**</span></span>1</font> </span></span>**

**Proliferation of verbal tenses**

The verbal tenses which are not “basic” are the following:

· anterior (has worked)

· anterior of the past (had worked)

· future of the past (would work)

· anterior of the future (will have worked)

· anterior of the future of the past (would have worked)

Each of them is presented in, at least, some known natural languages. This list of the verbal tenses is exhaustive for all the languages known presently (Cinque 1999, 82-83).

The basic feature of all these tense is introducing of another referential time (*R*) beside Present (*P*). Let us consider the corresponding topology. This means that the set *A* of the temporal universe (which corresponds to either Past or Future) is, in turn, divided into two parts by additional boundary *R*. Thus, the set *A* becomes a set containing subsets.

In the case of the anterior of the future of the past (*e.g.*, “She promised in November that they would have received her paper by the first day of term”) one of the subsets of *A* is, again, divided into two subsets.

This means that instead of the topological operators of the “basic” tenses with their generalized quantifiers of ‹1, 1› type, here, we have the topological operators with quantifiers of ‹1, 2› type and even (for the anterior of the future of the past) of ‹1, 3› type.

Each of these tenses could be presented in either perfect or imperfect, depending on inclusion or exclusion of the boundary, correspondingly.

The mathematical formalism is the same as has been first proposed by (Kurucz, Wolter, Zakharyaschev, 2005) for the distance spaces (let us remind that the time of human perception is also a distance space). The four topological operators of the square of *same* are quite similar, and so, it is enough for illustration to describe only for *same *(< means temporal precedence, = temporal coincidence):

· anterior: ∃^{<}^{R}^{=}* ^{P}* (imperfect), ∃

^{£}

^{R}^{=}

*(perfect)*

^{P}· anterior of the past: ∃^{ <}^{R}^{<}* ^{P }*(imperfect), ∃

^{ }

^{£}

^{R}^{<}

*(perfect)*

^{P }· future of the past: ∃^{R}^{<}^{x }^{<}* ^{P }*(imperfect), ∃

^{R}^{<}

^{x }^{£}

*(perfect), where*

^{P }*x*is a temporal variable,

· anterior of the future: ∃^{P}^{<}^{x}^{<}* ^{R }*(imperfect), ∃

^{P}^{<}

^{x}^{ }

^{£}

*(perfect)*

^{R }· anterior of the future of the past: ∃^{R1}^{<}^{x}^{<}^{R2}^{<}* ^{P }*(imperfect), ∃

^{R1}^{<}

^{x}^{£}

^{R2}^{<}

*(perfect), where*

^{P }*R1*and

*R2*are the first and the second reference points (topological boundaries) in Past.

(I am far from sure that all of the perfect forms above do really exist in the natural languages).

It is interesting to note the difference between past (“worked”) and anterior (“has worked”): both refer to Present but with different type quantifiers. This difference reflexes a difference in the complexity of the corresponding systems: in the anterior tense, the boundary of Present is not the only reference point but simply a singular value possessed by a mobile reference boundary. Thus, the anterior operators take into account the complex structure of the set *A* (which is considered in the “basic” past tense as having no internal structure).

The new boundary R is defined through the notion of distance (from P), that is, via temporal preference logic (either strong preference < or weak ) with the quantifiers of ‹1, 1, 1› type. In the particular case of the anterior of the future of the past, when there are two reference boundaries R1 and R2, they are defined not independently but in relation to each other. Thus, the corresponding quantifier is of ‹1, 1, 2› type (not ‹1, 1, 1, 1›).

One can conclude that the proliferation of verbal tenses is a result of combination of two kinds of temporal logic, topological and distance ones.