Chapter 40
_The Disjunctive Syllogism regarded as an Immediate Inference_.
§ 770. If no stress be laid on the transition from disjunctive hypothesis to fact, the disjunctive syllogism will run with the same facility as its predecessor into the moulds of immediate inference.
§ 771.
_Denial of Antecedent_. Subalternation.
Either A is B or C is D, Every case of A not being B is a case of C being D. .'. A not being B, C is D. .'. Some case of A not being B is a case of C being D.
§ 772.
_Denial of Consequent_. Conversion by Contraposition + Subalternation.
Either A is B or C is D. All cases of A not being B are cases of C being D. .'. C not being D, A is B .'. All cases of C not being D are cases of A being B. .'. Some case of C not being D is a case of A being B.
§ 773. Similarly the two invalid types of disjunctive syllogism will be found to coincide with fallacies of immediate inference.
§ 774.
_Affirmation of Antecedent_. Contraposition without Conversion.
Either A is B or C is D. All cases of A not being B are cases of C being D. .'. A being B, C is not D .'. All cases of A being B are cases of C not being D.
§ 775. The affirmation of the antecedent thus comes under the formula--
All not-A is B, .'. All A is not-B,
a form of inference which cannot hold except where A and B are known to be incompatible. Who, for instance, would assent to this?--
All non-boating men play cricket. .'. All boating men are non-cricketers.
§ 776.
_Affirmation of Consequent_. Simple Conversion of A.
Either A is B or C is D. All cases of A not being B are cases of C being D. .'.C being D, A is not B. .'. All cases of C being D are cases of A not being B.
§ 777. We may however argue in this way--
Conversion of A per accidens. Either A is B or C is D. All cases of A not being B are cases of C being D. .'. C being D, A is sometimes B. .'. Some cases of C being D are cases of A not being B.
The men who pass this examination must have either talent or industry. .'. Granting that they are industrious, they may be without talent.