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 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, 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 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 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 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 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 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 defined (or adopted as a primitive idea) the negation of propositions of the type to which belongs, we define: "The negation of ' always' is 'not- sometimes'; and the negation of ' sometimes' is 'not- 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 , where is any assigned finite number.
[38]The method of deduction is given in Principia Mathematica, vol. I. * 9.
[39]For linguistic reasons, to avoid suggesting either the plural or the singular, it is often convenient to say " is not always false" rather than " sometimes" or " is sometimes true."
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 is ." We will take it that is defined by a propositional function , and by a propositional function . E.g., if is men, will be " is human"; if is mortals, will be "there is a time at which dies." Then "all is " means: "' implies ' is always true." It is to be observed that "all is " does not apply only to those terms that actually are 's; it says something equally about terms which are not 's. Suppose we come across an of which we do not know whether it is an or not; still, our statement "all is " tells us something about , namely, that if is an , then is a . And this is every bit as true when is not an as when is an . 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 is ," it is not necessary to be able to enumerate what terms are 's; provided we know what is meant by being an and what by being a , we can understand completely what is actually affirmed by "all is ," however little we may know of actual instances of either. This shows that it is not merely the actual terms that are 's that are relevant in the statement "all is ," but all the terms concerning which the supposition that they are 's is significant, i.e. all the terms that are 's, together with all the terms that are not '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 " is human" is true for some values of . Here all values of (i.e. all values for which " 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 is those terms for which is true, and is those for which is true. (As we shall see in a later chapter, all classes are derived in this way from propositional functions.) Then:
"All is " means "' implies ' is always true."
"Some is " means "' and ' is sometimes true."
"No is " means "' implies not- ' is always true."
"Some is not " means "' and not- ' is sometimes true."
It will be observed that the propositional functions which are here asserted for all or some values are not and themselves, but truth-functions of and for the same argument . The easiest way to conceive of the sort of thing that is intended is to start not from and in general, but from and , where is some constant. Suppose we are considering "all men are mortal": we will begin with
"If Socrates is human, Socrates is mortal,"
and then we will regard "Socrates" as replaced by a variable wherever "Socrates" occurs. The object to be secured is that, although remains a variable, without any definite value, yet it is to have the same value in " " as in " " when we are asserting that " implies " is always true. This requires that we shall start with a function whose values are such as " implies ," rather than with two separate functions and ; for if we start with two separate functions we can never secure that the , while remaining undetermined, shall have the same value in both.
For brevity we say " always implies " when we mean that " implies " is always true. Propositions of the form " always implies " 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 is ," with which traditional logic begins. It is typical of the lack of analysis involved that traditional logic treats "all is " as a proposition of the same form as " is "—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 " always implies ," while the second is of the form " ." 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 is " and "no is " do not really differ in form, except by the substitution of not- for , and that the same applies to "some is " and "some is not ." 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 is " do not involve the "existence" of 's, i.e. do not require that there should be terms which are 's. The above definitions lead to the result that, if is always false, i.e. if there are no 's, then "all is " and "no is " will both be true, whatever may be. For, according to the definition in the last chapter, " implies " means "not- or " which is always true if not- 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 " always implies , and is sometimes true" is essentially composite, and it would be very awkward to give this as the definition of "all is ," for then we should have no language left for " always implies ," which is needed a hundred times for once that the other is needed. But, with our definitions, "all is " does not imply "some is ," since the first allows the non-existence of and the second does not; thus conversion per accidens becomes invalid, and some moods of the syllogism are fallacious, e.g. Darapti: "All is , all is , therefore some is ," which fails if there is no .
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 "satisfies" a function if is true; this is the same sense in which the roots of an equation are said to satisfy the equation. Now if is sometimes true, we may say there are 's for which it is true, or we may say "arguments satisfying 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 " 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 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 who happens to be a man. Generally, "terms satisfying exist" means " is sometimes true"; but " exists" (where is a term satisfying ) 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 three-fold division is obvious. If " " 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 is drawn from a bag which contains a number of balls: if all the balls are white, " is white" is necessary; if some are white, it is possible; if none, it is impossible. Here all that is known about is that it satisfies a certain propositional function, namely, " 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 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.
