Extensions of V.A. Smirnov’s temporal logic systems in the context of determinism

Abstract

The article is related to the consideration of a possible extension of V.A. Smirnov's systems in the context of the problem of determinism. The problem of determinism is traditionally closely connected with various systems of temporal logic. The considerations about determinism can be seen in the early works of Aristotle as well as in Prior's works on the foundations of temporal logic. The idea of branching time has a special place in temporal logic. It stands as one of the standard approaches to understanding determinism in temporal logic. At the present time there are several approaches to branching time logic. Each of them is not complete and has certain logico-philosophical limitations. This gives us a reason to appeal to some of Smirnov's works. In the 1980s, he proposed an alternative understanding of the logic of branching time, which allows us to eliminate some undesirable deterministic consequences in the interpretation of temporal statements. Smirnov's understanding of possibility is especially interesting. It allows us to display the ontological status of objects that Seymour Michael describes in his reconstruction of Diodorus Kronos' deterministic argument – if we shift to the past when we analyze histories, the $\Diamond A$ interpretation allows us to say that something is possible, but is not true at any point in the future. The paper discusses in detail Smirnov's original ideas in temporal logic as well as several systems proposed by him – $K_s$ and $K_r$. As a result, a possible extension of Smirnov's systems is proposed, the decidability of this extension is shown, and prospects for further study of the problem are identified.