[page 155]
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 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, [page 156] 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 A’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: “x is human” is a propositional function; so long as x remains undetermined, it is neither true nor false, but when a value is assigned to x 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 A is B” are propositional functions: A and B 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 x is a flash of lightning, x is followed by thunder.” The process of generalisation (with whose validity we are [page 157] fortunately not concerned) consists in passing from a number of such instances to the universal truth of the propositional function: “If x is a flash of lightning, x 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 not concerned with the truth or falsehood of what is said, but only with its logical analysis.) We are told that A is, in every instance, followed by B. Now if there are “instances” of A, A must be some general concept of which it is significant to say “x1 is A,” “x2 is A,” “x3 is A,” and so on, where x1, x2, x3 are particulars which are not identical one with another. This applies, e.g., to our previous case of lightning. We say that lightning (A) is followed by thunder (B). But the separate flashes are particulars, not identical, but sharing the common property of being lightning. The only way of expressing a [page 158] 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 A is in every instance followed by B, we mean that, whatever x may be, if x is an A, it is followed by a B; 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 “φx” is the function, and a is the right sort of object to be an argument to “φx,” then φa is to be true, however a may have been chosen. For example, “if a is human, a is mortal” is true whether a is human or not; in fact, every proposition of this form is true. Thus the propositional function “if x is human, x 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 ‘x is not a unicorn’ is true in all cases.” The assertions in the preceding chapter about propositions, e.g. “‘p or q’ implies ‘q or p,’” are really assertions [page 159] 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 p or q, but as being true of any p or q 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 “x is a man” is sometimes true. When we say “some men are Greeks,” that means that the propositional function “x is a man and a Greek” is sometimes true. When we say “cannibals still exist in Africa,” that means that the propositional function “x is a cannibal now in Africa” is sometimes true, i.e. is true for some values of x. To say “there are at least n individuals in the world” is to say that the propositional function “α is a class of individuals and a member of the cardinal number n” is sometimes true, or, as we may say, is true for certain [page 160] values of α. 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 “α is a class of n individuals,” contains two variables, α and n. The axiom of infinity, in the language of propositional functions, is: “The propositional function ‘if n is an inductive number, it is true for some values of α that α is a class of n individuals’ is true for all possible values of n.” Here there is a subordinate function, “α is a class of n individuals,” which is said to be, in respect of α, sometimes true; and the assertion that this happens if n is an inductive number is said to be, in respect of n, always true.
The statement that a function φx is always true is the negation of the statement that not-φx is sometimes true, and the statement that φx is sometimes true is the negation of the statement that not-φx is always true. Thus the statement “all men are mortals” is the negation of the statement that the function “x is an immortal man” is sometimes true. And the statement “there are unicorns” is the negation of the statement that the function “x is not a unicorn” is always true.1 We say that φx is “never true” or “always false” if not-φx 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: “‘φx is always true’ is to mean ‘it is false that not-φx is sometimes true.’” But for reasons connected with the theory of types it seems more correct to take both “always” and “sometimes” as primitive ideas, and define by their means the negation of propositions in which they occur. That is to say, assuming that we have already [page 161] defined (or adopted as a primitive idea) the negation of propositions of the type to which φx belongs, we define: “The negation of ‘φx always’ is ‘not-φx sometimes’; and the negation of ‘φx sometimes’ is ‘not-φx always.’” In like manner we can re-define disjunction and the other truth-functions, as applied to propositions containing apparent variables, in terms of the definitions and primitive ideas for propositions containing no apparent variables. Propositions containing no apparent variables are called “elementary propositions.” From these we can mount up step by step, using such methods as have just been indicated, to the theory of truth-functions as applied to propositions containing one, two, three … variables, or any number up to n, where n is any assigned finite number.2
