We observe that by the Law of Duality the term not-B is thus described *b* = A*b* ꖌ *ab*. (2) For A in this proposition we substitute its description as given in (1), obtaining *b* = AB*b* ꖌ *ab*. But according to the Law of Contradiction the term AB*b* must be excluded from thought, or AB*b* = 0. Hence it results that *b* is either nothing at all, or it is *ab*; and the conclusion is *b* = *ab*. As it will often be necessary to refer to a conclusion of this kind I shall call it, as is usual, the *Contrapositive Proposition* of the original. The reader need hardly be cautioned to observe that from all A’s are B’s it does not follow that all not-A’s are not-B’s. For by the Law of Duality we have *a* = *a*B ꖌ *ab*, and it will not be found possible to make any substitution in this by our original premise A = AB. It still remains doubtful, therefore, whether not-metal is element or not-element. The proof of the Contrapositive Proposition given above is exactly the same as that which Euclid applies in the case of geometrical notions. De Morgan describes Euclid’s process as follows[75]:--“From every not-B is not-A he produces Every A is B, thus: If it be possible, let this A be not-B, but every not-B is not-A, therefore this A is not-A, which is absurd: whence every A is B.” Now De Morgan thinks that this proof is entirely needless, because common logic gives the inference without the use of any geometrical reasoning. I conceive however that logic gives the inference only by an indirect process. De Morgan claims “to see identity in Every A is B and every not-B is not-A, by a process of thought prior to syllogism.” Whether prior to syllogism or not, I claim that it is not prior to the laws of thought and the process of substitutive inference, by which it may be undoubtedly demonstrated. [75] *Philosophical Magazine*, Dec. 1852; p. 437. *Employment of the Contrapositive Proposition.* We can frequently employ the contrapositive form of a proposition by the method of substitution; and certain moods of the ancient syllogism, which we have hitherto passed over, may thus be satisfactorily comprehended in our system. Take for instance the following syllogism in the mood Camestres:-- “Whales are not true fish; for they do not respire water, whereas true fish do respire water.” Let us take A = whale B = true fish C = respiring water The premises are of the forms A = A*c* (1) B = BC (2) Now, by the process of contraposition we obtain from the second premise *c* = *bc* and we can substitute this expression for *c* in (1), obtaining A = A*bc* or “Whales are not true fish, not respiring water.” The mood Cesare does not really differ from Camestres except in the order of the premises, and it could be exhibited in an exactly similar manner.