The Principles of Mathematics (1903)

§ 12

Symbolic Logic is essentially concerned with inference in general[7], and is distinguished from various special branches of mathematics by its generality. Neither mathematics nor symbolic logic will study such special relations as (say) temporal priority, but mathematics will deal explicitly with the class of relations possessing the formal properties of temporal priority—properties which are summed up in the notion of continuity[8]. And the formal properties of a relation may be defined as those that can be expressed interms of logical constants, or again as those which, while they are preserved, permit our relation to be varied without invalidating any inference in which the said relation is regarded in the light of a variable. But symbolic logic, in the narrower sense which is convenient, will not investigate what inferences are possible in respect of continuous relations (i.e. relations generating continuous series); this investigation belongs to mathematics, but is still too special for symbolic logic. What symbolic logic does investigate is the general rules by which inferences are made, and it requires a classification of relations or propositions only in so far as these general rules introduce particular notions. The particular notions which appear in the propositions of symbolic logic, and all others definable interms of these notions, are the logical constants. The number of indefinable logical constants is not great: it appears, in fact, to be eight or nine. These notions alone form the subject-matter of the whole of mathematics: no others, except such as are definable in terms of the original eight or nine, occur anywhere in Arithmetic, Geometry, or rational Dynamics. For the technical study of Symbolic Logic, it is convenient to take as a single indefinable the notion of a formal implication, i.e. of such propositions as x is a man implies x is mortal, for all values of x—propositions whose general type is: ϕ(x) implies ψ(x), for all values of x, where ϕ(x), ψ(x), for all values of x, are propositions. The analysis of this notion of formal implication belongs to the principles of the subject, but is not required for its formal development. In addition to this notion, we require as indefinables the following: Implication between propositions not containing variables, the relation of a term to the class of which it is a member, the notion of such that, the notion of relation, and truth. By means of these notions, all the propositions of symbolic logic can be stated.(§ 12 ¶ 1)

§ 12 n. 1. I may as well say at once that I do not distinguish between inference and deduction. What is called induction appears to me to be either disguised deduction or a mere method of making plausible guesses.