Aspect in Ancient Greek - Nijmegen Centre for Semantics

184 Appendix A: The language of Compositional DRTA.6 AxiomsIn order to formulate the axioms that reduce the class of models specified inA.3 to the desired class, it is convenient to have the following abbreviation:ABB0abbreviation full formi⌊δ 1 . . .δ n ⌋j ∀v[[δ 1 ≠ v ∧ . . . ∧ δ n ≠ v] → w(v)(i) = w(v)(j)]for all terms i and j of type s and all δ 1 . . .δ n of type r (both constants andvariables). In other words, ‘i⌊δ 1 . . .δ n ⌋j’ stands for ‘i and j differ at most inδ 1 , . . .,δ n ’.By adopting the following axioms, we impose the necessary structure onour models. VAR is a predicate of type 〈r,t〉 that singles out variable registers:AX1 a. ∀i∀v∀x[VAR(v) → ∃j[i⌊v⌋j ∧w(v)(j) = x]]b. ∀i∀v∀b[VAR(v) → ∃j[i⌊v⌋j ∧w(v)(j) = b]]c. ∀i∀v∀a[VAR(v) → ∃j[i⌊v⌋j ∧w(v)(j) = a]]d. ∀i∀v∀a[VAR(v) → ∃j[i⌊v⌋j ∧w(v)(j) = a]]AX2AX3VAR(u), if u is an unspecific discourse referent.u n ≠ u m for each different unspecific discourse referents u n and u mAX4 a. ∀i[w(t TT )(i) = a TT ]b. ∀i[w(n)(i) = a n ]The AX1 axiom says that, in each state, the value of each variable register canbe set to any variable while the values of other registers can remain unchanged.This axiom makes states and registers essentially behave as assignments andvariables in predicate logic. AX4 ensures that constant registers have a fixedinhabitant.Type logic enriched with these axioms has the unselective binding propertywhich is defined as follows:Unselective Binding Lemma. Let u 1 , ...u n be constants of type r, let y 1 ,...y n be distinct variables, such that y k is of type α if u k is of type r α ,let ϕ be a formula that does not contain j and write⌈w(u 1 )(j)/x 1 , . . .w(u n )(j)/x n ⌉ϕ for the simultaneous substitution ofw(u 1 )(j) for x 1 and ...and w(u n )(j) for x n in ϕ, then:|= AX ∀i[∃j[i⌊u 1 , . . .,u n ]j ∧ ⌈w(u 1 )(j)/x 1 , . . .w(u n )(j)/x n ⌉ϕ] ↔∃y 1 . . . ∃y n ϕ]In words, the Unselective Binding Lemma states that a series of existentialquantifications over ‘normal’ individuals can be replaced by a single existentialquantification over a variable of type s. This is not surprising given that a statecan be viewed as a list of values for registers.