Aspect in Ancient Greek - Nijmegen Centre for Semantics

A.3 Models 181• A model determines an entire domain function, mapping each type αonto a domain of individuals D α suitable for that type:(i) D e = D(ii) D b = E(iii) D a = T 0(iv) D re = R e(v) D rb = R b(vi) D ra = R a(vii) D s = S(viii) D t = {0, 1}(ix) D 〈α,β〉 = D DαβD = ⋃ α D αI, the interpretation function, interprets the constants Con of a language:I: Con → D such that for all α ∈ Con τ : I(α) ∈ D τ• R = R e ∪ R b ∪ R a• The model also satisfies AX1-AX8 defined in section A.6.We define:• T 1 , the set of times, is the powerset of the set of moments of time T 0(T 1 = P(T 0 ))• T 2 , the set of intervals, is a subset of the set of times (T 2 ⊂ T 1 ). Aninterval t is a non-empty set of time points such that for all moments oftime i, i ′ , i ′′ ∈ T 0 :(i) if i, i ′′ ∈ t and i ≼ i ′ and i ′ ≼ i ′′ then i ′ ∈ t convexity• t, t ′ ∈ T 2 and i, i ′ ∈ T 0 . Then we define(i) t ≺ t ′ iff for all i ∈ t, i ′ ∈ t ′ , i ≺ i ′(ii) t ⊃≺ t ′ iff t ≺ t ′ and there is no t ′′ such that t ≺ t ′′ and t ′′ ≺ t ′(iii) t ⊆ t ′ iff for all t ′′ if t ′′ ≺ t ′ then t ′′ ≺ t and if t ′ ≺ t ′′ then t ≺ t ′′(iv) t ⊂ t ′ iff t ⊆ t ′ and t ≠ t ′(v) t ○ t ′ iff there is a t ′′ such that t ′′ ⊆ t and t ′′ ⊆ t ′(vi) t ⊂· t ′ iff t ⊂ t ′ and there is a t ′′ such that t ≺ t ′′ and t ′′ ⊆ t ′