Bogdan Gapinski Semantics: Modal Logics / Applicative Categorical Grammars

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 prenominal adjectives postnominal modifiers preposition intransitive verb or verb phrase transitive verb ditransitive verb preverbal verb-phrase modifier aka adverb 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

Bogdan Gapinski Semantics: Modal Logics / Applicative Categorical Grammars