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))
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