A finite axiomatization of the set of strongly valid Ockhamist formulas

The subject of this paper is the tense and modal logic called 'Ockhamist' in [1, p. 574]. The main result is a completeness theorem for a finite axiomatization of validity relative to Ockhamist frames. This answers the question (left open in [1]) of the finite axiomatizability of this class of formulas. The Ockhamists' attitude towards tenses is an 'Actualist Indeterminist' point of view. Roughly speaking, Indeterminism pictures time as treelike; although a moment must have exactly one past it may have several possible futures. Actualism figures in the interpretation of the future tense, taking "' will happen" to mean "a holds at some moment of the 'actual' future". One consequence is that in general tensed formulas can be true or false only relative to a possible course of events, construed as the actual future. Ockhamist possibility and necessity are strictly connected with time, in that "necessary" is meant as "necessary given the past (including the present moment)": we say that c is (now) possible, whenever a is true in some possible world having the same past as the actual one. In particular, the principle of the "unpreventability of the past" holds: every proposition concerning only the past is necessarily true or necessarily false. Given a tree representing time, the actual future of a moment x is represented by a branch starting with x; it is natural to assume that the possible futures of x are represented by a set of branches satisfying suitable closure properties. Depending on how the closure properties are selected, various notions of validity can be defined: in particular, the two that in [1] are referred to as (Ockhamist) validity