or

'Some S is P' (Proposition I) we can therefore infer 'Some P is S' (Proposition I). Similar grounds can be found in the space relations of the figures for the conversion of E (No S is P) into E (No P is S); while the figure only allows us to convert O (Some S isn't P) into the worthless Proposition E, already referred to (No P is some S or other).

Thus when we draw the diagrams we can convert without reference to the formal rules, merely by observing what the relations of the circles must be under the given conditions. This is a much more natural and rational process than to blindly follow mechanical rules. The only rule involved in the construction of diagrams in conversion or syllogism is this: Try to make them represent the premise or premises without at the same time representing any conclusion you have in mind. If this cannot be done the conclusion follows. If it can be done it does not.

Euler's diagrams have rendered great service to logic; but it must not be forgotten that in using them or any other diagrams constructed on the same principle we assume that spatial relations can be relied upon to represent relations which are not spatial. Diagrams in logic are metaphors, and to reason in metaphors is usually extremely dangerous. Experience happens to show that in the case of Euler's diagrams the metaphor is not misleading, but we must not forget on that account that it is usually better and safer when we can do so to reason about the relations of things themselves directly than through the mutual relations of their symbols.

The reason that Euler's diagrams seem to make logical relations so clear is that they appeal directly to the senses, and that of all the relations perceived by sense those of space are the most constant, the most universal, and the most easily represented. Almost every conceivable relation thus comes to be symbolized in terms of space and seems to be better understood when it is expressed in spatial language. It is said that every preposition once expressed a spatial relation, and the same is true of very many words and phrases used with reference to the mental life (e.g., apprehend', ' movement of thought', 'idea in, or before the mind', 'convey an idea', 'express an emotion', 'impression', etc.).

The great objection to Euler's diagrams is that, like the rules which they were intended to supplement, they apply only to relations of inclusion or exclusion between classes. Both are wholly inapplicable to either dynamic or nondynamic relations between different individuals. Both, therefore, are of service within but a small portion of the whole sphere of thought.

A broader treatment.

There is no reason why the term conversion should not be broadened so as to include the transposition of subject and predicate when the copula is understood to express something other than mere identity or non-identity of things or classes. There are many propositions in which the subject and predicate name two different objects while the copula affirms or denies a dynamic or non-dynamic relation between them. The transposition of the subject and predicate of such propositions might fairly be called conversion. The difficulty connected with the traditional conversion is to settle the distribution of the new subject; and it arises from the fact that a predicate used descriptively is turned into a subject used demonstratively. With the kind of conversion just mentioned there is no such difficulty, for in dynamic and non-dynamic propositions the predicate is already used demonstratively. Whatever mechanical difficulty presented itself would come

from the copula. Sometimes it could remain unchanged and sometimes it would have to be altered so as to express a reversed relation. If John (subject) is-a-relative-of (copula) James (predicate), James is-a-relative-of John. Here the relation, so far at least as it is expressed, is the same for both parties and might be represented by an arrow pointed at both ends: John James; and the proposition can be converted by a mere transposition of subject and predicate. But if John is-the-father-of James we cannot infer that James is-the-father-of John. Here the relation expressed is different for the two parties and should be represented by an arrow pointing in one direction only: John James; and when we convert, the copula must be changed, so as to express the relation from the other side: James is-the-child-of John, James John.

→

When to reverse the relation expressed in the copula and when to leave it alone is a question that might be seriously considered if it were necessary or desirable to pay attention merely to our words and not to what they mean. But this is not necessary or desirable; and the question needs no serious consideration, for when we pay attention to the real object of discourse and understand the meaning of the words used there is no difficulty.

Whether we use the term conversion in this broad sense or in a still broader sense to include statements about any objects on the strength of statements about other objects in which the first objects were mentioned, there is no general rule for conversion which can be followed blindly and no set of symbols which is always applicable. The only thing to do is to turn from mechanical rules and from symbols to the things themselves, find out exactly what relations are asserted of the object spoken about, and then ask ourselves whether there are not corresponding relations of other objects mentioned or implied without which the relations asserted could not possibly or conceivably exist. To do this we must imagine not only a single state of affairs in which

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.

CHAPTER XII.

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 Limitations 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

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