Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/142



Summary of *4.

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 0 or 1; "$\scriptstyle{p\supset q}$" is to have the value 0 if $$\scriptstyle{p}$$ is 1 and $$\scriptstyle{q}$$ is 0; otherwise it is to have the value 1; $$\scriptstyle{\sim p}$$ is to be 1 if $$\scriptstyle{p}$$ is 0, and 0 if $$\scriptstyle{p}$$ is 1; $$\scriptstyle{p.q}$$ is to be 1 if $$\scriptstyle{p}$$ and $$\scriptstyle{q}$$ are both 1, and is to be 0 in any other case; $$\scriptstyle{p\or q}$$ is to be 0 if $$\scriptstyle{p}$$ and $$\scriptstyle{q}$$ are both 0, and is to be 1 in any other case; and the assertion-sign is to mean that what follows has the value 1. 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 pre-requisite of all the rest. For this reason, we shall only deal briefly with what is required for the algebra of symbolic logic.

When each of two propositions implies the other, we say that the two are equivalent, which we write "$\scriptstyle{p\equiv q}$." We put

$$\scriptstyle{p\equiv q.=.p\supset q\supset p\quad\text{Df}}$$

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 $\scriptstyle{p\equiv q}$, $$\scriptstyle{q}$$ may be substituted for $$\scriptstyle{p}$$ without altering the truth-value of any function of $$\scriptstyle{p}$$ 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