SYLLOGISM (Lat. syllogismus, from Gk. reasoning. conclusion, from cre?.?.o yiCeaBot, syllogizesthai, to infer, conclude, from syn, together + logizestha), to reason, from 2.6yoe, logos, word, reason). The name of a logical operation when expressed in a certain form in accordance with the principles of formal logic. When we reason, or get at truth by means of inference, we are said to pro ceed either inductively (see or de ductively. Deductive reasoning, when fully and methodically expressed, takes the form called the syllogism. 'This thing will sink in water, for it is a stone,' is a deductive argument, but not fully stated; this is called an enthymeme (q.v.). The complete form is, "'stones sink in water; this is a stone; therefore, this sinks in water'— which form is called a syllogism.
To a perfect syllogism it is necessary (1) that there should be three, and no more than three, propositions (sec Pnouosmox); these are the conclusion, o• the matter to be proved, and two others that are the means of proving it, called the premises. It is also necessary (2) that there should be three, and no more than three, terms, namely, the subject and the predicate of the con clusion, and one, called the middle which must occur in both premises, being the connect ing link for bringing the two other terms to gether in the conclusion. The predicate of the conclusion is called the major term, because it is in its extension (q.v.) the largest of the three; the subject of the conclusion is the minor term, as being the smallest in extension. The major and minor terms are called extremes. The three terms appear in the premises in this manner: the major term and the middle term appear in one premise, called the major premise: the middle term and the minor term in the minor premise. In the syllogism above stated, the terms are '(a thing that will) sink in water' (major), `this thing' (minor), 'stone' (middle) ; the premises are, 'stones sink in water' (major), 'this thing is a stone' (minor) ; the conclusion is. 'this thing sinks in water'; (3) one premise at least must he affirmative; (4) if one premise lie negative• the conclusion must be negative; (5) the middle term must be distributed (= taken in the whole of its extension) in at least one pre mise; (6) an extreme, if undistributed in a pre mise, may not be distributed in the conclusion. Any syllogism which violates any one or more of these six syllogistic rules is invalid. There are two other rules which are derivative: (7) one premise at least must be universal; (8) if one premise be particular, the conclusion must be particular also. Any syllogism which violates either of these rules violates also one or more of the first six rules given above. For the dis cussion of the question as to the evidence of these canons, see LOGIC.
Categorical syllogistic forms are divided into figures, and each figure into moods, which are the distinct syllogistic forms, the principle of division being as follows: The figure is deter mined by the function of the middle term, in the two premises: it may he the subject of the major premise, and the predicate of the minor (1st figure) : the predicate in both (2d figure), the subject in both (3d figure). the predicate of the major and the subject of the minor (4th figure). The 4th figure did not belong to the original scheme of Aristotle, and it is usually considered as both unnatural and unnecessary, being only an awkward inversion of the first. The syllogisms of each figure are said to differ in mood, or ac cording to the quality and the quantity of the propositions—that is, according as these are affirmative or negative (quality), universal or particular (quantity).
A conditional syllogism whose conditional major premise presents some sort of alternative is called a dilemma. We may have these cases: Either (a) alternative consequences may be as serted to follow upon a single condition (or com bination of conditions) or (b) alternative con ditions may be asserted to determine a single consequence (or combination of consequences) ; or (c) an alternative may be presented between a condition with its consequence and another con dition with its separate consequence. If alter native consequences are asserted to follow upon a single condition, then it is possible to have a valid conditional syllogism either when a minor premise affirms the condition, warranting as conclusion the affirmation of the alternative con sequence; or when a minor premise denies con junctively ('neither—no•') the alternative con sequences, warranting as conclusion the denial of the antecedent. If alternative conditions are asserted to determine a single consequence, it is possible to have a valid syllogism when a minor premise either categorically affirms one of the conditions or disjunctively affirms both con ditions, in either case justifying as conclusion the affirmation of the consequent ; or when the minor premise categorically denies the consequent, justifying as conclusion the conjunctive denial of the conditions. If the major premise presents an alternative between a condition with its con sequence and another condition with its separate consequence, a valid syllogism obtains when the minor premise either disjunctively affirms the two conditions, justifying the disjunctive affirma tion of the two consequences, or when the minor premise conjunctively or disjunctively denies both consequences, justifying a conjunctive or disjunctive denial respectively of the two ante cedents. For bibliography, see books mentioned under LOGIC. See also FALLACY; DILEMMA; CONVERSION; OBVERSION.