The Syntax of Time-Distinctions

THE SYNTAX OF TIME-DISTINCTIONS1 I. Truth and Time in Ancient and Modern Logic Truth, on the face of it, is a property of propositions which is liable to alter with the time at which they are put forward. Thus 'Socrates is sitting down' is true at any time at which he is in fact sitting down, and false at all other times. Against this, it is not uncommonly argued that the sentence 'Socrates is sitting down' does not express a complete proposition, but rather a function of a date. It is short for 'Socrates is sitting down at —,' where the blank is understood as being fiUed by some unambiguous indication of the date at which the sentence is uttered ('4 P. M. on April 3, 325 B. C.,' for example). It therefore expresses different propositions when uttered at different times, and each one of the propositions it expresses is either true always or false always. Modern exact logicians commonly operate with 'propositions' in the second (tunelessly true) sense, while ancient and medieval logicians had in mind 'propositions' of the first ('tensed') sort. It should be emphasised, however, that there are no grounds of a purely logical character for the current preference, and that 'propositions' in the ancient and medieval sense lend themselves as readily to the application of contemporary logical techniques and procedures as do 'propositions' in the modern sense. (At this point Strawson, who regards it as a limitation of modern methods that they cannot cope with 'propositions' in the ancient and medieval sense, and Quine, who objects to the use of such 'propositions' in logic because modern methods cannot handle them, would seem to be equaUy in error2.) Moreover, the actual application of these techniques and procedures to tensed propositions promises to yield results of considerable interest both logically and metalogically. This was dimly seen by C. S. Peirce, who 'never shared' the common opinion that time is an 'extra-logical' matter, though he thought, in 1903, that 'logic had not yet reached that state of development at which the introduction of 1 Presidential Address given at the New Zealand Congress of Philosophy, August 27, 1954. 2 For this dispute see W. V. Quine, 'Mr. Strawson on Logical Theory', Mind, October 1953, pp. 440—443. 8 Franciscan Studies. 1958105 Io6A. N. PRIOR temporal modifications of its forms would not result in great confusion.'3 "What the time was not ripe for in 1903, it may well be ripe for now, for in the intervening period we have acquired a vast fund of knowledge about the possible structures of modal systems, and (as the scholastic logocians knew4) tense and mood are species of the same genus. We have also begun to learn how to handle a logic of three truth-values, and we shall find this to the point too. Suppose we use the ordinary variables 'p', 'q', Y1 etc. for 'propositions ' in the ancient and medieval rather than the modern sense, and employ the usual truth-operators in the following way (admitting in the meantime only two truth-values): — 'Np' ('Not p') is true at any time at which 'p' is false, and false at aU other times. 'Kpq' ('Both ? and q') is true at any time at which both 'p' and 'q' are true, and false at aU other times. 'Apq' ('Either ? or q') is false at any time at which both 'p' and 'q' are false, and true at all other times. 'Cpq' (? ? then q') is false at any time at which 'p' is true and 'q' false, and true at all other times. 'Epq' ('If and only if ? then q') is true at any time at which 'p' and 'q' have the same truth-value, and false at all other times. The classical propositional calculus, with its symbols thus interpreted, will then hold in its entirety, unaltered. For in the formula 'Epp', for example, 'p' will be equivalent to 'p at x', where 'x' is the date of utterance , and the whole therefore to ? (p at x) (p at x)', in which the arguments are propositions in the modern sense, substitutable for the variables in the 'Epp' of the propositional calculus as currently interpreted...