The Principles of Mathematics (1903)

§ 11

Symbolic or Formal Logic—I shall use these terms as synonyms—is the study of the various general types of deduction. The word symbolic designates the subject by an accidental characteristic, for the employment of mathematical symbols, here as elsewhere, is merely a theoretically irrelevant convenience. The syllogism inall its figures belongs to Symbolic Logic, and would be the whole subject if all deduction were syllogistic, as the scholastic tradition supposed. It is from the recognition of asyllogistic inferences that modern Symbolic Logic, from Leibniz onward, has derived the motive to progress. Since the publication of Boole’s Laws of Thought (1854), the subject has been pursued with a certain vigour, and has attained to a very considerable technical development[4]. Nevertheless, the subject achieved almost nothing of utility either to philosophy or to other branches of mathematics, until it was transformed by the new methods of Professor Peano[5]. Symbolic Logic has now become not only absolutely essential to every philosophical logician, but also necessary for the comprehension of mathematics generally, and even for the successful practice of certain branches of mathematics. How useful it is in practice can only be judged by those who have experienced the increase of power derived from acquiring it; its theoretical functions must be briefly set forth in the present chapter[6].(§ 11 ¶ 1)

§ 11 n. 1. By far the most complete account of the non-Peanesque methods will be found in the three volumes of Schröder, Vorlesungen über die Algebra der Logik, Leipzig, 1890, 1891, 1895.

§ 11 n. 2. See Formulaire de Mathématique, Turin, 1895, with subsequent editions in later years; also Revue de Mathématiques, Vol. VII, No. 1 (1900). The editions of the Formulaire will be quoted as F. 1895 and so on. The Revue de Mathématiques, which was originally the Revisti di Matematica, will be referred to as R. d. M..

§ 11 n. 3. In what follows the main outlines are due to Professor Peano, except as regards relations; even in those cases where I depart from his views, the problems considered have been suggested to me by his works.