Principia mathematica, vol. 1 (of 3)

In this number, we shall be concerned with rules analogous, more or less, to those of ordinary algebra. It is from these rules that the usual "calculus of formal logic" starts. Treated as a "calculus," the rules of deduction are capable of many other interpretations. But all other interpretations depend upon the one here considered, since in all of them we deduce consequences from our rules, and thus presuppose the theory of deduction. One very simple interpretation of the "calculus" is as follows: The entities considered are to be numbers which are all either or ; "" is to have the value if is and is ; otherwise it is to have the value ; is to be if is , and if is ; is to be if and are both , and is to be in any other case; is to be if and are both , and is to be in any other case; and the assertion-sign is to mean that what follows has the value . Symbolic logic considered as a calculus has undoubtedly much interest on its own account; but in our opinion this aspect has hitherto been too much emphasized, at the expense of the aspect in which symbolic logic is merely the most elementary part of mathematics, and the logical prerequisite of all the rest. For this reason, we shall only deal briefly with what is required for the algebra of symbolic logic.

It is obvious that two propositions are equivalent when, and only when, both are true or both are false. Following Frege, we shall call the truth-value of a proposition truth if it is true, and falsehood if it is false. Thus two propositions are equivalent when they have the same truth-value.

It should be observed that, if , may be substituted for without altering the truth-value of any function of which involves no primitive ideas except those enumerated in *1. This can be proved in each separate case, but not generally, because we have no means of specifying (with our apparatus of primitive ideas) that a function is one which can be built up out of these ideas alone. We shall give the name of a truth-function to a function whose argument is a proposition, and whose truth-value depends only upon the truth-value of its argument. All the functions of propositions with which we shall be specially concerned will be truth-functions, i.e. we shall have The reason of this is, that the functions of propositions with which we deal are all built up by means of the primitive ideas of *1. But it is not a universal characteristic of functions of propositions to be truth-functions. For example, " believes " may be true for one true value of and false for another.