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

*3. The Logical Product of two Propositions

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 $\scriptstyle {\sim (\sim p\lor \sim q)}$, 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

*3·01.⁠$\scriptstyle {p.q.=.\sim (\sim p\lor \sim q)\quad {\text{Df}}}$

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

*3·02.⁠$\scriptstyle {p\supset q\supset r.=.p\supset q.q\supset r\quad {\text{Df}}}$

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, $\scriptstyle {\sim \phi p\lor \sim \psi p}$ is an elementary propositional function. Hence by *2·11,

$\scriptstyle {\vdash :\sim \phi p\lor \sim \psi p.\lor .\sim (\sim \phi p\lor \sim \psi p)}$.

Hence by *2·32 and *1·01,

$\scriptstyle {\vdash :.\phi p.\supset :\psi p.\supset .\sim (\sim \phi p\lor \sim \psi p)}$,

i.e. by *3·01,

$\scriptstyle {\vdash :.\phi p.\supset :\psi p.\supset .\phi p.\psi p}$.

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 *3·03. 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