Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/136



Summary of *3.

The logical product of two propositions $$\scriptstyle{p}$$ and $$\scriptstyle{q}$$ is practically the proposition "$\scriptstyle{p}$ and $$\scriptstyle{q}$$ are both true." But this as it stands would have to be a new primitive idea. We therefore take as the logical product the proposition $\scriptstyle{\sim(\sim p\or\sim q)}$, i.e. "it is false that either $$\scriptstyle{p}$$ is false or $$\scriptstyle{q}$$ is false," which is obviously true when and only when $$\scriptstyle{p}$$ and $$\scriptstyle{q}$$ are both true. Thus we put

$$\scriptstyle{p.q.=.\sim(\sim p\or\sim q)\quad\text{Df}}$$

where "$\scriptstyle{p.q}$" is the logical product of $$\scriptstyle{p}$$ and $\scriptstyle{q}$.

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

This definition serves merely to abbreviate proofs.

When we are given two asserted propositional functions "$\scriptstyle{\vdash.\phi x}$" and "$\scriptstyle{\vdash.\psi x}$," we shall have "$\scriptstyle{\vdash.\phi x.\psi x}$" whenever $$\scriptstyle{\phi}$$ and $$\scriptstyle{\psi}$$ take arguments of the same type. This will be proved for any functions in *9; for the present, we are confined to elementary propositional functions of elementary propositions. In this case, the result is proved as follows:

By *1·7, $$\scriptstyle{\sim\phi p}$$ and $$\scriptstyle{\sim\psi p}$$ are elementary propositional functions, and therefore, by *1·72, $$\scriptstyle{\sim\phi p\or\sim\psi p}$$ is an elementary propositional function. Hence by *2·11,

Hence by *2·32 and *1·01, i.e. by *3·01,

Hence by *1·11, when we have "$\scriptstyle{\vdash.\phi p}$" and "$\scriptstyle{\vdash.\psi p}$" we have "$\scriptstyle{\vdash.\phi p.\psi p}$." This proposition is. It is to be understood, like *1·72, as applying also to functions of two or more variables.

The above is the practically most useful form of the axiom of identification of real variables (cf. *1·72). In practice, when the restriction to elementary propositions and propositional functions has been removed, a convenient means by which two functions can often be recognized as taking arguments of the same type is the following:

If $$\scriptstyle{\phi x}$$ contains, in any way, a constituent $$\scriptstyle{\chi(x,y,z,\ldots)}$$ and $$\scriptstyle{\psi x}$$ contains, in any way, a constituent $\scriptstyle{\chi(x,u,v,\ldots)}$, then both $$\scriptstyle{\phi x}$$ and $$\scriptstyle{\psi x}$$ take arguments