By Clarence Irving Lewis, Cooper H. Langford

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That is, if we combine two equations, a = 0 and 6 = 0, by addi tion, we lose nothing of the force of the equations combined. But if we should combine them by multiplication, the result, a b = 0, would be valid but would not be equivalent to the separate equations combined. 2 In this type of diagram, classes are represented by circles or other areas, and the diagram is so drawn as always to represent ‘ the universe/ or 1 References are, of course, to the numbered theorems of the preceding chapter.

Such mathematical truths will be a priori, since they can be certified by logic alone. But they will be a priori precisely because they are not synthetic, but are merely instances of general logical principles. As has been said, this is implicit in the previously recognized ideal of mathematics as at once deductive and abstract; for example, in the recognition that non-Euclidean geometry has 22 SYM BOLIC LOGIC the same mathematical validity as Euclid. The importance of the logistic development, in this connection, is that the possibility of achieving the logistic form depends upon the elimination from mathematics of any deductive step which is not simply an instance of some universal logical principle; and that symbolic representation is a very reliable check upon this purely analytic character of the demonstration.

If all a is b and all b is c, then all a is c: the relation c is transitive. 3-8 a c b is equivalent to - 6 c - a . [2 *8 ] a c b is equivalent to a - 6 = 0 . [2*9] a- b = - ( - a )* - 6 = - 6 --(-a ). And [2 - 8 ] - 6 * -(-a ) = 0 is equivalent to - 6 c - a . ” The terms of any relation c may be transposed by negating both. This is called the Law of Transposition. These last two laws are the fundamental principles underlying the validity of the syllogism. The logical copula ‘ is’ may be replaced by any relation which is transitive and obeys the Law of Transposition: the validity of syllogistic reasoning will be unaffected by such substitution.