CONDITIONAL SYLLOGISMS
Sec. 1. Conditional Syllogisms may be generally described as those that
contain conditional propositions. They are usually divided into two
classes, Hypothetical and Disjunctive.
A Hypothetical Syllogism is one that consists of a Hypothetical Major
Premise, a Categorical Minor Premise, and a Categorical Conclusion. Two
Moods are usually recognised the _Modus ponens_, in which the antecedent
of the hypothetical major premise is affirmed; and the _Modus tollens_,
in which its consequent is denied.
(1) _Modus ponens_, or Constructive.
If A is B, C is D;
A is B:
.'. C is D.
If Aristotle's reasoning is conclusive, Plato's theory of Ideas is
erroneous;
Aristotle's reasoning is conclusive:
.'. Plato's theory of Ideas is erroneous.
Rule of the _Modus ponens_: The antecedent of the major premise being
affirmed in the minor premise, the consequent is also affirmed in the
conclusion.
(2) _Modus tollens_, or Destructive.
If A is B, C is D;
C is not D:
.'. A is not B.
If Pythagoras is to be trusted, Justice is a number;
Justice is not a number:
.'. Pythagoras is not to be trusted.
Rule of the _Modus tollens_: The consequent of the major premise being
denied in the minor premise, the antecedent is denied in the conclusion.
By using negative major premises two other forms are obtainable: then,
either by affirming the antecedent or by denying the consequent, we draw
a negative conclusion.
Thus (_Modus ponens_): (_Modus tollens_):
If A is B, C is not D; If A is B, C is not D;
A is B: C is D:
.'. C is not D. .'. A is not B.
Further, since the antecedent of the major premise, taken by itself, may
be negative, it seems possible to obtain four more forms, two in each
Mood, from the following major premises:
(1) If A is not B, C is D;
(2) If A is not B, C is not D.
But since the quality of a Hypothetical Proposition is determined by the
quality of its consequent, not at all by the quality of its antecedent,
we cannot get from these two major premises any really new Moods, that
is to say, Moods exhibiting any formal difference from the four
previously expounded.
It is obvious that, given the hypothetical major premise--
If A is B, C is D--
we cannot, by denying the antecedent, infer a denial of the consequent.
That A is B, is a mark of C being D; but we are not told that it is the
sole and indispensable condition of it. If men read good books, they
acquire knowledge; but they may acquire knowledge by other means, as by
observation. For the same reason, we cannot by affirming the consequent
infer the affirmation of the antecedent: Caius may have acquired
knowledge; but we cannot thence conclude that he has read good books.
To see this in another light, let us recall chap. v. Sec. 4, where it was
shown that a hypothetical proposition may be translated into a
categorical one; whence it follows that a Hypothetical Syllogism may be
translated into a Categorical Syllogism. Treating the above examples
thus, we find that the _Modus ponens_ (with affirmative major premise)
takes the form of Barbara, and the _Modus tollens_ the form of
Camestres:
_Modus ponens._ Barbara.
If A is B, C is D; The case of A being B is a case of C being D;
A is B: This is a case of A being B:
.'. C is D. .'. This is a case of C being D.
Now if, instead of this, we affirm the consequent, to form the new minor
premise,
This is a case of C being D,
there will be a Syllogism in the Second Figure with two affirmative
premises, and therefore the fallacy of undistributed Middle. Again:
_Modus tollens._ Camestres.
If A is B, C is D; The case of A being B is a case of C being D:
C is not D: This is not a case of C being D:
.'. A is not B. .'. This is not a case of A being B.
But if, instead of this, we deny the antecedent, to form the new minor
premise,
This is not a case of A being B,
there arises a syllogism in the First Figure with a negative minor
premise, and therefore the fallacy of illicit process of the major term.
By thus reducing the Hypothetical Syllogism to the Categorical form,
what is lost in elegance is gained in intelligibility. For, first, we
may justify ourselves in speaking of the hypothetical premise as the
major, and of the categorical premise as the minor; since in the
categorical form they contain respectively the major and minor terms.
