_Of Complex Syllogisms_.
§ 731. A Complex Syllogism is one which is composed, in whole or part,
of complex propositions.
§ 732. Though there are only two kinds of complex proposition, there
are three varieties of complex syllogism. For we may have
(1) a syllogism in which the only kind of complex proposition
employed is the conjunctive;
(2) a syllogism in which the only kind of complex proposition
employed is the disjunctive;
(3) a syllogism which has one premiss conjunctive and the other
disjunctive.
The chief instance of the third kind is that known as the Dilemma.
Syllogism
___________________|_______________
| |
Simple Complex
(Categorical) (Conditional)
_____________________|_______________
| | |
Conjunctive Disjunctive Dilemma
(Hypothetical)
_The Conjunctive Syllogism_.
§ 733. The Conjunctive Syllogism has one or both premisses conjunctive
propositions: but if only one is conjunctive, the other must be a
simple one.
§ 734. Where both premisses are conjunctive, the conclusion will be of
the same character; where only one is conjunctive, the conclusion will
be a simple proposition.
§ 735. Of these two kinds of conjunctive syllogisms we will first take
that which consists throughout of conjunctive propositions.
_The Wholly Conjunctive Syllogism_.
§ 736. Wholly conjunctive syllogisms do not differ essentially from
simple ones, to which they are immediately reducible. They admit of
being constructed in every mood and figure, and the moods of the
imperfect figures may be brought into the first by following the
ordinary rules of reduction. For instance--
Cesare. Celarent.
If A is B, C is never D. \ / If C is D, A is never B.
If E is F, C is always D. | = | If E is F, C is always D.
.'. If E is F, A is never B. / \ .'. If E is F, A is never B.
If it is day, the stars never shine.\ /If the stars shine, it is never day.
If it is night, the stars always \=/ If it is night, the stars always
shine. / \ shine.
.'. If it is night, it is never day / \.'. If it is night, it is never day.
Disamis. Darii.
If C is D, A is sometimes B. \ / If C is D, E is always F.
If C is D, E is always F. | = | If A is B, C is sometimes D.
If E is F, A is sometimes B. / \ .'. If A is B, E is sometimes F.
.'. If E is F, A is sometimes B.
If she goes, I sometimes go. \ / If she goes, he always goes,
If she goes, he always goes. | = | If I go, she sometimes goes.
.'. If he goes, I sometimes go. / \ .'. If I go, he sometimes goes.
.'. If he goes, I sometimes go.
_The Partly Conjunctive Syllogism._
§ 737. It is this kind which is usually meant when the Conjunctive or
Hypothetical Syllogism is spoken of.
§ 738. Of the two premisses, one conjunctive and one simple, the
conjunctive is considered to be the major, and the simple premiss the
minor. For the conjunctive premiss lays down a certain relation to
hold between two propositions as a matter of theory, which is applied
in the minor to a matter of fact.
§ 739. Taking a conjunctive proposition as a major premiss, there are
four simple minors possible. For we may either assert or deny the
antecedent or the consequent of the conjunctive.
Constructive Mood. Destructive Mood.
(1) If A is B, C is D. (2) If A is B, C is D.
A is B. C is not D.
.'. C is D. .'. A is not B.
(3) If A is B, C is D. (4) If A is B, C is D.
A is not B. C is D.
No conclusion. No conclusion.
§ 740. When we take as a minor 'A is not B ' (3), it is clear that we
can get no conclusion. For to say that C is D whenever A is B gives us
no right to deny that C can be D in the absence of that
condition. What we have predicated has been merely inclusion of the
case AB in the case CD.
[Illustration]
§ 741. Again, when we take as a minor, 'C is D' (4), we can get no
universal conclusion. For though A being B is declared to involve as a
consequence C being D, yet it is possible for C to be D under other
circumstances, or from other causes. Granting the truth of the
proposition 'If the sky falls, we shall catch larks,' it by no means
follows that there are no other conditions under which this result can
be attained.
§ 742. From a consideration of the above four cases we elicit the
following
_Canon of the Conjunctive Syllogism._
To affirm the antecedent is to affirm the consequent, and to deny the
consequent is to deny the antecedent: but from denying the antecedent
or affirming the consequent no conclusion follows.
§ 743. There is a case, however, in which we can legitimately deny the
antecedent and affirm the consequent of a conjunctive proposition,
namely, when the relation predicated between the antecedent and the
consequent is not that of inclusion but of coincidence--where in fact
the conjunctive proposition conforms to the type u.
For example--
_Denial of the Antecedent_.
If you repent, then only are you forgiven.
You do not repent.
.'. You are not forgiven.
_Affirmation of the Consequent_.
If you repent, then only are you forgiven.
You are forgiven.
.'. You repent.