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Montague Grammar

EECS 595 - Fall 2004

Amy Kao

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Montague Grammar

• Maps syntactic structure with semantic structure• Uses formal language to describe natural language

(1970) Universal Grammar

Theory of formal syntax and semantics applied to formal & natural languages

(1970) English as a formal language

Theory of English as a form of formal language

(1973) The Proper Treatment of Quantification in Ordinary English

Application of Universal Grammar theories of a fragment of English

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Relevant Contributors• Richard Montague (1930-1971)

• Student of Tarski

• Created Philosophies and Theories of Montague Grammar

• Taught at UCLA

• Barbara Hall Partee

• Student of Chomsky

• Wrote interpretations that made Montague’s work more understandable

• Teaches at U. Mass Amherst

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Syntactic CategoriesCategory Abbreviation PTQ Name Nearest linguistic equivalentt (primitive) Truth-value expression; or

declarative sentenceSentence

e (primitive) Entity expression; orindividual expression

(noun phrase)

t/e IV Intransitive verb phrase transitive verb, transitive verb andits object, or other verb phrases

t/IV T Term Noun phraseIV/T TV Transitive verb phrase Transitive verbIV/IV IAV IV-modifying adverb VP-adverb and prepositional

phrases containing in and about.t//e CN Common noun phrase Noun or NOMt/t None Sentence-modifying adverb Sentence-modifying adverbIAV/T None IAV-making preposition Locative, etc., prepositionIV/t None Sentence-taking verb

phraseV which takes that-COMP

IV/IV None IV-taking verb phrase V which takes infinitive COMP

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Category Definitions/Generation

• Categories are of form: X/Y – semantics of Y into the truth value of X

• Abbreviations for first 5 categories– IV = t/e– T = t/IV– TV = IV/T– IAV = IV/IV– CN = t//e

• Infinite number of possible categories– May use as many slashes as needed for new categories

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Example Expressions Category Basic Expression

1 BIV {run, walk, talk, rise, change}2 BT {John, Mary, Bill, ninety, he0, he1, he2, …}3 BTV {find lose, eat, love, date, be, seek, conceive}4 BIAV {rapidly, slowly, voluntarily, allegedly}5 BCN {man, woman, park, fish, pen, unicorn, price, temperature}6 Bt/t {necessarily}7 BIAV /T {in, about}8 BIV/t {believe that, assert that}9 BIV/IV {try to , wish to}

Partee 1973

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Example Rule F3 For BTV

F3(, ) = If first word of is a TV: if is not a variable

himi if is hei

If is 1 2 where 1 is a TV/T:

1 2 if is not a variable

himi 2 if is hei

F3(shave, a fish) = shave a fish

F3(seek, he1) = seek him1

F3(read a large book, Mary) = read Mary a large book

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Syntactic Rules

If X/Y and Y then Fi(,) X

Mary loves him, I, F1(love him, Mary)

love him, IV, F3(love, he) Mary, T

he, Tlove, TV

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Extensions and Intensions

Extension Category Appropriate Intension FunctionTruth-value of a sentence From indices to truth-values

Thing named by a name From indices to things

Set of objects a common noun or intransitive verbphrase applies to

From indices to sets

Extension: Semantic interpretation

Intension: Function generating an extension

Classmate of John = Graduate student at U MichFormer [Classmate of John] ≠ Former [Graduate student at U Mich]

Extension Problem

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Intensional Logic (IL)• IL = Intensions and Types

• Syntactic Category Rule = Semantic Rule = Type

• ^X = intension of X– Example: if J = John, ^J = function returning individual named

John

• If X/Y and Y and , translates into ’’, then Fi(,) translates into ’(^’).

• Semantic Primitive t = truth values

• Semantic type is function of model view # Rule Semantic Rule Semantic Set1 t Dt is {0,1}. all truth-values2 e De is A. all entities3 If a and b are types, <a, b> is a type D<a,b> is Db

Da all functions from Da to Db for Da

4 If a is a type, <s, a> is a type D<s,a> is Sa = SaWxT all functions from world-time pairs

to Da

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Truth DefinitionsConcept Truth Definition

Base X satisfies F(no conditions needed)

Recursive X satisfies “F and G”iff X satisfies F and X satisfies G

Consequence X is a consequence of sentences in class Kiff X is true in every model where every sentence in K is true

Equivalence X and Y are logically equivalentif X is a consequence of Y and Y is a consequence of X

Truth X is logically trueif for all models, X is true

Contradiction Class K is contradictoryif no model exists where all sentences in K are true

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Model Theoretic Semantics

• Semantics based on truth conditions– Tarski’s Model Theory (1954)

• IL is based on truth Conditions • 3 Levels of Symbols

– Logical Constants: =, , etc– Variables: As in traditional math– Non-logical Constants: , , relation symbols, function

symbols, and constant individual symbols

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General Quantification & Compositionality

• Compositionality: phrase’s meaning derived from meaning of constituents & syntactic structure

• General Quantification: allows for syntax and semantic structure to be equivalent

Sentence First Order Logic General QuantificationJohn talks Talks(j) John’ (^talk’)Every student talks ∀x [Studen(t )x -> Talks( )x ] Every man’ (^ talk’)

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Controversies• Formal Logic

– Opposition: differing views of what semantics is• Chomskyan: semantics is branch of psychology• Semantists: semantics different from knowledge of semantics

– Defense: Montague’s “English as a Formal Language”– Formal Semantics now mainstream

• Model Theory– Opposition: Prefer more concrete expressions– Defense: Understands sentences in terms of human mental models

• Truth Conditions– Opposition: Brings in too many irrelevant factors– Defense: Humans judge sentences based on context

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Developments Influenced by Montague Grammar

• Head-Driven Phrase Structure Grammar (HPSG)– Influenced by syntactic categories

• File Change Semantics(FCS)• Discourse Representation Semantics (DRS)• Situation Semantics• Extended Categorical Grammar• Generalized Phrase Structure Grammar (GPSG)• Lexical Semantics