The Principles of Mathematics (1903)

§ 38

At this point, it is necessary to consider a very difficult logical problem, namely, the distinction between a proposition actually asserted, and a proposition considered merely as a complex concept. One of our indemonstrable principles was, it will be remembered, that if the hypothesis of an implication is true, it may be dropped, and the consequent asserted. This principle, it was observed, eludes formal statement, and points to a certain failure of formalism in general. The principle is employed whenever a proposition is said to be proved; for what happens is, in all such cases, that the proposition is shown to be implied by some true proposition. Another form in which the principle is constantly employed is the substitution of a constant, satisfying the hypothesis, in the consequent of a formal implication. If ϕx implies ψx for all values of x, and if a is a constant satisfying ϕx, we can assert ψa, dropping the true hypothesis ϕa. This occurs, for example, whenever any of those rules of inference which employ the hypothesis that the variables involved are propositions, are applied to particular propositions.(§ 38 ¶ 1)

The independence of this principle is brought out by a consideration of Lewis Carroll's puzzle, What the Tortoise said to Achilles[35]. The principles of inference which we accepted lead to the proposition that, if p and q be propositions, then p together with p implies q implies q. At first sight, it might be thought that this would enable us to assert q provided p is true and implies q. But the puzzle in question shows that this is not the case, and that, until we have some new principle, we shall only be led into an endless regress of more and more complicated implications, without ever arriving at the assertion of q. We need, in fact, the notion of therefore, which is quite different from the notion of implies, and holds between different entities. In grammar, the distinction is that between a verb and a verbal noun, between, say, A is greater than B and A's being greater than B. In the first of these, a proposition is actually asserted, whereas in the second it is merely considered. But these are psychological terms, whereas the difference which I desire to express is genuinely logical. It is plain that, if I may be allowed to use the word assertion in a non-psychological sense, the proposition p implies q asserts an implication, though it does not assert p or q. The p and the q which enter into this proposition are not strictly the same as the p or the q which are separate propositions, at least, if they are true. The question is: How does a proposition differ by being actually true from what it would be as an entity if it were not true? It is plain that true and false propositions alike are entities of a kind, but that true propositions have a quality not belonging to false ones, a quality which, in a non-psychological sense, may be called being asserted. Yet there are grave difficulties in forming a consistent theory on this point, for if assertion in any way changed a proposition, no proposition which can possibly in any context be unasserted could be true, since when asserted it would become a different proposition. But this is plainly false; for in p implies q, p and q are not asserted, and yet they may be true. Leaving this puzzle to logic, however, we must insist that there is a difference of some kind between an asserted and an unasserted proposition[36]. When we say therefore, we state a relation which can only hold between asserted propositions, and which thus differs from implication. Wherever therefore occurs, the hypothesis may be dropped, and the conclusion asserted by itself. This seems to be the first step in answering Lewis Carroll's puzzle.(§ 38 ¶ 2)

§ 38 n. 2. Frege (loc. cit.) has a special symbol to denote assertion.