Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/41

 we can only legitimately assert "any value" if all values are true; for otherwise, since the value of the variable remains to be determined, it might be so determined as to give a false proposition. Thus in the above instance, since we have  we may infer   And generally, given an assertion containing a real variable $\scriptstyle{x}$, we may transform the real variable into an apparent one by placing the $$\scriptstyle{x}$$ in brackets at the beginning, followed by as many dots as there are after the assertion-sign.

When we assert something containing a real variable, we cannot strictly be said to be asserting a proposition, for we only obtain a definite proposition by assigning a value to the variable, and then our assertion only applies to one definite case, so that it has not at all the same force as before. When what we assert contains a real variable, we are asserting a wholly undetermined one of all the propositions that result from giving various values to the variable. It will be convenient to speak of such assertions as asserting a propositional function. The ordinary formulae of mathematics contain such assertions; for example does not assert this or that particular case of the formula, nor does it assert that the formula holds for all possible values of $\scriptstyle{x}$, though it is equivalent to this latter assertion; it simply asserts that the formula holds, leaving $$\scriptstyle{x}$$ wholly undetermined; and it is able to do this legitimately, because, however $$\scriptstyle{x}$$ may be determined, a true proposition results.

Although an assertion containing a real variable does not, in strictness, assert a proposition, yet it will be spoken of as asserting a proposition except when the nature of the ambiguous assertion involved is under discussion.

Definition and real variables. When the definiens contains one or more real variables, the definiendum must also contain them. For in this case we have a function of the real variables, and the definiendum must have the same meaning as the definiens for all values of these variables, which requires that the symbol which is the definiendum should contain the letters representing the real variables. This rule is not always observed by mathematicians, and its infringement has sometimes caused important confusions of thought, notably in geometry and the philosophy of space.

In the definitions given above of "$\scriptstyle{p.q}$" and "$\scriptstyle{p\supset q}$" and "$\scriptstyle{p\equiv q}$," $$\scriptstyle{p}$$ and $$\scriptstyle{q}$$ are real variables, and therefore appear on both sides of the definition. In the definition of "$\scriptstyle{\sim\{(x).\phi x\}}$"|undefined only the function considered, namely $\scriptstyle{\phi\hat x}$, is a real variable; thus so far as concerns the rule in question, $$\scriptstyle{x}$$ need not appear on the left. But when a real variable is a function, it is necessary to indicate