Deductive Logic

-- 753. It is the a.s.sertion of fact in the minor premiss, where we have the application of an abstract principle to a concrete instance, which alone ent.i.tles the partly conjunctive syllogism to be regarded as a syllogism at all. Apart from this the forms of semi-conjunctive reasoning run at once into the moulds of immediate inference.

-- 754. The constructive mood will then be read in this way--

If A is B, C is D,.'. A being B, C is D.

reducing itself to an instance of immediate inference by subaltern opposition--

Every case of A being B, is a case of C being D.

.'. Some particular case of A being B is a case of C being D.

-- 755. Again, the destructive conjunctive will read as follows--

If A is B, C is D,.'. C not being D, A is not B.

which is equivalent to

All cases of A being B are cases of C being D.

.'. Whatever is not a case of C being D is not a case of A being B.

.'. Some particular case of C not being D is not a case of A being B.

But what is this but an immediate inference by contraposition, coming under the formula

All A is B,.'. All not-B is not-A,

and followed by Subalternation?

-- 756. The fallacy of affirming the consequent becomes by this mode of treatment an instance of the vice of immediate inference known as the simple conversion of an A proposition. 'If A is B, C is D' is not convertible with 'If C is D, A is B' any more than 'All A is B' is convertible with 'All B is A.'

-- 757. We may however argue in this way

If A is B, C is D, C is D,.'.

which is equivalent to saying,

When A is B, C is always D,.'. When C is D, A is sometimes B,

and falls under the legitimate form of conversion of A per accidens--

All cases of A being B are cases of C being D.

.'. Some cases of C being D are cases of A being B.

-- 758. The fallacy of denying the antecedent a.s.sumes the following form--

If A is B, C is D,.'. If A is not B, C is not D,

equivalent to--

All cases of A being B are cases of C being D.

.'. Whatever is not a case of A being B is not a case of C being D.

This is the same as to argue--

All A is B,.'. All not-A is not-B,

an erroneous form of immediate inference for which there is no special name, but which involves the vice of simple conversion of A, since 'All not-A is not-B' is the contrapositive, not of 'All A is B,' but of its simple converse 'All B is A.'

-- 759. The above-mentioned form of immediate inference, however (namely, the employment of contraposition without conversion), is valid in the case of the U proposition; and so also is simple conversion. Accordingly we are able, as we have seen, in dealing with a proposition of that form, both to deny the antecedent and to a.s.sert the consequent with impunity--

If A is B, then only C is D,.'. A not being B, C is not D;

and again, C being D, A must be B.

CHAPTER XXIII.

_Of the Disjunctive Syllogism_.

-- 760. Roughly speaking, a Disjunctive Syllogism results from the combination of a disjunctive with a simple premiss. As in the preceding form, the complex proposition is regarded as the major premiss, since it lays down a hypothesis, which is applied to fact in the minor.

-- 761. The Disjunctive Syllogism may be exactly defined as follows--

A complex syllogism, which has for its major premiss a disjunctive proposition, either the antecedent or consequent of which is in the minor premiss simply affirmed or denied.

-- 762. Thus there are four types of disjunctive syllogism possible.

_Constructive Moods._

(1) Either A is B or C is D. (2) Either A is B or C is D.

A is not B. C is not D.

.'. C is D..'. A is B.

Either death is annihilation or we are immortal.

Death is not annihilation.

.'. We are immortal.

Either the water is shallow or the boys will be drowned.

The boys are not drowned.

.'. The water is shallow.

_Destructive Moods_.

(3) Either A is B or C is D. (4) Either A is B or C is D.

A is B. C is D.

.'. C is not D..'. A is not B.