mative and negative, of the following categorical syllogisms, be regarded as identical:— 1. Affirmative Categoricals:— (1) All men are mortal, All kings are men, .. All kings are mortal. 2. Negative (1) All men are mortal, All kings are not mortal, .. All kings are not men. 1. (2) All men are mortal, Categoricals : (2) All men are mortal, Corresponding Constructive Hypothetical-categoricals:— (1) If all kings are men, all (2) kings are mortal; All kings are men; .. All kings are mortal. If some kings are men, some kings are mortal; Some kings are men; .. Some kings are mortal. 2. Corresponding Destructive Hypothetical-categoricals: (1) If all kings are men, all (2) If some kings are men, some kings are mortal; All kings are not mortal, .. All kings are not men. kings are mortal; No kings are mortal, .. No kings are men. The minor premiss in one and the conclusion in the other of the affirmative and negative categoricals have the same subject and predicate, and stand to each other in the same relation in which the minor premiss in one and the conclusion in the other of the constructive and destructive hypothetical-categoricals stand to each other. But who would maintain that in those categorical syllogisms, "the minor and the conclusion indifferently change places,” or that "the same proposition is reciprocally medium or conclusion"? II. In a mixed syllogism there are three terms as in a pure syllogism. In the example taken above, the consequent as a many-worded term, is the major term, the antecedent as a manyworded term, is the middle term, and this case' or 'the case in question' understood, is the minor term. This will be evident, if the mixed syllogism is reduced to the pure form: (i) Categorical: Every case of the existence of A is a case of the existence of B; the case in question (or this case) is a case of the existence of A: therefore the case in question (or this case) is a case of the existence of B. Here the three terms are- -(1) case of the existence of B (major term), (2) case of the existence of A (middle term), and (3) the case in question or this case (minor term). (2) is the middle term to which (1) and (3), the two extremes, are related;—that is, a relation between (1) and (3) is established from a relation of each of them to a third (2) or middle term, as in the case of a categorical syllogism. (ii) Hypothetical: If A is, B is; if this case is, A is: therefore if this case is, B is. This is a pure hypothetical syllogism in Barbara. Here the middle term is the antecedent in the major premiss, and consequent in the minor, as it should be in that mood. From this it is evident, that the objection that a mixed syllogism has no middle term, and consists of two terms only, is entirely unfounded. It has arisen from a misunderstanding of the true nature of the hypothetical major premiss, which has been erroneously regarded as consisting of two propositions instead of two many-worded terms. It is also evident that the middle term is not, as Hamilton says, a proposition, but a manyworded term. III. If A is B, C is D; This is the form in which a mixed syllogism regarded as an immediate inference is stated; and it is argued that the conclusion follows immediately from the premiss, and that no minor premiss is necessary. Now, it can be shown that a categorical syllogism may likewise be stated in the above form; and should it, therefore, be regarded as an immediate inference ? All men are mortal, .. All kings, being men, are mortal. Here also the conclusion follows from the premiss. But it is evident that the conclusion is but a short or abridged statement of two propositions, namely, the minor premiss, ‘all kings are men,' and the conclusion, 'all kings are mortal.' Some logicians indeed actually maintained that even in the categorical syllogism, the minor premiss is unnecessary, that the conclusion follows from the major premiss. Thus they would regard categorical syllogisms as consisting of two propositions only, and consequently as immediate and not as mediate inferences. But we have seen (pp. 257-8) that the conclusion does not follow from the major premiss alone, nor from the minor alone, but from the major and the minor taken jointly. And this is true of mixed syllogisms as well as of categoricals. The conclusion ‘A being B, C is D,' is merely a short or abridged statement of two propositions, namely, the minor premiss 'A is B,' and the conclusion 'C is D.' Here may be noticed an objection raised by Professor Bain. He sees no real inference in mixed syllogisms. By real inference he means a proposition that is not contained in, or implied by, the premiss or premisses. This objection is founded on a misunderstanding of the true nature of deductive inference. It is equally applicable to categorical syllogisms. In these also the conclusion is not a real inference, but a proposition which is contained in, or implied by, the two premisses. Without disputing about words, it may be said that the inference is mediate and real in mixed syllogisms, if it is mediate and real in categoricals, D.-A NOTE ON THE REDUCTION OF INDUCTIVE REASONING TO THE SYLLOGISTIC FORM. The fundamental principles of Inductive Reasoning (whatever be their origin and nature) are the two Laws of Causation and Uniformity of Nature. The first law includes the two propositions—(1) every phenomenon has a cause, and (2) the cause of a phenomenon is the invariable, or, as Mill says, the unconditionally invariable antecedent of the phenomenon. The second law means that (3) the same cause or antecedent will, under the same circumstances, produce the same effect. All inductive reasonings are conducted either directly in accordance with one or other of these laws or with laws that follow from them. For example, from the second proposition of the first law follow such laws as the following given by Professor Bain1: (4) 'whatever antecedent can be left out, without prejudice to the effect, can be no part of the cause;' (5) 'when an antecedent can not be left out without the consequent disappearing, such antecedent must be the cause or a part of the cause;' (6) 'an antecedent and a consequent rising and falling together in numerical concomitance are to be held as cause and effect,' and also the following: (7) 'if two or more instances of a phenomenon under investigation have only one circumstance in common, that circumstance is the cause (or effect) of the phenomenon ;' (8) 'if an instance where a phenomenon occurs, and an instance where it does not occur, have every circumstance in common except one, that one occurring only in the first; the circumstance present in the first and absent in the second, is the cause, or a part of the cause, of the given phenomenon "2. 1 Bain's Induction, 2nd ed., pp. 47, 48, 57. 2 That the propositions marked (4), (5), (6), (7), and (8) follow from the proposition marked (2) can be shown as follows: (4) is the converse of the obverse of (2). Obvert (2), and then convert the obverse; the cause of a phenomenon is not the variable antecedent of the phenomenon-[E, obverse of (2)]. (4) That which Examples of Inductive Reasoning: (1) The antecedents A B C produce the consequents a b c .. The antecedent A is the cause of the phenomenon a according to the principle—a derivative one-marked (7) above, and called the Canon of the Method of Agreement. This inductive reasoning may be easily reduced to a syllogism which has for its major premiss the canon, and for its minor the data of the reasoning, that is, the instances of the phenomenon. The syllogism is a hypothetical-categorical one, and is as follows :— If two or more instances of a phenomenon under investigation have only one circumstance in common, that circumstance is the cause of the phenomenon (major premiss). The four instances given of the phenomenon a under investigation have only one circumstance, namely, A, in common (minor premiss). is the variable antecedent of a phenomenon, or, in other words, which 'can be left out without prejudice to the effect,' is not the cause of the phenomenon (E, converse of the obverse). (5) is the converse of (2), which, being a definition, may be converted simply. (5) That which is the invariable antecedent of a phenomenon, or, in other words, which 'can not be left out without the consequent disappearing,' is the cause of the phenomenon [A, converse of (2)]. (6) is a mathematical inference from (2). The cause and the effect increase or decrease together. A=B... 2A=2 B, or nA=nB. (7) follows from (4) and (5) taken together. By (4) the circumstances which are not common to all the instances of the phenomenon, that is, which can be left out without prejudice to the effect,' can be no part of the cause. By (5) the circumstance which is common to all the instances, that is, which 'cannot be left out without the consequent disappearing,' is the cause or a part of the cause of the phenomenon. (8) follows likewise from (4) and (5) taken together. |