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:

$\scriptstyle {(*4\cdot 73)\quad \vdash :.q.\supset :p.\equiv .p.q.}$

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

${\begin{array}{ll}\scriptstyle {(*4\cdot 4)}&\scriptstyle {\vdash :.p.q\lor r.\equiv :p.q.\lor .p.r,}\\\scriptstyle {(*4\cdot 41)\quad }&\scriptstyle {\vdash :.p.\lor .q.r:\equiv :p\lor q.p\lor r.}\end{array}}$

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 "$\scriptstyle {\hat {x}}$ 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 of $\scriptstyle {\phi {\hat {x}}}$, there is a range, or collection, of values, consisting of all the propositions (true or false) which can be obtained by giving