Bogdan Gapinski

Semantics: Modal Logics / Applicative Categorical Grammars

Presentation based on the book “Type-Logical Semantics” by Bob Carpenter

Modal Logic - Motivation• Problems with true-false logic

– The ancients believed [the morning star is the morning star]

– The ancients believed [the morning star is the evening star]• morning star = evening star = Venus

– Terry intentionally shot {the burglar / his best friend}• what if his best friend is the burglar

– Morgan swam the channel quickly

– Morgan crossed the channel slowly• swimming/crossing speed

– Francis is a good Broadway {dancer / singer}• comparison classes

Modal Logics – general idea

• p means “p is necessarily true”

• we want (p)p but not p(p)• Kripke’s idea:

– a possible world determines truth of falsehood of formulas

– worlds can be interpreted as points in time

– denotation of the formula depends on the world

– p is true iff p is true in every possible world

– define as not(not(p)) • A formula is possibly true if it is not necessarily false• jp can be true at a world even if p is false

Indexicality

• Expressions that have their interpretations determined by the context of utterance– personal pronouns: I, you, we

– temporal expressions now, yesterday

– locative expression here

• add parameters for speaker/hearer/location to the denotation function

• Generalized idea: single context index c with arbitrary number of properties that could be retrieved by functions, for instance

speak: Context Ind speak(c) = an individual who is speaking

General Modal Logics

• Notion of accessibility• Accessibility relation A World x World

– wAw’ means w’ is possible relative to w– p is true in a world w iff p is true in every world w’ such that

wAw’

• Logics can be defined by imposing conditions on A and specifying axioms they satisfy

• Example: =“is known” not(p) not(p) – “if p is not known, then it is known to be not known”– knowledge representation with for agents with full introspection

Implication and Counterfactuals

• If there were no cats, cats would eat mice.• If there were no dogs, cats would eat mice.

• Lewis: indicative conditional vs. subjunctive conditional– If Oswald did not kill Kennedy, then someone else did

– If Oswald had not killed Kennedy, then someone else would have.

– but…

– If Oswald has not killed Kennedy, someone else will have• said the next in line would-be assassin…

• Translate “p then q” as (p q)

Tense Logic

• Worlds = moments in time (Tim)• Accessibility = temporal precedence (<)• Fp is true at time t iff p is true at t’ such that t’>t• Pp is true at time t iff p is true at t’ such that t’<t

– Wp = not(F(not(p))) [Always Will]

– Hp = not(P(not(p))) [Always Has]• FHp p

• Different kind of logic systems result from conditions imposed on <

Tense and Aspect

• Tenses: past, present, future• Aspect: perfective, progressive, simple• Reichenback’s approach:

– event, reference, speech times– Tenses:

• Past: tr<ts

• Present tr=ts

• Future: tr>ts

• Past perfect: te<tr<ts

• Simple past: te=tr<ts

Calculus with Types

• Types – set Typ – BasType Typ– If p, q Typ then (p -> q) Typ

– For us, BasType ={Ind, Bool}– Ex. ((Ind -> Bool) -> (Ind -> Bool))

Calculus with Types• Terms – set Termp

– For each type p, we have a set of variables Varp and constants Consp

– Varp Termp

– Conp Termp

– a(b) Termp if a Termp->q and b Termp

x.a Termp->q if x Vatp and a Termq

– run: Ind -> Bool, lee: Ind quickly: (Ind->Bool)->Ind->Bool – run(lee): Bool– quickly(run): Ind -> Bool– quickly(run)(lee): Bool

– x: Ind x.(like(x)(ricky))

Calculus with Types

• Beta-reduction: (x.p)(q) -> p[q/x]

• (x.(x)(x)) (x.(x)(x)) -> ???

