Early Work on the Substitutional Theory [1905]

Here we print two texts on the substitutional theory. The first of them, 3a, which gives the earliest detailed explanation of that theory, is a letter of 15 December 1905 from Russell to G. H. Hardy. The second, 3b, entitled “On Substitution” and dated 22 December 1905 by a note on it in Russell’s hand, is the earliest extant paper exclusively on the theory. The manuscript of 3b was originally filed in the Russell Archives with Hardy’s letters, reflecting the fact that this 1905 paper was sent to Hardy-probably early the following year. (As we see below, it was first sent to another mathematician.)

The next indication, after the letter to Hardy on 15 December, that Russell was developing the substitutional theory was a letter which he wrote to Couturat four days later about existence theorems and the theory’s role in solving the logical paradoxes:

Assuredly, one cannot take consistency as a criterion for existence.… It is certain that, by the methods which have been used until now,

existence theorems have been proved too easily; that is what the contradictions show. The method of substitution, by which I dispense with classes, has almost the same effect in practice as the theory of types (in

my Appendix B 〈 in the Principles of Mathematics〉). I write p x a

or pa ‘x

for the proposition which results from the substitution of x for a in p. Then the operation pa replaces functions and classes. For example,

xy .: p x a

.p, a . p y a

Df

pa qb.: p x a

.x . q x b

Df

and so on. But pa is meaningless in itself. It is only a symbol whose usage is defined in certain contexts.1