Inference has been defined as both a product and a process. When used to indicate a process the term inference becomes synonomous with reasoning. If logicians could agree to confine inference to the product and reasoning to the process, it would remove an ambiguity which is more or less misleading. But since this has not become the custom, we shall use inference as indicating the process as well as the product. _Definitions――Middle Term Explained._ Inference is the thought process of deriving a judgment from one or two antecedent judgments. _Mediate inference is inference by means of a middle term._ Reasoning of this nature involves three terms, two of which are compared with a third or middle term, and then related to each other to form a new judgment. The middle term is the common unit, or the _standard_ by which the other terms are measured. To illustrate: If John and James are each six feet tall, then plainly, they are of the same height. The standard, or _middle_ term, is “_six feet tall_.” =2. THE SYLLOGISM.= Just as the judgment is expressed by means of the proposition, so mediate inference is best expressed by means of the syllogism.[9] The following are syllogisms: (1) James is six feet tall, John is six feet tall, Hence James is as tall as John. (2) All true teachers are just, You are a true teacher, Hence you are just. (3) All men are mortal, You are a man, Hence you are mortal. =3. THE RULES OF THE SYLLOGISM.= All syllogistic reasoning is conditioned by the following eight rules: (1) A syllogism must have three, and only three, different terms. (2) A syllogism must have three, and only three, propositions. (3) The middle term must be distributed at least once. (4) No term must be distributed in the conclusion which is not also distributed in a premise. (5) No conclusion can be drawn from two negative premises. (6) If one premise be negative, the conclusion must be negative; and conversely, to prove a negative conclusion, one of the premises must be negative. (7) No conclusion can be drawn from two particular premises. (8) If one premise be particular, the conclusion must be particular. These rules are exceedingly important, as their observance is necessary in all mediate reasoning. The student needs, not only to understand the meaning of these rules, but he needs to commit them to memory so thoroughly that they may be recalled without hesitation or mistake. To aid the memory, the eight rules may be divided into these four groups: I. Rules one and two relate to the _composition of the syllogism_. II. Rules three and four pertain to the _distribution of terms_. III. Rules five and six have reference to _negative premises_. IV. Rules seven and eight concern _particular premises_. =4. RULES OF THE SYLLOGISM EXPLAINED.= (1) _A syllogism must have three and only three terms._