The Category System

• Basic Categories:– np noun phrase– n noun – s sentence

Syntactic Categories - Formal Definition

• The collection of syntactic categories determined by the collection BasCat– BasCat Cat– if A, B Cat then (A/B) and (B\A) Cat

A/B – forward functor

B\A – backward functor

Examples

• np/n • n/n• n\n• (n\n)/np • np\s• (np\s)/np• ((np\s)/np)/np• (np/s)/(np/s)

• determiners• prenominal adjectives• postnominal modifiers• preposition• intransitive verb or verb phrase• transitive verb• ditransitive verb• preverbal verb-phrase modifier

aka adverb

Type Assignment

• Type assignment function Typ– Typ(A/B)=Typ(B\A)= Typ (B) Typ(A)

– Typ(np) = Ind– Typ(n) = Ind Bool– Typ(s) = Bool

Categorical Lexicon

• Relation between basic expressions of a language, syntactic category and meaning

• Meaning = -term

• Categorical Lexicon – relation Lex BaseExp x (Cat x Term) such that if <e,<A,a>> Lex then a Term Typ(A)

• Notation e a : A

Phase-structure Denotation

• Function: [ . ]Lex

– a:A [e] if e a:A Lex

– a(b):A [e1 e2] if a:A/B [e1] and b:B [e2]

– a(b):A [e1 e2] if a:B\A [e2] and b:B [e1]

Lexicon: Example

• Sandy sandy:np• the L: np/p• kid kid:n• tall tall:n/n (P.x.P(x))• outside outside:n\n• in in:n\n/np• runs run:np\s• loves love:np\s/np• gives give:np\s/np/np• outside outside:(np\s)\np\s• in in:(np\s)\np\s/np

Example of a derivation: the tall kid runs

• tall:n/n [tall]

• kid:n [kid]

• tall(kid):n [tall kid]

• L:np/n [the]

• L(tall(kid)):np [the tall kid]

• run: np\s [runs]

• run(L(tall(kid))): s [the tall kid runs]

The tall kid runs

tall:n/nL:np/n kid:n run:np\s

tall(kid):n

L(tall(kid)):np

run(L(tall(kid))):s

Derivation Tree

Type Soundness

• If a : A [e] then a Term Typ(A)

• This is a big deal!

• Similarity to typing schemes of functional languages

Ambiguity• Lexical syntactic ambiguity: an expression has two

lexical entries with different syntactic categories (kiss)• Lexical semantical ambiguity: two different lambda-terms

assigned to the same category (bank)• Vagueness: sister-in-law, glove• Negation test:

– Gerry went to the bank.– No, he didn’t, he went to the river.– Robin is wearing a glove.– * No he isn’t, that is a left glove.

Derivational Ambiguity – two parse trees for the same set of words having the same lexical entries

the nearpyramid box on the table

pyr:n

near:n\n/np

L:np/n box:n on(L(table)):n\n

L(box):np

near(L(box)):n\n

near(L(box))(pyr):n

on(L(table))(near(L(box))(pyr)):n

the nearpyramid box on the table

pyr:n

near:n\n/np

L:np/n box:n on(L(table)):n\n

on(L(table))(box):n

L(on(L(table))(box)):n

near(on(L(table))(box)):n\n

near(on(L(table))(box))(pyr):n\n

Local and Global Ambiguity

• Local ambiguity – a subexpression is ambiguous– The tall kid in Pittsburg run

– The horse raced past the barn fell.– The cotton clothing is made with comes from Egypt.

• garden-path effect in psycholinguistics

Meaning postulates

red= P. x.P(x) and red2(x)

in = y. P. x.P(x) and in2 (y)(x)

red(in(chs)(car))=x.((car(x) and in2 (chs)(x)) and red2 (x))

in(chs)(red(car))=x.((car(x) and red2 (x)) and in2 (chs)(x))

red car in Chester

red:n/n in(chs):n\ncar:n

red(car):n

in(chs)(red(car)):n

red car in Chester

red:n/n in(chs):n\ncar:nn

in(chs)(car):n

red(in(chs)(car)):n

CoordinationTerry andjumps Francis runs

t:np jump:np\s CoorBool(and):s\s/s f:np run:np\s

jump(t):s run(f):s

and(jump(t))(run(f)):s

jumps and runsFrancis

:CoorInd->Bool(and):(np\s)\(np\s)/(np\s)

jump:np\sf:np

Lx.and(jump(x))(run(x)):np\s

and(jump(f))(run(f)):s

Coorp(and):A\A/A

run:np\s