The forms which are taken as simplest in traditional formal logic are really far from being so, and all involve the assertion of all values or some values of a compound propositional function. Take, to begin with, “all S is P.” We will take it that S is defined by a propositional function φx, and P by a propositional function ψx. E.g., if S is men, φx will be “x is human”; if P is mortals, ψx will be “there is a time at which x dies.” Then “all S is P” means: “‘φx implies ψx’ is always true.” It is to be observed that “all S is P” does not apply only to those terms that actually are S’s; it says something equally about terms which are not S’s. Suppose we come across an x of which we do not know whether it is an S or not; still, our statement “all S is P” tells us something about x, namely, that if x is an S, then x is a P. And this is every bit as true when x is not an S as when x is an S. If it were not equally true in both cases, the reductio ad absurdum would not be a valid method; for the essence of this method consists in using implications in cases where (as it afterwards turns out) the hypothesis is false. We may put the matter another way. In order to understand “all S is P,” it is not necessary to be able to enumerate what terms are S’s; provided we know what is meant by being an S and what by being a P, we can understand completely what is actually affirmed [page 162] by “all S is P,” however little we may know of actual instances of either. This shows that it is not merely the actual terms that are S’s that are relevant in the statement “all S is P,” but all the terms concerning which the supposition that they are S’s is significant, i.e. all the terms that are S’s, together with all the terms that are not S’s—i.e. the whole of the appropriate logical “type.” What applies to statements about all applies also to statements about some. “There are men,” e.g., means that “x is human” is true for some values of x. Here all values of x (i.e. all values for which “x is human” is significant, whether true or false) are relevant, and not only those that in fact are human. (This becomes obvious if we consider how we could prove such a statement to be false.) Every assertion about “all” or “some” thus involves not only the arguments that make a certain function true, but all that make it significant, i.e. all for which it has a value at all, whether true or false.
We may now proceed with our interpretation of the traditional forms of the old-fashioned formal logic. We assume that S is those terms x for which φx is true, and P is those for which ψx is true. (As we shall see in a later chapter, all classes are derived in this way from propositional functions.) Then:
“All S is P” means “‘φx implies ψx’ is always true.”
“Some S is P” means “‘φx and ψx’ is sometimes true.”
“No S is P” means “‘φx implies not-ψx’ is always true.”
“Some S is not P” means “‘φx and not-ψx’ is sometimes true.”
It will be observed that the propositional functions which are here asserted for all or some values are not φx and ψx themselves, but truth-functions of φx and ψx for the same argument x. The easiest way to conceive of the sort of thing that is intended is to start not from φx and ψx in general, but from φa and ψa, where a is some constant. Suppose we are considering “all men are mortal”: we will begin with
“If Socrates is human, Socrates is mortal,”
[page 163]
and then we will regard “Socrates” as replaced by a variable x wherever “Socrates” occurs. The object to be secured is that, although x remains a variable, without any definite value, yet it is to have the same value in “φx” as in “ψx” when we are asserting that “φx implies ψx” is always true. This requires that we shall start with a function whose values are such as “φa implies ψa,” rather than with two separate functions φx and ψx; for if we start with two separate functions we can never secure that the x, while remaining undetermined, shall have the same value in both.
For brevity we say “φx always implies ψx” when we mean that “φx implies ψx” is always true. Propositions of the form “φx always implies ψx” are called “formal implications”; this name is given equally if there are several variables.
The above definitions show how far removed from the simplest forms are such propositions as “all S is P,” with which traditional logic begins. It is typical of the lack of analysis involved that traditional logic treats “all S is P” as a proposition of the same form as “x is P”—e.g., it treats “all men are mortal” as of the same form as “Socrates is mortal.” As we have just seen, the first is of the form “φx always implies ψx,” while the second is of the form “ψx.” The emphatic separation of these two forms, which was effected by Peano and Frege, was a very vital advance in symbolic logic.
