Attention is directed to the fact that by this process of alternate conversion and obversion we are able on the strength of a given proposition to make assertions not only about objects to which the subject and predicate terms are applicable, but about objects—if we assume them to exist— to which one or both are wholly inapplicable (ie., to non-A's and non-P's). We must, however, be very careful not to jump at conclusions of this sort. From the proposition All S is P (All men are mortal) we can infer No non-P is S (No immortals are men), or All non-P is non-S (All immortals are non-human); but we cannot infer All non-S is non-P (All non-humans are immortals).

The principal difficulty in conversion is due, as we have seen, to the fact that a descriptive predicate has to be turned into a demonstrative subject with the proper quantity. From this difficulty exclusive and exceptive propositions are free, since they always distribute their predicates. If Europeans alone are capable of self-government it must be that all races capable of self-government are Europeans. Indeed it is a rather curious fact that exclusive and exceptive propositions imply something about all the objects mentioned in the predicate and about all those not specially mentioned in the subject, but not necessarily about all those specially mentioned in the subject; so that the objects to which they seem to call special attention are those about which they say the least. From the supposed fact that Europeans alone are capable of self-government it follows, as we have seen, that all races capable of self-government are European; it also follows that no non-European races are capable of selfgovernment; but it does not necessarily follow that all European races are capable of self-government. Because of this I am inclined to think that to transpose the subject and predicate of an exclusive or exceptive proposition is an analysis of meaning rather than a conversion. We cannot be sure that European races alone are capable of self-government unless we already know that all the races capable of

self-government are European.

The value of the exclusive form seems to lie in the contrast it brings out between the objects specially mentioned (to which the attribute in question at least may belong), and the rest of the class to which they belong-the kind of contrast which serves as the basis for all classification.

The subject of conversion has been discussed so far from as mechanical a standpoint as possible. Following the older logicians we have given rules for the manipulation of words which can be followed blindly.

The treatment by diagrams.

In the latter half of the eighteenth century Leonard Euler invented, or rather revived, a set of simple diagrams by which the relations between classes of objects could be so easily and so well symbolized that it was no longer necessary to follow rules mechanically, or even to remember them at all. If the members of each class of objects are supposed to be enclosed in a circle, the visible relations of the circles can be relied upon to indicate the relations of the classes. When one class is included in (or is identical with) another (Proposition A), the circle S, supposed to enclose the members of the first, must be drawn inside of (or coincident with) the circle P, supposed to enclose the members of the second; when some members of the first class are also members of the second (Proposition I), at least a part of the circle S must lie inside of the circle P; when there are some members of the first class which are not members of the second (Proposition O), at least a part of the circle S must lie outside of the circle P; and when no member of the first class is a member of the second (Proposition E), all of the circle S must lie outside of the circle P.

So far it has been the circle S we have discussed, as wholly or partly within or without the circle P. But we can neither draw nor conceive of figures so constructed that a circle S lies wholly or partly within another circle P without part of P's area lying within S. From All S is P' (Proposition A)

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