Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/69

 is important philosophically, and is relevant at certain points in symbolic logic.

The judgments we have been dealing with hitherto are such as are of the same form as judgments of perception, i.e. their subjects are always particular and definite. But there are many judgments which are not of this form. Such are "all men are mortal," "I met a man," "some men are Greeks." Before dealing with such judgments, we will introduce some technical terms.

We will give the name of "a complex" to any such object as "$\scriptstyle{a}$ in the relation $$\scriptstyle{R}$$ to $\scriptstyle{b}$" or "$\scriptstyle{a}$ having the quality $\scriptstyle{q}$," or "$\scriptstyle{a}$ and $$\scriptstyle{b}$$ and $$\scriptstyle{c}$$ standing in the relation $\scriptstyle{S}$." Broadly speaking, a complex is anything which occurs in the universe and is not simple. We will call a judgment elementary when it merely asserts such things as "$\scriptstyle{a}$ has the relation $$\scriptstyle{R}$$ to $\scriptstyle{b}$," "$\scriptstyle{a}$ has the quality $\scriptstyle{q}$" or "$\scriptstyle{a}$ and $$\scriptstyle{b}$$ and $$\scriptstyle{c}$$ stand in the relation $\scriptstyle{S}$." Then an elementary judgment is true when there is a corresponding complex, and false when there is no corresponding complex.

But take now such a proposition as "all men are mortal." Here the judgment does not correspond to one complex, but to many, namely "Socrates is mortal," "Plato is mortal," "Aristotle is mortal," etc. (For the moment, it is unnecessary to inquire whether each of these does not require further treatment before we reach the ultimate complexes involved. For purposes of illustration, "Socrates is mortal" is here treated as an elementary judgment, though it is in fact not one, as will be explained later. Truly elementary judgments are not very easily found.) We do not mean to deny that there may be some relation of the concept man to the concept mortal which may be equivalent to "all men are mortal," but in any case this relation is not the same thing as what we affirm when we say that all men are mortal. Our judgment that all men are mortal collects together a number of elementary judgments. It is not, however, composed of these, since (e.g.) the fact that Socrates is mortal is no part of what we assert, as may be seen by considering the fact that our assertion can be understood by a person who has never heard of Socrates. In order to understand the judgment "all men are mortal," it is not necessary to know what men there are. We must admit, therefore, as a radically new kind of judgment, such general assertions as "all men are mortal." We assert that, given that $$\scriptstyle{x}$$ is human, $$\scriptstyle{x}$$ is always mortal. That is, we assert "$\scriptstyle{x}$ is mortal" of every $$\scriptstyle{x}$$ which is human. Thus we are able to judge (whether truly or falsely) that all the objects which have some assigned property also have some other assigned property. That is, given any propositional functions $$\scriptstyle{\phi\hat x}$$ and $\scriptstyle{\psi\hat x}$, there is a judgment asserting $$\scriptstyle{\psi x}$$ with every $$\scriptstyle{x}$$ for which we have $\scriptstyle{\phi x}$. Such judgments we will call general judgments.

It is evident (as explained above) that the definition of truth is different