Page images
PDF
EPUB

$74. ANALYSIS OF THE SYLLOGISM.-The proposition is but the expression of a significative relation between its terms. Hence the premises of a syllogism are merely statements of the significative relations of the terms of the conclusion (the major and the minor) respectively with the middle term; and the conclusion the significative relation thereby inferred between its terms. The essential elements of the process consist, therefore, in the comparison of the two terms of the conclusion. respectively with the third, or middle term, and in inferring a direct relation between them.

$75. DEFINITION OF THE SYLLOGISM.Hence syllogistic inference may be more specifically defined as consisting in the inference of a significative relation between two terms from their known significative relations to a third term with which they are respectively compared.'

$76. THE PRINCIPLE OF THE SYLLOGISM. -The principle of the syllogism (by which is meant the principle or axiom on which depends the illative force or conclusiveness of syllogistic inference) is expressed in the Dictum of Aristotle, or, as it is technically called, the

1 The definition in the text is taken substantially from that of De Morgan; who defines the syllogism as "the inference of the relation of two names from the relation of each of those names to a third" (Formal Log., p. 176).

Dictum de Omni et Nullo. It is variously stated by logicians, but the several forms are all, in effect, identical. Its best expression is as follows:

DICTUM DE OMNI ET NULLO.-"Where three terms (which we will call the middle and the two extremes) so subsist with relation to each other that the one extreme is contained in the middle, and the middle is contained in [or excluded from] the other extreme, then [as the case may be] the extreme included in the middle will be included in [or excluded from] the other extreme." Where the predication is affirmative the principle is called the Dictum de Omni; where negative, the Dictum de Nullo.

Omitting in the form given above the words. in brackets, it becomes the Dictum de Omni; substituting the words in brackets, marked as quoted, for the corresponding expressions, it becomes the Dictum de Nullo.

The two forms of the Dictum (affirmative and negative) correspond precisely to the two forms of syllogisms called Barbara and Celarent,' viz. :

'This is substantially the form given to the Dictum by Aristotle, Prior Analytics, i., iv.

Forms of the Syllogism. There are nineteen forms of valid syllogisms recognized by logicians, which are explained in the next chapter. But if we reject the use of particular propositions (§ 52 n.) all may be reduced to the two forms

[blocks in formation]

THE PRINCIPLE OF SUBSTITUTION

877. RULES OF INFERENCE.-The following practical rules may be deduced from the Dictum :

(1) In any affirmative proposition we may always (without affecting its illative force or conclusiveness) substitute for the subject any other term denoting the same, or part of the same, significates; and for the predicate any term denoting the same significates, or a class that contains them.

Or, more briefly, we may always in the subject substitute species for genus; and in the predicate, genus for species.

(2) So, in any negative proposition, we may, without affecting its illative force, substitute for either subject or predicate any term denoting the same, or part of the same, significates.

Or, more briefly, we may always, in the negative proposition, either in the subject or the predicate, substitute species for genus.

above given, which are called Barbara and Celarent. In these forms the several terms may be represented indifferently by any letters; and the order of the propositions is immaterial. In the traditional Logic the order of the propositions is always as in the examples given in the text.

(3) To which may be added the following: In any affirmative proposition we may always substitute for the predicate any other term that denotes the same significates as the subject, or a class containing them.'

$78. EQUIVALENCE OF TERMS DEFINED. In the above rules, it will be observed, the term substituted is not necessarily equivalent in signification to the term for which it is substituted; but it is equivalent so far as the force of the inference is concerned, or, as the lawyers say, quoad the argument. It may be said, therefore, briefly, that mediate, or syllogistic inference consists simply in substituting for the terms of propositions other terms equivalent in ratiocinative value.

$79. CONVERSIONS OF PROPOSITIONS.The case of conversion of propositions seems indeed, to be an exception; for here the process seems to consist, not in the substitution of terms, but in the substitution of a new

1 The deduction of these rules from the Dictum is perhaps sufficiently obvious, but as it may not be apparent to all, we subjoin the demonstration:

In the first syllogism (Barbara) it will be perceived, as expressed in the minor premise, that Z is a species, and X the genus, of Y, and that the conclusion is arrived at by substituting for Y, in the major premise, its species Z; or, for Y in the minor premise, its genus X.

In the latter syllogism (Celarent) the process consists in substituting for Y, in the major premise, its species Z; and so it is obvious we may substitute for X in the major premise any

proposition containing the same terms as the original with the order of terms transposed. But the exception, in the case of negative and equational propositions, is more apparent than real; for the two forms of the proposition (i. e., the converted and the original proposition) are precisely the same in effect, and there is, in fact, neither term nor proposition substituted. For when we say " Y is not X," we equally and as explicitly say "X is not Y"-the meaning of either proposition being simply that the two classes denoted by X and Y are mutually exclusive; and so in the equational proposition (Y = X) we say, in the same breath, both that Y is equal to X, and that X is equal to Y. So,

species of the genus X, as, e. g., A, B, or C, and thus conclude that " Z is not A, B, or C" (as the case may be); as may be illustrated by appropriate diagrams:

[ocr errors]

So, in the major premise in Barbara, we may substitute for X the expression YX, or any species of X containing Y, as, e. g., A, and thus conclude that Z is YX, or Z is A, as the case may be.

« PreviousContinue »