Deductive Logic Part 42

CHAPTER XXV.

_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 a.s.sent 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 pa.s.s this examination must have either talent or industry.

.'. Granting that they are industrious, they may be without talent.

CHAPTER XXVI.

_Of the Mixed Form of Complex Syllogism_.

-- 778. Under this head are included all syllogisms in which a conjunctive is combined with a disjunctive premiss. The best known form is

_The Dilemma_.

-- 779. The Dilemma may be defined as--

A complex syllogism, having for its major premiss a conjunctive proposition with more than one antecedent, or more than one consequent, or both, which (antecedent or consequent) the minor premiss disjunctively affirms or denies.

-- 780. It will facilitate the comprehension of the dilemma, if the following three points are borne in mind--

(1) that the dilemma conforms to the canon of the partly conjunctive syllogism, and therefore a valid conclusion can be obtained only by affirming the antecedent or denying the consequent;

(2) that the minor premiss must be disjunctive;

(3) that if only the antecedent be more than one, the conclusion will be a simple proposition; but if both antecedent and consequent be more than one, the conclusion will itself be disjunctive.

-- 781. The dilemma, it will be seen, differs from the partly conjunctive syllogism chiefly in the fact of having a disjunctive affirmation of the antecedent or denial of the consequent in the minor, instead of a simple one. It is this which const.i.tutes the essence of the dilemma, and which determines its possible varieties. For if only the antecedent or only the consequent be more than one, we must, in order to obtain a disjunctive minor, affirm the antecedent or deny the consequent respectively; whereas, if there be more than one of both, it is open to us to take either course. This gives us four types of dilemma.

-- 782.

(1). _Simple Constructive._

If A is B or C is D, E is F.

Either A is B or C is D.

.'. E is F.

(2). _Simple Destructive._

If A is B, C is D and E is F.

Either C is not D or E is not F.