Part 3 — CHAPTER VI.
Reading edition
DISJUNCTIVE SYLLOGISMS.
316. The Disjunctive Syllogism.—A disjunctive (or alternative) syllogism may be defined as a formal reasoning in which a categorical premiss is combined with a disjunctive (alternative) premiss so as to yield a conclusion which is either categorical or else disjunctive (alternative) with fewer alternants than are contained in the disjunctive premiss.389
For example,
A is either B or C,
A is not B,
therefore, A is C ;
Either P or Q or R is true,
P is not true,
therefore, Either Q or R is true.
The categorical premiss in each of the above syllogisms denies one of the alternants of the alternative premiss, and the conclusion affirms the remaining alternant or alternants. Reasonings of this type are accordingly described as examples of the modus tollendo ponens.
It follows from the resolution of disjunctive propositions into conditionals or hypotheticals given in section 193 that (questions of modality being left on one side) the force of a disjunctive as a premiss in an argument is equivalent either to that of a conditional or to that of a hypothetical proposition.
Thus,
Either A is B or C is D,
A is not B,
therefore, C is D ;
may be resolved into the form
If A is not B, C is D,
A is not B,
therefore, C is D ;
or into the form
If C is not D, A is B,
A is not B,
therefore, C is D.
A corollary from the above is that those who deny the character of mediate reasoning to the mixed hypothetical syllogism must also deny it to the disjunctive syllogism, or else must refuse to recognise the resolution of the disjunctive proposition into one or more hypotheticals.
In the above example it is not quite clear from the form of the major premiss whether we have a true hypothetical or a conditional. But in the following examples, which are added to illustrate the distinction, it is evident that the alternative propositions are equivalent to a true hypothetical and to a conditional respectively:
Either all A’s are B’s or all A’s are C’s,
This A is not B,
therefore, All A’s are C’s ;
All A’s are either B or C,
This A is not B,
therefore, This A is C.390
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
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 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
The following table of the ponendo ponens, &c., in their valid and invalid forms may be useful:
ValidInvalid
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.
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
Dilemmas are called constructive or destructive according as the minor premiss alternatively affirms the antecedents, or denies the consequents, of the major.395
Since it is a distinguishing characteristic of the dilemma that the minor should be alternative, it follows that the hypotheticals into which the major premiss of a constructive dilemma may be resolved must contain at least two distinct antecedents. They may, however, have a common consequent. The conclusion of the dilemma will then categorically affirm this consequent, and will correspond with it in form.396 The dilemma itself is in this case called simple. If, on the other hand, the major premiss contains more than one consequent, the conclusion will necessarily be alternative, and the dilemma is called complex.
Similarly, in a destructive dilemma the hypotheticals into which the major can be resolved must have more than one consequent, but they may or may not have a common antecedent; and the dilemma will be simple or complex accordingly.
We have then four forms of dilemma as follows: (i) The simple constructive dilemma.
If A is B, E is F ; and if C is D, E is F ; but Either A is B or C is D ; therefore, E is F.
(ii) The complex constructive dilemma.
If A is B, E is F ; and if C is D, G is H ; but Either A is B or C is D ; therefore, Either E is F or G is H.397
(iii) The simple destructive dilemma.
If A is B, C is D ; and if A is B, E is F ; but Either C is not D or E is not F ; therefore, A is not B.
(iv) The complex destructive dilemma.
If A is B, E is F ; and if C is D, G is H ; but Either E is not F or G is not H ; therefore, Either A is not B or C is not D.398
In the case of dilemmas, as in the case of mixed hypothetical syllogisms, the constructive form may be reduced to the destructive form, and vice versâ. All that has to be done is to contraposit the hypotheticals which constitute the major premiss. One example will suffice. Taking the simple constructive dilemma given above, and contrapositing the major, we have,—
If E is not F, A is not B ; and if E is not F, C is not D ; but Either A is B or C is D ; therefore, E is F ;
and this is a dilemma in the simple destructive form.
The definition of the dilemma given above is practically identical with that given by Fowler (Deductive Logic, p. 116). Mansel (Aldrich, p. 108) defines the dilemma as “a syllogism having a conditional (hypothetical) major premiss with more than one antecedent, and a disjunctive minor.” Equivalent definitions are given by Whately and Jevons. According to this view, while the constructive dilemma may be either simple or complex, the destructive dilemma must always be complex, since in the corresponding simple form (as in the example given on page 364) there is only one antecedent in the major. This exclusion seems arbitrary and is a ground for rejecting the definition in question. Whately, indeed, regards the name dilemma as necessarily implying two antecedents ; but it should rather be regarded as implying two alternatives, either of which being selected a conclusion follows that is unacceptable. Whately goes on to assert that the excluded form is merely a destructive hypothetical syllogism, similar to the following,
If A is B, C is D ;
C is not D ;
therefore, A is not B.
But the two really differ precisely as the simple constructive dilemma given on page 364 differs from the constructive hypothetical syllogism,—
If A is B, E is F ;
A is B ;
therefore, E is F.
Besides, it is clear that the form under discussion is not merely a destructive hypothetical syllogism such as has been already discussed, since the premiss which is combined with the hypothetical premiss is not categorical but alternative.
The following definition is sometimes given:—“The dilemma (or trilemma or polylemma) is an argument in which a choice is allowed between two (or three or more) alternatives, but it is shewn that whichever alternative is taken the same conclusion follows.” This definition, which no doubt gives point to the expression “the horns of a dilemma,” includes the simple constructive dilemma and the simple destructive dilemma; but it does not allow that either of the complex dilemmas is properly so-called, since in each case we are left with the same number of alternants in the conclusion as are contained in the alternative premiss. On the other hand, it embraces forms that are excluded by both the preceding definitions; for example, the following reasoning—which should rather be classed simply as a destructive hypothetico-categorical syllogism—
If A is, either B or C is ;
but Neither B nor C is ;
therefore, A is not.399
Jevons (Elements of Logic, p. 168) remarks that “dilemmatic arguments are more often fallacious than not, because it is seldom possible to find instances where two alternatives exhaust all the possible cases, unless indeed one of them be the simple negative of the other.” In other words, many dilemmatic arguments will be found to contain a premiss involving a fallacy of incomplete alternation. It should, however, be observed that in strictness an argument is not itself to be called fallacious because it contains a false premiss.
EXERCISES.
319. What can be inferred from the premisses, Either A is B or C is D, Either C is not D or E is F? Exhibit the reasoning (a) in the form of a hypothetical syllogism, (b) in the form of a dilemma. [K.]
320. Reduce the following argument, consisting of three disjunctive propositions, to the form of an ordinary categorical syllogism: Everything is either M or P, Everything is either not S or not M, therefore, Everything is either P or not S. [K.]
321. Discuss the logical conclusiveness of fatalistic reasoning like this:—If I am fated to be drowned now, there is no use in my struggling; if not, there is no need of it. But either I am fated to be drowned now or I am not; so that it is either useless or needless for me to struggle against it. [B.]