A. R U L E S O F F O R M A T IO N FO R L A N G U A G E I
The syntactical method will here be developed in connection with two particular symbolic languages taken as object-languages. The first of these languages-we shall call it Language I, or, briefly, I-includes, on the mathematical side, the elementary arithmetic of the natural numbers to a certain limited extent, roughly corresponding to those theories which are designated as constructivist, finitist, or intuitionist. The limitation consists pri marily in the fact that only definite number-properties occurthat is to say, those of which the possession or non-possession by any number whatsoever can be determined in a finite number of steps according to a fixed method. It is on account of this limita tion that we call I a definite language, although it is not a definite language in the narrower sense of containing only definite, that is to say, resoluble (i.e. either demonstrable or refutable) sentences. Later on, we shall be dealing with Language II, which includes Language I within itself as a sub-language. Language II contains in addition indefinite concepts, and embraces both the arithmetic of the real numbers and mathematical analysis to the extent to which it is developed in classical mathematics, and further the theory of aggregates. Languages I and II do not only include mathematics, however; above all, they afford the possibility of constructing empirical sentences concerning any domain of objects. In II, for instance, both classical and relativistic physics can be formulated. We attach special importance to the syntactical treat ment of the synthetic (not purely logico-mathematical) sentences, which are usually ignored in modern logic. The mathematical sentences, considered from the point of view of language as a whole, are only aids to operation with empirical, that is to say, non-mathematical, sentences.