Aspect in Ancient Greek - Nijmegen Centre for Semantics

4.9 The conative and likelihood interpretations 109Consider (136), which gives the semantics for (135):(136) PAST(IMP(λe d buy(e)))= λQ[Q(t TT ) ⊕tTT ≺ n ](λPλt[P(e) ⊕ eτ(e) ·⊃ t TT](λe d buy(e)))≡ed buy(e)τ(e) ·⊃ t TTt TT ≺ nIn contrast to the natural language sentence (135), the logical form (136) doesentail that there is a (complete) eventuality e of which the predicate holds,which is clearly not what we want. We hit here upon the notorious problemof the imperfective paradox, which I have already briefly discussed in section3.2.2 and which was shown to be a challenge for the English progressive aswell.I don’t have a solution to the imperfective paradox. I believe that such asolution goes beyond the scope of this work and deserves a study of its own, aswitnessed by the many attempts found in the literature. However, in order notto neglect the imperfective paradox completely, I will show how one specificproposal to solve the imperfective paradox (for the English progressive) can beintegrated into my account of imperfective aspect. This is the account of Dowty(1979:145–150). The solution involves an ‘intensionalisation’ of the semanticsof imperfective aspect. To put it simply, the imperfective now indicates thatthere is an eventuality to which the predicate applies in the normal, not theactual, course of eventualities. The formalisation of this new imperfectiveoperator IMP ′ is given in (137):(137) IMP ′ = λPλt w ′Inert t (w 0 ,w ′ ) → [eτ(e) ·⊃ t ⊕P(w′ )(e)]A crucial part of IMP ′ is the notion of inertia worlds. Inert t (w 0 ,w ′ ) readsas: w ′ is an inertia world for the actual world w 0 at time t, which meansthat w ′ is exactly like world w 0 up to and including t and after t the course