(2) It is a fallacy to regard the denial of the antecedent as justifying the denial of the consequent. For example,
| If P is true then Q is true, | |
| P is not true, | |
| therefore, | Q is not true. |
These fallacies may be regarded as corresponding respectively to undistributed middle and illicit major in the case of categorical syllogisms.[381]
[381] Given “If P and only if P then Q,” then we may of course argue from Q to P or from not-P to not-Q; and no doubt in the case of ordinary hypotheticals it is often tacitly understood that the consequent is true only if the antecedent is true. This must, however, be expressly stated if the argument based upon it is to be formally valid.
354 The results reached in this and the preceding section may be summed up in the following canon for the mixed hypothetical syllogism: Given a hypothetical premiss expressed affirmatively, then the affirmation of the antecedent justifies the affirmation of the consequent; and the denial of the consequent justifies the denial of the antecedent; but not conversely in either case.
307. The Reduction of Mixed Hypothetical Syllogisms.—Any case of the modus tollens may be reduced to the modus ponens, and vice versâ.
Thus,
| If P is true then Q is true, | |
| Q is not true, | |
| therefore, | P is not true, |
becomes, by contraposition of the hypothetical premiss,
| If Q is not true then P is not true, | |
| Q is not true, | |
| therefore, | P is not true ; |