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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 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
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
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
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
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
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
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
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
 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 types1
 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 types2
 Calculus with Types
  • Beta-reduction: (x.p)(q) -> p[q/x]
  • (x.(x)(x)) (x.(x)(x)) -> ???
the category system
The Category System
  • Basic Categories:
    • np noun phrase
    • n noun
    • s sentence
syntactic categories formal definition
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

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

Examples
type assignment
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
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
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
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
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]
slide20

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
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
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
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 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
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
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