Lesson Seventeen
The Conditional Syllogism
Selections from Aristotle’s Prior Analytics
41a21 – 41b5
It is clear then that the ostensive syllogisms are
effected by means of the aforesaid figures; these
considerations will show that reductions ad
absurdum also are effected in the same way. For all
who effect an argument per impossibile infer
syllogistically what is false, and prove the original
conclusion conditionally when something impossible
results from the assumption of its contradictory; e.g.,
that the diagonal of the square is incommensurate
with the side, because odd numbers are equal to
evens if it is supposed to be commensurate. One
infers syllogistically that odd numbers come out
equal to evens, and one proves conditionally the
incommensurability of the diagonal, since a
falsehood results through contradicting this. For this
we found to be reasoning per impossibile, viz.,
proving something impossible by means of an
hypothesis conceded at the beginning. Consequently,
since the falsehood is established in reductions ad
impossibile by an ostensive syllogism, and the
original conclusion is proved conditionally, and we
have already stated that ostensive syllogisms are
effected by means of these figures, it is evident that
syllogisms per impossibile also will be made through
these figures.
Likewise all the other conditional syllogisms: for
in every case the syllogism leads up to the
proposition that is substituted for the original thesis;
but the original thesis is reached by means of a
concession or some other condition. But if this is
true, every demonstration and every syllogism must
be formed by means of the three figures mentioned
above. But when this has been shown it is clear that
every syllogism is perfected by means of the first
figure and is reducible to the universal syllogisms in
this figure.
Definitions
conditional syllogism – syllogism one of whose premises is a conditional statement.
reduction to the absurd (ad absurdum) – argument which proves a conclusion by showing that
its opposite leads to an absurdity.
modus ponens – conditional syllogism which asserts the antecedent.
modus tollens – conditional syllogism which denies the consequent.
Lesson
After Aristotle has reduced abbreviated syllogisms to syllogisms of the three figures, he
claims that he can in fact show that every syllogism is reduced to one of these three figures. Most
modern logicians disagree. They claim that Aristotle has not accounted for the conditional, or
hypothetical, syllogism, and that this kind is in fact more basic than the syllogisms which
Aristotle gives. As we shall see, however, Aristotle’s principles do account for the conditional
syllogism. In this lesson, we will examine Aristotle’s account of the conditional syllogism and a
particular variety of it – the reduction to the absurd.
The Conditional Syllogism
In Lesson Eight we briefly examined the conditional statement. As you may recall, the
conditional statement has two parts, the antecedent and the consequent. The whole statement is
true only if the consequent follows from the antecedent. Thus, even if both parts of the
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statements are true, if the second does not follow from the first, the whole statement is false. On
the other hand, the whole statement can be true even if one or both of the parts are false, as long
as the second follows from the first.
We must notice that the conditional statement, according to this explanation, seems very
similar to the definition of the syllogism. The syllogism, as you recall, is a complex expression in
which, the premises being given, the conclusion necessarily follows from them. In fact, when
Aristotle gives the syllogisms, he gives them in the form of conditional statements with two
antecedents: “If A belongs to every B, and B belongs to every C, then A also belongs to every
C.” This is a sign that the conditional syllogism will be related to the syllogisms in the three
figures.
We can find clearer evidence of that relation by considering the conditional statement in
itself. The consequent must follow from the antecedent in order for the whole statement to be
true. But we can only prove that it follows by making a syllogism of one of the three figures,
using the antecedent as a premise. We can conclude, then, that the conditional statement is
usually just an abbreviated syllogism in which the explicit premise is not asserted, but merely
proposed. An example will help to explain what we mean.
Take the conditional statement “If man were a plant, he would lack sensation.” If we
apply the rules of the abbreviated syllogism, we can see that the conclusion “Every man lacks
sensation” follows from the explicit premise “Every man is a plant” and the implicit premise
“Every plant lacks sensation.” We see that the conditional is true because the implied syllogism
is valid, even though its conclusion is false. In the same way, a conditional statement can be true
even if its consequent and antecedent are false. Such a conditional statement still stands as true
because it does not assert the antecedent as a truth. Rather, it asserts only that if the antecedent
were true, the consequent would follow from it. That men are plants is only supposed; the
consequent, men lack sensation, follows from that supposition.
A second derivation of the conditional statement, however, occurs when it is substituted
for a confusing or elaborate simple universal statement. As we saw before, a statement is simple
if the subject and predicate each form an essential unity, no matter how many words they
contain. For example, “bodily, living, sensitive, rational substance” is a simple noun because it
forms an essential unity, usually signified by the word “man.” The statement “A bodily, living,
sensitive, and rational substance is a man” is therefore a simple statement. To express such an
elaborate simple statement, however, it is sometimes easier to use a conditional sentence whose
subject is the remote genus of the thing being explained. Thus we say that, in the antecedent, the
subject has certain properties, and in the consequent, it has other “properties” (e.g., the name of a
thing we want to define) which follow from it having the first ones. For example, instead of
stating, “Every bodily, living, sensitive, rational substance is a man,” we might find it easier to
state, “If a substance is bodily, sensitive, and rational, then it is a man.” In this way we can
substitute a conditional statement for a very long and complicated simple statement. The
meaning is the same, but the conditional expression is more easily understood.
