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noun Logic A form of deductive reasoning consisting of a major premise, a minor premise, and a conclusion; for example, All humans are mortal, the major premise, I am a human, the minor premise, therefore, I am mortal, the conclusion.
noun Reasoning from the general to the specific; deduction.
noun A subtle or specious piece of reasoning.

Century Dictionary (31)

A logical formula consisting of two premises and a conclusion alleged to follow from them, in which a term contained in both premises disappears: but the truth of neither the premises nor the conclusion is necessarily asserted. This definition includes the modus ponens (which see, under modus), the formula of which is that from the following from an antecedent of a consequent, together with the antecedent, follows the consequent. This depends upon two principles—first, the principle of identity, that anything follows from itself; and, secondly, the principle that to say that from A it follows that from B follows C is the same as to say that from A and B follows C. Under the former principle comes the formula that the following from an antecedent of a consequent follows from itself, and this, according to the second principle, is identical with the principle of the modus panens. But the syllogism is often restricted to those formula; which embody the nota notæ (or maxim, nota notæ est nota rei ipsius), which may be stated under the form—from the following of anything from a consequent follows the following of the same thing from the antecedent of that consequent. Under this form it is the principle of contraposition. The simplest possible of such syllogisms is like this: Enoch was a man; hence, since being mortal is a consequence of being a man, Enoch was mortal. All syllogisms except the modus ponens involve this principle. A syllogism which involves only this principle, and that in the simplest and directest manner, like the last example, is called a syllogism in Barbara. In such a syllogism the premise enunciating a general rule is called the major premise, while that which subsumes a case under that rule is called the minor premise. A syllogism whose cogency depends only upon what is within the domain of consciousness is called an explicatory (or analytic) syllogism. A syllogism which supposes (though only problematically) a generalizing character in nature is called an ampliative (or synthetic) syllogism. (See explicative inference (under inference), and induction, 5.) Analytic syllogisms are either necessary or probable. Necessary syllogisms are either non-relative or relative. Non-relative syllogisms are either categorical or hypothetical, but that is a trifling distinction. They are also either direct or indirect. A direct syllogism is one which applies the principle of contraposition in a direct and simple manner. An indirect syllogism is either minor or major. A minor indirect syllogism is one which from the major premise of a direct (or less indirect) syllogism and a consequence which would follow from its conclusion infers that the same consequence would follow from the minor premise. The following is an example; All men are mortal; but if Enoch and Elijah were mortal, the Bible errs; hence, if Enoch and Elijah were men, the Bible errs. A major indirect syllogism is one which from the minor premise of another syllogism and a consequence from the conclusion infers that the same thing would follow from the major premise. Example: All patriarchs are men; but if all patriarchs die, the Bible errs; hence, if all men die, the Bible errs. Such inversions may be much complicated: thus, No one translated is mortal; but if no mortals go to heaven, I am much mistaken; hence, if all who go to heaven are translated, I am much mistaken. To say that from a proposition it would follow that I err when I know I am right would amount to denying that proposition, and, conversely, to deny it positively would amount to saying that, if it were true, I should be wrong when I know I am right. A denial is thus the precise logical equivalent of that consequence. An indirect syllogism in which the contraposition involves such a consequence is said to be of the second or third figure, according as its indirection is of the minor or major kind. The fourth figure, admitted by some logicians, depends upon contraposition of the same sort, but more complicated, like the last example. The first figure comprises, in some sects of logic, the direct syllogism only; in others, the direct syllogisms together with those which are otherwise assigned to the fourth figure. (See figure, 9.) The names of the different varieties, called moods of syllogism, are given by Petrus Hispanus in these hexameters: Barbara: Celarent: Darii: Ferio: Baralipton: Celantes: Dabitis: Fapesmo: Frisesomorum. Cesare: Camestres: Festino: Baroco: Darapti: Felapton: Disamis: Datisi: Bocardo: Ferison. (See these words, and mood, 2.) Probable deductive syllogisms are really direct statistical inferences (which see, under inference). The following is an example: In the African race there are more female than male births; the colored children under one year of age in the United States at the time of the census of 1880 form a random sample of births of Africans; hence, there should be more females than males under one year of age among the colored population of the United States in 1880. The conditions of the validity of such a syllogism are two: first, the character forming the major term (here that of the relative numbers of females and males) must be taken at random—that is, it must not be one which is likely to be subject to peculiar uniformities which could affect the conclusion; second, the minor term, or sample taken, must be numerous and a random sample—that is, not likely to be of a markedly different character from that which is general in the class sampled. The conclusion is probable and approximate—that is, the larger the sample is the smaller will be the probable error of the predicted ratio. Synthetical or ampliative syllogisms are indirect probable syllogisms. The major indirect probable syllogism is induction (which see). The following is an example: The colored children under one year of age in the United States in 1880 form a random sample of births of Africans; but if there ought to have been more males than females among those children, the colored population of the United States is very different from the bulk of Africans; hence, if in the African race in general there are more male than female births, the colored population of the United States is very different from the bulk of Africans. It must be remembered that an observation of a ratio is never exact, but merely admits some values and excludes others; its denial excludes the former, and admits the latter. The denial of a statistical rule is thus itself a statistical rule; and hence such forms as the following are indirect probable syllogisms: American colored children under one year of age in 1880 form a sample of African births; among these the females are in excess; hence, in African births generally the females are probably in excess. The minor indirect probable syllogism is hypothetic inference. (See hypothesis, 4.) Relative syllogisms are those which involve other than merely transitive relations. These were first studied by De Morgan, and afterward by an American logician, but were involved in much difficulty until another American student, O. H. Mitchell, furnished in 1882 the clue to their unravelment. Every relative syllogism has at its core a non-relative syllogism, but this is generalized in a peculiar way—namely, every relative term refers to two or more universes, which may be coextensive, or may be entirely unlike as universes of material things, of space, of time, of qualities, etc. A relative proposition refers to some or all of each of several universes, and the order of the reference is material. (See proposition, 3.) Transpositions, identifications, and diversifications are performed upon principles now clearly made out. An important circumstance in regard to relative syllogism is that the same premise may be repeatedly introduced with new effect. Among relative syllogisms are comprised all the elements of mathematical reasoning, especially the Fermatian inference, the syllogism of transposed quantity, and the peculiar reasoning of the differential calculus. Many times, when she wol make A fulle good silogisme, I drede That aftirward there shall indede Follow an evell conclusioun. Rom. of the Rose, l. 4457. The doctrine of syllogisms comprehendeth the rules of judgment upon that which is invented. Bacon, Advancement of Learning, it.
Deductive or explicatory reasoning as opposed to induction and hypothesis: a use of the term which has been common since Aristotle. Allow some principles or axioms were rightly induced, yet nevertheless certain it is that middle propositions cannot be deduced from them in subject of nature by syllogism—that is, by touch and reduction of them to principles in a middle term. Bacon, Advancement of Learning, ii.
Affirmative syllogism a syllogism the conclusion of which is an affirmative proposition.

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With regard to the matter of M. A——, and putting all personal consideration upon one side, my syllogism is as follows. — Recollections of My Youth
It's called a "syllogism," you prissy, yapping lapdog. — The Jawa Report
But the syllogism is, for me or least, my political conservatism is informed by my religious faith and from that comes a philosophy which forms opinions about issues like abortion. — Latest Articles
The Liberals can barely read the cereal boxes, let alone classic SicFi, or History, or Economics, or Logic, or Latin, or what a syllogism is or, sigh … — Hot Air » Top Picks
I mean if this syllogism is true your very salvation is in danger unless you, yourself, are a community organizer. — THE IRATE NATION

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