Introduction to Mathematical Philosophy

true or " or " is true, then either is true or " or " is true. (The twist in this proposition serves to increase its deductive power.) [Pg 149] (5) If implies , then " or " implies " or ." These are the formal principles of deduction employed in Principia Mathematica. A formal principle of deduction has a double use, and it is in order to make this clear that we have cited the above five propositions. It has a use as the premiss of an inference, and a use as establishing the fact that the premiss implies the conclusion. In the schema of an inference we have a proposition , and a proposition " implies ," from which we infer . Now when we are concerned with the principles of deduction, our apparatus of primitive propositions has to yield both the and the " implies " of our inferences. That is to say, our rules of deduction are to be used, not only as rules, which is their use for establishing " implies " but also as substantive premisses, i.e. as the of our schema. Suppose, for example, we wish to prove that if implies , then if implies it follows that implies . We have here a relation of three propositions which state implications. Put implies , implies , and implies . Then we have to prove that implies that implies . Now take the fifth of our above principles, substitute not- for , and remember that "not- or " is by definition the same as " implies ." Thus our fifth principle yields: "If implies , then ' implies ' implies 'implies ,'" i.e. "implies that implies ." Call this proposition . But the fourth of our principles, when we substitute not-, not-, for and , and remember the definition of implication, becomes: "If implies that implies , then implies that implies ." Writing in place of , in place of , and in place of , this becomes: "If implies that implies , then implies that implies ." Call this . [Pg 150] Now we proved by means of our fifth principle that " implies that implies ," which was what we called . Thus we have here an instance of the schema of inference, since represents the of our scheme, and represents the " implies ." Hence we arrive at , namely, " implies that implies ," which was the proposition to be proved. In this proof, the adaptation of our fifth principle, which yields , occurs as a substantive premiss; while the adaptation of our fourth principle, which yields , is used to give the form of the inference. The formal and material employments of premisses in the theory of deduction are closely intertwined, and it is not very important to keep them separated, provided we realise that they are in theory distinct. The earliest method of arriving at new results from a premiss is one which is illustrated in the above deduction, but which itself can hardly be called deduction. The primitive propositions, whatever they may be, are to be regarded as asserted for all possible values of the variable propositions , , which occur in them. We may therefore substitute for (say) any expression whose value is always a proposition, e.g. not-, " implies ," and so on. By means of such substitutions we really obtain sets of special cases of our original proposition, but from a practical point of view we obtain what are virtually new propositions. The legitimacy of substitutions of this kind has to be insured by means of a non-formal principle of inference.[36] [36]No such principle is enunciated in Principia Mathematica, or in M. Nicod's article mentioned above. But this would seem to be an omission. We may now state the one formal principle of inference to which M. Nicod has reduced the five given above. For this purpose we will first show how certain truth-functions can be defined in terms of incompatibility. We saw already that means " implies ." [Pg 151] We now observe that means " implies both and ." For this expression means " is incompatible with the incompatibility of and ," i.e. " implies that and are not incompatible," i.e. " implies that and are both true"—for, as we saw, the conjunction of and is the negation of their incompatibility. Observe next that means " implies itself." This is a particular case of . Let us write for the negation of ; thus will mean the negation of , i.e. it will mean the conjunction of and . It follows that expresses the incompatibility of with the conjunction of and ; in other words, it states that if and are both true, is false, i.e. and are both true; in still simpler words, it states that and jointly imply and jointly. Now, put Then M. Nicod's sole formal principle of deduction is in other words, implies both and . He employs in addition one non-formal principle belonging to the theory of types (which need not concern us), and one corresponding to the principle that, given , and given that implies , we can assert . This principle is: "If is true, and is true, then is true." From this apparatus the whole theory of deduction follows, except in so far as we are concerned with deduction from or to the existence or the universal truth of "propositional functions," which we shall consider in the next chapter. There is? if I am not mistaken, a certain confusion in the [Pg 152] minds of some authors as to the relation, between propositions, in virtue of which an inference is valid. In order that it may be valid to infer from , it is only necessary that should be true and that the proposition "not- or " should be true. Whenever this is the case, it is clear that must be true. But inference will only in fact take place when the proposition "not- or " is known otherwise than through knowledge of not- or knowledge of . Whenever is false, "not- or " is true, but is useless for inference, which requires that should be true. Whenever is already known to be true, "not- or " is of course also known to be true, but is again useless for inference, since is already known, and therefore does not need to be inferred. In fact, inference only arises when "not- or " can be known without our knowing already which of the two alternatives it is that makes the disjunction true. Now, the circumstances under which this occurs are those in which certain relations of form exist between and . For example, we know that if implies the negation of , then implies the negation of . Between " implies not-" and " implies not-" there is a formal relation which enables us to know that the first implies the second, without having first to know that the first is false or to know that the second is true. It is under such circumstances that the relation of implication is practically useful for drawing inferences. But this formal relation is only required in order that we may be able to know that either the premiss is false or the conclusion is true. It is the truth of "not- or " that is required for the validity of the inference; what is required further is only required for the practical feasibility of the inference. Professor C. I. Lewis[37] has especially studied the narrower, formal relation which we may call "formal deducibility." He urges that the wider relation, that expressed by "not- or " should not be called "implication." That is, however, a matter of words. [Pg 153] Provided our use of words is consistent, it matters little how we define them. The essential point of difference between the theory which I advocate and the theory advocated by Professor Lewis is this: He maintains that, when one proposition is "formally deducible" from another , the relation which we perceive between them is one which he calls "strict implication," which is not the relation expressed by "not- or " but a narrower relation, holding only when there are certain formal connections between and . I maintain that, whether or not there be such a relation as he speaks of, it is in any case one that mathematics does not need, and therefore one that, on general grounds of economy, ought not to be admitted into our apparatus of fundamental notions; that, whenever the relation of "formal deducibility" holds between two propositions, it is the case that we can see that either the first is false or the second true, and that nothing beyond this fact is necessary to be admitted into our premisses; and that, finally, the reasons of detail which Professor Lewis adduces against the view which I advocate can all be met in detail, and depend for their plausibility upon a covert and unconscious assumption of the point of view which I reject. I conclude, therefore, that there is no need to admit as a fundamental notion any form of implication not expressible as a truth-function. [37]See Mind, vol. XXI., 1912, pp. 522-531; and vol. XXIII., 1914, pp. 240-247. [Pg 154]