Conclusion: "Socrates" (minor term) is mortal (major term).
The reason for the rule that there shall be "only three" terms is that reasoning consists in comparing two terms with each other through the medium of a third term. There must be three terms; if there are more than three terms, we form two syllogisms instead of one.
III. That one premise, at least, must be affirmative. This, because "from two negative propositions nothing can be inferred." A negative proposition asserts that two things differ, and if we have two propositions so asserting difference, we can infer nothing from them. If our Syllogism stated that: (1) "Man is not mortal;" and (2) that "Socrates is not a man;" we could form no Conclusion, either that Socrates was or was not mortal. There would be no logical connection between the two premises, and therefore no Conclusion could be deduced therefrom. Therefore, at least one premise must be affirmative.
IV. If one premise is negative, the conclusion must be negative. This because "if one term agrees and another disagrees with a third term, they must disagree with each other." Thus if our Syllogism stated that: (1) "Man is not mortal;" and (2) that: "Socrates is a man;" we must announce the Negative Conclusion that: (3) "Socrates is not mortal."
V. That the Middle Term must be distributed; (that is, taken universally) in at least one premise. This "because, otherwise, the Major Term may be compared with one part of the Middle Term, and the Minor Term with another part of the latter; and there will be actually no common Middle Term, and consequently no common ground for an inference." The violation of this rule causes what is commonly known as "The Undistributed Middle," a celebrated Fallacy condemned by the logicians. In the Syllogism mentioned as an example in this chapter, the proposition "Man is mortal," really means "All men," that is, Man in his universal sense. Literally the proposition is "All men are mortal," from which it is seen that Socrates being "a man" (or some of all men) must partake of the quality of the universal Man. If the Syllogism, instead, read: "Some men are mortal," it would not follow that Socrates must be mortal—he might or might not be so. Another form of this fallacy is shown in the statement that (1) White is a color; (2) Black is a color; hence (3) Black must be White. The two premises really mean "White is some color; Black is some color;" and not that either is "all colors." Another example is: "Men are bipeds; birds are bipeds; hence, men are birds." In this example "bipeds" is not distributed as "all bipeds" but is simply not-distributed as "some bipeds." These syllogisms, therefore, not being according to rule, must fail. They are not true syllogisms, and constitute fallacies.
To be "distributed," the Middle Term must be the Subject of a Universal Proposition, or the Predicate of a Negative Proposition; to be "undistributed" it must be the Subject of a Particular Proposition, or the Predicate of an Affirmative Proposition. (See chapter on Propositions.)
VI. That an extreme, if undistributed in a Premise, may not be distributed in the Conclusion. This because it would be illogical and unreasonable to assert more in the conclusion than we find in the premises. It would be most illogical to argue that: (1) "All horses are animals; (2) no man is a horse; therefore (3) no man is an animal." The conclusion would be invalid, because the term animal is distributed in the conclusion, (being the predicate of a negative proposition) while it is not distributed in the premise (being the predicate of an affirmative proposition).
As we have said before, any Syllogism which violates any of the above six syllogisms is invalid and a fallacy.
There are two additional rules which may be called derivative. Any syllogism which violates either of these two derivative rules, also violates one or more of the first six rules as given above in detail.
The Two Derivative Rules of the Syllogism are as follows: