Basic Confusions in Current Notions of Propositional Calculi

Abstract

BASIC CONFUSIONS IN CURRENT NOTIONS OF PROPOSITIONAL CALCULI IN a preceding article,1 we took note of three salient features in the so-called propositional calculi of modern mathematical logic: (I) the use of eonstants and variables to represent the possibility of propositional composition; (2) the tautologous character of all theorems in the calculus; (3) the truth-functionalinterpretation of such propositional compounds. Nevertheless, it may be recalled that of these three features only the last one was the subject of arty real discussion. Thus we sought to show how, when propositional compounds are interpl'.eted truth-functionally, it is impossible to give any sort of adequate account of so-called implicative compounds. And yet at the same time, such compounds, so far from being dispensable or capable of being explained away, are actually presupposed and employed in any conceivable development of propositional calculi. Accordingly, interpreting the types of propositional composition that enter into these calculi truthfunctionally , the calculi themselves are thereby rendered both inadequate and inconsistent. Nor did there seem to be any way out of the impasse, save that of frankly recognizing that these calculi are. not descriptive of logic at all, and that their subject matter consists not of objects of second intention, but of first intention. But why might not this third feature of the calculi be simply omitted? After all, it being only an interpretation, why would it not be possible merely to interpret implicative compounds in a way other than the truth-functional one, and thereby make it assured that the calculi were yeritably proposi,tional calculi? Indeed, the only reason we gave for our suggestion that the 1 Veatch, H., "Aristotelian and Mathematical Logic," The Thomiat, XIII (Jan. 1950)' pp. 50-96. 288 BASIC CONFUSIONS OF PROPOSITIONAL CALCULI fl89 calculi might have to be regarded as pertaining to objects of first intention rather than second intention was that, on the truth-functional interpretation of the theorems, these theorems simply ceased to be either adequately or consistently descriptive of propositional relations. Eliminate the truth-functional interpretation, therefore, and the whole difficulty might seem to vanish. Not only that, but a random glance at the theorems in any one of these calculi would certainly seem to indicate that they were descriptive of propositions. Thus it simply is true of propositions that: p ::) q . ::) . ,.._, q ::) ,,.._, p pVq·::)·qVp p · V · q V r:::) :p V q · V · r ,..._, p v ,.._, q . ::) . ,.._, (p . q) ......., (p . ,.._, p) In other words, upon examination, theorems of this sort do turn out to be tautologous in the sense defined,2 i. e. they are necessarily true, no matter what .proposi,tional values be substituted for the variables p, q, r, etc. Moreover, to clinch the point that it might be possible to interpret the implicative compounds in the calculus in a way other than truth-functionally, .we need only remark that this has actually been accomplished. For Prof. C. I. Lewis, in his so-called calculus of strict implication, has in effect done just this. Thus to follow Prof. Lewis' own account, he says, speaking of the truth-functional interpretation of propositional compounds : " In the usual terms, this means that the truth or falsity of p and q being given, the truth or falsity of,.__, p, of p ::) q, of p V ,....., q, of p · ::) · q ::) p, and of every other function of p and q, which can figure in the system is thereby categorically determined. ·Any system having this character-that there is no function or relation of the elements which is in the system except such as are categorically determined to be true 9 Ibid., p. 60. 240 HENRY VEATCH or to be false by the truth or falsity of their elementary terms-may be called a ' truth-value system.' " 3 On this basis, then, continues Prot Lewis," p :) q definitely holds except when p is true and q false, in which case it definitely fails to hold. This is, in fact, exactly what is expressed by its definition. By contrast p >-3 q 4 is definitely false if p is true and q false, but is not determined to be either true...