1911 Encyclopædia Britannica/Syllogism

SYLLOGISM (Gr. avWoyiaixos, from avv, and X670S, an argument resulting from combination, i.e. of premises), in logic, an argument consisting of premises and a conclusion. Aris- totle's definition is (Anal. Pr. a i. 24 b 18; cf. Top. a i. 100 a 25): ffuXXoTicrjuis eori Xo*yos h> $ Ttdevnov tlvcov erepov ri toiv tceifiivoov «£ ava.yK.ns ovfifialvei tQ touto. elvai, " a syllogism is an argument in which, certain things being posited (the premises), something other than the premises necessarily results from their being true." This definition, though it contains the really important facts, is too wide in two respects. (1) Aristqtle himself and subsequent logicians restrict the term to arguments in which there are but two premises. (2) In point of fact, all logicians further confine the syllogism to arguments in which the terms are related as subject and predicate (or attribute in the widest sense) . A fortiori arguments, for example, wherein relations of quantity are brought together, though syllogistic in type, are generally excluded. Owing largely to the simplicity and symmetry of the syllogism it has been a commonplace of logic to make the syllogistic form the type of all thought. Modern logicians (cf. especially F. H. Bradley in his Logic) have, however, shown that in practice its importance is greatly exaggerated.

A. The Deductive Syllogism. — This argument is the simplest form of "mediate" inference, i.e. an argument in which two terms are brought into a necessary relation by the aid of a "middle" term which serves as a bridge. It requires, therefore, two propositions known as premises^[1] (also spelled premisses, as being more in accordance with the Lat. praemissae [propositiones sententiae], things put or posited in advance) which contain one common term and one other term each. In the conclusion the middle term disappears and the other two are brought together. The premises are assumed: whether true or false, the conclusion follows necessarily. If the premises are true, the conclusion must be true: if they are false the great' probability is that the conclusion is false. The predicate of the conclusion is called the major term, the subject the minor term; the term which is common to the premises and disappears in the conclusion is the middle term. Hence the premise which contains the major term is called the major premise: that which contains the minor, the minor premise. The form of the syllogism is therefore:—

A is B Major premise

C is A Minor

.'. C is B Conclusion

Syllogisms differ in (a) "figure" and (6) "mood." (a) Difference of figure depends on the order of the terms in the premises. The above is the scheme of figure I. If the middle term is the predicate in both premises, the syllogism is in figure II.: if the subject in both, figure III. These are the only figures recognized by Aristotle, though he points out that the premises in figure I. may justify a conclusion in which the predicate is not, as normally, the major term, but the minor. This possibility, according to Averroes, led to the adoption by the physician Galen of the so-called fourth figure, in which the middle term is predicate of the major and subject of the minor. This, however, destroys the appropriateness of the phrases major and minor term which are specially chosen because in fact the major term does imply the more comprehensive notion. The conclusion is an artificial proposition which would be stated naturally in the converse.

b. The distinction of moods is according to the quantity or quality of the propositions of the syllogism (universal, particular, affirmative, negative, in all the possible combinations). So far as mere form goes, each mocd may occur in every figure, though in many cases the conclusion apparently yielded from the premises is invalid. A simple calculation shows that formally there are 64 possible moods. Investigation shows that of these nineteen 2 only are valid, and rules have been formulated which give the reasons for the invalidity of the remaining 45.

The rules which govern syllogistic arguments thus described are: —

i. A syllogism must contain three and three terms only, (a) Four terms would mean the absence of any connecting link. (6) If the middle term is ambiguous there are really four not three terms. The violation of (a) is the fallacy " Quaternio terminorum "; of (b) " ambiguous middle."

ii. The middle teirn must be distributed in one premise at least, i.e. it must be taken universally, as including all the particulars over which it extends (see Extension). Violation of this is the fallacy of " undistributed middle."

iii. No inference can be made from two negative premises.

iv. If either premise is negative, the conclusion is negative.

v. The conclusion cannot be negative, if both premises are affirmative.

vi. No term may be distributed in this conclusion which was not distributed in the premise in which it occurs. Violation of this rule is called an "illicit process of the major (or the minor) term."

vii. From two particular premises nothing can be inferred.^[2]

