# syllogism

**syllogism,**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.

*The Columbia Electronic Encyclopedia,* 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.

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