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)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: Context 6 Ind speak(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 6 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: t r <t s Present t r =t s Future: t r >t s Past perfect: t e <t r <t s Simple past: t e =t r <t s

Calculus with Types Types – set Typ –BasType f 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 Term p –For each type p, we have a set of variables Var p and constants Cons p – Var p 0Term p – Con p 0Term p –a(b) 0Term p if a 0Term p->q and b 0Term p – x.a 0Term p->q if x 0 Vat p and a 0Term q –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 f Cat –if A, B 0 Cat then (A/B) and (B\A) 0 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) 6 Typ(A) –Typ(np) = Ind –Typ(n) = Ind 6 Bool –Typ(s) = Bool

Categorical Lexicon Relation between basic expressions of a language, syntactic category and meaning Meaning = -term Categorical Lexicon – relation Lex f BaseExp x (Cat x Term) such that if > 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 [e 1 e 2 ] if a:A/B 0 [e 1 ] and b:B 0 [e 2 ] –a(b):A 0 [e 1 e 2 ] if a:B\A 0 [e 2 ] and b:B 0 [e 1 ]

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

The tall kid runs tall:n/n L:np/n kid:nrun:np\s tall(kid):n L(tall(kid)):np run(L(tall(kid))):s Derivation Tree

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 the near pyramid box on the table pyr:n near:n\n/np L:np/n box:non(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:non(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 red 2 (x) in = y. P. x.P(x) and in 2 (y)(x) red(in(chs)(car))= x.((car(x) and in 2 (chs)(x)) and red 2 (x)) in(chs)(red(car))= x.((car(x) and red 2 (x)) and in 2 (chs)(x)) redcarin Chester red:n/n in(chs):n\ncar:n red(car):n in(chs)(red(car)):n redcarin Chester red:n/n in(chs):n\ncar:nn in(chs)(car):n red(in(chs)(car)):n

Coordination TerryandjumpsFrancisruns t:np jump:np\s Coor Bool (and):s\s/s f:nprun:np\s jump(t):s run(f):s and(jump(t))(run(f)):s jumpsand runs Francis : Coor Ind->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 Coor p (and):A\A/A run:np\s