viii. If either premise is particular, the conclusion must be particular.^[3] The general criticism of the syllogism as a means of discovering truth is that it is a petitio principii, or begging of the question. This accusation is based to some extent on the Aristotelian "Dictum de omni et nullo" {Anal. Pri. a i. 24, b 26-30), generally stated as " That which is affirmed or denied of any whole may be affirmed or denied of anything contained within (or 'any part of') that whole." To take a concrete instance of a valid mood: all men are mortal, all Frenchmen are men, therefore all Frenchmen are mortal (the mood Barbara). It is argued that either there is here no real discovery (i.e. new truth) or the major premise is improperly used (begs the question) inasmuch as unless we knew that all Frenchmen are mortal we could not state that all men are mortal. The problem raised is a real one, and has been discussed by all logicians, from the time of Mill especially. In brief, the solution depends upon the view we take of the major premise, "all men are mortal." If that judgment is taken as a mere enumeration of particulars, i.e. in extension, as meaning that all men have been investigated and found to be mortal, clearly it could not be used to make the new discovery that a particular group of men are mortal; the syllogism so understood is a petitio principii. If, however, we take the true view of the major premise, namely, that it is not a mere summary of observed particulars but the enunciation of a necessary connexion between two concepts or universals, then the conclusion assumes a different character. The " whole " (omne) of the dictum, the major term, ceases to be taken in extension, and becomes intensive or connotative, and the infer- ence consists in subsuming the minor under (bringing it into connexion with) the major. This is the true view of the scientific or inductive universal (as opposed to that of nominalism or pure empiricism). It remains true that in fact the conclusion is contained in the premises—this is essential to the validity of the syllogism—but the inference is a real one because it brings out and shows the necessity of a conclusion which was not before in our minds.

Hypothetical and Disjunctive Syllogisms.—The term syllogism has been extended to cover certain forms of ratiocination which are not based on categorical propositions. The propriety of _ this extended use is open to question and is denied by some logicians.

a. Hypothetical "Syllogisms" are those in which one premise is a hypothetical proposition, the other a categorical. Two forms are possible (i.) modus ponens (which establishes the consequent set down in the major premise) : if A is B, it is C (or C is D) ; Ais B; therefore A is C (or C is D), and (ii.) modus Miens (which dis- proves the antecedent) : if A is B, it is C (or C is D) ; A is not C (or C is not D) ; therefore it is not B (or A is not B). In (i.) a valid conclusion follows from the affirmation of the antecedent: in (ii.) from denying the consequent, but in neither case conversely. The distinction is of greater importance than would appear when one realizes how obvious the facts really are, and in practice it happens frequently that speakers claim with success to disprove a proposition by disproving the fact alleged in support of_ it, or to establish a hypothesis by showing that facts agree with its consequences.

b. Disjunctive "Syllogisms" are those in which one premise^ is a disjunctive proposition, the other a categorical proposition which states or denies one of the two alternatives set forth. Again two forms occur: (i.) modus ponendo Miens which by the affirmation of one alternative denies the other (A is either B or C; A is B; therefore it is not C : or either A is B, or C is D ; A is B ; therefore C is not D : or either A or B is C ; A is C ; therefore B is not C); (ii.) modus tollendo ponens which by the denial of the one, establishes the validity of the other alternative (A is either B or C ; A is not B; therefore it is C : or either A or B is C ; A is not C ; therefore B is C : or either A is B, or C is D ; A is not B ; therefore C is D). The validity of such arguments depends upon the sense in which we understand the disjunctive proposition: we must assume that the alternatives are mutually exclusive.^[4]

Sorites.—Finally it is necessary to mention a complex syllogistic argument known as the Sorites (Gr. aup6s, heap). It has been de- fined as a syllogism in Fig. 1 (see above) having many middle terms; it is really a series of syllogisms (a polysyllogism), each one proving a premise of another, the intermediate conclusions being suppressed. Its form is A is B, B is C, C is D . . . . Y is Z, therefore A is Z. Each syllogism of the series is called a prosyllogism"^[5] in relation to the one that succeeds, and an " episyllogism " in relation to its predecessors. Resolution of the sorites into its con- stituent elements gives the rules (a) that no premise except the first may be particular and (jS) that no premise except the last may be negative.

B. The Inductive Syllogism, like the deductive, is first systematized by Aristotle, who described it as 6 k% kirayoryrjs mXhoyioixds. Unlike the deductive it consists in establishing a conclusion from particular premises, i.e. of referring the major term to the middle by means of the minor. The form is " A B C D, &c, are P; A B C D are all M; thus all M are P." This so-called syllogism has been much criticized by modern logicians on various grounds (see Logic).

Discussions of the syllogism will be found in all textbooks on Logic, and the more elaborate syllogistic forms are discussed in the article Logic.

1. Aristotle Trpor&reis, originally translated propositions; praemissae dates from 12th century Latin translations of Arabic versions of Aristotle. The term "premises" (a house, &c), is derived loosely from the legal phase denoting that which has already been mentioned in a document, and is etymologically the same.
2. The following mnemonic hexameter verses are generally given (first apparently in Aldrich's Artis logicae rudimenta) to aid in remembering these moods. The vowels in the words, A, E, I, O, show the quantity and quality of the premises:—
   Barbara Celarent Darii Ferioque prions;
   Cesare Camestres Festino Baroco secundi;
   Tertia Darapti Disamis Datisi Felapton
   Bocardo Ferison habet: quarta insuper addit
   Bramantip Camenes Dimaris Fesapo Fresison.
3. These latter are corollaries of previous rules.
4. For a dilemma which includes both hypothetical and disjungtive reasoning see Dilemma.
5. Where one premise of a prosyllogism is omitted (see Enthymeme), this argument is sometimes called an " epicheirema."