Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/145

SECTION A] Note. The above three propositions show that the relation of equivalence is reflexive (*4·2), symmetrical (*4·21), and transitive (*4·22). Implication is reflexive and transitive, but not symmetrical. The properties of being symmetrical, transitive, and (at least within a certain field) reflexive are essential to any relation which is to have the formal characters of equality.


 * 4·24.$$\scriptstyle{\vdash:p.\equiv.p.p}$$

Dem.


 * 4·25.$$\scriptstyle{\vdash:p.\equiv.p\or p\quad\left[\text{Taut}.\text{Add}\frac{p}{q}\right]}$$

Note. *4·24·25 are two forms of the law of tautology, which is what chiefly distinguishes the algebra of symbolic logic from ordinary algebra.


 * 4·3.$$\scriptstyle{\vdash:p.q.\equiv.q.p\quad[*3\cdot22]}$$

Note. Whenever we have, whatever values $$\scriptstyle{p}$$ and $$\scriptstyle{q}$$ may have, we have also


 * 4·31.$$\scriptstyle{\vdash:p\or q.\equiv.q\or p\quad[\text{Perm}]}$$


 * 4·32.$$\scriptstyle{\vdash:(p.q).r.\equiv.p.(q.r)}$$

Dem.

Note. Here "(1)" stands for "$\scriptstyle{\vdash:.p.q.\supset.\sim r:\equiv:p.\supset.\sim(q.r)}$," which is obtained from the above steps by *4·22. The use of *4·22 will often be tacit, as above. The principle is the same as that explained in respect of implication in *2·31.


 * 4·33.$$\scriptstyle{\vdash:(p\or q)\or r.\equiv.p\or(q\or r)\quad[*2\cdot31\cdot32]}$$

The above are the associative laws for multiplication and addition. To avoid brackets, we introduce the following definition: