(1) In the modus ponens (also called the constructive hypothetical syllogism) the categorical premiss affirms the antecedent of the hypothetical premiss, thereby justifying as a conclusion the affirmation of its consequent. For example,

If P is true then Q is true,
P is true,
therefore, Q is true.

(2) In the modus tollens (also called the destructive hypothetical syllogism) the categorical premiss denies the consequent of the hypothetical premiss, thereby justifying as a conclusion the denial of its antecedent. For example,

If P is true then Q is true,
Q is not true,
therefore, P is not true.

These moods fall into line respectively with the first and second figures of the categorical syllogism. For we have seen that in figure 1 we pass from ground to consequence, and in figure 2 from denial of consequence to denial of ground.[380] It has, however, been shewn in section [266] that to every syllogism in figure 1 there corresponds not only a syllogism in figure 2, but also a syllogism in figure 3; and the question may therefore be asked what the mixed hypothetical syllogism 353 yields that will fall into line with figure 3. The answer is that, taking the place of figure 3, we have a reasoning which consists in disproving a connexion of ground and consequence by shewing that the supposed ground holds true but not the supposed consequence. This may be illustrated by writing down the two other reasonings corresponding to the ordinary modus ponens. We have

(1)If P, Q ;(a)
but P ;(b)
∴ Q.(c)
(2)If P, Q ;(a)
but not Q ;contradictory of (c)
∴ not P.contradictory of (b)
(3)P ;(b)
but not Q ;contradictory of (c)
∴ Q is not anecessary consequence of P.contradictory of (a)

[380] The mixed hypothetical syllogism may be reduced to the form of a pure hypothetical syllogism by writing the categorical P is true in the form If anything is true, P is true. If this is done, it will be seen from another point of view that the modus ponens may be regarded as belonging to figure 1 and the modus tollens to figure 2.

If (1) is considered to be in figure 1, then (2) is in figure 2, and (3) in figure 3. It is true that (3) departs too much from the ordinary type of the mixed hypothetical syllogism to justify us in calling it by that name. But it is a form of reasoning that may well receive definite recognition.

306. Fallacies in Mixed Hypothetical Syllogisms.—There are two principal fallacies that may be committed in arguing from a hypothetical major premiss:
(1) It is a fallacy to regard the affirmation of the consequent as justifying the affirmation of the antecedent. For example,

If P is true then Q is true,
Q is true,
therefore, P is true.