This article addresses the asymmetry between the two main aspectual paradigms in the Classical Greek verbal system: the imperfective and the aorist (perfective). Whereas the imperfective has separate indicative forms for present and past time reference, i.e. the ‘primary’ and the ‘secondary’ indicative, the aorist only has a secondary (‘past’) indicative. I argue that this asymmetry is not only morphological but also semantic. That is, while the secondary imperfective indicative (the ‘imperfect’) is confined to past time reference, the secondary aorist indicative is used not only to refer to the past but also to the present. It then enters into aspectual competition with the primary imperfective indicative (the ‘present’). Based on R.W. Langacker's (2011) Cognitive Grammar account of aspect, I distinguish five types of context in which a present tense form with perfective aspect is a desideratum, and argue that here the secondary aorist indicative is used to fulfil this function. Moreover, I present a diachronic account of the origin of this remarkable asymmetry, arguing that the aorist indicative was never a past tense to begin with.