Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/35

 Primitive propositions. Some propositions must be assumed without proof, since all inference proceeds from propositions previously asserted. These, as far as they concern the functions of propositions mentioned above, will be found stated in *1, where the formal and continuous exposition of the subject commences. Such propositions will be called "primitive propositions." These, like the primitive ideas, are to some extent a matter of arbitrary choice; though, as in the previous case, a logical system grows in importance according as the primitive propositions are few and simple. It will be found that owing to the weakness of the imagination in dealing with simple abstract ideas no very great stress can be laid upon their obviousness. They are obvious to the instructed mind, but then so are many propositions which cannot be quite true, as being disproved by their contradictory consequences. The proof of a logical system is its adequacy and its coherence. That is: (1) the system must embrace among its deductions all those propositions which we believe to be true and capable of deduction from logical premisses alone, though possibly they may require some slight limitation in the form of an increased stringency of enunciation; and (2) the system must lead to no contradictions, namely in pursuing our inferences we must never be led to assert both $$\scriptstyle{p}$$ and not-$\scriptstyle{p}$, i.e. both "$\scriptstyle{\vdash . p}$" and "$\scriptstyle{\vdash . \sim p}$" cannot legitimately appear.

The following are the primitive propositions employed in the calculus of propositions. The letters "$\scriptstyle{\text{Pp}}$"|undefined stand for "primitive proposition."

(1) Anything implied by a true premiss is true $\scriptstyle{\text{Pp}}$.|undefined

This is the rule which justifies inference.

(2) $$\scriptstyle{\vdash : p \or p. \supset. p \quad \text{Pp}}$$, i.e. if $$\scriptstyle{p}$$ or $$\scriptstyle{p}$$ is true, then $$\scriptstyle{p}$$ is true.

(3) $$\scriptstyle{\vdash : q. \supset. p \or q \quad \text{Pp}}$$, i.e. if $$\scriptstyle{q}$$ is true, then $$\scriptstyle{p}$$ or $$\scriptstyle{q}$$ is true.

(4) $$\scriptstyle{\vdash : p \or q. \supset. q \or p \quad \text{Pp}}$$, i.e. if $$\scriptstyle{p}$$ or $$\scriptstyle{q}$$ is true, then $$\scriptstyle{q}$$ or $$\scriptstyle{p}$$ is true.

(5) $$\scriptstyle{\vdash : p \or ( q \or r ) \quad \text{Pp}}$$, i.e. 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.

(6) $$\scriptstyle{\vdash :. q \supset r. \supset : p \or q. \supset. p \or r \quad \text{Pp}}$$, i.e. if $$\scriptstyle{q}$$ implies $\scriptstyle{r}$, then "$\scriptstyle{p}$ or $\scriptstyle{q}$" implies "$\scriptstyle{p}$ or $\scriptstyle{r}$."

(7) Besides the above primitive propositions, we require a primitive proposition called "the axiom of identification of real variables." When we have separately asserted two different functions of $\scriptstyle{x}$, where $$\scriptstyle{x}$$ is undetermined, it is often important to know whether we can identify the $$\scriptstyle{x}$$ in one