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.