This result is in strict accordance with the fundamental principles of inference, and it may be a question whether it is not a self-evident result, independent of the steps of deduction by which we have reached it. For where two classes are coincident like A and B, whatever is true of the one is true of the other; what is excluded from the one must be excluded from the other similarly. Now as *a* bears to A exactly the same relation that *b* bears to B, the identity of either pair follows from the identity of the other pair. In every identity, equality, or similarity, we may argue from the negative of the one side to the negative of the other. Thus at ordinary temperatures Mercury = liquid-metal, hence obviously Not-mercury = not liquid-metal; or since Sirius = brightest fixed star, it follows that whatever star is not the brightest is not Sirius, and *vice versâ*. Every correct definition is of the form A = B, and may often require to be applied in the equivalent negative form. Let us take as an illustration of the mode of using this result the argument following: Vowels are letters which can be sounded alone, (1) The letter *w* cannot be sounded alone; (2) Therefore the letter *w* is not a vowel. (3) Here we have a definition (1), and a comparison of a thing with that definition (2), leading to exclusion of the thing from the class defined. Taking the terms A = vowel, B = letter which can be sounded alone, C = letter *w*, the premises are plainly of the forms A = B, (1) C = *b*C. (2) Now by the Indirect method we obtain from (1) the Contrapositive *b* = *a*, and inserting in (2) the equivalent for *b* we have C = *a*C, (3) or “the letter *w* is not a vowel.” *Miscellaneous Examples of the Method.* We can apply the Indirect Method of Inference however many may be the terms involved or the premises containing those terms. As the working of the method is best learnt from examples, I will take a case of two premises forming the syllogism Barbara: thus Iron is metal (1) Metal is element. (2) If we want to ascertain what inference is possible concerning the term *Iron*, we develop the term by the Law of Duality. Iron must be either metal or not-metal; iron which is metal must be either element or not-element; and similarly iron which is not-metal must be either element or not-element. There are then altogether four alternatives among which the description of iron must be contained; thus Iron, metal, element, (α) Iron, metal, not-element, (β) Iron, not-metal, element, (γ) Iron, not-metal, not-element. (δ)