# The Principles of Mathematics (1903)

## § 478

There are some difficulties in the above theory which it will be well to discuss. In the first place, it seems doubtful whether the introduction of truth-values marksany real analysis. If we consider, say, Caesar died, it would seem that what is asserted is the propositional concept the death of Caesar, not the truth of the death of Caesar. This latter seems to be merely another propositional concept, asserted in the death of Caesar is true, which is not, I think, the same proposition as Caesar died. There is great difficulty in avoiding psychological elements here, and it would seem that Frege has allowed them to intrude in describing judgment as the recognition of truth (Gg. p. x). The difficulty is due to the fact that there is a psychological sense of assertion, which is what is lacking to Meinong's Annahmen, and that this does not run parallel with the logical sense. Psychologically, any proposition, whether true or false, may be merely thought of, or may be actually asserted: but for this possibility, error would be impossible. But logically, true propositions only are asserted, though they may occur in an unasserted form as parts of other propositions. In p implies q, either or both of the propositions p, q may be true, yet each, in this proposition, is unasserted in a logical, and not merely in a psychological, sense. Thus assertion has a definite place among logical notions, though there is a psychological notion of assertion to which nothing logical corresponds. But assertion does not seem to be a constituent of an asserted proposition, although it is, in some sense, contained in an asserted proposition. If p is a proposition, p's truth is a concept which has being even if p is false, and thus p's truth is not the same as p asserted. Thus no concept can be found which is equivalent to p asserted, and therefore assertion is not a constituent in p asserted. Yet assertion is not a term to which p, when asserted, has an external relation; for any such relation would need to be itself asserted in order to yield what we want. Also a difficulty arises owing to the apparent fact, which may however be doubted, that an asserted proposition can never be part of another proposition: thus, if this be a fact, where any statement is made about p asserted, it is not really about p asserted, but only about the assertion of p. This difficulty becomes serious in the case of Frege's one and only principle of inference (Bs. p. 9): p is true and p implies q; therefore q is true[119]. Here it is quite essential that there should be three actual assertions, otherwise the assertion of propositions deduced from asserted premises would be impossible; yet the three assertions together form one proposition, whose unity is shown by the word therefore, without which q would not have been deduced, but would have been asserted as a fresh premise.(§ 478 ¶ 1)

It is also almost impossible, at least to me, to divorce assertion from truth, as Frege does. an asserted proposition, it would seem, must be the same as a true proposition. We may allow that negation belongs to the content of a proposition (Bs. p. 4), and regard every assertion as asserting something to be true. We shall then correlate p and not-p as unasserted propositions, and regard p is false as meaning not-p is true. But to divorce assertion from truth seems only possible by taking assertion in a psychological sense.(§ 478 ¶ 2)