The conditional statement, then, is either 1) an abbreviated syllogism which does not
positively assert its explicit premise or 2) a substitute for the universal statement. In either case,
the conditional syllogism, of which the conditional statement forms the principle part, follows
clear rules.
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Now, the conditional syllogism has one conditional and one asserting premise, and it
comes in two valid moods, called modus ponens (“the way of positing”) and modus tollens (“the
way of removing”). The first, modus ponens, works by asserting the antecedent, which was only
supposed in the conditional statement. For example, “If man is an animal, then he has sensation.
But man is an animal. Therefore, he has sensation” proceeds according to modus ponens. The
second, modus tollens, works by denying the consequent and thus denying the antecedent from
which it follows. For example, “If man is a plant, then he lacks sensation. But man does not lack
sensation, therefore he is not a plant” works by modus tollens.
The conditional syllogism is invalid if the consequent is asserted, or the antecedent
denied. For example, the statement “If man is a beast, he will have sensation” is true, because the
consequent follows from the antecedent. If I were to assert that man is not a beast, it does not
follow that man does not have sensation. Similarly, if it is raining, the ground will be wet, but the
ground being wet does not imply that it is raining, since the sprinklers can also make the ground
wet. Thus, the only two valid moods of the conditional syllogism are modus ponens and modus
tollens.
The following are the moods of the conditional syllogism:
CAUTION: In this chart, X and Y represent propositions, not terms.
Conditional Syllogisms
Modus Ponens
Modus Tollens
If X is true, then Y is true.
X is true.
Therefore, Y is true.
If X is true, then Y is true.
Y is false.
Therefore, X is false.
Reduction to the Absurd
Reduction to the absurd is a kind of syllogism that proves something true by showing that
its contradictory is false. Euclid often uses this method in his books on geometry. Aristotle
teaches that the reduction to the absurd uses the conditional syllogism. Here is an example of
such a reduction:
Either every two lines have a unit that measures both evenly, or some two lines do
not have such a unit. If every two lines have such a unit, then the number of times
that the unit that measures both the side of the square and its diagonal measures
the diagonal is both even and odd. But no number can be both even and odd.
Therefore, [by modus tollens] it is false that every two lines have such a unit.
Thus, the contradictory, some two lines do not have a common unit, is a true
statement.
In this example, Aristotle assumes the contradictory of what he wished to prove, using it
as the antecedent in the conditional statement. Since the consequent of the conditional is false,
the antecedent must also be false, by modus tollens. And, since the antecedent is the
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contradictory of what he wished to prove, the intended conclusion must be true. Thus, the
intended conclusion has been proven by a reduction to absurdity.
Since the conditional syllogism reduces to the syllogisms of the three figures, so does the
reduction to the absurd. Thus, Aristotle states that every reduction to the absurd can be
transformed into a direct proof, that is, into a syllogism of one of the three figures. Also, recall
that the reduction of syllogisms by contradiction is an application of the method of reduction to
the absurd. Both begin by assuming the opposite of what they intend to prove, and then show
that that opposite is false.
Exercises
Exercise 1: State whether the following syllogisms are valid or invalid.
1.
2.
3.
4.
If triangles have angles equal to 180
degrees, then squares have angles
equal to 360 degrees.
Triangles have angles equal to 180
degrees.
Therefore, squares have angles equal
to 360 degrees.
Should all goods come from virtue,
no evil man possesses the good.
Some evil men possess the good.
Therefore, some goods do not come
from virtue.
If every triangle has angles equal to
180º, then every square has angles
equal to 360º.
Every square does have angles equal
to 360º.
Therefore, every triangle has angles
equal to 180º.
5.
If mathematics is wisdom, then
children can be wise.
Children cannot be wise.
Therefore, mathematics is not
wisdom.
6.
Things are in a species when they
have an essence.
Nothing has an essence.
Therefore, nothing is in a species.
7.
If some logician is emotional, then
some logician is not logical.
Every logician is logical.
Therefore, no logician is emotional.
8.
When cats have nine lives, then they
have immaterial souls.
Cats have nine lives.
Cats have immaterial souls.
9.
If a lion is an animal, then it has
sensation.
Lions are animals.
Therefore, lions have sensation.
10.
If a square were a circle, it would be
a plane figure.
The square is not a circle.
Therefore, it is not a plane figure.
If virtue is knowledge, then virtue is
teachable.
But virtue is not knowledge.
Therefore, virtue is not teachable.
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