And, secondly, we may justify ourselves in treating the Hypothetical
Syllogism as a kind of Mediate Inference, in spite of the fact that it
does not exhibit two terms compared by means of a third; since in the
Categorical form such terms distinctly appear: a new term ('This')
emerges in the position of the minor; the place of the Middle is filled
by the antecedent of the major premise in the _Modus ponens_, and by the
consequent in the _Modus tollens_.
The mediate element of the inference in a Hypothetical Syllogism
consists in asserting, or denying, the fulfilment of a given condition;
just as in a Categorical syllogism to identify the minor term with the
Middle is a condition of the major term's being predicated of it. In the
hypothetical proposition--
If A is B, C is D--
the Antecedent, _A is B_, is the _conditio sufficiens_, or mark, of the
Consequent, _C is D_; and therefore the Consequent, _C is D_, is a
_conditio sine qua non_ of the antecedent, _A is B_; and it is by means
of affirming the former condition, or else denying the latter, that a
conclusion is rendered possible.
Indeed, we need not say that the element of mediation consists in
affirming, _or denying_, the fulfilment of a given condition: it is
enough to say 'in affirming.' For thus to explain the _Modus tollens_,
reduce it to the _Modus ponens_ (contrapositing the major premise and
obverting the minor):
Celarent.
If A is B, C is D: The case of C being not-D is
.'. If C is not-D, A is not B; not a case of A being B;
C is not-D: This is a case of C being
.'. A is not B. not-D:
.'. This is not a case of A
being B.
The above four forms commonly treated of as Hypothetical Syllogisms, are
called by Ueberweg and Dr. Keynes 'Hypothetico-Categorical.' Ueberweg
restricts the name 'Hypothetical' simply (and Dr. Keynes the name
'Conditional') to such Syllogisms as the following, having two
Hypothetical Premises:
If C is D, E is F;
If A is B, C is D:
.'. If A is B, E is F.
If we recognise particular hypothetical propositions (see chap. v. Sec. 4),
it is obvious that such Syllogisms may be constructed in all the Moods
and Figures of the Categorical Syllogism; and of course they may be
translated into Categoricals. We often reason in this hypothetical way.
For example:
If the margin of cultivation be extended, rents will rise;
If prices of produce rise, the margin of cultivation will be extended:
.'. If prices of produce rise, rents will rise.
But the function of the Hypothetical Syllogism (commonly so called), as
also of the Disjunctive Syllogism (to be discussed in the next section)
is to get rid of the conditional element of the premises, to pass from
suspense to certainty, and obtain a decisive categorical conclusion;
whereas these Syllogisms with two hypothetical premises leave us still
with a hypothetical conclusion. This circumstance seems to ally them
more closely with Categorical Syllogisms than with those that are
discussed in the present chapter. That they are Categoricals in disguise
may be seen by considering that the above syllogism is not materially
significant, unless in each proposition the word 'If' is equivalent to
'Whenever.' Accordingly, the name 'Hypothetical Syllogism,' is here
employed in the older usage.
Sec. 2. A Disjunctive Syllogism consists of a Disjunctive Major Premise, a
Categorical Minor Premise, and a Categorical Conclusion.
How many Moods are to be recognised in this kind of argument depends on
whether the alternatives of the Disjunctive Premise are regarded as
mutually exclusive or possibly coincident. In saying '_Either_ A is B,
_or_ C is D,' do we mean 'either, but not both,' or 'either, it may be
both'? (See chap. v. Sec. 4.)
When the alternatives of the Disjunctive are not exclusive, we have only
the
_Modus tollendo ponens._
Either A is B, or C is D;
A is not B (or C is not D):
.'. C is D (or A is B).
Either wages fall, or the weaker hands are dismissed;
Wages do not fall:
.'. The weaker hands are dismissed.
But we cannot argue--
Wages fall:
.'. The weaker hands are not dismissed;
since in 'hard times' both events may happen together.
Rule of the _Modus tollendo ponens_: If one alternative be denied, the
other is affirmed.
When, however, the alternatives of the Disjunctive are mutually
exclusive, we have also the
_Modus ponendo tollens._
Either A is B, or C is D;
A is B (or C is D):
.'. C is not D (or A is not B).
Either the Tories or the Whigs win the election;
The Tories win:
.'. The Whigs do not win.
