Language and Logic: Language and Logic: tools for the semantic study of tools for the semantic study of

natural languagenatural language

Henriëtte de Swart

Barcelona, May 2005

Ingredients of first-order logicIngredients of first-order logic

Individual constants: a, b, c, ..Individual variables: x,y,z,..Predicate constants: P(s), R(x,y), ..Connectives: ,,,,…Quantifiers: , .

AchievementsAchievements

Translation InterpretationInterpretation of 1st order logic is fixed

(w.r.t. model, variable assignment).E.g. proper namesEve kissed PeterKiss(e,p) <e,p> K

QuantifiersQuantifiers

Every student read a bookx(Stud(x) y(Book(y) Read(x,y)))Not everyone is happy x(Happy(x))The Queen of Holland is happy x (QoH(x) H(x) y (QoH(y) y=x))

ScopeScope

A letter was posted at every ambassy.yx (Let(y) & Amb(x) Post(x,y))Direct scope (=semantics mirrors syntax)xy (Amb(x) Let(y) & Post(x,y))Inverse scopeAt every ambassy a letter was posted.

Limits of first-order logic ILimits of first-order logic I

Not enough types: individual arguments not enough.

Mary believes that she is a genius (proposition)

Believe(m, Genius(m))Mary likes skating/likes to skate (property)Like(m,S)These are not wff’s in 1st order logic!

Predication and modificationPredication and modification

A red sweater, a Dutch semanticist. x (Sem(x) Dutch(x)) intersective A large mouse, a small elephant. Not: x (Mouse(x) Large(x)) But: x (Ms(x) (Large(Ms))(x) subsective A fake gun, fake fur Not: x (Gun(x) Fake(x)) intensional

Limits of first-order logic IILimits of first-order logic II

Second order quantification: John has all the properties of Santa Claus P (P(s) P(j)) Most students are happy. Not: x or x. Not: Mx(St(x) H(x)) Not: Mx(St(x) H(x)) More than 80% of the Democrats voted for Kerry.

Limits of first-order logic IIILimits of first-order logic III

Not enough compositionalityPrinciple of Compositionality of

meaning: the meaning of a complex whole is a function of the meaning of the composing parts, and the way in which they are put together.

Interpretation of NPsInterpretation of NPs

A student left.x(St(x) L(x)).Every student left.x (St(x) L(x))Most students left.No first-order representation available.What is the semantics of the NP??

Higher order logicHigher order logic

Type theoryLambda abstractionMontague Grammar: fully compositional

interpretation of natural language.Map categories onto types, interpret types

with lambda abstraction.NPs interpreted as generalized quantifiers

(Barwise and Cooper 1981)

IntermezzoIntermezzo

Type theory (handout)Lambda abstraction (handout)

GQ theoryGQ theory

Generalized quantifier theory: interpret all NPs as expressions of type <<e,t>,t>.

N and VP denote properties (= sets of individuals).

NP denotes set of properties (= set of sets of individuals) VP NP

Det establishes relation between two sets (denoted by N and VP): Q(A,B).

Proper names in GQ theoryProper names in GQ theory

Jenny is happy.Happy(j) j H (1st order)Happy {P| P(j)} HGQ-theorie

j

First-order quantifiers IFirst-order quantifiers I

All students are intelligent.Intelligent All studentsIntelligent {P|x (St(x) P(x))}

S I

S I

First-order quantifiers IIFirst-order quantifiers II

No student is rich.Rich No student.Rich {P| x St(x) P(x)}

S R

RS=

2nd order quantifiers2nd order quantifiers

Most students are happyHappy most students

S H

|SH| >

|S-H|

RelationsRelations

All students are intelligentStudent IntelligentNo student is richStudent Rich = Most students are happy|Student Happy| > |Student – Happy|

Advantages of GQ IAdvantages of GQ I

Unified type for all NPs: <<e,t>,t>Unified interpretation for all NPs:PQ(P).Unified type for all Determiners:

<<e,t>,<<e,t>,t>>>. Unified interpretation for all Dets:PQ Det(P)(Q).

Advantages of GQ theory IIAdvantages of GQ theory II

Apply theory of relations to determiners.Determiners: two-place relations between

sets.Properties of all natural language

determiners: conservativity, extension, quantity.

Properties of subclasses of determiners: symmetry, transitivity, monotonicity, etc.

Conservativity I Conservativity I

Conservativity: Q(A,B) Q(A,AB)All students are intelligent Alle students are intelligent studentsNo child is unhappy No child is an unhappy child. Most teachers are motivated Most teachers are motivated teachers.

