54. Rules of the Syllogism. — Besides the special rules of each of the figures, logicians have been wont to formulate eight rules applicable to the syllogism in general, expressing the nature of the reasoning.
FIRST RULE. — Terminus esto triplex: medius, majorque, minorque. — The syllogism must have three terms, neither more nor fewer. To reason is in fact to compare two terms with one and the same third, so as to see what logical relation exists between the two terms so compared.
This rule may be violated by defect, in using only two terms, or by excess, in using more than three.
(1) A syllogism with two terms is, e. g., where one of the premises is tautological. E. g.: Every effect has a cause. But the universe is an effect. Therefore the universe has a cause.
This first rule is violated by the form of sophism called petitio principii, which resolves the qnestion by the question (begging the question).
(2) A syllogism contains more than three terms when one term is equivocal and is taken in different acceptations. E. g.,: The operations of thought have the brain as organ. An operation which has the brain as organ is material. Therefore the operations of thought are material.
In this syllogism the middle term, has the brain as organ, is equivocal.
SECOND RULE. — Latius hoc (terminos extremos) quam praemissae conclusio non vult, or: AEque ac praemissae extendat conclusio voces. — The extremes must be the same in the conclusion as in the premises.
The conclusion expresses the results of the comparison made in the premises. It cannot go beyond that; otherwise it would pass from the terms compared in the premises to other terms, and thus would violate the first rule, the essential condition of reasoning.
THIRD RULE. — Aut semel aut iterum medius generaliter esto. — The middle term must be taken as universal in one premise at least.
The analysis of the process of reasoning (50) has made this third rule intelligible. If the middle term were taken twice in a restricted sense, that part of its extension which it represents might possibly be different in the two cases, and there would be four terms in the syllogism (first rule). E. g.: Every metal is heavy. This substance is heavy. Therefore this substance is a metal. The middle term, heavy, is not universal in either of the premises.
This very common sophism is characterized by the adage: Ab uno disce omnes.
FOURTH RULE. — Nequaquam medium capiat conclusio fas est. — The middle term may not enter into the conclusion.
It is for the conclusion to apply to the two extremes the result of the comparison made in the premises between them and the middle term. To introduce the middle term into the conclusion, then, would be to miss the aim of the reasoning.
FIFTH RULE — Ambae affirmantes nequeunt generare negatem. — Two affirmative premises cannot beget a negative conclusion.
If two ideas agree with one and the same third idea, the other rules of the syllogism being observed, they cannot but agree with each other; and the identity affirmed in the premises cannot be denied in the conclusion.
SIXTH RULE. — Utraque si praemissa neget, nil inde sequetur. — With two negative premises no conclusion is possible.
Two extremes both excluded from one middle term cannot be connected with each other on account of this exclusion.
But on the other hand, it is possible that two terms excluded from one given middle term may be comparable with another middle term with which both must be coupled, or else one coupled and the other separated. The use of this other middle term would give a conclusion.
The fact, then, that two extremes are excluded from a given middle term warrants no assertion as to the relation of the extremes.
SEVENTH RULE. — Pejorem. sequitur semper conclusio partem. — The conclusion should follow the premise of lower rank. This formula has a double application:
(1) If one of the premises is negative, the conclusion must be negative. If, of two ideas A and B, A agrees with a third idea, C, while B does not, it is impossible to conclude therefrom that A agrees with B.
(2) If one of the premises is particular, the conclusion cannot be universal.
As the premises cannot both be negative (sixth rule), only two cases are to be considered:
(a) Both the premises are affirmative. (b) One is affirmative; the other, negative.
In case (a) both the predicates are particular; one of the two subjects is by hypothesis particular: there is, then, only one universal term in the premises. As this must be the middle term (third rule), neither of the extremes is universal in the premises and, consequently, cannot be so in the conclusion. So that the conclusion, since it necessarily has a particular subject, is particular.
In case (b) the premises include two universal terms: the predicate of the negative premise and the subject of the proposition which, by hypothesis, is universal.
But the conclusion is negative, so that its predicate is universal. This term. which is the predicate in the conclusion, is not the middle term (fourth rule). The second universal term of the premises is therefore the middle term. Hence the extreme which becomes the subject of the conclusion is particular in the premises, and, consequently, in the conclusion. Therefore the conclusion is particular.
For example: Every man is corporeal. But A is not corporeal. Therefore A is not a man.
The result would be the same if one proposition were both universal and negative, as: No man is spiritual. But A is a man. Therefore A is not spiritual. — Or: But B is spiritual. Therefore B is not a man. When one premise is particular, then, the conclusion must be particular.
EIGHTH RULE — Nil sequitur geminis ex particularibus unquam. — No conclusion follows from two particular premises.
As both the premises cannot be negative (sixth rule), the only possible cases are: (a) Both premises are affirmative. (b) One is affirmative; the other, negative.
In case (a) all the terms are particular: the two predicates, because the propositions are affirmative; the two subjects, by hypothesis. The middle term, therefore, is not once taken universally. The third rule is violated. No conclusion.
Example: Some men are rich. Some men are ignorant. Therefore some rich men are ignorant.
If this syllogism were valid, it might be proved in the same way that some rich men are poor, which exposes the sophism.
In case (b) the premises contain only one universal term, the predicate of the negative premiss. But the conclusion being negative, its predicate is universal; being so in the conclusion, it must also be universal in the premises. Consequently, the middle term, which cannot be identical with the predicate of the conclusion (fourth rule), is twice particular in the premises. Once more, the third rule is violated. No conclusion. Example: Some men are learned. But some men are not virtuous. Therefore some learned men are not virtuous.
The inconsequence is manifest.