# Introduction to Mathematical Philosophy

By Bertrand Russell | Go to book overview

CHAPTER XV
PROPOSITIONAL FUNCTIONS

WHEN, in the preceding chapter, we were discussing propositions, we did not attempt to give a definition of the word "proposition." But although the word cannot be formally defined, it is necessary to say something as to its meaning, in order to avoid the very common confusion with "propositional functions," which are to be the topic of the present chapter.

We mean by a "proposition" primarily a form of words which expresses what is either true or false. I say "primarily," because I do not wish to exclude other than verbal symbols, or even mere thoughts if they have a symbolic character. But I think the word "proposition" should be limited to what may, in some sense, be called "symbols," and further to such symbols as give expression to truth and falsehood. Thus "two and two are four" and "two and two are five" will be propositions, and so will "Socrates is a man" and "Socrates is not a man." The statement: "Whatever numbers a and b may be, (a+b)2= a2+2ab+ b2" is a proposition; but the bare formula "(a+b)2= a2+2ab+b2" alone is not, since it asserts nothing definite unless we are further told, or led to suppose, that a and b are to have all possible values, or are to have such-and-such values. The former of these is tacitly assumed, as a rule, in the enunciation of mathematical formul¦, which thus become propositions; but if no such assumption were made, they would be "propositional functions." A "propositional function," in fact, is an expression containing one or more undetermined constituents,

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