The Principles of Mathematics (1903)

§ 493

Implication and Symbolic Logic. The relation which Frege employs as fundamental in the logic of propositions is not exactly the same as what I have called implication: it is a relation which holds between p and q whenever q is true or p is not true, whereas the relation which I employ holds whenever p and q are propositions, and q is true or p is false. That is to say, Frege's relation holds when p is not a proposition at all, whatever q may be; mine does not hold unless p and q are propositions. His definition has the formal advantage that it avoids the necessity for hypotheses of the form p and q are propositions; but it has the disadvantage that it does not lead to a definition of proposition and of negation. In fact, negation is taken by Frege as indefinable; proposition is introduced by means of the indefinable notion of a truth-value. Whatever x may be, the truth-value of x is to indicate the true if x is true, and the false in all other cases. Frege's notation has certain advantages over Peano's, in spite of the fact that it is exceedingly cumbrous and difficult to use. He invariably defines expressions for all values of the variable, whereas Peano's definitions are often preceded by a hypothesis. He has a special symbol for assertion, and he is able to assert for all values of x a propositional function not stating an implication, which Peano's symbolism will not do. He also distinguishes, by the use of Latin and German letters respectively, between any proposition of a certain propositional function and all such propositions. By always using implications, Frege avoids the logical product of two propositions, and therefore has no axioms corresponding to Importation and Exportation[136]. Thus the joint assertion of p and q is the denial of p implies not-q.(§ 493 ¶ 1)