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a mode of argument that forms the core of the body of Western logical thought. Aristotle defined syllogistic logic, and his formulations were thought to be the final word in logic; they underwent only minor revisions in the subsequent 2,200 years. Every syllogism is a sequence of three propositions such that the first two imply the third, the conclusion. There are three basic types of syllogism: hypothetical, disjunctive, and categorical. The hypothetical syllogism, modus ponens, has as its first premise a conditional hypothesis: If p then q; it continues: p, therefore q. The disjunctive syllogism, modus tollens, has as its first premise a statement of alternatives: Either p or q; it continues: not q, therefore p. The categorical syllogism comprises three categorical propositions, which must be statements of the form all x are y, no x is y, some x is y, or some x is not y. A categorical syllogism contains precisely three terms: the major term, which is the predicate of the conclusion; the minor term, the subject of the conclusion; and the middle term, which appears in both premises but not in the conclusion. Thus: All philosophers are men (middle term); all men are mortal; therefore, All philosophers (minor term) are mortal (major term). The premises containing the major and minor terms are named the major and minor premises, respectively. Aristotle noted five basic rules governing the validity of categorical syllogisms: The middle term must be distributed at least once (a term is said to be distributed when it refers to all members of the denoted class, as in all x are y and no x is y); a term distributed in the conclusion must be distributed in the premise in which it occurs; two negative premises imply no valid conclusion; if one premise is negative, then the conclusion must be negative; and two affirmatives imply an affirmative. John Venn, an English logician, in 1880 introduced a device for analyzing categorical syllogisms, known as the Venn diagram. Three overlapping circles are drawn to represent the classes denoted by the three terms. Universal propositions (all x are y, no x is y) are indicated by shading the sections of the circles representing the excluded classes. Particular propositions (some x is y, some x is not y) are indicated by placing some mark, usually an "X," in the section of the circle representing the class whose members are specified. The conclusion may then be read directly from the diagram.



a type of deductive inference, the two premises and the conclusion of which have the same subject-predicate structure.

The term “syllogism” is most often applied to the “categorical” syllogisms, the premises and conclusions of which are statements (judgments) expressed in simple sentences, with the verb “to be” (indicative mood, singular or plural, with or without negation) as the grammatical predicate linking the terms of the sentence—the subject and the predicate in the logical sense (the designation of a class). The sentences in categorical syllogisms are formed with quantifying words, such as “all,” “any,” “each,” “every,” and “some” (or “there is,” “there exists,” and so forth). They may take any of four forms (the terms of the sentences are designated by capital letters): all R are Q (a universal affirmative, usually designated by the letter A); no R is Q (universal negative, designated by E); some R are Q (particular affirmative, designated by I), and some R are not Q (particular negative, designated by O).

The following judgment is an example of a categorical syllogism: no P is M; some S are M; therefore some S are not P. This may also be stated as a conditional sentence: if no P is M and some S are M, then some S are not P. The following is also an example of a categorical syllogism: every M is P; every S is M; therefore every S is P. (The latter type of syllogism is represented by the literary example “All humans are mortal; all Greeks are humans; therefore all Greeks are mortal.”) The premise containing the predicate of the conclusion (the major term P) is called the major premise, and the premise containing the subject of the conclusion (the minor term S), the minor premise.

Syllogisms are divided into four figures, depending on the middle term M, which is included only in the premises of the syllogism. In the first, M serves as the subject in the major premise and as the predicate in the minor premise; in the second, as the predicate in both premises; in the third, as the subject in both premises; and in the fourth, as the predicate in the major premise and the subject in the minor premise.

Different moods of syllogisms are distinguished, depending on the form of the syllogistic sentences (A, E, I, or O). In each figure there are 4 X 4 X 4 = 64 conceivable moods, or a total of 256 moods. However, only 24 prove to be valid (that is, guaranteeing a true conclusion from true premises). Of these, five are weakened moods—that is, it is possible to strengthen them by the substitution of a particular sentence for a universal one in the conclusion. Thus, there are 19 unweakened, valid syllogistic moods in the four syllogistic figures. (In the following list, the first letter denotes the type of major premise; the second, the type of minor premise; and the third, the type of conclusion.) The 19 unweakened moods are AAA, EAE, All, and EIO in the first figure; EAE, AEE, EIO, and AOO in the second; AAI, IA I, All, EAO, OAO, and EIO in the third; and AAI, AEE, IAI, EA O, and EIO in the fourth. (See for a discussion of the validity of these moods and the invalidness of the others.)

The term “syllogism” is also used in a broader sense to refer to deductions drawn from other types of sentences. Thus, there are conditional, conditional-categorical, disjunctive-categorical, and conditional disjunctive syllogisms. Sometimes, the term “syllogism” is used as a synonym for “deduction.”


A statement together with a conclusion; this usually has the form “if p then q.”


1. a deductive inference consisting of two premises and a conclusion, all of which are categorial propositions. The subject of the conclusion is the minor term and its predicate the major term; the middle term occurs in both premises but not the conclusion. There are 256 such arguments but only 24 are valid. Some men are mortal; some men are angelic; so some mortals are angelic is invalid, while some temples are in ruins; all ruins are fascinating; so some temples are fascinating is valid. Here fascinating, in ruins, and temples are respectively major, middle, and minor terms
2. a deductive inference of certain other forms with two premises, such as the hypothetical syllogism, if P then Q; if Q then R; so if P then R
3. a piece of deductive reasoning from the general to the particular
4. a subtle or deceptive piece of reasoning
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In response, realists might embrace the inferentialist functional thesis that a mathematical sentence can perform the function of facilitating deductive inferences from concrete sentences to other concrete sentences, while adhering to the realist semantic thesis that a mathematical sentence performs the function of describing the abstract world.
I have said that deductive inferences are justified by their conformity to valid general rules, and that general rules are justified by their conformity to valid inferences.
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