Conservativity IIConservativity II

Q(A,B) Q(A,AB)

Cons:

A-B AB B-A |B-A| is

irrelevant

Reduction of possible denotations for dets.

QuantityQuantity

Quantity: closure under permutation Q(A,B) = Q(A,B)

Number is important, quality is not. Obvious for: all, no, two, most, at most ten,

etc.But: John’s bike has been stolen.

ExtensionExtension

Extension: QE(A,B) en E E’, dan QE’(A,B).

E.g.no quantifier ‘blik’Blik (A,B) = 1 iff |-A| = 3BlikE (A,B) = 1 iff |E-A| = 3

‘Blik’ is conservative, and quantitative, but not extensional.

Everything togetherEverything together

Cons+Quant+Ext:only |A|, |AB| matters.

E’

E

A-B AB B-A

Weak/strong distinctionWeak/strong distinction

Distribution in existential contexts.There is a child/someone/no one in the

garden.There are children/two/at most

five/many/no/few children in the garden.*There is every/neither child in the garden. *There are all/most/both/neither children in

the garden.

Symmetry ISymmetry I

Weak determiners are symmetric: Q(A,B) Q(B,A).

Two lawyers are doctors Twee doctors are lawyers.|AB| determines truthconditions.

Symmetry IISymmetry II

Symmetric: two, some, many, few, at most two, more than three, no, ..

|AB| > 2, > n, < n, =, , …Not-symmetric: all, most, both, neither,

80% of, sommige (Dutch) …Truth conditions: |A| or |A-B| involved

besides |AB|.

Symmetry and existentialitySymmetry and existentiality

Keenan (1987, 1989): Symmetric quantifiers are not ‘truly’ quantificational (one-place, rather than two-place).

Existential context offers predicate of existence that fills the slot of the ‘other’ argument.

This will be exploited in type-shifting analyses.

MonotonicityMonotonicity

Inferences.MON: Q(A,B) and B B’, then Q(A,B’).MON: Q(A,B) and B’ B, then Q(A,B’). MON: Q(A,B) and A A’, then Q(A’,B). MON: Q(A,B) and A’ A, then Q(A’,B).

Increasing monotonicityIncreasing monotonicity

MON: Q(A,B) an B B’, then Q(A,B’).All children came home late All children came home.No child came home late -/->No child came home.Exactly five children came home late -/->

Exactly five children came home.

Downward monotonicity Downward monotonicity

MON: Q(A,B) and B’ B, then Q(A,B’).No child came home No child came home late.All children came home -/->All children came home late.Exactly five children came home -/->Exactly five children came home late.

Left monotonicity (up)Left monotonicity (up)

MON: Q(A,B) and A A’, then Q(A’,B).Some boys were asleep Some children were asleepNo boys were asleep -/->No small children were asleep.All boys were asleep -/->All children were asleep.

Left monotonicity (down)Left monotonicity (down)

MON: Q(A,B) and A’ A, then Q(A’,B).No children were asleep No boys were asleepAll children were asleep All boys were asleep.Some children were asleep -/->Some boys were asleep.

Licensing of NPIsLicensing of NPIs

Negative Polarity items: expressions that only occur in ‘negative’ environments:

John didn’t eat any apples.Nobody said anything.#Mary ate any apples.#Someone ate any apples.

NPIs and monotonicity INPIs and monotonicity I

Ladusaw (1979), Zwarts (1981): negative polarity items are licensed in monotone decreasing environments.

I did not read any papers. Nobody can stand him No student who saw anything went to the police. Everyone who saw anything should report to the

police.

NPIs and monotonicity IINPIs and monotonicity II

At most three students read any papers. Few students ever noticed. Few students can stand the dean. At most five students have written a report yet. Less than five students who noticed anything went

to the police. If you saw anything, you should report to the

police.

Disadvantages of GQ theoryDisadvantages of GQ theory

Not fine-grained enough to account for traditional classifications in terms of quantificational, predicative, referential.

No account of discourse anaphora (indefinites vs. quantifiers).

No account of predication, incorporation.No account of the special status of bare

NPs.

Type-shiftingType-shifting

Type-shifting (Partee (1987) has been proposed as a possible way to merge insights from GQ theory with other views.

Many type-shifting analyses have focussed on the availability of a type <e,t> or <<e,t>,<e,t>>reading for indefinites besides a type e or <<e,t>,t> reading.