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 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 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 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 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 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 defined (or adopted as a primitive idea) the negation of propositions of the type to which belongs, we define: "The negation of ' always' is 'not- sometimes'; and the negation of ' sometimes' is 'not- 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 , where is any assigned finite number.
[38]The method of deduction is given in Principia Mathematica, vol. I. * 9.
[39]For linguistic reasons, to avoid suggesting either the plural or the singular, it is often convenient to say " is not always false" rather than " sometimes" or " is sometimes true."
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 is ." We will take it that is defined by a propositional function , and by a propositional function . E.g., if is men, will be " is human"; if is mortals, will be "there is a time at which dies." Then "all is " means: "' implies ' is always true." It is to be observed that "all is " does not apply only to those terms that actually are 's; it says something equally about terms which are not 's. Suppose we come across an of which we do not know whether it is an or not; still, our statement "all is " tells us something about , namely, that if is an , then is a . And this is every bit as true when is not an as when is an . 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 is ," it is not necessary to be able to enumerate what terms are 's; provided we know what is meant by being an and what by being a , we can understand completely what is actually affirmed by "all is ," however little we may know of actual instances of either. This shows that it is not merely the actual terms that are 's that are relevant in the statement "all is ," but all the terms concerning which the supposition that they are 's is significant, i.e. all the terms that are 's, together with all the terms that are not '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 " is human" is true for some values of . Here all values of (i.e. all values for which " 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 is those terms for which is true, and is those for which is true. (As we shall see in a later chapter, all classes are derived in this way from propositional functions.) Then:
"All is " means "' implies ' is always true."
"Some is " means "' and ' is sometimes true."
"No is " means "' implies not- ' is always true."
"Some is not " means "' and not- ' is sometimes true."
It will be observed that the propositional functions which are here asserted for all or some values are not and themselves, but truth-functions of and for the same argument . The easiest way to conceive of the sort of thing that is intended is to start not from and in general, but from and , where is some constant. Suppose we are considering "all men are mortal": we will begin with
"If Socrates is human, Socrates is mortal,"
and then we will regard "Socrates" as replaced by a variable wherever "Socrates" occurs. The object to be secured is that, although remains a variable, without any definite value, yet it is to have the same value in " " as in " " when we are asserting that " implies " is always true. This requires that we shall start with a function whose values are such as " implies ," rather than with two separate functions and ; for if we start with two separate functions we can never secure that the , while remaining undetermined, shall have the same value in both.
For brevity we say " always implies " when we mean that " implies " is always true. Propositions of the form " always implies " 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 is ," with which traditional logic begins. It is typical of the lack of analysis involved that traditional logic treats "all is " as a proposition of the same form as " is "—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 " always implies ," while the second is of the form " ." 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 is " and "no is " do not really differ in form, except by the substitution of not- for , and that the same applies to "some is " and "some is not ." 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 is " do not involve the "existence" of 's, i.e. do not require that there should be terms which are 's. The above definitions lead to the result that, if is always false, i.e. if there are no 's, then "all is " and "no is " will both be true, whatever may be. For, according to the definition in the last chapter, " implies " means "not- or " which is always true if not- 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 " always implies , and is sometimes true" is essentially composite, and it would be very awkward to give this as the definition of "all is ," for then we should have no language left for " always implies ," which is needed a hundred times for once that the other is needed. But, with our definitions, "all is " does not imply "some is ," since the first allows the non-existence of and the second does not; thus conversion per accidens becomes invalid, and some moods of the syllogism are fallacious, e.g. Darapti: "All is , all is , therefore some is ," which fails if there is no .
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 "satisfies" a function if is true; this is the same sense in which the roots of an equation are said to satisfy the equation. Now if is sometimes true, we may say there are 's for which it is true, or we may say "arguments satisfying 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 " 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 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 who happens to be a man. Generally, "terms satisfying exist" means " is sometimes true"; but " exists" (where is a term satisfying ) 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 three-fold division is obvious. If " " 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 is drawn from a bag which contains a number of balls: if all the balls are white, " is white" is necessary; if some are white, it is possible; if none, it is impossible. Here all that is known about is that it satisfies a certain propositional function, namely, " 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 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.