The Principles of Mathematics (1903)

§ 18

The ten axioms are the following. (1) If p implies q, then p implies q[14]: in other words, whatever p and q may be, p implies q is a proposition. (2) If p implies q, then p implies p: in other words, whatever implies anything is also a proposition. (3) If p implies q, then q implies q; in other words, whatever is implied by anything is a proposition. (4) A true hypothesis in an implication may be dropped, and the consequent asserted. This is a principle incapable of formal symbolic statement, and illustrating the essential limitations of formalism—a point to which I shall return at a later stage. This definition is highly artificial, and illustrates the great distinction between mathematical and philosophical definitions. It is as follows: If p implies q, then, if q implies q, pq (the logical product of p and q) means that if p implies that q implies r, then r is true. In other words, if p and q are propositions, their joint assertion is equivalent to saying that every proposition is true which is such that the first implies that the second implies it. We cannot, with formal correctness, state our definition in this shorter form, for the hypothesis p and q are propositions is already the logical product of p is a proposition and q is a proposition. We cannow state the six main principles of inference, to each of which, owing to its importance, a name is to be given; of these all except the last will be found in Peano’s accounts of the subject. (5) If p implies p and q implies q, then pq implies p. this is called simplification, and asserts merely that the joint assertion of two propositions implies the assertion of the first of the two. (6) If p implies q and q implies r, then p implies r. This will be called the syllogism. (7) If q implies q and r implies r, and if p implies that q implies r, then pq implies r. This is the principle of importation. In the hypothesis, we have a product of three propositions; but this can of course be defined by means of the product of two. The principle states that if p implies that q implies r, then r follows from the joint assertion of p and q. For example: If I call on so-and-so, then if she is at home I shall be admitted implies If I call on so-and-so and she is at home, I shall be admitted. (8) If p implies p and q implies q, then, if pq implies r, then p implies that q implies r. This is the converse of the preceding principle, and is called exportation[15]. The previous illustration reversed will illustrate the principle. (9) If p implies q and p implies r, then p implies qr: in other words, a proposition which implies each of two propositions implies them both. This is called the principle of composition. (10) If p implies p and q implies q, then p implies q implies p implies p. This is called the principle of reduction; it has less self-evidence than the previous principles, but is equivalent to many propositions that are self-evident. I prefer it to these, because it is explicitly concerned, like its predecessors, with implication, and has the same kind of logical character as they have. If we remember that p implies q is equivalent to q or not-p, we can easily convince ourselves that the above principle is true; for p implies q implies p is equivalent to p or the denial of q or not-p, i.e. to p or p and not q, i.e. to p. But this way of persuading ourselves that the principle of reduction is true involves many logical principles which have not yet been demonstrated, and cannot be demonstrated except by reduction or some equivalent.The principle is especially useful in connection with negation. Without its help, by means of the first nine principles, we can prove the law of contradiction; we can prove, if p and q be propositions, that p implies not-not-p; that p implies not-q is equivalent to q implies not-p and to not-pq; that p implies q implies not-q implies not-p; that p implies that not-p implies p; that not-p is equivalent to p implies not-p; and that p implies not-q is equivalent to not-not-p implies not-q. But we cannot prove without reduction or some equivalent (so far at least as I have been able to discover) that p or not-p must be true (the law of the excluded middle); that every proposition is equivalent to the negation of some other proposition; that not-not-p implies p; that not-q implies not-p implies p implies q; that not-p implies p implies p, or that p implies q implies q or not-p. Each of these assumptions is equivalent to the principle of reduction, and may, if we choose, be substituted for it. Some of them—especially excluded middle and double negation—appear to have far more self-evidence. Bu when we have seen how to define disjunction and engation in terms of implication, we shall see that the supposed simplicity vanishes, and that, for formal purposes at any rate, reduction is simpler than any of the possible alternatives. For this reason I retain it among my premisses in preference to more usual and more superficially obvious propositions.(§ 18 ¶ 1)

§ 18 n. 1. Note that the implications denoted by if and then, in these axioms, are formal, while those denoted by implies are material.

§ 18 n. 2. (7) and (8) cannot (I think) be deduced from the definition of the logical product, because they are required for passing from If p is a proposition, then q is a proposition implies etc. to If p and q are propositions, then etc.