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FIGURE III

All men are mortal;

All men are fallible;

M-P

M-S

Therefore some fallible beings are mortal. .'. S—P

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In the first figure the middle term is the subject of the major premise and the predicate of the minor premise. In the second figure it is the predicate of both premises. In the third figure it is the subject of both premises. In the fourth figure it is the predicate of the major premise and the subject of the minor premise. It may be remarked that while the fourth figure is theoretically possible, it is of no particular practical importance and is not recognized in Aristotle's doctrine of the syllogism.

Reduction. The first figure was regarded by Aristotle as the most direct and convincing, and was called by him the perfect figure. In the first figure the major premise is a universal proposition, and the minor points out something which this universal includes. The other figures were called the imperfect figures, and the process of changing these to the first figure is called reduction. Elaborate rules governing the process of reduction were formulated during the Middle Ages. Reduction is accomplished through certain processes of obversion and conversion, and through the transposition of the premises whenever necessary. The syllogism, for example, in the third

figure above, may be reduced to the first figure by simply converting the minor premise so as to make it read, 'Some fallible beings are men'; and the syllogism in the fourth figure may be reduced by transposing the premises and converting the conclusion. Reduction is of interest chiefly because it shows that whenever we put an argument in the form of a syllogism, the particular form of syllogism that we adopt is more or less a matter of accident.

Sorites, or Chain of Reasoning.-A Sorites is a chain of reasoning in which the two terms of the conclusion are united through the mediation of more than one intervening or connecting term. It may assume either of the two following forms:

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C is D;

D is E;

All vertebrates are animals;

All animals are mortal;

.. A is E... All negroes are mortal.

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It is possible to treat a Sorites as an abbreviated form of syllogistic inference, because the chain of reasoning may be resolved into a series of syllogisms, each of which, except the last, yields a conclusion

that serves as a premise in the succeeding syllogism.. From this point of view, the first of the above inferences is equivalent to three complete syllogisms,

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Inferences in Quantitative Relations.-Certain inferences that deal with quantitative relations give valid conclusions, in spite of the fact that they seem. to violate the rules of the syllogism. The following are examples:

I

A is greater than B; B is greater than C; .. A is greater than C.

II

A is north of B;

B is north of C;

.'. A is north of C.

In form these arguments are exactly the same as:

A is the landlord of B;

B is the landlord of C;

.. A is the landlord of C.

Yet this latter conclusion does not follow from the premises. All of these syllogisms, it will be noticed,

have four terms. What requires explanation, therefore, is the fact that, in spite of this apparent irregularity, it is possible to draw valid conclusions when the subject-matter concerns relations of quantity. The explanation of this fact is, in brief, that the valid conclusions are possible because they rest upon a true major premise which does not appear in the argument. If A is north of B, and B is north of C, we can infer the relation of A and C, because we are familiar with the nature of space relations. To state the law or the generalization which underlies the inference is a matter of some difficulty. According to some writers the inference, in correct syllogistic form, would read about like this:

Whatever is north of that which is north of another is north of that other;

A is something that is north of that which is north of C; .. A is north of C.

It is true that we never formulate the major premise of this inference, and that we usually do not even suspect its presence. But, as we shall see a little later (see Chapter VIII), the suppression of one of our premises is a frequent occurrence in everyday reasoning. This major premise is not formulated, just because the relationship which it expresses is so simple and obvious. This relationship is peculiar to the realm of quantity, and so the recognition of this relationship enables us to make inferences in this realm which have no precise parallel in other fields.

CHAPTER VII

HYPOTHETICAL AND DISJUNCTIVE
SYLLOGISMS

In a previous chapter propositions were distin guished as categorical and conditional. The latter kind again presents two forms, the hypothetical and the disjunctive. Corresponding to the two kinds of conditional propositions, we have two kinds of conditional syllogisms, the hypothetical and the disjunctive syllogism, just as the syllogism discussed in the preceding chapter corresponds to the categorical proposition.

The Hypothetical Syllogism.-It was pointed out that the hypothetical proposition expresses a condition and a result, e.g., ' If it storms, the boat will capsize.' The part that expresses the condition,' if it storms,' is called the antecedent; the part that expresses the result, the boat will capsize,' is called the consequent. The reason for this distinction will appear in a moment.

In hypothetical syllogisms both hypothetical and categorical propositions are employed. The hypo thetical syllogism consists of a hypothetical major premise and a categorical minor premise. The following is an example:

If the strike has been called off, the men are back at work; The strike has been called off;

Therefore the men are back at work.

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