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

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.

${\begin{aligned}&{\begin{array}{llr}\scriptstyle {\vdash .*3\cdot 26.}&\scriptstyle {\supset \vdash :p.p.\supset .p\qquad \qquad \qquad \qquad }&\scriptstyle {(1)}\\\scriptstyle {\vdash .*3\cdot 2.}&\scriptstyle {\supset \vdash :.p.\supset :p.\supset .p.p:}\\\scriptstyle {[*2\cdot 43]}&\scriptstyle {\supset \vdash :p.\supset .p.p}&\scriptstyle {(2)}\end{array}}\\&\scriptstyle {~\vdash .(1).(2).*3\cdot 2.\supset \vdash .{\text{Prop}}}\end{aligned}}$

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

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.$\scriptstyle {\vdash :p.q.\equiv .q.p\quad [*3\cdot 22]}$

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

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

we have also

$\scriptstyle {\phi (p,q).\equiv .\phi (q,p)}$.

For

$\scriptstyle {\{\phi (p,q).\supset .\phi (q,p)\}{\frac {q,p}{p,q}}.\supset :\phi (q,p).\supset .\phi (p,q)}$.

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

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

Dem.

${\begin{array}{lllr}\scriptstyle {\vdash .*4\cdot 15.}&\scriptstyle {\supset \vdash :.p.q.\supset .\sim r:}&\scriptstyle {\equiv :q.r.\supset .\sim p:}\\\scriptstyle {[*4\cdot 12]}&&\scriptstyle {\equiv :p.\supset .\sim (q.r)\qquad }&\scriptstyle {(1)}\\\scriptstyle {\vdash .(1).*4\cdot 11.}&\scriptstyle {\supset \vdash :\sim (p.q.\supset .\sim }&\scriptstyle {r).\equiv .\sim \{p.\supset .\sim (q.r)\}:}\\\scriptstyle {[(*1\cdot 01.*3\cdot 01)]}&\scriptstyle {\supset \vdash .{\text{Prop}}}\end{array}}$

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.$\scriptstyle {\vdash :(p\lor q)\lor r.\equiv .p\lor (q\lor r)\quad [*2\cdot 31\cdot 32]}$

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