Bogdan Gapinski Semantics: Modal Logics / Applicative Categorical Grammars

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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]

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### Bogdan GapinskiSemantics: 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)6p but not p6(~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 L as not(~not(p))
• A formula is possibly true if it is not necessarily false
• jLp 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: Context6 Indspeak(c) = an individual who is speaking
General Modal Logics
• Notion of accessibility
• Accessibility relation A f 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) 6~ 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 6 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 6p
• 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
• BasTypef Typ
• If p, q 0Typ then (p -> q) 0 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
• Varp0Termp
• Conp0Termp
• a(b) 0Termp if a 0Termp->q and b 0Termp
• x.a 0Termp->q if x 0 Vatp and a 0Termq
• 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
• BasCatf Cat
• if A, B 0 Cat then (A/B) and (B\A) 0 Cat

A/B – forward functor

B\A – backward functor

np/n

n/n

n\n

(n\n)/np

np\s

(np\s)/np

((np\s)/np)/np

(np/s)/(np/s)

determiners

postnominal modifiers

preposition

intransitive verb or verb phrase

transitive verb

ditransitive verb

Examples
Type Assignment
• Type assignment function Typ
• Typ(A/B)=Typ(B\A)= Typ (B) 6 Typ(A)
• Typ(np) = Ind
• Typ(n) = Ind6Bool
• Typ(s) = Bool
Categorical Lexicon
• Relation between basic expressions of a language, syntactic category and meaning
• Meaning = -term
• Categorical Lexicon – relation Lexf BaseExp x (Cat x Term) such that if <e,<A,a>> 0 Lex then a 0 Term Typ(A)
• Notation e Y a : A
Phase-structure Denotation
• Function: [ . ]Lex
• a:A 0 [e] if e Y a:A 0 Lex
• a(b):A 0 [e1 e2] if a:A/B 0 [e1]andb:B 0 [e2]
• a(b):A 0 [e1 e2] if a:B\A 0 [e2]andb:B 0 [e1]
Lexicon: Example
• Sandy Ysandy:np
• the YL: np/p
• kid Ykid:n
• tall Ytall:n/n (P.x.P(x))
• outside Youtside:n\n
• in Yin:n\n/np
• runs Yrun:np\s
• loves Ylove:np\s/np
• gives Ygive:np\s/np/np
• outside Youtside:(np\s)\np\s
• in Yin:(np\s)\np\s/np
Example of a derivation: the tall kid runs
• tall:n/n 0 [tall]
• kid:n 0 [kid]
• tall(kid):n 0 [tall kid]
• L:np/n 0 [the]
• L(tall(kid)):np 0 [the tall kid]
• run: np\s 0 [runs]
• run(L(tall(kid))): s 0 [the tall kid runs]

Derivation Tree

The

tall

kid

runs

L:np/n

tall:n/n

kid:n

run:np\s

tall(kid):n

L(tall(kid)):np

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

Type Soundness
• If a : A 0 [e] then a 0 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

near

the

pyramid

box

on the table

L:np/n

pyr:n

box:n

on(L(table)):n\n

near:n\n/np

L(box):np

near(L(box)):n\n

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

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

pyramid

near

the

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

car

in Chester

red

car

in Chester

red:n/n

red:n/n

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))

car:n

in(chs):n\n

car:nn

in(chs):n\n

red(car):n

in(chs)(car):n

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

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

Coordination

Terry

jumps

and

Francis

runs

t:np

jump:np\s

f:np

run:np\s

CoorBool(and):s\s/s

run(f):s

jump(t):s

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

runs

Francis

jumps

and

f:np

jump:np\s

:

run:np\s

CoorInd->Bool(and):

(np\s)\(np\s)/(np\s)

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

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

Coorp(and):A\A/A