Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/123

*1·5. $\scriptstyle {\vdash :p\lor (q\lor r).\supset .q\lor (p\lor r)\quad {\text{Pp.}}}$

This principle states: "If either $\scriptstyle {p}$ is true, or '$\scriptstyle {q}$ or $\scriptstyle {r}$' is true, then either $\scriptstyle {q}$ is true, or '$\scriptstyle {p}$ or $\scriptstyle {r}$' is true." It is a form of the associative law for logical addition, and will be called the "associative principle." It will be referred to as "Assoc." The proposition

$\scriptstyle {p\lor (q\lor r).\supset .(p\lor q)\lor r}$,

which would be the natural form for the associative law, has less deductive power, and is therefore not taken as a primitive proposition.

*1·6. $\scriptstyle {\vdash :.q\supset r.\supset :p\lor q.\supset .p\lor r\quad {\text{Pp.}}}$

This principle states: "If $\scriptstyle {q}$ implies $\scriptstyle {r}$, then '$\scriptstyle {p}$ or $\scriptstyle {q}$' implies '$\scriptstyle {p}$ or $\scriptstyle {r}$.'" In other words, in an implication, an alternative may be added to both premiss and conclusion without impairing the truth of the implication. The principle will be called the "principle of summation," and will be referred to as "Sum."

*1·7. If $\scriptstyle {p}$ is an elementary proposition, $\scriptstyle {\sim p}$ is an elementary proposition. $\scriptstyle {\text{Pp.}}$

*1·71. If $\scriptstyle {p}$ and $\scriptstyle {q}$ are elementary propositions, $\scriptstyle {p\lor q}$ is an elementary proposition. $\scriptstyle {\text{Pp.}}$

*1·72. If $\scriptstyle {\phi p}$ and $\scriptstyle {\psi p}$ are elementary propositional functions which take elementary propositions as arguments, $\scriptstyle {\phi p\lor \psi p}$ is an elementary propositional function. $\scriptstyle {\text{Pp.}}$

This axiom is to apply also to functions of two or more variables. It is called the "axiom of identification of real variables." It will be observed that if $\scriptstyle {\phi }$ and $\scriptstyle {\psi }$ are functions which take arguments of different types, there is no such function as "$\scriptstyle {\phi x\lor \psi x}$," because $\scriptstyle {\phi }$ and $\scriptstyle {\psi }$ cannot significantly have the same argument. A more general form of the above axiom will be given in *9.

The use of the above axioms will generally be tacit. It is only through them and the axioms of *9 that the theory of types explained in the Introduction becomes relevant, and any view of logic which justifies these axioms justifies such subsequent reasoning as employs the theory of types.

This completes the list of primitive propositions required for the theory of deduction as applied to elementary propositions.