Introduction to Mathematical Philosophy

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 [Pg 160] 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 [Pg 161] 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," [Pg 162] 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 [Pg 163] 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 [Pg 164] 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 [Pg 165] 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. [Pg 166]