We may also, of course, argue as above in the _Modus tollendo ponens_--
The Tories do not win:
.'. The Whigs do.
But in this example, to make the _Modus tollendo ponens_ materially
valid, it must be impossible that the election should result in a tie.
The danger of the Disjunctive Proposition is that the alternatives may
not, between them, exhaust the possible cases. Only contradictory
alternatives are sure to cover the whole ground.
Rule of the _Modus ponendo tollens:_ If one alternative be affirmed, the
other is denied.
Since a disjunctive proposition may be turned into a hypothetical
proposition (chap. v. Sec. 4,) a Disjunctive Syllogism may be turned into a
Hypothetical Syllogism:
_Modus tollendo ponens._ _Modus ponens._
Either A is B, or C is D; If A is not B, C is D;
A is not B: A is not B:
.'. C is D. .'. C is D.
Similarly the _Modus ponendo tollens_ is equivalent to that kind of
_Modus ponens_ which may be formed with a negative major premise; for if
the alternatives of a disjunctive proposition be exclusive, the
corresponding hypothetical be affirmative or negative:
_Modus ponendo tollens._ _Modus ponens._
Either A is B, or C is D; If A is B, C is not D;
A is B: A is B:
.'. C is not D. .'. C is not D.
Hence, finally, a Disjunctive Syllogism being equivalent to a
Hypothetical, and a Hypothetical to a Categorical; a Disjunctive
Syllogism is equivalent and reducible to a Categorical. It is a form of
Mediate Inference in the same sense as the Hypothetical Syllogism is;
that is to say, the conclusion depends upon an affirmation, or denial,
of the fulfilment of a condition implied in the disjunctive major
premise.
Sec. 3. The Dilemma is perhaps the most popularly interesting of all forms
of proof. It is a favourite weapon of orators and wits; and "impaled
upon the horns of a dilemma" is a painful situation in which every one
delights to see his adversary. It seems to have been described by
Rhetoricians before finding its way into works on Logic; and Logicians,
to judge from their diverse ways of defining it, have found some
difficulty in making up their minds as to its exact character.
There is a famous Dilemma employed by Demosthenes, from which the
general nature of the argument may be gathered:
If AEschines joined in the public rejoicings, he is
inconsistent; if he did not, he is unpatriotic;
But either he joined, or he did not join:
Therefore he is either inconsistent or unpatriotic.
That is, reduced to symbols:
If A is B, C is D; and if E is F, G is H:
But either A is B, or E is F;
.'. Either C is D or G is H (_Complex Constructive_).
This is a compound Conditional Syllogism, which may be analysed as
follows:
Either A is B or E is F.
Suppose that E is not F: Suppose that A is not B:
Then A is B. Then E is F.
But if A is B, C is D; But if E is F, G is H;
(A is B): (E is F):
.'. C is D. .'. G is H.
.'. Either C is D or G is H.
A Dilemma, then, is a compound Conditional Syllogism, having for its
Major Premise two Hypothetical Propositions, and for its Minor Premise a
Disjunctive Proposition, whose alternative terms either affirm the
Antecedents or deny the Consequents of the two Hypothetical Propositions
forming the Major Premise.
The hypothetical propositions in the major premise, may have all four
terms distinct (as in the above example); and then the conclusion is a
disjunctive proposition, and the Dilemma is said to be Complex. Or the
two hypothetical propositions may have a common antecedent or a common
consequent; and then the conclusion is a categorical proposition, and
the Dilemma is said to be Simple.
Again, the alternatives of the disjunctive minor premise may be
affirmative or negative: if affirmative, the Dilemma is called
Constructive; and if negative, Destructive.
Using, then, only affirmative hypothetical propositions in the major
premise, there are four Moods:
1. The Simple Constructive--
If A is B, C is D; and if E is F, C is D:
But either A is B, or E is F:
.'. C is D.
If the Tories win the election, the Government will avoid
innovation; and if the Whigs win, the House of Lords will
prevent them innovating:
But either the Tories or the Whigs will win:
.'. There will be no innovation.
2. The Complex Constructive--
If A is B, C is D; and if E is F, G is H:
But either A is B, or E is F:
.'. Either C is D or G is H.
