The following table of the ponendo ponens, &c., in their valid and invalid forms may be useful:
| Valid | Invalid | |
| Ponendo Ponens | If P then Q, but P, ∴ Q. | If P then Q, but Q, ∴ P. |
| Tollendo Tollens | If Q then P, but not P, ∴ not Q. | If Q then P, but not Q, ∴ not P. |
| Tollendo Ponens | Either P or Q, but not P, ∴ Q. | Not both P and Q, but not Q, ∴ P. |
| Ponendo Tollens | Not both P and Q, but P, ∴ not Q. | Either P or Q, but Q, ∴ not P. |
The above valid forms are mutually reducible to one another and the same is true of the invalid forms.
363 318. The Dilemma.—The proper place of the dilemma amongst hypothetical and disjunctive arguments is difficult to determine, inasmuch as conflicting definitions are given by different logicians. The following definition may be taken as perhaps on the whole the most satisfactory:—A dilemma is a formal argument containing a premiss in which two or more hypotheticals are conjunctively affirmed, and a second premiss in which the antecedents of these hypotheticals are alternatively affirmed or their consequents alternatively denied.[393] These premisses are usually called the major and the minor respectively.[394]
[393] In the strict use of the term, a dilemma implies only two alternants in the alternative premiss; if there are more than two alternants we have a trilemma, or a tetralemma, or a polylemma, as the case may be.
[394] This application of the terms major and minor is somewhat arbitrary. The dilemmatic force of the argument is indeed made more apparent by stating the alternative premiss (i.e., the so-called minor premiss) first.
Dilemmas are called constructive or destructive according as the minor premiss alternatively affirms the antecedents, or denies the consequents, of the major.[395]
[395] A further form of argument may be distinguished in which the alternation contained in the so-called minor premiss is affirmed only hypothetically, and in which, therefore, the conclusion also is hypothetical. For example,
| If A is B, E is F ; and if C is D, E is F ; | |
| If X is Y, either A is B or C is D ; | |
| therefore, | If X is Y, E is F. |
This might be called the hypothetical dilemma. It admits of varieties corresponding to the varieties of the ordinary dilemma; but no detailed treatment of it seems called for.