Formal Semantics. 1 Type Logic

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1 Formal Semantics Principle of Compositionality The meaning of a sentence is determined by the meanings of its parts and the way they are put together. 1 Type Logic Types (a measure on expressions) The smallest set T ype such that: Basic types: e T ype (entities), t T ype (truth values) Functional types: If σ T ype and τ T ype, then σ, τ T ype. Vocabulary (the set of primitive symbols) Connectives:,,,, Quantifiers:, The mighty lambda: λ Brackets: (, ), [, ] Typed variables and constants: V ar σ, Con σ, for any type σ Syntax/Meaningful Expressions (the set ME) Variables and constants: V ar σ, Con σ ME σ, for any type σ Functional terms: If A ME σ,τ and B ME σ, then A(B) ME τ. Lambda terms: If u V ar σ and A ME τ, then λu[a] ME σ,τ. Sentential terms: If ϕ, ψ ME t, then ϕ, (ϕ ψ), (ϕ ψ), (ϕ ψ), (ϕ ψ) ME t. Quantified terms: If u V ar and ϕ ME t, then u ϕ, u ϕ ME t. Domains (the set of all meanings) 1

2 Domain of entities: D e Domain of truth-values: D t = {0, 1} Functional domains: D σ,τ, the set of all functions from D σ to D τ Full domain: D = D σ σ T ype Models, Assignment functions Models: M = D M, I M, where D M is the full domain I M : Con σ D σ is an interpretation function Assignment functions: g : V ar σ D σ If u V ar σ and d D σ (for some type σ), then g[u/d](u) = d. Semantics Variables and constants: [[α]] M,g = { g(α) I M (α) if α V ar if α Con Functional terms: [[A(B)]] M,g = [[A]] M,g ([[B]] M,g ) Lambda terms: [[λu[a]]] M,g (d) = [[A]] M,g[u/d], where u V ar σ, d D σ (for some type σ) Sentential terms: [[ ϕ]] M,g = 1 iff [[ϕ]] M,g = 0 [[ϕ ψ]] M,g = 1 iff [[ϕ]] M,g = [[ψ]] M,g = 1 [[ϕ ψ]] M,g = 1 iff [[ϕ]] M,g = 1 or [[ψ]] M,g = 1 [[ϕ ψ]] M,g = 1 iff [[ϕ]] M,g = 0 or [[ψ]] M,g = 1 [[ϕ ψ]] M,g = 1 iff [[ϕ]] M,g = [[ψ]] M,g Quantified terms: [[ u ϕ]] M,g = 1 iff for some d D σ : [[ϕ]] M,g[u/d] = 1, where u V ar σ [[ u ϕ]] M,g = 1 iff for all d D σ : [[ϕ]] M,g[u/d] = 1, where u V ar σ Translation Rule If A ME σ,τ and B ME σ, then [A B] = [B A] = A (B ) ME τ, where X is the translation of the lexical item X into Type Logic. 2

3 Example (1) [ S P aris [ V P is-bigger-than D-Dorf ]] Lexical translations: P aris = p ME e D-Dorf = d ME e is-bigger-than = λy e [λx e [B(y)(x)]] ME e, e,t Translating the entire sentence: [ P aris [ is-bigger-than D-Dorf ]] = [ is-bigger-than D-Dorf ] (P aris ) (TR) = is-bigger-than (D-Dorf )(P aris ) (TR) = λy e [λx e [B(y)(x)]](d)(p) (Lex) Interpretation of constants in some model M: I M (p) = Paris D e I M (d) = Düsseldorf [ D e ] Paris 0 Paris I M (B) = Düsseldorf 0 [ ] Paris 1 D e, e,t Düsseldorf Düsseldorf 0 Interpretation of the translation of the entire sentence: [[[ P aris [ is-bigger-than D-Dorf ]] ]] M,g = [[λy e [λx e [B(y)(x)]](d)(p)]] M,g (above) = [[λy e [λx e [B(y)(x)]](d)]] M,g ([[p]] M,g ) (functional terms) = [[λy e [λx e [B(y)(x)]]]] M,g ([[d]] M,g )([[p]] M,g ) (functional terms) = [[λy e [λx e [B(y)(x)]]]] M,g (I M (d))(i M (p)) (constants 2) = [[λy e [λx e [B(y)(x)]]]] M,g (Düsseldorf)(Paris) (model 2) = [[λx e [B(y)(x)]]] M,g[y/Düsseldorf] (Paris) (lambda terms) = [[B(y)(x)]] M,g[y/Düsseldorf][x/Paris] (lambda terms) = [[B]] M,g[y/Düsseldorf][x/Paris] ([[y]] M,g[y/Düsseldorf][x/Paris] )([[x]] M,g[y/Düsseldorf][x/Paris] ) (functional terms) = I M (B)([[y]] M,g[y/Düsseldorf][x/Paris] )([[x]] M,g[y/Düsseldorf][x/Paris] ) (constants) = I M (B)(g[y/Düsseldorf][x/Paris](y))(g[y/Düsseldorf][x/Paris](x)) (variables) = I M (B)(Düsseldorf)(Paris) (g) = 1 (model) 3

4 2 Semantics: Using Type Logic in the metalanguage Sample lexicon lexical meaning logical type [[John]] = John e [[left]] = λy e [y left] e, t [[likes]] = λy e [λx e [x likes y]] e, e, t [[student]] = λy e [y is a student] e, t [[every]] = λp e,t [λq e,t [ x e (P (x) Q(x))]] e, t, e, t, t [[some]] = λp e,t [λq e,t [ x e (P (x) Q(x))]] e, t, e, t, t Function Application (Composition Rule) If [[A]] is of type σ, τ and [[B]] is of type σ, then [[A B]] = [[B A]] = [[A]]([[B]]), which is of type τ. Lambda Conversion λx σ,τ [...x...](a) =...a..., where a is of type σ. Example (2) Every student lef t. (surface string) a. [ S [ DP [ D every] [ NP student]][ V P left]] (parsed structure) b. [every student] lef t (simplified parse) Bottom-up derivation: [[every student]] = [[every]]([[student]]) = λp e,t [λq e,t [ x e (P (x) Q(x))]]([[student]]) (Lex) = λp e,t [λq e,t [ x e (P (x) Q(x))]](λy e [y is a student ]) (Lex) = λq e,t [ x e (λy e [y is a student](x) Q(x))] (LC) = λq e,t [ x e (x is a student Q(x))] (LC) [[[every student] left]] = [[every student]]([[left]]) = λq e,t [ x e (x is a student Q(x))]([[left]]) (above) = λq e,t [ x e (x is a student Q(x))](λy e [y left]) (Lex) = x e (x is a student λy e [y left](x)) (LC) = x e (x is a student x left) (LC) Top-down derivation: [[[every student] left]] = [[every student]]([[left]]) = [[every]]([[student]])([[left]]) 4

5 = λp e,t [λq e,t [ x e (P (x) Q(x))]](λy e [y is a student])(λy e [y left]) (Lex 3) = λq e,t [ x e (λy e [y is a student](x) Q(x))](λy e [y left]) (LC) = λq e,t [ x e (x is a student Q(x))](λy e [y left]) (LC) = x e (x is a student λy e [y left](x)) (LC) = x e (x is a student x left) (LC) Exercise fashion. Derive the meaning for John [ likes Mary ] in both a bottom-up and top-down 5

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