Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/51

The following are some propositions concerning classes which are analogues of propositions previously given concerning propositions:

$\scriptstyle {\vdash .\alpha \cap \beta =\lnot (\lnot \alpha \cup \lnot \beta )}$,

i.e. the common part of $\scriptstyle {\alpha }$ and $\scriptstyle {\beta }$ is the negation of "$\scriptstyle {{\text{not}}-\alpha }$ or $\scriptstyle {{\text{not}}-\beta }$";

$\scriptstyle {\vdash .x\in (\alpha \cup \lnot \alpha )}$,

i.e. "$\scriptstyle {x}$ is a member of $\scriptstyle {\alpha }$ or $\scriptstyle {{\text{not}}-\alpha }$";

$\scriptstyle {\vdash .x\sim \in (\alpha \cap \lnot \alpha )}$,

i.e. "$\scriptstyle {x}$ is not a member of both $\scriptstyle {\alpha }$ and $\scriptstyle {{\text{not}}-\alpha }$";

${\begin{aligned}&\scriptstyle {\vdash .\alpha =\lnot (\lnot \alpha ),}\\&\scriptstyle {\vdash :\alpha \subset \beta .\equiv .\lnot \beta \subset \lnot \alpha },\\&\scriptstyle {\vdash :\alpha =\beta .\equiv .\lnot \alpha =\lnot \beta },\\&\scriptstyle {\vdash :\alpha =\alpha \cap \alpha },\\&\scriptstyle {\vdash :\alpha =\alpha \cup \alpha }.\end{aligned}}$

The two last are the two forms of the law of tautology.

The law of absorption holds in the form

$\scriptstyle {\vdash :\alpha \subset \beta .\equiv .\alpha =\alpha \cap \beta }$.

Thus for example "all Cretans are liars" is equivalent to "Cretans are identical with lying Cretans."

Just as we have

$\scriptstyle {\vdash :p\supset q.q\supset r.\supset .p\supset r}$,

so we have

$\scriptstyle {\vdash :\alpha \subset \beta .\beta \subset \gamma .\supset .\alpha \subset \gamma }$.

This expresses the ordinary syllogism in Barbara (with the premisses interchanged); for "$\scriptstyle {\alpha \subset \beta }$" means the same as "all $\scriptstyle {\alpha }$'s are $\scriptstyle {\beta }$'s," so that the above proposition states: "If all $\scriptstyle {\alpha }$'s are $\scriptstyle {\beta }$'s, and all $\scriptstyle {\beta }$'s are $\scriptstyle {\gamma }$'s, then all $\scriptstyle {\alpha }$'s are $\scriptstyle {\gamma }$'s." (It should be observed that syllogisms are traditionally expressed with "therefore," as if they asserted both premisses and conclusion. This is, of course, merely a slipshod way of speaking, since what is really asserted is only the connection of premisses with conclusion.)

The syllogism in Barbara when the minor premiss has an individual subject is

$\scriptstyle {\vdash :x\in \beta .\beta \subset \gamma .\supset .x\in \gamma }$,

e.g. "if Socrates is a man, and all men are mortals, then Socrates is a mortal." This, as was pointed out by Peano, is not a particular case of "$\scriptstyle {\alpha \subset \beta .\beta \subset \gamma .\supset .\alpha \subset \gamma }$," since "$\scriptstyle {x\in \beta }$" is not a particular case of "$\scriptstyle {\alpha \subset \beta }$." This point is important, since traditional logic is here mistaken. The nature and magnitude of its mistake will become clearer at a later stage. For relations, we have precisely analogous definitions and propositions. We put

$\scriptstyle {R{\dot {\cap }}S={\hat {x}}{\hat {y}}(xRy.xSy)\quad {\text{Df,}}}$

which leads to

$\scriptstyle {\vdash :x(R{\dot {\cap }}S)y.\equiv .xRy.xSy}$.