(2) The syllogism is the common mode of expression for mediate inference. (3) Valid syllogistic reasoning is conditioned by eight rules. The first and second relate to the composition of the syllogism; the third and fourth to the distribution of terms; the fifth and sixth to negative premises; the seventh and eighth to particular premises. (4) All syllogisms must have three terms: the major, the minor, and the middle. The middle term occurs twice in the premises but never appears in the conclusion. The minor term is always the subject, and the major term the predicate of the conclusion. The major term is usually broader than the minor. No conclusion can be drawn from four terms. To attempt this gives rise to the fallacy of four terms. All syllogisms must have three propositions, the major and the minor premises, and the conclusion. The major premise first and the minor second is the more logical arrangement, although the common conversational form is to use the minor premise first. Ambiguous middle amounts to the fallacy of four terms. Unless the middle term is distributed at least once in the syllogism, it fails to become a common standard. Distributing a term in the conclusion, without its being distributed in its premise, is equivalent to asserting that, “What is true of a part is true of the whole.” This error results in the fallacies of illicit major and minor. A conclusion from two negatives is impossible, because of the total exclusion of the middle term. Of two terms, if one is affirmed and the other denied of a third term, then they must be denied of each other; and, conversely, if two terms are to be denied of each other, one must be affirmed and the other denied of a given third term. This fundamental principle necessitates deriving a negative conclusion from two premises when one is negative. It, likewise, compels the converse of this. A valid conclusion from two particulars is untenable because of the two negative fallacies, or some fallacy relative to the distribution of terms. One particular premise forces a particular conclusion because of the fallacies of two negatives, two particulars, and illicit minor. (5) Aristotle’s dictum simplified means, “What is true of the whole is true of the part.” (6) The canons of the syllogism, three in number, are: (1) “Two terms agreeing with one and the same third term agree with each other.” (2) “Two terms of which one agrees and the other does not agree with one and the same third term do not agree with each other.” (3) “Two terms both disagreeing with one and the same third term may or may not agree with each other.” (7) The foregoing canons may be stated as mathematical axioms. =10. ILLUSTRATIVE EXERCISES.= (1a) Make use of the proper symbols and indicate the three terms of each of the attending syllogisms: (1) All fixed stars twinkle, Vega is a fixed star, ―――――――――――――――――――――――― ∴ Vega twinkles.