Our principle of inference then includes the rule of negative premises whenever it is true, and discriminates correctly between the cases where it does and does not hold true. The paralogism, anciently called *the Fallacy of Undistributed Middle*, is also easily exhibited and infallibly avoided by our system. Let the premises be Hydrogen is an element, (1) All metals are elements. (2) According to the syllogistic rules the middle term “element” is here undistributed, and no conclusion can be obtained; we cannot tell then whether hydrogen is or is not a metal. Represent the terms as follows A = hydrogen, B = element, C = metal. The premises then become A = AB, (1) C = CB. (2) The reader will here, as in a former page (p. 62), find it impossible to make any substitution. The only term which occurs in both premises is B, but it is differently combined in the two premises. For B we must not substitute A, which is equivalent to AB, not to B. Nor must we confuse together CB and AB, which, though they contain one common letter, are different aggregate terms. The rule of substitution gives us no right to decompose combinations; and if we adhere rigidly to the rule, that if two terms are stated to be equivalent we may substitute one for the other, we cannot commit the fallacy. It is apparent that the form of premises stated above is the same as that which we obtained by translating two negative premises into the affirmative form. The old fallacy, technically called the *Illicit Process of the Major Term*, is more easy to commit and more difficult to detect than any other breach of the syllogistic rules. In our system it could hardly occur. From the premises All planets are subject to gravity, (1) Fixed stars are not planets, (2) we might inadvertently but fallaciously infer that, “Fixed stars are not subject to gravity.” To reduce the premises to symbolic form, let A = planet B = fixed star C = subject to gravity; then we have the propositions A = AC (1) B = B*a*. (2) The reader will try in vain to produce from these premises by legitimate substitution any relation between B and C; he could not then commit the fallacy of asserting that B is not C. There remain two other kinds of paralogism, commonly known as the fallacy of Four Terms and the Illicit Process of the Minor Term. They are so evidently impossible while we obey the rule of the substitution of equivalents, that it is not necessary to give any illustrations. When there are four distinct terms in two propositions as in A = B and C = D, there could evidently be no opening for substitution. As to the Illicit Process of the Minor Term it consists in a flagrant substitution for a term of another wider term which is not known to be equivalent to it, and which is therefore not allowed by our rule to be substituted for it. CHAPTER V. DISJUNCTIVE PROPOSITIONS.