§1. Our first theory concerns a set of elements, denoted by letters a, b, c, … and words derived from them. The simplest word, apart from a single letter, has the form ab; other words are formed by adjoining a letter on the right to a previous word. Thus ab, abc, abcd are words. We shall say e.g. d is “hooked on” to abc.

The only statements about words in the theory are of the form X = Y, where X, Y are words, and the sign = has the usual properties. The statements are general in the sense that if any word be substituted for a letter in X or Y, and X', Y' be the new words so got, then X' = Y'. Furthermore if X = Y, then Xa = Ya. Note that if e be hooked on to abcd, we get (abcd)e, which is written abcde.