It will be seen that “all S is P” and “no S is P” do not really differ in form, except by the substitution of not-ψx for ψx, and that the same applies to “some S is P” and “some S is not P.” It should also be observed that the traditional rules of conversion are faulty, if we adopt the view, which is the only technically tolerable one, that such propositions as “all S is P” do not involve the “existence” of S’s, i.e. do not require that there should be terms which are S’s. The above definitions lead to the result that, if φx is always false, i.e. if there are no S’s, then “all S is P” and “no S is P” will both be true, [page 164] whatever P may be. For, according to the definition in the last chapter, “φx implies ψx” means “not-φx or ψx,” which is always true if not-φx is always true. At the first moment, this result might lead the reader to desire different definitions, but a little practical experience soon shows that any different definitions would be inconvenient and would conceal the important ideas. The proposition “φx always implies ψx, and φx is sometimes true” is essentially composite, and it would be very awkward to give this as the definition of “all S is P,” for then we should have no language left for “φx always implies ψx,” which is needed a hundred times for once that the other is needed. But, with our definitions, “all S is P” does not imply “some S is P,” since the first allows the non-existence of S and the second does not; thus conversion per accidens becomes invalid, and some moods of the syllogism are fallacious, e.g. Darapti: “All M is S, all M is P, therefore some S is P,” which fails if there is no M.
The notion of “existence” has several forms, one of which will occupy us in the next chapter; but the fundamental form is that which is derived immediately from the notion of “sometimes true.” We say that an argument a “satisfies” a function φx if φa is true; this is the same sense in which the roots of an equation are said to satisfy the equation. Now if φx is sometimes true, we may say there are x’s for which it is true, or we may say “arguments satisfying φx exist.” This is the fundamental meaning of the word “existence.” Other meanings are either derived from this, or embody mere confusion of thought. We may correctly say “men exist,” meaning that “x is a man” is sometimes true. But if we make a pseudo-syllogism: “Men exist, Socrates is a man, therefore Socrates exists,” we are talking nonsense, since “Socrates” is not, like “men,” merely an undetermined argument to a given propositional function. The fallacy is closely analogous to that of the argument: “Men are numerous, Socrates is a man, therefore Socrates is numerous.” In this case it is obvious that the conclusion is nonsensical, but [page 165] in the case of existence it is not obvious, for reasons which will appear more fully in the next chapter. For the present let us merely note the fact that, though it is correct to say “men exist,” it is incorrect, or rather meaningless, to ascribe existence to a given particular x who happens to be a man. Generally, “terms satisfying φx exist” means “φx is sometimes true”; but “a exists” (where a is a term satisfying φx) is a mere noise or shape, devoid of significance. It will be found that by bearing in mind this simple fallacy we can solve many ancient philosophical puzzles concerning the meaning of existence.
Another set of notions as to which philosophy has allowed itself to fall into hopeless confusions through not sufficiently separating propositions and propositional functions are the notions of “modality”: necessary, possible, and impossible. (Sometimes contingent or assertoric is used instead of possible.) The traditional view was that, among true propositions, some were necessary, while others were merely contingent or assertoric; while among false propositions some were impossible, namely, those whose contradictories were necessary, while others merely happened not to be true. In fact, however, there was never any clear account of what was added to truth by the conception of necessity. In the case of propositional functions, the threefold division is obvious. If “φx” is an undetermined value of a certain propositional function, it will be necessary if the function is always true, possible if it is sometimes true, and impossible if it is never true. This sort of situation arises in regard to probability, for example. Suppose a ball x is drawn from a bag which contains a number of balls: if all the balls are white, “x is white” is necessary; if some are white, it is possible; if none, it is impossible. Here all that is known about x is that it satisfies a certain propositional function, namely, “x was a ball in the bag.” This is a situation which is general in probability problems and not uncommon in practical life—e.g. when a person calls of whom we know nothing except that he brings a letter of introduction from our friend so-and-so. In all such [page 166] cases, as in regard to modality in general, the propositional function is relevant. For clear thinking, in many very diverse directions, the habit of keeping propositional functions sharply separated from propositions is of the utmost importance, and the failure to do so in the past has been a disgrace to philosophy.
[Chapter XV notes]
1. For linguistic reasons, to avoid suggesting either the plural or the singular, it is often convenient to say “φx is not always false” rather than “φx sometimes” or “φx is sometimes true.”
2. The method of deduction is given in Principia Mathematica, vol. i. *9.