A further Variation.—There is still another form of statement, in which the terms compared are not, as above, severally subject and predicate, but, in the same proposition, are both subject, or both predicate, as when we say, A and B are equal; B and C are equal; therefore, A and C are equal. This is a valid synthetic syllogism, though not recognized by logicians previously to the New Analytic of Hamilton. It is termed by him the unfigured syllogism.
Hypothetical reasoning not syllogistic.—It has been customary to treat of hypothetical reasoning, in its two forms of conditional and disjunctive, as forms or kinds of syllogism. As when we say, if A is B, C is D; but A is B, therefore C is D; or, disjunctively, either A is B, or C is D; but A is not B, therefore C is D. These, however, are not properly syllogisms. The inference is not mediate, through comparison with a common or middle term, but immediate, whereas the syllogism is, in all its forms, a process of mediate inference.
Summary of Distinctions.—To sum up the distinctions now pointed out. All inference is either immediate, as in the case of hypothetical reasoning, whether conjunctive or disjunctive, or else mediate, as in the syllogism. The latter may be inductive or deductive; and, as to form, analytic or synthetic, figured or unfigured.
VI. Laws of Thought on which the Syllogism depends.
Statement.—There are certain universal laws of thought on which all reasoning, and, of course, all syllogisms, depend. These laws, according to Hamilton, are the principles of identity, of contradiction, and of excluded middle; from which primary laws results a fourth, that of reason and consequent.
Law of Identity, what.—The principle of identity compels us to recognize the equivalence of a whole and its several parts taken together, as applied to any conception and its distinctive characters. As, for example, the sameness or equivalence of the notion man with the aggregate of qualities or characters that constitute that notion.
Law of Contradiction, what.—The law of contradiction is the principle that what is contradictory is unthinkable: as, for example, that A has, and yet has not, a given quality, B.
Law of excluded Middle.—The principle of excluded middle is this, that of two contradictory notions, we must think one or the other to be true; as, that A either has or has not the quality B.
Law of Reason and Consequent.—From these primary principles results the law of reason and consequent. All logical inference is based on that law of our nature, that one notion shall always depend on another. This inference is of two kinds, from the whole to the parts, or from the parts to the whole, respectively called deductive and inductive, as already explained.
Certain Points not included in the preceding Synopsis.—I have presented, as was proposed, in brief outline, a synopsis of the forms of reasoning. For a full treatment of these forms, and the laws which govern them, the treatises on logic must be consulted.