Transition Semantics and Determinism in Temporal Logic

Abstract

This article discusses the preconditions for the formation of transition semantic in the context of determinism. The influence of ideas about determinism can be traced in various branches of philosophy, including temporal logic. Considering some aspects of determinism allows us to describe in detail the reasons for the formation in the logic of certain approaches to evaluating temporal statements. Thus, this paper considers Arthur Prior's analysis of the deterministic arguments underlying the classical variants of Ockhamist and Peircean-type semantics and the “wait and see” principle. Further study of these semantics, for a number of logical and philosophical reasons, diverges into two independent and competing directions — Branching Space-Time and the Thin Red Line theory. These systems are variants of the evaluation of modern statements through pairs <moment, history>, which still need to be completed due to some of their shortcomings. The latter has prompted some researchers to look for alternative evaluation options. In particular, the work of John McFarlane is noteworthy, where special attention is paid to determining the conditions for the adequate representation of the idea of an open future. In his understanding, the key condition is the combination of the two philosophical intuitions about determinism, which can be realized by relativizing the evaluation of statements through the context of the statement. Subsequently, these works were brought to the attention of Thomas Muller, who, dealing with the issue of mapping the local aspects of modal consistency (also from a philosophical point of view), put forward the idea of the transition semantic. Its current development is reflected in the works of Antje Rumberg, who proposes a variant of semantics with the presence of the stability operator. The resulting variant of semantics allows us to solve several basic questions concerning determinism. In particular, the presence of the stability operator makes it possible to show the specific nature of the expression of statements about accidental events. Also, the evaluation of statements allows us to accurately and consistently implement the “wait and see” principle proposed by Prior. Nevertheless, this variant of semantics has a number of problems with the previously described notions about the essence of the future.