Y," which is called simple conversion; and so with all definitions, and other equational propositions; and also with the particular affirmative proposition, "Some Y. is X." But the affirmative proposition, Y is X," cannot be thus simply converted; for the subject class is identical with only "some" of the predicate class, and in conversion the predicate must be qualified by that particle, thus substituting a new term. Or, symbolically, the proposition, "Y is X," can be converted only into the proposition," Some X is Y"; which is called conversion per accidens.

II

SEVERAL THEORIES OF PREDICATION

855. THE COPULA.-In the logical proposition, as we have seen, the copula is interpreted as meaning "is contained in," or the contrary; and this is the traditional, or, as it may be called, orthodox, theory of predication. But the copula may be otherwise interpreted; and from these several interpretations several theories of predication will result. Of these, two may be distinguished as requiring some remark, namely, the Equational Theory, in which the copula is interpreted as meaning, “is equivalent to," and is expressed by the sign of equivalence (=); and the Intensive Theory, where it

is interpreted as meaning, "has the quality or attribute." Thus, e. g., the proposition, " Man is rational," is interpreted according to the Traditional Theory as meaning, "the class man is contained in the class rational"; according to the Equational Theory, as meaning, the class man is the same as the class rational"; and according to the Intensive, as meaning, the individuals constituting the class man have the quality or attribute, rational, or of rationality.'

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8 56. THE EQUATIONAL THEORY.-In the logical proposition, the classes denoted by the subject and predicate may be equal; for, where this is the case, each may be said to be contained in the other. Hence in such cases the proposition is always convertible, as, e. g., we may say indifferently that " man is a rational animal," or that "a rational animal is a man,' or, generally, if Y = X, either that "Y is X" or 'X is Y." Such propositions are recognized and used in the traditional Logic, as in the case of definitions, and in other cases, but it is not thought necessary to express the equivalence of the terms. Hence in the affirmative proposition " Y is X" it cannot be determined from the form of the proposition whether X is of greater extension than Y, or of the same extension.

$57. QUANTIFICATION OF THE PREDICATE.

-The modern doctrine of " the quantification of the predicate" has for its object to remedy this supposed defect by expressing in every proposition by an appropriate sign the quantity of the predicate, or, in other words, by indicating whether it is distributed or not'; and this is effected by prefixing to the predicate a sign indicating the relation of quantity between it and the subject, and giving to the proposition an equational form. Thus, e. g., the proposition, " Y is X," may be expressed in the form " Y = vX," which is the method of Boole; or in the form " Y = YX," which is the form proposed by Jevons, and is read, "Y = the part of X that is Y," or "the Y's are the X's that are also Y's." Or, more simply, instead of the proposition, “ Y is X,” we may say, "Y is a certain species of X"; or, to take a concrete example, instead of the proposition, "Man is an animal," we may say, "Man is a certain species or kind of animal." Hence, whether an equational proposition shall be expressed in the traditional or in the equational form is a matter of choice to be determined by convenience. Generally the

1 A term is said to be "distributed" when it is taken universally, i. e., where the other term of the proposition is, or may be, predicated of all the individuals denoted by it, as, e. g., the subject of a universal affirmative, or either subject or predicate of a universal negative proposition (see § 87).

traditional form is sufficient, as we can readily determine from the matter of the proposition whether it is to be regarded as equational or otherwise. But in the mathematics the equational form is much the more efficient, and is therefore always used.

$58. THE INTENSIVE THEORY.-The difference between the traditional and the intensive theory of predication is that, in construing the proposition, we have regard in the former to the extension of the terms only; but in the latter, in construing the predicate, we have regard to its intension. Thus, when we say " Man is mortal," we mean, in the former case, that the class man is contained in the class mortal; but in the latter, that man has the quality or attribute of mortality. But the latter expression means nothing more than that "the quality of mortality is contained in, or among, the qualities of man "; which is itself an extensive proposition. Hence the intensive interpretation of the proposition simply results in an extensive proposition in which the qualities of the original terms are substituted for its original significates, and the terms inverted. Thus, e. g., if we denote by Y' the qualities of Y, and by X' the qualities of X, the proposition, Y is X, may be converted into X' is Y'; which may be called Intensive Conversion, or conversion by Intensive Interpretation.

$59. TRADITIONAL THEORY OF PREDICATION. Even under this theory the proposition seems to be susceptible of several interpretations. Thus, e. g., we have interpreted the copula as meaning " is contained in " or " is a species of"; and again we may interpret it as meaning that the significates constituting the subject class may each and all be called by the name constituting the predicate — or, in other words, that the name predicated belongs to the significates of the subject term, or of any of them; which has been called interpreting the judgment "in its denomination" (Thompson's Laws of Thought, § 195). But for all logical purposes these interpretations are practically the same, and it will make no difference whether the proposition be interpreted in the one way or the other. This is sufficiently obvious with regard to the expressions, "is contained in,' and" is a species of "; and is equally true of the interpretation suggested by Dr. Thompson. For, taking as an example the proposition, "Man is an animal," it is obviously indifferent whether we construe it as meaning "the class man is included in the class animal," or that "it is a species of the class animal," or that the name animal is applicable to all significates of the name man.” These varieties of interpretation will, therefore, not demand a further consideration.