[390] When the alternative major premiss is equivalent not to a true hypothetical but to a conditional (as in the second of the above examples), the syllogism may be reduced to pure categorical form (unless the categorical and conditional forms of proposition are in some way differentiated from one another). Thus,
| Every A which is not B is C, | |
| This A is an A which is not B, | |
| therefore, | This A is C. |
361 317. The modus ponendo tollens.—In addition to the modus tollendo ponens, some logicians recognise as valid a modus ponendo tollens in which the categorical premiss affirms one of the alternants of the disjunctive premiss, and the conclusion denies the other alternant or alternants. Thus,
| A is either B or C, | |
| A is B, | |
| therefore, | A is not C. |
The argument here proceeds on the assumption that the alternants are mutually exclusive; but this, on the interpretation of alternative propositions adopted in section [191], is not necessarily the case. Hence the recognition or denial of the validity of the modus ponendo tollens in its ordinary form depends upon our interpretation of the alternative form of proposition.[391]
[391] It will be observed that, interpreting the alternants as not necessarily exclusive of one another, the modus ponendo tollens in the above form is equivalent to one of the fallacies in the mixed hypothetical syllogism mentioned in section [306].
No doubt exclusiveness is often intended to be implied and is understood to be implied. For example, “He was either first or second in the race, He was second, therefore, He was not first.” This reasoning would ordinarily be accepted as valid. But its validity really depends not on the expressed major premiss, but on the understood premiss, “No one can be both first and second in a race.” The following reasoning is in fact equally valid with the one stated above, “He was second in the race, therefore, He was not first.” The alternative premiss is, therefore, quite immaterial to the reasoning; we could do just as well without it, for the really vital premiss, “No one can be both first and second in a race,” is true, and would be accepted as such, quite irrespective of the truth of the alternative proposition, “He was either first or second.” In other 362 cases the mutual exclusiveness of the alternants may be tacitly understood, although not obvious à priori as in the above example. But in no case can a special implication of this kind be recognised when we are dealing with purely symbolic forms. If we hold that the modus ponendo tollens as above stated is formally valid, we must be prepared to interpret the alternants as in every case mutually exclusive.
If, however, we take a major premiss which is disjunctive, not in the ordinary sense (in which disjunctive is equivalent to alternative), but in the more accurate sense explained in section [189], then we may have a formally valid reasoning which has every right to be described as a modus ponendo tollens. Thus,
| P and Q are not both true ; | |
| but P is true ; | |
| therefore, | Q is not true.[392] |
[392] This is in the stricter sense a disjunctive syllogism, the modus tollendo ponens being an alternative syllogism. The reader must, however, be careful to remember that the latter is what is ordinarily meant by the disjunctive syllogism in logical text-books.