192 Partial identities are of the form S = SP, and are the expression equationally of ordinary universal affirmative propositions. If we take the proposition All S is P, it is clear that we cannot write it S = P, since the class P, instead of being coextensive with the class S, may include it and a good deal more besides. Since, however, by the law of identity All S is S, it follows from All S is P that All S is SP. We can also pass back from the latter of these propositions to the former. Hence the two propositions are equivalent. But All S is SP may at once be reduced to the equational form S = SP. For this breaks up into the two propositions All S is SP and All SP is S, and since the second of these is a mere formal proposition based on the law of identity, the equation must necessarily hold good if All S is SP is given. To take a concrete example, the proposition All men are mortal becomes equationally Men = mortal men. Similarly the universal negative proposition SeP may be expressed in the equational form S = Sp (where p = not-P).
Limited identities are of the form VS = VP, which may be interpreted “Within the sphere of the class V, all S is P and all P is S,” or “The S’s and P’s, which are V’s, are identical.” So far as V represents a determinate class, there is little difference between these limited identities and simple identities. This is shewn by the fact that Jevons himself gives Equilateral triangles = equiangular triangles as an instance of a simple identity, whereas its proper place in his classification would appear to be amongst the limited identities, for its interpretation is that “within the sphere of triangles—all the equilaterals are all the equiangulars.”
The equation VS = VP is, however, used by Boole—and also by Jevons subsequently—as the expression equationally of the particular proposition, and if it can really suffice for this, its recognition as a distinct type is justified. If we take the proposition Some S is P, we find that the classes S and P are affirmed to have some part in common, but no indication is given whereby this part can be identified. Boole accordingly indicates it by the arbitrary symbol V. It is then clear that All VS is VP and also that All VP is VS, and we have the above equation.
193 It is no part of our present purpose to discuss systems of symbolic logic; but it may be briefly pointed out that the above representation of the particular proposition is far from satisfactory. In order to justify it, limitations have to be placed upon the interpretation of V which altogether differentiate it from other class-symbols. Thus, the equation VS = VP is consistent with No S is P (and, therefore, cannot be equivalent to Some S is P) provided that no V is either S or P, for in this case we have VS = 0 and VP = 0. V must, therefore, be limited by the antecedent condition that it represents an existing class and a class that contains either S or P, and it is in this condition quite as much as in the equation itself that the real force of the particular proposition is expressed.[189]
[189] Compare Venn, Symbolic Logic, pp. 161, 2.
If particular propositions are true contradictories of universal propositions, then it would seem to follow that in a system in which universals are expressed as equalities, particulars should be expressed as inequalities. This would mean the introduction of the symbols > and <, related to the corresponding mathematical symbols in just the same way as the logical symbol of equality is related to the mathematical symbol of equality; that is to say, S > SP would imply logically more than mere numerical inequality, it would imply that the class S includes the whole of the class SP and more besides. Thus interpreted, S > SP expresses the particular negative proposition, Some S is not P.[190] If we further introduce the symbol 0 as expressing nonentity, No S is P may be written SP = 0, and its contradictory, i.e., Some S is P, may be written SP > 0. We shall then have the following scheme (where p = not-P):
| All S is P | expressed by S = SP or by Sp = 0; |
| Some S is not P | ″ ″ S > SP ″ Sp > 0; |
| No S is P | ″ ″ SP = 0 ″ S = Sp ; |
| Some S is P | ″ ″ SP > 0 ″ S > Sp. |
[190] Similarly X > Y expresses the two statements “All Y is X, but Some X is not Y,” just as X = Y expresses the two statements “All Y is X and All X is Y.”
194 This scheme, it will be observed, is based on the assumption that particulars are existentially affirmative while universals are existentially negative. This introduces a question which will be discussed in detail in the following [chapter]. The object of the present section is merely to illustrate the expression of propositions equationally, and the symbolism involved has, therefore, been treated as briefly as has seemed compatible with a clear explanation of its purport. Any more detailed treatment would involve a discussion of problems belonging to symbolic logic.
139. The expression of Propositions as Equations.—There are rare cases in which propositions fall naturally into what is practically an equational form; for example, Civilization and Christianity are co-extensive. But, speaking generally, the equational relation, as implicated in ordinary propositions, is not one that is spontaneously, or even easily, grasped by the mind. Hence as a psychological account of the process of judgment the equational rendering may be rejected. It is, moreover, not desirable that equations should supersede the generally recognised propositional forms in ordinary logical doctrine, for such doctrine should not depart more than can be helped from the forms of ordinary speech. But, on the other hand, the equational treatment of propositions must not be simply put on one side as erroneous or unworkable. It has been shewn in the preceding section that it is at any rate possible to reduce all categorical propositions to a form in which they express equalities or inequalities; and such reduction is of the greatest importance in systems of symbolic logic. Even for purposes of ordinary logical doctrine, the enquiry how far propositions may be expressed equationally serves to afford a more complete insight into their full import, or at any rate their full implication. Hence while ordinary formal logic should not be entirely based upon an equational reading of propositions, it cannot afford altogether to neglect this way of regarding them.