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the asserted relations exist, but many; to find out whether there is not at least one (conceivable, or possible, or actual, as the case may be) in which the other relations that seem to be involved are not really involved. This is thinking, and no mechanical rules can save us the trouble.
MEDIATE INFERENCE AND SYLLOGISM.
It has already been explained that mediate inference takes place when we recognize some new aspect of the total state of affairs in which alone all the relations asserted by two or more premises can exist together. To put the matter more concretely: Mediate inference takes place when we conclude anything about the relations of two or more objects to each other from the relations of each to some third object, the word 'object ' being used in the broadest possible sense to include qualities and relations as well as things. From the fact that A is larger than B and that C is smaller than B we can conclude that A is larger than C; and this is mediate inference.
No inference can be drawn about the relations of two objects to each other, unless the object with which each of them is compared is in both cases of deduction. the same.
From the fact that A is larger than B and that C is smaller than D, nothing can be inferred about the relations of A and C.
Moreover no inference can usually be drawn unless each of the two objects is compared with the third in the same respect; unless the relations discussed are homogeneous, or at least unless they belong to the same unitary system. From the fact that A is larger than B and that C is lighter than B, no inference can be drawn about the relations of A and C. Where Euclid says
Things which are equal to the same thing are equal to one another”, we must understand
him to mean: “ Things which are equal to the same thing in any given respect are equal to each other in that respect A may be equal to C in physical strength, and B equal to C in intelligence without A and B being equal to each other in any respect whatever.
In the same way the line AB may be equal in length and the line 'EF equal in color to the line GH, without their being equal to each other in either one or the other.
While no inference is usually possible when the relations dealt with are heterogeneous, the inferences drăwn when all the relations under discussion are homogeneous do not belong to formal logic, If A is B's landlord or creditor or agent, and B is C's, it is a lawyer's business, not that of the formal logician, to say whether A is in any sense C's landlord or creditor or agent; if a substance D has a chemical affinity for E, and E for F, nobody but a chemist can tell whether D necessarily has or has not an affinity for F; if G is four times as large as H, and H is seven times as large as K, the relative sizes of G and K is a question of mathematics, and the traditional field of deduction is so limited that the formal logician as such is debarred from drawing a conclusion.
In syllogism or the kind of mediate inference discussed in formal logic, two of the three relations usually involved in the premises and the conclusion are homogeneous, and the third is (or may reasonably be treated as) a relation of identity.* If we say ‘G is four times as large as H and H
* I say “two of the three relations usually involved' because in a sorites there may be an indefinite number of premises. In this case the rela. tions stated in all the premises except the last must be relations of identity.
The rule that two of the three relations must be homogeneous and the third a relation of identity does not exclude the case where all three are relations of identity, e.g., •A is identical with B, and B is identical with C, therefore A is identical with C'. The most serious objection that I can think of to the rule as I have stated it seems to come from such cases as this: • A is larger, taller, sweeter, heavier, better, prettier than B, B is larger, etc., than C, therefore A is larger, etc., than Ç’, Here the rela.
1. is seven times as large as K; therefore G is twenty-eight
times as large as K', the reasoning belongs to mathematics; C but when we say . G is four times as large as H, and H and
K are one and the same thing; therefore G is four times as IT large as K', the reasoning is syllogistic and belongs to deduc
tive logic. Similarly if we say “A is four miles due west of B and C is three miles due north of B, therefore A is five miles southwest of C’, the reasoning is geometrical; but if we say “ A is four miles due west of B, and C is three miles
due north of B (i.e., not at all west of B), therefore A and C C are not the same', the reasoning is syllogistic. Or to put 0
it somewhat differently, if we say “ A is two miles west of B and three miles east of C, therefore B is five miles east of C', the reasoning is geometrical or arithmetical and beyond the sphere of formal logic (not of course 'illogical '); but if we say “ A is two miles west of B and three miles east of C, therefore something two miles west of B is three miles east of C (i.e., one and the same thing is both two miles west of B and three miles east of C)', the reasoning is syllogistic. In the examples of arithmetical and geometrical reasoning here given all the relations affirmed or denied were relations of size, number, or direction; but in the examples of syllogistic reasoning this was not the case. In the first of these three examples of syllogistic reasoning the second premise
tions are all homogeneous and no one of them is a relation of identity ; and yet the conclusion seems to follow in each case from pure logic, and without reference to any special science. I
the answer to such an objection would be that before we speak of a thing as having more of a given quality than anything else we recognize that things can be arranged in a series with reference to that quality, so that whatever goes beyond something else goes still more beyond the things which that something else goes beyond. All that is implied by the use of the comparative -er or more — So that when we say “A is taller than B, and B is taller than C, therefore A is taller than C', we might have said • A is taller than B, and whatever B is taller than, C is a thing that B is taller than, therefore A is taller than C’. In the syllogism thus stated the second premise asserts a relation of identity and the rule holds good.
affirmed a relation of identity and thus made the conclusion possible; in the second example the conclusion denied that two objects were identical, because they possessed incompatible relations; in the third example the conclusion called attention to the fact that the objects described in the two premises were identical, so that the relations which they affirmed coexisted, or both belonged to the same object.
The three examples of syllogistic reasoning which we have just given to illustrate the difference between such reasoning
and that which is not syllogistic can also be used • Figures'.
to illustrate the difference between three different kinds of syllogism, for there are certain respects in which they are quite different from each other. The examples were these:
1) G is four times as large as H;
H and K are one and the same;
2) A is four miles due west of B;
C is three miles due north of B;
3) A is two miles west of B;
A is three miles east of C;
miles east of C.
In each of the syllogisms there is a term (called the Middle Term) which occurs in each of the premises but not in the conclusion; but in the different syllogisms this middle term does not occur in the same place. In the first syllogism the middle term H is the predicate of the first premise and the subject of the second; in the second the middle term B is the predicate of both premises; and in the third the middle term A is subject of both. This difference of order is accompanied by a corresponding difference of thought; and each one of the three syllogisms may be regarded as an example