1 Basic verbal tenses and aspects
I call “basic” the verbal tenses governed by the ‹1, 1› type temporal modal operators. These are the following: Present (e.g., “works”), Past (e.g., “worked”), Future (e.g., “will work”) and perfect/imperfect.
Let us consider a set A within the temporal universe. Let us suppose that it corresponds to Past (alternatively, we can suppose that it corresponds to Future: the topological formalism is the same in both cases). Thus, the set corresponding to Future is the complement of A.
· Imperfect corresponds to the interior of A, that is, A without its boundary.
· Perfect corresponds to the closure of A, that is, A together with its boundary.
These topological interpretations correspond to the linguistic facts
· that the imperfect presents an unfinished action, having no term (in the same manner as the interior of the set contains only internal elements of the set, without elements of its boundary), and
· that the perfect presents a finished action, having a defined term in Present (in the same manner as the closure of the set contains, beside its internal elements, the elements of its boundary).
The modal operators of the basic tenses and aspects form the complete square and are the following (supposing that A corresponds to Past, which is the natural approach where the perfect and the imperfect are defined “for the past”):
· Past perfect: temporal some (“somewhere”),
· Past imperfect: temporal all (“anywhere”) [cf. Cinque’s (1999, 91, 96) difficulties in discerning between the imperfect and the habitual aspect: the answer is that they correspond to different types of generalized quantifiers, ‹1, 1› and ‹1›],
· Future: temporal none (“nowhere in Past”),
· Present: temporal not all (“not anywhere in Past”).
Note that these ‹1, 1› type temporal modal operators are distinct from the ‹1› type spatiotemporal operators described above (sect. 27).