TABLE OF THE MUTUAL RELATIONS OF THE EIGHT PROPOSITIONAL FORMS O EITHER SYSTEM OF PARTICULARITY. (For Generals only.) Inconsistents; Int. or Integr. — In¦ Contrar. Contraries Incons. = Compossibles; in II. No inference. alternation; Blanks, in I. = Sul It appears from this Table, that Afi and Ina (A and O), which, on the Aristotelic doctrine, are Contradictories, become only Contraries when we admit the semi-definite meaning of some; for by sublating Ina, which denies only a part (some only), we know not whether to posit Afi, which affirms the whole, or Ifi, which affirms only some (other) part, or Ana, which denies the whole; since each of these three is incompossible with Ina. For the same reason, Ifa and Ani, which are only A and O converted, are merely Contraries on this system, though Contradictories on the other, wherein some means perhaps all. Indeed, there can be no Contradiction on this system, wherein whole and part negative each other, just as much as affirmation and negation. The only Contradictories are those in which the distinction of whole and part does not exist; — Judgments about Singulars or Individuals, for instance, and about Universals regarded as Singulars or as undivided wholes. Thus, Common salt is chloride of sodium contradicts Common salt is not chloride of sodium; for Common salt, though really a General Term, is here actually thought as undivided, so that the two Judgments contradict each other as directly as do these two Singulars, John is sick, John is not sick. If either Judgment in one of these pairs is sublated, the other is necessarily posited. "The propositional form If is consistent with all the affirmatives; Ini is not only consistent with all the negatives, but is compossible with every other form in universals. It is useful only to divide a class, and is opposed only by the negation of divisibility." The whole scheme of Opposition upon this system may be safely characterized as too complex to be of any practical use, though the learner may be profited by some study of its details. CHAPTER VII. THE DOCTRINE OF MEDIATE INFERENCE: THE ARISTOTELIC ANALYSIS OF SYLLOGISMS. 1. Figure and Mood. 2. Conditional Syllogisms. 3. Defective and Complex Syllogisms. EDIATE Inference is that act of Pure Thought, MED whereby the relation of the two Terms of a possible Judgment to each other is ascertained by comparing each of them separately with a third Term. Thus, if I cannot immediately determine whether A is, or is not, B, I can compare each with M. If, as the result of such comparison, it is found that A is M and B is M, then we infer mediately that is, through this relation of each to a third —that A is B. But if this comparison shows that one of these Terms is, and the other is not, M, then we infer mediately that A is not B. The affirmative conclusion is evidently governed by the Axiom of Identity, which declares that A is B, if it is that (M) which is the equivalent of B; or to use language more consonant with the phraseology hitherto employed, and converting B is M into M is B, we say that B is a Mark of A, when it is a Mark of that (M) which is a Mark of A, — nota notæ est nota rei ipsius. The negative conclusion results from the Axiom of Non-Contradiction, which declares that A is not B, when it is equivalent to that (or has for a Mark that) (M), which is not B; or, what is the same thing, when it is not equivalent to that (M) which is B. The fundamental principle of Mediate Inference or Syl logism is thus traced to those Axioms which, as we have already seen, must govern all the processes of Pure Thought; or rather, Mediate Inference itself is but one of the special applications of those Axioms. Instead of using these Primary Axioms themselves, logicians have usually, in order to demonstrate the processes of syllogistic reasoning, preferred to employ certain intermediate principles or maxims, one of which we have just mentioned, — that the Mark of a Mark is a Mark of the thing itself. But as these maxims can be directly deduced from the original Axioms, to which, indeed, they owe all their validity, it seems better to test the legitimacy of each step by a reference to the primary, rather than to any derivative, principle. Thus far, A and B, in their comparison with M, have been regarded simply as undivided wholes; but it is evident that the same considerations will hold good if we substitute, for either or both of them, all, or any indefinite part, of a divided Universal. Thus, if we find that Some A are M, and Some B are M, we are compelled to conclude, by the Axiom of Identity, that Some A are (some) B; or, taking a negative instance, if Some A are M, and Not any B is M, then we infer that Some A are not (any) B. Hence we see the correctness of the derivative or intermediate principle which Sir W. Hamilton enounces as "the supreme Canon of Categorical Syllogisms," IN so FAR AS two Notions (Concepts or Individuals), either both agree, or, one agreeing, the other does not agree, with a common third Notion, IN SO FAR these Notions do or do not agree with each other. But if, by calling it "supreme," he means that it is the But if, by calling it " ultimate and original Canon, his position may be doubted; for it is evidently a compound statement, embracing, with an unimportant change of phraseology, the two Primary Axioms of Identity and Non-Contradiction, and guarding them with those limitations under which alone are they ever applicable. 66 66 We have seen that, though either or both of the two Terms be quantified Particularly, the Syllogism still holds good, at least, to the extent to which the two Terms are quantified. But the third Term must be taken Universally at least once in comparing it with the other Notions; otherwise, we have no security that these others are compared with the same, or a common," third Term. Though we know, for instance, that A is some M, and B is some M, still we cannot conclude that A is B; for the some M" which is A may not be the same some M " which is B. Though Some learned men are pedants, and Some learned men are wise, it does not follow that Pedants are wise; for two very different classes of learned persons are here spoken of. Hence we have this general rule for all Syllogisms, that the Middle Term must be distributed (i. e. taken Universally) in at least one of the comparisons which are instituted between it and the other two Terms. We say, "at least one" of the two comparisons; for the other may be quantified Particularly without injury to the reasoning. Thus, if All men are mortal, and X, Y, and Z are (some) men, we may legitimately conclude that X, Y, and Z are mortals; for to whatever class these " some men belong, they are necessarily included under "all men,' who are declared to be mortal. A Syllogism evidently comprises three Judgments, one of which affirms the agreement or non-agreement of its two Terms with each other to be the necessary consequence of two other Judgments, in which a common third Term is affirmed to agree with both, or with one only, of these two Terms. The main Judgment is called the Conclusion; the two subsidiary Judgments, on which it depends, are termed the Premises; and the necessary connection between the Premises and the Conclusion entitles us to infer the one from the other that which is the Con sequence. The essence of the Syllogism, and all that is |