In order to be able validly to infer the truth of a proposition, we must know that some other proposition is true, and that there is between the two a relation of the sort called "implication," i.e. that (as we say) the premiss "implies" the conclusion. (We shall define this relation shortly.) Or we may know that a certain other proposition is false, and that there is a relation between the two of the sort called "disjunction," expressed by " or ,"[32] so that the knowledge that the one is false allows us to infer that the other is true. Again, what we wish to infer may be the falsehood of some proposition, not its truth. This may be inferred from the truth of another proposition, provided we know that the two are "incompatible," i.e. that if one is true, the other is false. It may also be inferred from the falsehood of another proposition, in just the same circumstances in which the truth of the other might have been inferred from the truth of the one; i.e. from the falsehood of we may infer the falsehood of , when implies . All these four are cases of inference. When our minds are fixed upon inference, it seems natural to take "implication" as the primitive fundamental relation, since this is the relation which must hold between and if we are to be able to infer the truth of from the truth of . But for technical reasons this is not the best primitive idea to choose. Before proceeding to primitive ideas and definitions, let us consider further the various functions of propositions suggested by the above-mentioned relations of propositions. [32]We shall use the letters , , , , to denote variable propositions. The simplest of such functions is the negative, "not-." This is that function of which is true when is false, and false when is true. It is convenient to speak of the truth of a proposition, or its falsehood, as its "truth-value"[33]; i.e. truth is the "truth-value" of a true proposition, and falsehood of a false one. Thus not has the opposite truth-value to . [Pg 146] [33]This term is due to Frege. We may take next disjunction, " or ." This is a function whose truth-value is truth when is true and also when is true, but is falsehood when both and are false. Next we may take conjunction, " and " This has truth for its truth-value when and are both true; otherwise it has falsehood for its truth-value. Take next incompatibility, i.e. " and are not both true." This is the negation of conjunction; it is also the disjunction of the negations of and , i.e. it is "not- or not-." Its truth-value is truth when is false and likewise when is false; its truth-value is falsehood when and are both true.