Aspect in Ancient Greek - Nijmegen Centre for Semantics

3.2 The perfective-imperfective distinction 45in the extension of a telic predicate are already maximal with respect to thispredicate).Unfortunately, in its simple form, this perfective operator won’t do thejob. The problem is the following: imagine that John sleeps from 1 to 2o’clock and from 3 to 4 o’clock. Let’s translate the non-quantised John sleepas the eventuality predicate j sleep. The extension of AOR ′ (j sleep) shouldinclude the two eventualities, the one from 1 to 2, e 1 , and the one from 3 to 4,e 2 , for both are locally maximal. Just like any other two eventualities, e 1 ande 2 together constitute a third eventuality, e 3 . Now the question is: is e 3 in theextension of j sleep? Krifka seems to assume that atelic predicates, like Johnsleep, are not only non-quantised, but also cumulative (see footnote 15):(63) A property P is cumulative iff for all e, e ′ if P(e) and P(e ′ ) thenP(e ⊔ e ′ )If one assumes that atelic predicates are cumulative, one has to accept thatj sleep holds of e 3 , too. But if j sleep holds of e 3 and e 1 ⊏ e 3 , thenAOR ′ (j sleep) does not hold of e 1 . But if AOR ′ (j sleep) does not hold ofe 1 , AOR ′ does not do what it should do, since e 1 is locally maximal with respectto j sleep and therefore we want it to be in the extension of AOR ′ (j sleep).To fix this, Krifka (1989b:180) proposes AOR instead of AOR ′ :(64) AOR = λPλe[P(e) ∧ ∀e ′ [(P(e ′ ) ∧e ⊏ e ′ ) → ¬ECONV(e ′ )]] 16AOR(P) holds of an eventuality e if P holds of this eventuality and all eventualitiese ′ of which e is a proper part and of which P holds are not convex(ECONV). This revision is meant to ensure that in the above scenario e 1and e 2 are in the extension of AOR(j sleep), by disregarding e 3 because itis not convex. Krifka does not define ECONV, the property convexity foreventualities, but he does define it in the temporal domain (Krifka 1989b:155):(65) t is convex iff for all t ′ , t ′′ if t ′ ⊑ t and t ′′ ⊑ t then for all t ′′′ such thatt ′ ≼ t ′′′ ≼ t ′′ it holds that t ′′′ ⊑ tAs this definition shows, a convex time is a time without interruptions, i.e. atime interval. 17This concludes the discussion of the AOR operator. Keep in mind that theproblem with the simpler version, AOR ′ , arises because atelic predicates areassumed to be not only non-quantised, but also cumulative. I will return tothis in section 4.5.16 I assume that “ECONV(e)” in the definition of Krifka is a typo and should be“ECONV(e ′ )”.17 Krifka assumes that the domain of times is structured as a join semi-lattice withoutbottom element, just like the domain of eventualities. As a consequence, not all times areintervals.