(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.” Making use of the symbols as explained on a previous page of this chapter, it will be seen that the first canon conforms to this syllogistic type: All M is G All S is M ―――――――――― ∴ All S is G The two terms are S and G, while M is the third term. The attending symbolizations illustrate, respectively, the second and third canons: No M is G All S is M ―――――――――― ∴ No S is G No M is G No S is M Conclusion indeterminate. =7. THREE MATHEMATICAL AXIOMS.= Analogous to the three canons treated in “6,” there are certain mathematical axioms which are here stated: (1) “Things equal to the same thing are equal to each other.” (2) “One thing equal to and the other thing not equal to the same third thing are not equal to each other.” (3) “Things not equal to the same thing may or may not equal each other.” Illustrations of the three axioms: (1) If x equals 5, and y equals 5, then x equals y. (2) If x equals 5, and y does not equal 5, then x does not equal y. (3) If x does not equal 5, and y does not equal 5, then x may or may not equal y. =8. OUTLINE.= MEDIATE INFERENCE. (1) Inference and reasoning. Definitions. Middle term explained. (2) The analogy between the judgment and the syllogism. (3) Rules of the syllogism given. Eight in number. (4) Rules of the syllogism explained: Rule 1. Syllogistic symbols. Major, minor, and middle terms; how found. Fallacy of four terms. Rule 2. Major and minor premises and conclusion, how determined. Logical arrangement. Reason for three propositions. Rule 3. Reason for omitting “ambiguous middle” from rule. Undistributed and distributed middle explained. Rule 4. Illicit major and minor explained and illustrated. Rule 5. Fallacy of two negatives. Rule 6. Fallacy of negative conclusion. Rule 7. Fallacy of two particulars. Rule 8. Fallacy of particular conclusion. (5) Aristotle’s dictum. (6) Canons of the syllogism. (7) Mathematical axioms. =9. SUMMARY.= (1) Inference is a term used to denote a process as well as a product. As a process reasoning and inference are in reality synonomous terms. Inference is a thought process of deriving a judgment from one or two antecedent judgments. Mediate inference is inference by means of a middle term. Mediate inference makes use of three terms, two of which are compared with a third term as a standard. This third term is called the middle term.