WE have now explored, somewhat hastily it is true, that part of the philosophy of mathematics which does not demand a critical examination of the idea of class. In the preceding chapter, however, we found ourselves confronted by problems which make such an examination imperative. Before we can undertake it, we must consider certain other parts of the philosophy of mathematics, which we have hitherto ignored. In a synthetic treatment, the parts which we shall now be concerned with come first: they are more fundamental than anything that we have discussed hitherto. Three topics will concern us before we reach the theory of classes, namely: (1) the theory of deduction, (2) propositional functions, (3) descriptions. Of these, the third is not logically presupposed in the theory of classes, but it is a simpler example of the kind of theory that is needed in dealing with classes. It is the first topic, the theory of deduction, that will concern us in the present chapter.
Mathematics is a deductive science: starting from certain premisses, it arrives, by a strict process of deduction, at the various theorems which constitute it. It is true that, in the past, mathematical deductions were often greatly lacking in rigour; it is true also that perfect rigour is a scarcely attainable ideal. Nevertheless, in so far as rigour is lacking in a mathematical proof, the proof is defective; it is no defence to urge that common sense shows the result to be correct, for if we were to rely upon that, it would be better to dispense with argument altogether, rather than bring fallacy to the rescue of common sense. No appeal to common sense, or "intuition," or anything except strict deductive logic, ought to be needed in mathematics after the premisses have been laid down.
Kant, having observed that the geometers of his day could not prove their theorems by unaided argument, but required an appeal to the figure, invented a theory of mathematical reasoning according to which the inference is never strictly logical, but always requires the support of what is called "intuition." The whole trend of modern mathematics, with its increased pursuit of rigour, has been against this Kantian theory. The things in the mathematics of Kant's day which cannot be proved, cannot be known—for example, the axiom of parallels. What can be known, in mathematics and by mathematical methods, is what can be deduced from pure logic. What else is to belong to human knowledge must be ascertained otherwise—empirically, through the senses or through experience in some form, but not a priori. The positive grounds for this thesis are to be found in Principia Mathematica, passim; a controversial defence of it is given in the Principles of Mathematics. We cannot here do more than refer the reader to those works, since the subject is too vast for hasty treatment. Meanwhile, we shall assume that all mathematics is deductive, and proceed to inquire as to what is involved in deduction.
In deduction, we have one or more propositions called premisses, from which we infer a proposition called the conclusion. For our purposes, it will be convenient, when there are originally several premisses, to amalgamate them into a single proposition, so as to be able to speak of the premiss as well as of the conclusion. Thus we may regard deduction as a process by which we pass from knowledge of a certain proposition, the premiss, to knowledge of a certain other proposition, the conclusion. But we shall not regard such a process as logical deduction unless it is correct, i.e. unless there is such a relation between premiss and conclusion that we have a right to believe the conclusion if we know the premiss to be true. It is this relation that is chiefly of interest in the logical theory of deduction.
In order to be able validly to infer the truth of a proposition, we must know that some other proposition is true, and that there is between the two a relation of the sort called "implication," i.e. that (as we say) the premiss "implies" the conclusion. (We shall define this relation shortly.) Or we may know that a certain other proposition is false, and that there is a relation between the two of the sort called "disjunction," expressed by "
or
,"[32] so that the knowledge that the one is false allows us to infer that the other is true. Again, what we wish to infer may be the falsehood of some proposition, not its truth. This may be inferred from the truth of another proposition, provided we know that the two are "incompatible," i.e. that if one is true, the other is false. It may also be inferred from the falsehood of another proposition, in just the same circumstances in which the truth of the other might have been inferred from the truth of the one; i.e. from the falsehood of