If appearance is all that exists, reality is a delusion; and
if there is a substance beyond consciousness, knowledge of
reality is impossible:
But either appearance is all, or there is a substance beyond
consciousness:
.'. Either reality is a delusion, or a knowledge of it is
impossible.
3. Simple Destructive--
If A is B, C is D; and if A is B, E is F:
But either C is not D, or E is not F:
.'. A is not B.
If table-rappers are to be trusted, the departed are spirits;
and they also exert mechanical energy:
But either the departed are not spirits, or they do not
exert mechanical energy:
.'. Table-rappers are not to be trusted.
4. Complex Destructive--
If A is B, C is D; and if E is F, G is H:
But either C is not D, or G is not H:
.'. Either A is not B, or E is not F.
If poetic justice is observed, virtue is rewarded; and if the
mirror is held up to Nature, the villain triumphs:
But either virtue is not rewarded, or the villain does not
triumph:
.'. Either poetic justice is not observed, or the mirror is
not held up to Nature.
Such are the four Moods of the Dilemma that emerge if we only use
affirmative hypotheticals for the major premise; but, certainly, it is
often quite as natural to employ two negative hypotheticals (indeed,
one might be affirmative and the other negative; but waive that); and
then four more moods emerge, all having negative conclusions. It is
needless to intimidate the reader by drawing up these four moods in
battle array: they always admit of reduction to the foregoing moods by
obverting the hypotheticals. Still, by the same process we may greatly
decrease the number of moods of the Categorical Syllogism; and just as
some Syllogisms are most simply expressed in Celarent or Cesare, so some
Dilemmas are most simply stated with negative major premises--e.g.,
The example of a Simple Constructive Dilemma above given would run more
naturally thus: _If the Tories win, the Government will not innovate;
and if the Whigs, the Lords will not let them_: and similarly
Demosthenes' Dilemma--_If AEschines joined, he is not consistent; and if
he did not, he is not patriotic_. Moreover, the propriety of recognising
Dilemmas with negative major premises, follows from the above analysis
of the Dilemma into a combination of Conditional Syllogisms, even if (as
in Sec. 1 of this chapter) we take account of only four Moods of the
Hypothetical Syllogism.
In the rhetorical use of the Dilemma, it may be observed that the
disjunction in the minor premise ought to be obvious, or (at any rate)
easily acceptable to the audience. Thus, _Either the Tories or the Whigs
will win; Either AEschines joined in the rejoicings, or he did not_; such
propositions are not likely to be disputed. But if the orator must stop
to prove his minor premise, the smacking effect of this figure (if the
expression be allowed) will be lost. Hence the minor premises of other
examples given above are only fit for a select audience. That _Either
ghosts are not spirits, or they do not exert mechanical energy_,
supposes a knowledge of the principle, generally taught by physical
philosophers, that only matter is the vehicle of energy; and that
_Either appearance is all, or there is substance beyond consciousness_,
is a doctrine which only metaphysical philosophers could be expected to
understand, and upon which they could not be expected to agree. However,
the chief danger is that a plausible disjunction may not be really such
as to exclude any middle ground: _Either the Tories or the Whigs win_,
is bad, if a tie be possible; though in the above argument this is
negligible, seeing that a tie cannot directly cause innovations. _Either
AEschines joined in the rejoicings, or he did not_, does not allow for a
decent conformity with the public movement where resistance would be
vain; yet such conformity as need not be inconsistent with subsequent
condemnation of the proceedings, nor incompatible with patriotic reserve
founded on a belief that the rejoicings are premature and ominous.
Another rhetorical consideration is, that the alternatives of the
disjunctive conclusion of a Complex Dilemma should both point the same
way, should be equally distasteful or paradoxical. 'Either inconsistent
or unpatriotic': horrid words to a politician! 'Either no reality or no
possible knowledge of it': very disappointing to an anxious inquirer!
Thus the disjunctive conclusion is as bad for an opponent as the
categorical one in a Simple Dilemma.
Logicians further speak of the Trilemma, with three Hypotheticals and a
corresponding triple Disjunction; and of a Polylemma, with any further
number of perplexities. But anyone who has a taste for logical forms may
have it amply gratified in numerous text-books.