Part 3 — CHAPTER V.

CONDITIONAL AND HYPOTHETICAL SYLLOGISMS.

301. The Conditional Syllogism, the Hypothetical Syllogism, and the Hypothetico-Categorical Syllogism.—The forms of reasoning in which conditional or hypothetical conclusions are inferred from two conditional or two hypothetical premisses are apparently overlooked by some logicians; at any rate, they frequently receive no distinct recognition, the term “hypothetical syllogism” being limited to the case in which one premiss only is hypothetical.

The following definitions may be given: (1) A conditional syllogism is a reasoning consisting of two conditional premisses and a conditional conclusion;376

e.g., If any A is C, it is D,

If any A is B, it is C,

therefore, If any A is B, it is D.

(2) A hypothetical syllogism (or, more distinctively, a pure hypothetical syllogism) is a reasoning consisting of two hypothetical premisses and a hypothetical conclusion;376

e.g.,— If Q is true, R is true,

If P is true, Q is true,

therefore, If P is true, R is true.

(3) A hypothetico-categorical syllogism (or, as it may also be called, a mixed hypothetical syllogism) is a reasoning consisting of three propositions in which one of the premisses is hypothetical in form, while the other premiss and the conclusion are categorical;377

e.g.,— If P is true, Q is true,

P is true,

therefore, Q is true.

This nomenclature, so far as concerns the distinction between the hypothetical and the hypothetico-categorical syllogism, is adopted by Spalding and Ueberweg. Sigwart uses the terms “pure hypothetical syllogism,” and “mixed hypothetical syllogism.” Some logicians (e.g., Fowler) give the name “hypothetical syllogism” to all the above forms of reasoning without distinction. Others (e.g., Jevons) define the hypothetical syllogism so as to include the last form only, the others not being recognised as distinct forms of reasoning at all. This view may be to some extent justified by the very close analogy that exists between the syllogism with two conditional or two hypothetical premisses and the categorical syllogism: but the difference in form is worth at least a brief discussion.

302. Distinctions of Mood and Figure in the case of Conditional and Hypothetical Syllogisms.—In the conditional, and in the hypothetical, syllogism, the antecedent of the conclusion is equivalent to the minor term of the categorical syllogism, the consequent of the conclusion to the major term, and the element which does not appear in the conclusion at all to the middle term. Distinctions of mood and figure may be recognised in precisely the same way as in the case of the categorical syllogism. Thus, the conditional syllogism given in the preceding section is in Barbara. The following are examples of other moods:

Festino,—Never when E is F, is it the case that C is D,

Sometimes when A is B, C is D,

therefore, Sometimes when A is B, it is not the case that E is F.

Darapti,—Whenever C is B, E is F,

Whenever C is D, A is B,

therefore, Sometimes when A is B, E is F.

Camenes,—Whenever E is F, C is D,

Never when C is D, is it the case that A is B,

therefore, Never when A is B, is it the case that E is F.

In these three examples the form in which the propositions are expressed suggests an assertoric interpretation. On the modal interpretation, either of conditionals or of hypotheticals, the problematic proposition may be regarded as taking the place of the particular, and we shall then again have all the ordinary distinctions of mood and figure. We may illustrate from hypotheticals:

Darii,—If Q is true, R is true,

If P is true, Q may be true,

therefore,If P is true, R may be true.

Baroco,—If R is true, Q is true,

If P is true, Q may be false,

therefore,If P is true, R may be false.

Disamis,—If Q is true, R may be true,

If Q is true, P is true,

therefore,If P is true, R may be true.378

Camenes,—If R is true, Q is true,

If Q is true, P is not true,

therefore,If P is true, R is not true.

