The Principles of Mathematics (1903)

§ 475

The work of Frege, which appears to be far less known than it deserves, contains many of the doctrines set forth in Parts I and II of the present work, and where it differs from the views which I have advocated, the differences demand discussion. Frege's work abounds in subtle distinctions, and avoids all the usual fallacies which beset writers on Logic. His symbolism, though unfortunately so cumbrous as to be very difficult to employ in practice, is based upon an analysis of logical notions much more profound than Peano's, and is philosophically very superior to its more convenient rival. In what follows, I shall try briefly to expound Frege's theories on the most important points, and to explain my grounds for differing where I do differ. But the points of disagreement are very few and slight compared to those of agreement. They all result from difference on three points: (1) Frege does not think there is a contradiction in the notion of concepts which cannot be made logical subjects (see § 49 supra); (2) he thinks that, if a term a occurs in a proposition, the proposition can always be analysed into a and an assertion about a (see Chapter VII); (3) he is not aware of the contradiction discussed in Chapter X. These are very fundamental matters, and it will be well here to discuss them afresh, since the previous discussion was written in almost complete ignorance of Frege's work.(§ 475 ¶ 1)

Frege is compelled, as I have been, to employ common words in technical senses which depart more or less from usage. As his departures are frequently different from mine, a difficulty arises as regards the translation of his terms. Some of these, to avoid confusion, I shall leave untranslated, since every English equivalent that I can think of has been already employed by me in a slightly different sense.(§ 475 ¶ 2)

The principal heads under which Frege's doctrines may be discussed are the following: (1) meaning and indication; (2) truth-values and judgment; (3) Begriff and Gegenstand; (4) classes; (5) implication and symbolic logic; (6) the definition of integers and the principle of abstraction; (7) mathematical induction and the theory of progressions. I shall deal successively with these topics.(§ 475 ¶ 3)