Last take implication, i.e. " implies ," or "if , then ." This is to be understood in the widest sense that will allow us to infer the truth of if we know the truth of . Thus we interpret it as meaning: "Unless is false, is true," or "either is false or is true." (The fact that "implies" is capable of other meanings does not concern us; this is the meaning which is convenient for us.) That is to say, " implies " is to mean "not- or ": its truth-value is to be truth if is false, likewise if is true, and is to be falsehood if is true and is false. We have thus five functions: negation, disjunction, conjunction, incompatibility, and implication. We might have added others, for example, joint falsehood, "not- and not-," but the above five will suffice. Negation differs from the other four in being a function of one proposition, whereas the others are functions of two. But all five agree in this, that their truth-value depends only upon that of the propositions which are their arguments. Given the truth or falsehood of , or of and (as the case may be), we are given the truth or falsehood of the negation, disjunction, conjunction, incompatibility, or implication. A function of propositions which has this property is called a "truth-function." The whole meaning of a truth-function is exhausted by the statement of the circumstances under which it is true or false. "Not-," for example, is simply that function of which is true when is false, and false when is true: there is no further [Pg 147] meaning to be assigned to it. The same applies to " or " and the rest. It follows that two truth-functions which have the same truth-value for all values of the argument are indistinguishable. For example, " and " is the negation of "not- or not-" and vice versa; thus either of these may be defined as the negation of the other. There is no further meaning in a truth-function over and above the conditions under which it is true or false. It is clear that the above five truth-functions are not all independent. We can define some of them in terms of others. There is no great difficulty in reducing the number to two; the two chosen in Principia Mathematica are negation and disjunction. Implication is then defined as "not- or "; incompatibility as "not- or not-"; conjunction as the negation of incompatibility. But it has been shown by Sheffer[34] that we can be content with one primitive idea for all five, and by Nicod[35] that this enables us to reduce the primitive propositions required in the theory of deduction to two non-formal principles and one formal one. For this purpose, we may take as our one indefinable either incompatibility or joint falsehood. We will choose the former. [34]Trans. Am. Math. Soc., vol. XIV. pp. 481-488. [35]Proc. Camb. Phil. Soc., vol. XIX., i., January 1917.