Aorist aspect
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The aorist aspect was one of the three original aspects that defined the Indo-European verbal paradigm. Unlike tenses, aspects have no reference to the time of the verbal action; instead, they describe the action's state of completion, or singularity. In Indo-European, this distinction manifests itself as a tripartite division between the aorist, imperfect and perfect. The aorist indicates a completed, singular action, while the imperfect indicates an incomplete, sometimes repeated action, and the perfect indicates actions of varying completion that in some manner affect the present.
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Usage
The critical fact to understand about the aorist is that it is not a tense, but an aspect. In Indo-European, the aorist generally functions alongside the imperfect, the two forming a semantic pair. One can easily appreciate the difference between them in English verb pairs such as: to see and to look, to find and to search, and to hear and to listen; in each of these pairs, the former, the aorist-like form, indicates a singular, momentary, finite action, with a clearly demarcated beginning and end, while the latter, the imperfect like form, indicates an ongoing process that cannot be isolated to one specific moment or action, and may also connote a repeated or habitual action.
Far more than tense, this aspectual duality provided the basis of verbal meaning in Proto-Indo-European, as is well demonstrated by Ancient Greek. For example, in Greek's subjunctive and optative moods (roughly equivalent to "may" and "might") there is no tense distinction made (such as English's "I might leave" and "I might have left") but only an aspect distinction, such that ακουωμεν and ακουσωεν mean "we may listen" and "we may hear", and not "we may listen" and "we may have listened."
Morphology
In Latin, Greek, and Sanskrit, the aorist is marked by several morphological devices, but three stand out as most common.
The s-aorist
The first is the s-aorist, so called because an 's' is inserted between the root and the personal ending. In Latin, for example, "dico" means "I say", while "dixi" (from dic-s-i) means "I said"; in Greek, "ακουω" means "I hear", while "ηκουσα" means "I heard." (Grammatical note: the first letter of "ηκουσα" is an eta, and not an alpha, because of a Greek verbal augment that marks the past indicative tense.)
Reduplication
The second marker of the aorist is reduplication. While a reduplication is more commonly associated with the morphology of the perfect, there are sporadic verbs which use it in the aorist. An example from Latin is "parco", which means "I pardon", while "peperci" means "I pardoned"; the Greek verb "αγω" — "I lead" — has the aorist "ηγαγον": "I led". (Grammatical note: the first letter of "ηγαγον" is an eta, and not an alpha, because of a Greek verbal augment that marks the past indicative tense.)
Ablaut
The aorist's third marker is a change in vowel grade, a process known as ablaut. Indo-European made great use of ablaut to express semantic changes morphologically, in fact, English uses ablaut abundantly, creating such verb forms as: swim, swam, swum; come, came, come; and take, took, taken. English further uses ablaut in extended forms, such as: sit, seat, sat, set (etymologically, to set is to cause to sit); and sing, sang, sung, song. In Latin, ablaut was a common marker of the aorist, for example: "capio", "I take"; but "cepi", "I took"; and Greek "λειπω", "I leave", but "ελιπον", "I left."