syllogism

American Heritage® Dictionary of the English Language, Fourth Edition

1. n. 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.
2. n. Reasoning from the general to the specific; deduction.
3. n. A subtle or specious piece of reasoning.

Century Dictionary and Cyclopedia

1. n. 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:
2. n. Deductive or explicatory reasoning as opposed to induction and hypothesis: a use of the term which has been common since Aristotle.
3. n. See the adjectives.

Wiktionary

1. n. logic An inference in which one proposition (the conclusion) follows necessarily from two other propositions, known as the premises.

GNU Webster's 1913

1. n. (Logic) The regular logical form of every argument, consisting of three propositions, of which the first two are called the premises, and the last, the conclusion. The conclusion necessarily follows from the premises; so that, if these are true, the conclusion must be true, and the argument amounts to demonstration.

WordNet 3.0

1. n. deductive reasoning in which a conclusion is derived from two premises