Aspect in Ancient Greek - Nijmegen Centre for Semantics

182 Appendix A: The language of Compositional DRTA.4 SyntaxThe set of well-formed expressions, Exp:(i)(ii)Basic expressions (of a certain type):a. Con α is the (possibly empty) set of constants of type αb. V ar α is the (infinite) set of variables of type αc. Exp α ⊇ Con α ∪ V ar αComplex expressions:a. If µ, ν ∈ Exp t and ξ ∈ V ar, then ¬µ, [µ∧ν], [µ∨ν], [µ → ν], ∃ξµ, ∀ξµ∈ Exp tb. If µ, ν ∈ Exp α , then [µ = ν] ∈ Exp tc. If µ ∈ Exp α , ν ∈ Exp 〈α,β〉 , then [ν(µ)] ∈ Exp βd. If µ ∈ Exp α , ξ ∈ V ar β , then λξµ ∈ Exp 〈β,α〉(204) Con = ⋃ α Con α, V ar = ⋃ α V ar α, Exp = ⋃ α Exp αI omit superfluous brackets.The constants and variables that I use are given in Table A.1. w in thistable is a fixed non-logical constant of type 〈r, 〈s,e〉〉. w(v)(i) stands for ‘thevalue of register v in a state i’.A.5 SemanticsSemantic values of arbitrary expressions are given relative to an assignmentfunction f that maps variables on objects from the domain: f : V ar → Dwith for each ξ ∈ V ar α , f(ξ) ∈ D α .Interpretation is defined as follows:(205) a. Basic expressions:(i) If µ ∈ Con, then µ M,f = I(µ)(ii) If ξ ∈ V ar, then ξ M,f = f(ξ)b. Complex expressions:(i) ¬µ M,f = 1 iff µ M,f = 0(ii) µ ∧ ν M,f = 1 iff µ M,f = ν M,f = 1(iii) µ ∨ ν M,f = 1 iff µ M,f = 1 or ν M,f = 1(iv) µ → ν M,f = 0 iff µ M,f = 1 and ν M,f = 0(v) ∃ξµ α M,f = 1 iff there is a d ∈ D α s.t. µ M,f[ξ/d] = 1(vi) ∀ξµ α M,f = 1 iff for all d ∈ D α µ M,f[ξ/d] = 1(vii) α = β M,f = 1 iff α f = β M,f(viii) β(α) M,f = β M,f (α M,f )(ix) λξ α µ β M,f is the function h : D α → D β s.t. for all d ∈ D α ,h(d) = µ M,f[ξ/d]