_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.