Introduction to Mathematical Philosophy

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 and may be, " is a proposition; but the bare formula "" alone is not, since it asserts nothing definite unless we are further told, or led to suppose, that and 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 formulæ, 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, [Pg 155] such that, when values are assigned to these constituents, the expression becomes a proposition. In other words, it is a function whose values are propositions. But this latter definition must be used with caution. A descriptive function, e.g. "the hardest proposition in 's mathematical treatise," will not be a propositional function, although its values are propositions. But in such a case the propositions are only described: in a propositional function, the values must actually enunciate propositions. Examples of propositional functions are easy to give: " is human" is a propositional function; so long as remains undetermined, it is neither true nor false, but when a value is assigned to it becomes a true or false proposition. Any mathematical equation is a propositional function. So long as the variables have no definite value, the equation is merely an expression awaiting determination in order to become a true or false proposition. If it is an equation containing one variable, it becomes true when the variable is made equal to a root of the equation, otherwise it becomes false; but if it is an "identity" it will be true when the variable is any number. The equation to a curve in a plane or to a surface in space is a propositional function, true for values of the co-ordinates belonging to points on the curve or surface, false for other values. Expressions of traditional logic such as "all is " are propositional functions: and have to be determined as definite classes before such expressions become true or false. The notion of "cases" or "instances" depends upon propositional functions. Consider, for example, the kind of process suggested by what is called "generalisation," and let us take some very primitive example, say, "lightning is followed by thunder." We have a number of "instances" of this, i.e. a number of propositions such as: "this is a flash of lightning and is followed by thunder." What are these occurrences "instances" of? They are instances of the propositional function: "If is a flash of lightning, is followed by thunder." The process of generalisation (with whose validity we are fortunately [Pg 156] not concerned) consists in passing from a number of such instances to the universal truth of the propositional function: "If is a flash of lightning, is followed by thunder." It will be found that, in an analogous way, propositional functions are always involved whenever we talk of instances or cases or examples. We do not need to ask, or attempt to answer, the question: "What is a propositional function?" A propositional function standing all alone may be taken to be a mere schema, a mere shell, an empty receptacle for meaning, not something already significant. We are concerned with propositional functions, broadly speaking, in two ways: first, as involved in the notions "true in all cases" and "true in some cases"; secondly, as involved in the theory of classes and relations. The second of these topics we will postpone to a later chapter; the first must occupy us now. When we say that something is "always true" or "true in all cases," it is clear that the "something" involved cannot be a proposition. A proposition is just true or false, and there is an end of the matter. There are no instances or cases of "Socrates is a man" or "Napoleon died at St Helena." These are propositions, and it would be meaningless to speak of their being true "in all cases." This phrase is only applicable to propositional functions. Take, for example, the sort of thing that is often said when causation is being discussed. (We are net concerned with the truth or falsehood of what is said, but only with its logical analysis.) We are told that is, in every instance, followed by . Now if there are "instances" of , must be some general concept of which it is significant to say " is ," " is ," " is ," and so on, where , , are particulars which are not identical one with another. This applies, e.g., to our previous case of lightning. We say that lightning () is followed by thunder (). But the separate flashes are particulars, not identical, but sharing the common property of being lightning. The only way of expressing a [Pg 157] common property generally is to say that a common property of a number of objects is a propositional function which becomes true when any one of these objects is taken as the value of the variable. In this case all the objects are "instances" of the truth of the propositional function—for a propositional function, though it cannot itself be true or false, is true in certain instances and false in certain others, unless it is "always true" or "always false." When, to return to our example, we say that is in every instance followed by , we mean that, whatever may be, if is an , it is followed by a ; that is, we are asserting that a certain propositional function is "always true." Sentences involving such words as "all," "every," "a," "the," "some" require propositional functions for their interpretation. The way in which propositional functions occur can be explained by means of two of the above words, namely, "all" and "some." There are, in the last analysis, only two things that can be done with a propositional function: one is to assert that it is true in all cases, the other to assert that it is true in at least one case, or in some cases (as we shall say, assuming that there is to be no necessary implication of a plurality of cases). All the other uses of propositional functions can be reduced to these two. When we say that a propositional function is true "in all cases," or "always" (as we shall also say, without any temporal suggestion), we mean that all its values are true. If "" is the function, and is the right sort of object to be an argument to "," then is to be true, however may have been chosen. For example, "if is human, is mortal" is true whether is human or not; in fact, every proposition of this form is true. Thus the propositional function "if is human, is mortal" is "always true," or "true in all cases." Or, again, the statement "there are no unicorns" is the same as the statement "the propositional function ' is not a unicorn' is true in all cases." The assertions in the preceding chapter about propositions, e.g. "' or ' implies ' or ,'" are really assertions [Pg 158] that certain propositional functions are true in all cases. We do not assert the above principle, for example, as being true only of this or that particular or , but as being true of any or concerning which it can be made significantly. The condition that a function is to be significant for a given argument is the same as the condition that it shall have a value for that argument, either true or false. The study of the conditions of significance belongs to the doctrine of types, which we shall not pursue beyond the sketch given in the preceding chapter. Not only the principles of deduction, but all the primitive propositions of logic, consist of assertions that certain propositional functions are always true. If this were not the case, they would have to mention particular things or concepts—Socrates, or redness, or east and west, or what not,—and clearly it is not the province of logic to make assertions which are true concerning one such thing or concept but not concerning another. It is part of the definition of logic (but not the whole of its definition) that all its propositions are completely general, i.e. they all consist of the assertion that some propositional function containing no constant terms is always true. We shall return in our final chapter to the discussion of propositional functions containing no constant terms. For the present we will proceed to the other thing that is to be done with a propositional function, namely, the assertion that it is "sometimes true," i.e. true in at least one instance. When we say "there are men," that means that the propositional function " is a man" is sometimes true. When we say "some men are Greeks," that means that the propositional function " is a man and a Greek" is sometimes true. When we say "cannibals still exist in Africa," that means that the propositional function " is a cannibal now in Africa" is sometimes true, i.e. is true for some values of . To say "there are at least individuals in the world" is to say that the propositional function " is a class of individuals and a member of the cardinal number " is sometimes true, or, as we may say, is true for certain [Pg 159] values of . This form of expression is more convenient when it is necessary to indicate which is the variable constituent which we are taking as the argument to our propositional function. For example, the above propositional function, which we may shorten to " is a class of individuals," contains two variables, and . The axiom of infinity, in the language of propositional functions, is: "The propositional function 'if is an inductive number, it is true for some values of that is a class of individuals' is true for all possible values of ." Here there is a subordinate function, " is a class of individuals," which is said to be, in respect of , sometimes true; and the assertion that this happens if is an inductive number is said to be, in respect of , always true. The statement that a function is always true is the negation of the statement that not- is sometimes true, and the statement that is sometimes true is the negation of the statement that not- is always true. Thus the statement "all men are mortals" is the negation of the statement that the function " is an immortal man" is sometimes true. And the statement "there are unicorns" is the negation of the statement that the function " is not a unicorn" is always true.[38] We say that is "never true" or "always false" if not- is always true. We can, if we choose, take one of the pair "always," "sometimes" as a primitive idea, and define the other by means of the one and negation. Thus if we choose "sometimes" as our primitive idea, we can define: "' is always true' is to mean 'it is false that not- is sometimes true.'"[39] But for reasons connected with the theory of types it