Lit2Go

Deductive Logic

by George William Joseph Stock, M.A.

Part 3: Chapter 22

• Year Published: 1888
• Language: English
• Country of Origin: England
• Source: Stock, G. W. J. (1888). Deductive Logic. Oxford, England; Pembroke College.
• Flesch–Kincaid Level: 11.0
• Word Count: 606
• Genre: Informational
• Keywords: math, math history

PART III.—OF INFERENCES

CHAPTER XXII.

_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.