Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/37

 absorbed by the factor $\scriptstyle{p}$, if $$\scriptstyle{p}$$ implies $\scriptstyle{q}$. This principle enables us to replace an implication $$\scriptstyle{(p \supset q)}$$ by an equivalence $$\scriptstyle{(p . \equiv . p . q)}$$ whenever it is convenient to do so.

An analogous and very important principle is the following:

Logical addition and multiplication of propositions obey the associative and commutative laws, and the distributive law in two forms, namely

The second of these distinguishes the relations of logical addition and multiplication from those of arithmetical addition and multiplication.

Propositional functions. Let $$\scriptstyle{\phi x}$$ be a statement containing a variable $$\scriptstyle{x}$$ and such that it becomes a proposition when $$\scriptstyle{x}$$ is given any fixed determined meaning. Then $$\scriptstyle{\phi x}$$ is called a "propositional function"; it is not a proposition, since owing to the ambiguity of $$\scriptstyle{x}$$ it really makes no assertion at all. Thus "$\scriptstyle{x}$ is hurt" really makes no assertion at all, till we have settled who $$\scriptstyle{x}$$ is. Yet owing to the individuality retained by the ambiguous variable $\scriptstyle{x}$, it is an ambiguous example from the collection of propositions arrived at by giving all possible determinations to $$\scriptstyle{x}$$ in "$\scriptstyle{x}$ is hurt" which yield a proposition, true or false. Also if "$\scriptstyle{x}$ is hurt" and "$\scriptstyle{y}$ is hurt" occur in the same context, where $$\scriptstyle{y}$$ is another variable, then according to the determinations given to $$\scriptstyle{x}$$ and $\scriptstyle{y}$, they can be settled to be (possibly) the same proposition or (possibly) different propositions. But apart from some determination given to $$\scriptstyle{x}$$ and $\scriptstyle{y}$, they retain in that context their ambiguous differentiation. Thus "$\scriptstyle{x}$ is hurt" is an ambiguous "value" of a propositional function. When we wish to speak of the propositional function corresponding to "$\scriptstyle{x}$ is hurt," we shall write "$\scriptstyle{\hat{x}}$|undefined is hurt." Thus "$$\scriptstyle{\hat{x}}$$ is hurt" is the propositional function and "$\scriptstyle{x}$ is hurt" is an ambiguous value of that function. Accordingly though "$\scriptstyle{x}$ is hurt" and "$\scriptstyle{y}$ is hurt" occurring in the same context can be distinguished, "$$\scriptstyle{\hat{x}}$$ is hurt" and "$$\scriptstyle{\hat{y}}$$ is hurt" convey no distinction of meaning at all. More generally, $$\scriptstyle{\phi x}$$ is an ambiguous value of the propositional function $$\scriptstyle{\phi \hat{x}}$$, and when a definite signification $$\scriptstyle{a}$$ is substituted for $\scriptstyle{x}$, $$ \scriptstyle{\phi a}$$ is an unambiguous value of $$\scriptstyle{\phi \hat{x}}$$.

Propositional functions are the fundamental kind from which the more usual kinds of function, such as "$\scriptstyle{\sin x}$" or "$\scriptstyle{\log x}$" or "the father of $\scriptstyle{x}$," are derived. These derivative functions are considered later, and are called "descriptive functions." The functions of propositions considered above are a particular case of propositional functions.

The range of values and total variation. Thus corresponding to any propositional function offunction [sic] $\scriptstyle{\phi\hat{x}}$,|undefined there is a range, or collection, of values, consisting of all the propositions (true or false) which can be obtained by giving