Aspect in Ancient Greek - Nijmegen Centre for Semantics

180 Appendix A: The language of Compositional DRTA.2 TypesThe set of types is defined as follows:(203) a. e is a type (the type for referring to entities like John and Mary)b. b is a type (the type for referring to eventualities)c. a is a type (the type for referring to times)d. r is a type (the type for referring to registers; r e for registers fortype e objects, r b for registers for type b objects, r a for registersfor type a objects)e. s is a type (the type for referring to states)f. t is a type (the type for referring to truth values)g. if α and β are types then so is 〈α, β〉A.3 ModelsM = 〈D, 〈E, ⊔〉, 〈T 0 , ≼〉, R, S, I〉• with D a set of normal individuals, E a set of eventualities, T 0 a set ofmoments of time (the set of real numbers), R a set of registers, and S aset of states, and τ a relation, and where• 〈E, ⊔〉 is a join semi-lattice without bottom element, i.e. ⊔ is an operationon E (i.e. ⊔ : E × E → E) such that for all e, e ′ , e ′′ ∈ E:(i) e ⊔ e ′ = e ′ ⊔ e commutativity(ii) e ⊔ e = e idempotency(iii) e ⊔ (e ′ ⊔ e ′′ ) = (e ⊔ e ′ ) ⊔ e ′′ associativity(iv) There is no e such that for all e ′ , e ⊔ e ′ = e ′ no bottomelement• 〈T 0 , ≼〉 is a dense linear ordering, i.e. ≼ is a binary relation on T 0 suchthat for all i, i ′ , i ′′ ∈ T 0 :(i) i ≼ i reflexivity(ii) if i ≼ i ′ and i ′ ≼ i ′′ then i ≼ i ′′ transitivity(iii) if i ≼ i ′ and i ′ ≼ i then i = i ′ antisymmetry(iv) i ≼ i ′ or i ′ ≼ i totality(v) if i ≺ i ′ then there is a i ′′′ such that densityi ≺ i ′′′ and i ′′′ ≺ i ′where i ≺ i ′ iff i ≼ i ′ and i ≠ i ′