303. Fallacies in Hypothetical Syllogisms.—On the mistaken supposition that a pure hypothetical proposition is equivalent to a categorical proposition in which both the subject and the predicate are singular terms, and therefore ipso facto distributed, it has been argued that the syllogistic rules relating to the distribution of terms have no application to hypothetical syllogisms; and that the only rules which need be considered in testing such syllogisms are those relating to quality, namely, the rule forbidding two negative premisses, and the rule insisting that a negative premiss and a negative conclusion must always be found together. But it is clearly an error to regard the consequent of a hypothetical proposition as equivalent to a singular term occurring as the predicate of a categorical proposition. An affirmative hypothetical is not simply convertible, and in respect of distribution, its consequent practically corresponds to the undistributed predicate of an affirmative categorical in which the terms are general. On the other hand, a negative hypothetical is simply convertible; and its consequent corresponds to the distributed predicate of a negative categorical. We may accordingly have fallacies in hypothetical syllogisms corresponding to (1) undistributed middle, (2) illicit major, (3) illicit minor. The following are examples of these fallacies respectively:— (1) If R then Q, If P then Q, therefore, If P then R ; (2) If Q then R, If P then not Q, therefore, If P then not R ; (3) If Q then R, If Q then P, therefore, If P then R.

304. The Reduction of Conditional and Hypothetical Syllogisms.—Conditional and hypothetical syllogisms in figures 2, 3, and 4 may be reduced to figure 1 just as in the case of categorical syllogisms. Thus the conditional syllogism in Camenes given in section 302 may be reduced as follows to Celarent:

Never when C is D, is it the case that A is B,

Whenever E is F, C is D,

therefore, Never when E is F, is it the case that A is B,

therefore, Never when A is B, is it the case that E is F.

According to the ordinary rule as indicated in the mnemonic, the premisses have here been transposed, and the conclusion of the new syllogism is converted in order to obtain the original conclusion.

Similarly the hypothetical syllogism in Baroco given in section 302 may be reduced as follows to Ferio:

If Q is false, R is false,

If P is true, Q may be false,

therefore, If P is true, R may be false.

305. The Moods of the Mixed Hypothetical Syllogism.—It is usual to distinguish two moods of the mixed hypothetical syllogism, the modus ponens and the modus tollens.379

(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 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 a necessary consequence of P.contradictory of (a)

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.

(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

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 ;

and this is the modus ponens.382

308. Is the reasoning contained in the mixed hypothetical syllogism mediate or immediate?383—Kant, Hamilton,384 Bain, and others argue that inferences of the kind that we have just been considering are properly to be regarded not as mediate, but as immediate, inferences.

Now, taking the syllogism—

If P is true then Q is true,

but P is true,

therefore, Q is true,

the conclusion is at any rate apparently obtained by a combination of two premisses, and the process is moreover one of elimination, namely, of the proposition P is true. Hence the burden of proof certainly lies with those who deny the claims of such an inference as this to be called mediate.

Bain (Logic, Deduction, p. 117) seems to argue that the so-called hypothetical syllogism is not really mediate inference, because it is “a pure instance of the law of consistency”; in other words, because “the conclusion is implied in what has already been stated.” But is not this the case in all formal mediate inference? It cannot be maintained that the categorical syllogism is more than a pure instance of the law of consistency; or that the conclusion in such a syllogism is not implied in what has been already stated. But possibly Bain may mean that the conclusion is implied in the hypothetical premiss alone. Indeed he goes on to say, “’If the weather continues fine, we shall go into the country’ is transformable into the equivalent form ‘The weather continues fine, and so we shall go into the country.’ Any person affirming the one, does not, in affirming the other, declare a new fact, but the same fact.” Surely this is not intended to be understood literally. Take the following:—If war is declared, I must return home; If the sun moves round the earth, modern astronomy is a delusion. Are these respectively equivalent to the statements, War has been declared, and so I must return home; The sun moves round the earth, and so modern astronomy is a delusion? Besides, if the proposition If P is true then Q is true implies the truth of P, what becomes of the possible reasoning, “But Q is not true, therefore, P is not true”?

Further arguments that have been adduced on the same side are as follows:— (1) “There is no middle term in the so-called hypothetical syllogism”.385 The answer is that there is an element in the premisses which does not appear in the conclusion, and that this corresponds to the middle term of the categorical syllogism.

