp loves (nounpawoman))) =∀(λu.man u ⇒ (λx .∃(λz.woman z ∧ love x z)) u)

sentence

every man

loves nounp

a woman

Lsem(sentence(nounp everyman)(verbp loves (nounpawoman))) =∀(λu.man u ⇒ ∃(λz.woman z ∧ love u z))

Subject wide scope and object wide scope

The sentence every man loves a woman has two possible meanings:

I ∀(λu.man u ⇒ ∃(λz .woman z ∧ love u z))

I ∃(λz .woman z ∧ ∀(λu.man u ⇒ love u z))

The second meaning can be obtained by using the term

λO S .O(λy .S(λx .love x y))

instead ofλO S .S(λx .O(λy .love x y))

This technique also allows to account for the de Re/de Dictoambiguity.

A larger grammar

sentence S → NP VP a Det → averbp VP → V NP dog N → dognounp NP → Det N cat N → catnadj N → N CP rat N → ratcpadj CP → C VP cheese N → cheesecprel CP → C S/NP saw V → sawrel S/NP → NP V chased V → chasedthat C → that ate V → ate

To obtain a semantics interpretation for this grammar, it suffices to use similarmeaning recipes as the one we have seen before.We need to define:

I Lsem(S/NP), Lsem(CP) and C,

I Lsem(rel), Lsem(cpadj), Lsem(cprel), Lsem(that), Lsem(nadj)

I Lsem(S/NP) = Lsem(VP) = ((e → t) → t) → t, Lsem(CP) = e → t andC = e → (e → t) → t

I Lsem(rel) = λV S O.V O S , Lsem(cpadj) = Lsem(cprel) = λC ,V .VC ,Lsem(that) = λx P.P x and Lsem(nadj) = λNCx .N x ∧ C x

A larger grammar

To obtain a semantics interpretation for this grammar, it suffices to use similarmeaning recipes as the one we have seen before.

We need to define:

A larger grammar

The semantics of a sentence

SENTENCE

A NADJ

RAT CPREL

THAT REL

A NADJ

CAT CPREL

THAT REL

CHASED

ATE NOUNP

A CHEESE

∃(λx .rat x∧(∃(λy .cat y∧∃(λz.dog z∧saw z y)∧chased y x∧∃(λu.chees u∧ate x u))))

The limits of Montague’s approach

Donkey sentence:

Every farmers who owns a donkey beats it.

A similar approach as the previous one fails:

[[who owns a donkey]] = λx .∃(λy .donkey y ∧ owns x y)

Some proposals to fix the problem

I Discourse Representation Theory (Kamp 1981)

I Dynamic Predicate Logic (Groenendijk and Sokhof 1991)

I abstracting over the context (de Groote 2006)

course 5: montague's approach to semantics · 2016. 1. 11. · montague semantics the...

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