# The Principles of Mathematics (1903)

## § 39

It is commonly said that an inference must have premisses and a conclusion, and it is held, apparently, that two or more premisses are necessary, if not to all inferences, yet to most. This view is borne out, at first sight, by obvious facts: every syllogism, for example, is held to have two premisses. Now such a theory greatly complicates the relation of implication, since it renders it a relation which may have any number of terms, and is symmetrical with respect to all but one of them, but not symmetrical with respect to that one (the conclusion). This complication is, however, unnecessary, first, because every simultaneous assertion of a number of propositions is itself a single proposition, and secondly, because, by the rule which we called exportation, it is always possible to exhibit an implication explicitly as holding between single propositions. To take the first point first: if k be a class of propositions, all the propositions of the class k are assered by the single proposition for all values of x, if x implies x, then x is a k implies x, or in moder ordinary language, every k is true. And as regards the second point, which assumes the number of premisses to be finite, pq implies r is equivalent, if q be a proposition, to p implies that q implies r, in which latter form the implications hold explicitly between single propositions. Hence we may afely hold implication to be a relation between two propositions, not a relation of an arbitrary number of premisses to a single conclusion.(§ 39 ¶ 1)