(2) “In the so-called hypothetical syllogism, the minor and the conclusion indifferently change places”.386 This statement is erroneous. Taking the valid syllogism given at the commencement of this section and transposing the so-called minor and the conclusion, we have a fallacy.

(3) “The major in a so-called hypothetical syllogism consists of two propositions, the categorical major of two terms.” This merely tells us that a hypothetical syllogism is not the same in form as a categorical syllogism, but seems to have no bearing on the question whether the so-called hypothetical syllogism is a case of mediate or of immediate inference.

Turning now to the other side of the question no satisfactory answers seem possible to the following arguments in favour of regarding the mixed hypothetical syllogism as a case of mediate inference. In any such syllogism, the two premisses are quite distinct, neither can be inferred from the other, but both are necessary in order that the conclusion may be obtained. Again if we compare with it the inferences which are on all sides admitted to be immediate inferences from the hypothetical proposition, the difference between the two cases is apparent. From If P is true then Q is true, I can infer immediately If Q is not true then P is not true ; but I require also to know that Q is not true in order to be able to infer that P is not true.

And whether the mixed hypothetical syllogism can or can not be actually reduced to pure categorical form, it can at least be shewn to be analogous to the ordinary categorical syllogism, which is admitted to be a case of mediate reasoning. Moreover there are distinct forms—the modus ponens and the modus tollens—which are analogous to distinct forms of the categorical syllogism; and fallacies in the mixed hypothetical syllogism correspond to certain fallacies in the categorical syllogism.

The argument in favour of regarding the modus tollens—If P is true then Q is true, but Q is not true, therefore, P is not true—as mediate inference is still more forcible; but of course the modus ponens and the modus tollens stand or fall together.387

Professor Croom Robertson (Mind, 1877, p. 264) has suggested an explanation as to the manner in which this controversy may have arisen. He distinguishes the hypothetical “if” from the inferential “if,” the latter being equivalent to since, seeing that, because. No doubt by the aid of a certain accentuation the word “if” may be made to carry with it this force. Professor Robertson quotes a passage from Clarissa Harlowe in which the remark, “If you have the value for my cousin that you say you have, you must needs think her worthy to be your wife,” is explained by the speaker to mean, “Since you have &c.” Using the word in this sense, the conclusion C is D certainly follows immediately from the bare statement If A is B, C is is D; or rather this statement itself affirms the conclusion. When, however, the word “if” carries with it this inferential implication, we cannot regard the proposition in which it occurs as merely hypothetical. We have rather a condensed mode of expression including two statements in one; it may indeed be argued that in the single statement thus interpreted we have a hypothetical syllogism expressed elliptically.388

EXERCISES.

309. Shew how the modus ponens may be reduced to the modus tollens. [K.]

310. Test the following: “If all men were capable of perfection, some would have attained it; but none having done so, none are capable of it.” [V.]

311. Examine technically the following argument:— If you needed food, I would give you money; but as you do not care to work, you cannot need food; therefore, I will give you no money. [J.]

312. Shew what conclusion can be inferred from the premisses: He always stays in when it rains, but he often goes out when it is cold. [J.]

313. Construct conditional and hypothetical syllogisms in Cesare, Bocardo, Dimaris and reduce them to the first figure. [K.]

314. Name the mood and figure of the following, and shew that either one may be reduced to the other form:

(1)If R is true, Q is true,

If P is true, Q is not true,

therefore, If P is true, R is not true ;

(2)If Y is true, Z is not true,

If Y is true, X may be true,

therefore, If X is true, Z need not be true.

[K.]

315. Let X, Y, Z, P, Q, R be six propositions. Given (1) If X is true, P is true ; (2) If Y is true, Q is true ; (3) If Z is true, R is true ; (4) Of X, Y, Z one at least is true ; (5) Of P, Q, R not more than one is true ; prove syllogistically (i) If P is true, X is true ; (ii) If Q is true, Y is true ; (iii) If R is true, Z is true ; (iv) Of P, Q, R, one at least is true ; (v) Of X, Y, Z, not more than one is true. [K.]