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

La bibliothèque libre.
Cette page n’a pas encore été corrigée

errata : p. 15, last line but one, for H function of f/JX" read "function rpx."

I] PROPOSITIONAL FUNCTIONS

absorbed by the factor p, if p implies q. This principle enables us to replace an implication (p::> q) by an equivalence (p . = . p . q) whenever it is con- venient to do so. An analogous and very important principle is the following: (*4’73) 1-:. q.::>:p. = .p. q. Logical addition and multiplication of propositions obey the associative and comnlutative laws, and the distributive law in two forms, namely (*4’4) I- :. p . q v r . = : p . q . v . p . r, (*4’41) 1-:. p . v . q . r : = : p v q . p v r. The second of these distinguishes the relations of logical addition and multiplication from those of arithmetical addition and multiplication. Propositional functions. Let cpx be a statement containing a vari- able x and such that it becolnes a proposition when x is given any fixed determined nleaning. Then x is caned a "propositional function"; it is not a proposition, since owing to the ambiguity of x it really makes no assertion at all. Thus" x is hurt" really makes no assertion at all, till we have settled who x is. Yet owing to the individuality retained by the ambiguous variable x, it is an ambiguous example from the collection of propositions arrived at by giving all possible determinations to x in "x is hurt" which yield a proposition, true or false. Also if "x is hurt" and "y is hurt" occur in the same context, where y is another variable, then according to the determinations given to x and y, they can be settled to be (possibly) the same proposition or (possibly) different propositions. But apart from some determination given to x and y, they retain in that context their ambiguous differentiation. Thus" x is hurt" is an ambiguous" value" of a propositional function. When we wish to speak of the propositional function corresponding to "x is hurt," we shall write "5; is hurt." Thus "5; is hurt" is the propositional function and "x is hurt" is an ambiguous value of that function. Accordingly though "x is hurt JJ and "y is hurt" occurring in the same context can be distinguished, "5; is hurt" and" fJ is hurt" convey no distinction of meaning at all. More generally, x is an ambiguous value of the propositional function , and when a definite signification a is substituted for x, a is an unambiguous value of ;. Propositional functions are the fundanlental kind from which the more usual kinds of function,. such as " sin x " or H log x’ or " the father of x," are derived. ’rhese 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 , there is a range, or collection, of values, con- sisting of all the propositions (true or false) which can be obtained by giving ), > -, "".’4 it ’.