CHAPTER XXII.. _of the Partly Conjunctive Syllogism Regarded As an Immediate

Of the Partly Conjunctive Syllogism regarded as an Immediate Inference.

§ 753. It is the assertion of fact in the minor premiss, where we have the application of an abstract principle to a concrete instance, which alone entitles 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, .'. A may be B,

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 assumes 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 assert 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.