| Darii,— | If Q is true, R is true, |
| If P is true, Q may be true, | |
| therefore, | If P is true, R may be true. |
| Baroco,— | If R is true, Q is true, |
| If P is true, Q may be false, | |
| therefore, | If P is true, R may be false. |
| Disamis,— | If Q is true, R may be true, |
| If Q is true, P is true, | |
| therefore, | If P is true, R may be true.[378] |
| Camenes,— | If R is true, Q is true, |
| If Q is true, P is not true, | |
| therefore, | If P is true, R is not true. |
[378] The reader may possibly hesitate to admit the validity of this reasoning, although he feels no difficulty in regard to the validity of an ordinary categorical syllogism in Disamis. This apparent anomaly is connected with the problem of existential import. It will be shewn in section [342] that the validity of Disamis depends on our interpretation of propositions as regards their existential import, and we may perhaps not regard categoricals and hypotheticals as analogous in this respect.
303. Fallacies in Hypothetical Syllogisms.—On the mistaken supposition that a pure hypothetical proposition is equivalent to a categorical proposition in which both the subject 351 and the predicate are singular terms, and therefore ipso facto distributed, it has been argued that the syllogistic rules relating to the distribution of terms have no application to hypothetical syllogisms; and that the only rules which need be considered in testing such syllogisms are those relating to quality, namely, the rule forbidding two negative premisses, and the rule insisting that a negative premiss and a negative conclusion must always be found together. But it is clearly an error to regard the consequent of a hypothetical proposition as equivalent to a singular term occurring as the predicate of a categorical proposition. An affirmative hypothetical is not simply convertible, and in respect of distribution, its consequent practically corresponds to the undistributed predicate of an affirmative categorical in which the terms are general. On the other hand, a negative hypothetical is simply convertible; and its consequent corresponds to the distributed predicate of a negative categorical. We may accordingly have fallacies in hypothetical syllogisms corresponding to (1) undistributed middle, (2) illicit major, (3) illicit minor. The following are examples of these fallacies respectively:—
(1) If R then Q, If P then Q, therefore, If P then R ;
(2) If Q then R, If P then not Q, therefore, If P then not R ;
(3) If Q then R, If Q then P, therefore, If P then R.
304. The Reduction of Conditional and Hypothetical Syllogisms.—Conditional and hypothetical syllogisms in figures 2, 3, and 4 may be reduced to figure 1 just as in the case of categorical syllogisms. Thus the conditional syllogism in Camenes given in section [302] may be reduced as follows to Celarent:
| Never when C is D, is it the case that A is B, | |
| Whenever E is F, C is D, | |
| therefore, | Never when E is F, is it the case that A is B, |
| therefore, | Never when A is B, is it the case that E is F. |
According to the ordinary rule as indicated in the mnemonic, the premisses have here been transposed, and the conclusion of the new syllogism is converted in order to obtain the original conclusion.
352 Similarly the hypothetical syllogism in Baroco given in section [302] may be reduced as follows to Ferio:
| If Q is false, R is false, | |
| If P is true, Q may be false, | |
| therefore, | If P is true, R may be false. |
305. The Moods of the Mixed Hypothetical Syllogism.—It is usual to distinguish two moods of the mixed hypothetical syllogism, the modus ponens and the modus tollens.[379]
[379] Ueberweg remarks that it would be more accurate to speak of the modus ponens as the modus ponendo ponens, and of the modus tollens as the modus tollendo tollens (Logic, p. 452).