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its conclusion were negative, and destructive though affirma
If A is B, C is not D;
A is B; ... C is not D (Constructive).
If A is not B, C is not D;
C is D; ... A is B (Destructive).
Hypothetical syllogisms look like an entirely new sort; but the novelty lies altogether in the verbal form, not in the relations expressed; for we have seen (page 109) that the relations expressed by hypothetical propositions can be expressed about as well in universal categorical propositions, and when the hypothetical major premise of a hypothetical syllogism is put into categorical form only a slight change is required in the minor to make the syllogism also categorical. Making these changes, we get
A state of affairs in which A is B is a state of affairs
in which C is D; This is a state of affairs in which A is B; .:. This is a state of affairs in which C is D.
A Christian forgives;
J. S. is a Christian; .: J. S. forgives. A state of affairs in which A is B is a state of affairs in
which C is D; This is a state of affairs in which C is not D; ... This is a state of affairs in which A is not B.
A Christian forgives;
J. S. does not forgive; .:: J. S. is not a Christian.
The constructive hypothetical syllogism thus resolves itself into an ordinary syllogism in the first figure; the destructive into one in the second.
The formal rule for hypothetical syllogisms is that the
minor premise must either affirm the antecedent of the hypothetical proposition in the major (as in the first two of these last examples) or deny the consequent (as in the second two). In the first of the following examples we commit the Fallacy of Denying the Antecedent; in the second, the Fallacy of Affirming the Consequent.
If a man is a Christian, he forgives;
J. S, is not a Christian; .. J. S. does not forgive.
If a man is a Christian, he forgives;
J. S. forgives; .:. J. S. is a Christian.
Turning from words to things, the meaning of the formal rule is as follows:
If the presence of one state of affairs (AB) always involves the presence of another (CD), and if the first state of affairs (AB) is present, the second state of affairs (CD) must also be present; if the second state of affairs (CD) is absent, the first state of affairs (AB) cannot be there to involve its presence; but the first state of affairs (AB) can be absent without involving the absence of the second (CD); and the second (CD) can be present without involving the presence of the first (AB).
These fallacies of 'denying the antecedent' and 'affirming the consequent' would not be fallacies at all if the world were so constituted that there was only one cause capable of producing a given effect or one premise capable of involving a given conclusion. If a person had to be drowned in order to be killed, we could not only say, He is drowned, therefore he is killed', ‘He is not killed, therefore he is not drowned’; but we could also say, 'He is not drowned, therefore he is not killed ’, and ‘He is killed, therefore he is drowned'. To avoid the fallacy we should think of what we are saying and remember that the world is not constituted in this way, but that any one of several causes may pioduce essentially the same result and any one of several premises involve the same conclusion,
Disjunctive Syllogisms are those which conDisjunctive syllogisms.
tain a disjunctive major premise and a categorical minor, e.g.:
A is either B or C; Either J is not K or L is M;
J is K; .:. A is C.
.:. L is M. In order that any conclusion should be justified it is necessary that the minor premise deny the existence of one of the alternatives mentioned in the major. We cannot say 'A is either B or C; it is B; therefore it is not C'.*
What we must be most careful about in the case of these syllogisms is to see that the major premise is really true; that there is no alternative which it does not mention. We should not say “This man must be either wise or foolish, he is not wise, therefore he is foolish’; for there are many persons of medium intelligence who cannot fairly be called either wise or foolish.
A Dilemma is 'a syllogism having a hypothetical major premise with more than one antecedent and a disjunctive minor'. “ In common speech ..
we are said Dilemmas.
to be in a dilemma when we have only two courses open to us and both of them are attended by unpleasant consequences. In arguments we are in this position when we are shut into a choice between two admissions and either admission leads to a conclusion which we do not like.” †
According to Jevons the Dilemma takes at least three different forms. The first form is called the Simple Constructive Dilemma:
If A is B, C is D; and if E is F, C is D;
* See p. 106 on meaning of either'.
+ Minto, p. 222.
Thus “if a science furnishes useful facts, it is worthy of being cultivated; and if the study of it exercises the reasoning powers, it is worthy of being cultivated; but either a science furnishes useful facts, or its study exercises the reasoning powers; therefore it is worthy of being cultivated.'
The second form of dilemma is the Complex Constructive Dilemma, which is as follows:
If A is B, C is D; and if E is F, G is H;
Therefore either C is D, or G is H. “ It is called complex because the conclusion is in the disjunctive form. As an instance we may take the argument, “If a statesman who sees his former opinions to be wrong does not alter his course, he is guilty of deceit; and if he does alter his course, he is open to a charge of inconsistency; but either he does not alter his course or he does; therefore he is either guilty of deceit or open to a charge of inconsistency.' In this case as in the greater number of dilemmas the terms A, B, C, D, etc., are not different.”
The third form--the Destructive Dilemma—“ is always complex, because it could otherwise be resolved into two unconnected destructive hypothetical syllogisms. It is in the following form:
If A is B, C is D; and if Eis F, G is H;
Therefore either A is not B, or E is not F. “For instance, ' If this man were wise, he would not speak irreverently of Scripture in jest; and if he were good, he would not do so in earnest; but he does it either in jest or in earnest; therefore he is either not wise or not good' (Whately).
“ Dilemmatic arguments are, however, more often fallacious than not, because it is seldom possible to find instances where two alternatives exhaust all the possible
cases, unless indeed one of them be the simple negative of the other in accordance with the law of excluded middle. Thus if we were to argue that “if a pupil is fond of learning he needs no stimulus, and that if he dislikes learning no stimulus will be of any avail, but that, as he is either fond of learning or dislikes it, a stimulus is either needless or of no avail', we evidently assume improperly the disjunctive minor premise. Fondness and dislike are not the only two possible alternatives, for there may be some who are neither fond of learning nor dislike it, and to these a stimulus in the shape of rewards may be desirable.” *
Principles of logic have reference to the relations of objects, not to the words in which those relations are
expressed. From this it follows that variations
can be introduced into the wording of an arguabbreviated
ment without affecting its validity. One variaargument.
tion that has seemed important enough to be discussed in almost every text-book since Aristotle consists in taking for granted certain of the relations involved, without any explicit mention of them. * Enthymemes', ' Epicheiremata' and 'Sorites’ are names for arguments abridged in different ways.
An Enthymeme (from év, in, and Avuos, the mind) is a syllogism-usually categorical—in which one of the premises or the conclusion is not expressed.
Supposing the syllogism in question to be this:
Three forms of
All men are mortal;
our reasoning would have been almost as clear and more effective rhetorically if we had merely said:
(1) Socrates is a man,
Therefore he is mortal;
* « Lessons in Logic”, p. 167.