The necessity of the past and modal-tense logic incompleteness.
MICHAEL J. WHITE · Notre Dame Journal of Formal Logic · 1984
The idea that there is a variety of necessity, i.e., the necessity of unpreventability, unalterability, or irrevocability, for which it is true to say that the past is necessary is a notion of great antiquity 1 which still possesses considerable intuitive appeal.However, this idea proves difficult to express adequately in a modal propositional logic that possesses both tense and alethic modal operators.The obvious candidate for a thesis expressing the necessity of the past, and the one normally so employed (see [7], p. 117), is Al PpDLPp.In this paper I explore several problems connected with the use of Thesis Al to express the concept of the necessity of the past in a mixed modal-tense logic.Section 1 consists of a brief rehearsal of a "philosophical" difficulty encountered in employing Al to express the necessity of the past: it proves difficult to isolate this necessity from the remainder of time, i.e., to avoid a form of fatalism.This problem with Al has been previously recognized.In Sections 2 and 3, I discuss several more strictly logical problems with Al.Section 2 pertains to Arthur Prior's use of Al in his modal-tense logical reconstruction of the famous "Master" argument of Diodorus Cronus.It is shown that Prior's modal-tense logical version of the conclusion of the Master can be *I should like to thank Professors John David Stone and J. F. A. K. van Benthem and the, at most, three anonymous readers of this paper for their helpful comments on and criticisms of several more-or-less defective and awkward earlier versions.Any remaining errors or infelicities are those which I have all but insisted on.