Aspect in Ancient Greek - Nijmegen Centre for Semantics

A.7 DRSs as abbreviations 185Apart from the axioms stated above which allow us to mimic DRT in typelogic, we need some axioms to ensure that the temporal predicates have theright properties:AX5 ∀b∀a 1 ∀a 2 [[τ(b,a 1 ) ∧ τ(b,a 2 )] → a 1 = a 2 ] functionality of τAX6 ∀b∃a[τ(b,a)] totality of τAX7 a. ∀a 1 ∀a 2 [a 1 ∪ a 2 = a 2 ∪ a 1 ] commutativity of ∪b. ∀a 1 [a 1 ∪ a 1 = a 1 ] idempotency of ∪c. ∀a 1 ∀a 2 ∀a 3 [[a 1 ∪ [a 2 ∪ a 3 ] = [[a 1 ∪ a 2 ] ∪ a 3 ] associativity of ∪AX8 ∀b 1 ∀b 2 ∀a 1 ∀a 2 ∀a 3 [τ(b 1 ,a 1 ) ∧ τ(b 2 ,a 2 ) ∧ τ(b 1 ⊔b 2 ,a 3 )]→ a 1 ∪ a 2 = a 3 ]]homomorphismA.7 DRSs as abbreviationsDRSs do not get a direct interpretation. Instead they are viewed as abbreviationsof type-logical expressions:abbreviation full formABB1 Π{δ 1 , . . .,δ n } λiΠ(w(δ 1 )(i)) . . .(w(δ n )(i))δ 1 = δ 2 λiw(δ 1 )(i) = w(δ n )(i)ABB2 ¬K λi¬∃jK(i)(j)K ∨K ′ λi∃j[K(i)(j) ∨K ′ (i)(j)]K → K ′ λi∀j[K(i)(j) → ∃kK ′ (j)(k)]ABB3u 1 . . .u nγ 1. . .γ mλiλj[i⌊u 1 , . . .,u n ⌋j ∧ γ 1 (j) ∧ . . . ∧ γ m (j)]ABB4 K ⊕K ′ λiλj∃k[K(i)(k) ∧K ′ (k)(j)]I have, however, used the following conventions:Rewrite rule 1 (RWR1): Π{δ 1 , δ 2 } as δ 1 Πδ 2 .Rewrite rule 2 (RWR2): Π{δ 1 } as Πδ 1 . 33 By abuse of notation I have written Π(δ 1 ) rather than Π{δ 1 } or Πδ 1 for the sake ofreadability throughout this thesis (p king(e) reads better than p kinge), apart from thisappendix.