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

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

The law of absorption holds in the form

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

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 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