5.5303 Roughly speaking, to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing at all.

5.531 Thus I do not write ‘f(a, b) . a = b’, but ‘f(a, a)’ (or ’f(b, b)); and not ‘f(a,b) .  Pa = b’, but ‘f(a, b)’.

5.532 And analogously I do not write ‘(dx, y) . f(x, y) . x = y’, but ’(dx) . f(x, x)’; and not ‘(dx, y) . f(x, y) .  Px = y’, but ‘(dx, y) . f(x, y)’. 5.5321 Thus, for example, instead of ‘(x) :  fx z x = a’ we write ’(dx) . fx . z :  (dx, y) . fx. fy’.  And the proposition, ‘Only one x satisfies f( )’, will read ‘(dx) . fx :  P(dx, y) . fx . fy’.

5.533 The identity-sign, therefore, is not an essential constituent of conceptual notation.

5.534 And now we see that in a correct conceptual notation pseudo-propositions like ‘a = a’, ‘a = b . b = c . z a = c’, ‘(x) . x = x’, ’(dx) . x = a’, etc. cannot even be written down.

5.535 This also disposes of all the problems that were connected with such pseudo-propositions.  All the problems that Russell’s ‘axiom of infinity’ brings with it can be solved at this point.  What the axiom of infinity is intended to say would express itself in language through the existence of infinitely many names with different meanings.

5.5351 There are certain cases in which one is tempted to use expressions of the form ‘a = a’ or ‘p z p’ and the like.  In fact, this happens when one wants to talk about prototypes, e.g. about proposition, thing, etc.  Thus in Russell’s Principles of Mathematics ’p is a proposition’—­which is nonsense--was given the symbolic rendering ‘p z p’ and placed as an hypothesis in front of certain propositions in order to exclude from their argument-places everything but propositions. (It is nonsense to place the hypothesis ‘p z p’ in front of a proposition, in order to ensure that its arguments shall have the right form, if only because with a non-proposition as argument the hypothesis becomes not false but nonsensical, and because arguments of the wrong kind make the proposition itself nonsensical, so that it preserves itself from wrong arguments just as well, or as badly, as the hypothesis without sense that was appended for that purpose.)

5.5352 In the same way people have wanted to express, ’There are no things ’, by writing ‘P(dx) . x = x’.  But even if this were a proposition, would it not be equally true if in fact ‘there were things’ but they were not identical with themselves?

5.54 In the general propositional form propositions occur in other propositions only as bases of truth-operations.

5.541 At first sight it looks as if it were also possible for one proposition to occur in another in a different way.  Particularly with certain forms of proposition in psychology, such as ’A believes that p is the case’ and A has the thought p’, etc.  For if these are considered superficially, it looks as if the proposition p stood in some kind of relation to an object A. (And in modern theory of knowledge (Russell, Moore, etc.) these propositions have actually been construed in this way.)