Logic: Deductive and Inductive Part 16

For other examples it is enough to refer to the _Critique of Pure Reason_, where Kant sets out the Antinomies of Rational Cosmology. But even if we do not agree with Kant that the human understanding, in attempting to deal with certain subjects beyond its reach, inevitably falls into such contradictory reasonings; yet it can hardly be doubted that we not unfrequently hold opinions which, if logically developed, result in Antinomies. And, accordingly, the Antinomy, if it cannot be imputed to Reason herself, may be a very fair, and a very wholesome _argumentum ad hominem_. It was the favourite weapon of the Pyrrhonists against the dogmatic philosophies that flourished after the death of Aristotle.

CHAPTER XII

CONDITIONAL SYLLOGISMS

-- 1. Conditional Syllogisms may be generally described as those that contain conditional propositions. They are usually divided into two cla.s.ses, Hypothetical and Disjunctive.

A Hypothetical Syllogism is one that consists of a Hypothetical Major Premise, a Categorical Minor Premise, and a Categorical Conclusion. Two Moods are usually recognised the _Modus ponens_, in which the antecedent of the hypothetical major premise is affirmed; and the _Modus tollens_ [sic], in which its consequent is denied.

(1) _Modus ponens_, or Constructive.

If A is B, C is D; A is B: ? C is D.

If Aristotle's reasoning is conclusive, Plato's theory of Ideas is erroneous;

Aristotle's reasoning is conclusive: ? Plato's theory of Ideas is erroneous.

Rule of the _Modus ponens_: The antecedent of the major premise being affirmed in the minor premise, the consequent is also affirmed in the conclusion.

(2) _Modus tollens_, or Destructive.

If A is B, C is D; C is not D: ? A is not B.

If Pythagoras is to be trusted, Justice is a number; Justice is not a number: ? Pythagoras is not to be trusted.

Rule of the _Modus tollens_: The consequent of the major premise being denied in the minor premise, the antecedent is denied in the conclusion.

By using negative major premises two other forms are obtainable: then, either by affirming the antecedent or by denying the consequent, we draw a negative conclusion.

Thus (_Modus ponens_): (_Modus tollens_):

If A is B, C is not D; If A is B, C is not D; A is B: C is D: ? C is not D. ? A is not B.

Further, since the antecedent of the major premise, taken by itself, may be negative, it seems possible to obtain four more forms, two in each Mood, from the following major premises:

(1) If A is not B, C is D; (2) If A is not B, C is not D.

But since the quality of a Hypothetical Proposition is determined by the quality of its consequent, not at all by the quality of its antecedent, we cannot get from these two major premises any really new Moods, that is to say, Moods exhibiting any formal difference from the four previously expounded.

It is obvious that, given the hypothetical major premise--

If A is B, C is D--

we cannot, by denying the antecedent, infer a denial of the consequent.

That A is B, is a mark of C being D; but we are not told that it is the sole and indispensable condition of it. If men read good books, they acquire knowledge; but they may acquire knowledge by other means, as by observation. For the same reason, we cannot by affirming the consequent infer the affirmation of the antecedent: Caius may have acquired knowledge; but we cannot thence conclude that he has read good books.

To see this in another light, let us recall chap. v. -- 4, where it was shown that a hypothetical proposition may be translated into a categorical one; whence it follows that a Hypothetical Syllogism may be translated into a Categorical Syllogism. Treating the above examples thus, we find that the _Modus ponens_ (with affirmative major premise) takes the form of Barbara, and the _Modus tollens_ the form of Camestres:

_Modus ponens._ Barbara.

If A is B, C is D; The case of A being B is a case of C being D; A is B: This is a case of A being B: ? C is D. ? This is a case of C being D.

Now if, instead of this, we affirm the consequent, to form the new minor premise,

This is a case of C being D,

there will be a Syllogism in the Second Figure with two affirmative premises, and therefore the fallacy of undistributed Middle. Again:

_Modus tollens._ Camestres.

If A is B, C is D; The case of A being B is a case of C being D: C is not D: This is not a case of C being D: ? A is not B. ? This is not a case of A being B.

But if, instead of this, we deny the antecedent, to form the new minor premise,

This is not a case of A being B,

there arises a syllogism in the First Figure with a negative minor premise, and therefore the fallacy of illicit process of the major term.

By thus reducing the Hypothetical Syllogism to the Categorical form, what is lost in elegance is gained in intelligibility. For, first, we may justify ourselves in speaking of the hypothetical premise as the major, and of the categorical premise as the minor; since in the categorical form they contain respectively the major and minor terms.

And, secondly, we may justify ourselves in treating the Hypothetical Syllogism as a kind of Mediate Inference, in spite of the fact that it does not exhibit two terms compared by means of a third; since in the Categorical form such terms distinctly appear: a new term ('This') emerges in the position of the minor; the place of the Middle is filled by the antecedent of the major premise in the _Modus ponens_, and by the consequent in the _Modus tollens_.

The mediate element of the inference in a Hypothetical Syllogism consists in a.s.serting, or denying, the fulfilment of a given condition; just as in a Categorical syllogism to identify the minor term with the Middle is a condition of the major term's being predicated of it. In the hypothetical proposition--

If A is B, C is D--

the Antecedent, _A is B_, is the _conditio sufficiens_, or mark, of the Consequent, _C is D_; and therefore the Consequent, _C is D_, is a _conditio sine qua non_ of the antecedent, _A is B_; and it is by means of affirming the former condition, or else denying the latter, that a conclusion is rendered possible.

Indeed, we need not say that the element of mediation consists in affirming, _or denying_, the fulfilment of a given condition: it is enough to say 'in affirming.' For thus to explain the _Modus tollens_, reduce it to the _Modus ponens_ (contrapositing the major premise and obverting the minor):

Celarent.

If A is B, C is D: The case of C being not-D is ? If C is not-D, A is not B; not a case of A being B; C is not-D: This is a case of C being ? A is not B. not-D: ? This is not a case of A being B.

The above four forms commonly treated of as Hypothetical Syllogisms, are called by Ueberweg and Dr. Keynes 'Hypothetico-Categorical.' Ueberweg restricts the name 'Hypothetical' simply (and Dr. Keynes the name 'Conditional') to such Syllogisms as the following, having two Hypothetical Premises:

If C is D, E is F; If A is B, C is D: ? If A is B, E is F.

If we recognise particular hypothetical propositions (see chap. v. -- 4), it is obvious that such Syllogisms may be constructed in all the Moods and Figures of the Categorical Syllogism; and of course they may be translated into Categoricals. We often reason in this hypothetical way.

For example:

If the margin of cultivation be extended, rents will rise; If prices of produce rise, the margin of cultivation will be extended: ? If prices of produce rise, rents will rise.

But the function of the Hypothetical Syllogism (commonly so called), as also of the Disjunctive Syllogism (to be discussed in the next section) is to get rid of the conditional element of the premises, to pa.s.s from suspense to certainty, and obtain a decisive categorical conclusion; whereas these Syllogisms with two hypothetical premises leave us still with a hypothetical conclusion. This circ.u.mstance seems to ally them more closely with Categorical Syllogisms than with those that are discussed in the present chapter. That they are Categoricals in disguise may be seen by considering that the above syllogism is not materially significant, unless in each proposition the word 'If' is equivalent to 'Whenever.' Accordingly, the name 'Hypothetical Syllogism,' is here employed in the older usage.

-- 2. A Disjunctive Syllogism consists of a Disjunctive Major Premise, a Categorical Minor Premise, and a Categorical Conclusion.