The Principles of Mathematics (1903)

§ 13

The subject of Symbolic Logic consists of three parts, the calculus of propositions, the calculus of classes, and the calculus of relations. Between the first two, there is, within limits, a certain parallelism, which arises as follows: In any symbolic expression, the letters may be interpreted as classes or as propositions, and the relation of inclusion in the one case may be replaced by that of formal implication in the other. Thus, for example, in the principle of the syllogism, if a, b, c be classes, and a is contained in b, b in c, then a is contained in c; but if a, b, c be propositions, and a implies b, b implies c, then a implies c. A great deal has been made of this duality, and in the later editions of the Formulaire, Peano appears to have sacrificed logical precision to its preservation[9]. But, as a matter of fact, there are many ways in which the calculus of propositions differs from that of classes. Consider, for example, the following: If p, q, r are propositions, and p implies q or r, then p implies q or p implies r. This proposition is true; but its correlative is false, namely: If a, b, c are classes, and a is contained in b or c, then a is contained in b or a is contained in c. For example, English people are all either men or women, not are not all men nor yet all women. The fact is that the duality holds for propositions asserting of a variable term that it belongs to a class, i.e. such propositions as x is a man, provided that the implication involved be formal, i.e. one which holds for all values of x. But x is a man is itself not a proposition at all, being neither true nor false; and it is not with such entities that we are concerned in the propositional calculus, but with genuine propositions. To continue the above illustration: It is true that for all values of x, x is a man or a woman either implies x is a man or x is a woman. But it is false that x is a man or woman either implies x is a man or x is a woman for all values of x. Thus the implication involved, which is always one of the two, is not formal, since it does not hold for all values of x, being not always the same one of the two. The symbolic affinity of the propositional and the class logic is, in fact, something of a snare, and we have to decide which of the two we are to make fundamental. Mr McColl, in an important series of papers[10] , has contended for the view that implication and propositions are more fundamental than inclusion and classes; and in this opinion I agree with him. But he does not appear to me to realize adequately the distinction between genuine propositions and such as contain a real variable: thus he is led to speak of propositions as sometimes true and sometimes false, which of course is impossible with a genuine proposition. As the distinction involved is of very great importance, I shall dwell on it before proceeding further. A proposition, we may say, is anything that is true or that is false. An expression such as x is a man is therefore not a proposition, for it is neither true nor false. If we give to x any constant value whatever, the expression becomes a proposition: it is thus as it were a schematic form standing for any one of a whole class of propositions. And when we say x is a man implies x is mortal for all values of x, we are not asserting a single implication, but a class of implications; we have now a genuine proposition, in which, though the letter x appears, there is no real variable: the variable is absorbed in the same kind of way as the x under the integral sign in a definite integral, so that the result is no longer a function of x. Peano distinguishes a variable which appears in this way as apparent, since the proposition does not depend upon the variable; whereas in x is a man there are different propositions for different values of the variable, and the variable is what Peano calls real[11]. I shall speak of propositions exclusively where there is no real variable: where there are one or more real variables, and for all values of the variables the expression involved is a proposition, I shall call the expression a propositional function. The study of genuine propositions is, in my opinion, more fundamental than that of classes; but the study of propositional functions appears to be strictly on a par with that of classes, and indeed scarcely distinguishable therefrom. Peano, like McColl, at first regarded propositions as more fundamental than classes, but he, even more definitely, considered propositional functions rather than propositions. From this criticism, Schröder is exempt: his second volume deals with genuine propositions, and point out their formal differences from classs.(§ 13 ¶ 1)

§ 13 n. 1. On the points where the duality breaks down, cf. Schröder, op. cit., Vol. II, Lecture 21.

§ 13 n. 2. Cf. The Calculus of Equivalent Statements, Proceedings of the London Mathematical Society, Vol. IX and subsequent volumes; Symbolic Reasoning, Mind, Jan. 1880, Oct. 1897, and Jan. 1900; La Logique Symbolique et ses Applications, Bibliothèque du Congrès International de Philosophie, Vol. III (Paris, 1901). I shall in future quote the proceedings of the above Congress by the title Congrès.

§ 13 n. 3. F. 1901, p. 2.