LOGICAL EQUATIONS AND THE QUANTIFICATION OF THE PREDICATE.
137. The employment of the symbol of Equality in Logic.—The symbol of equality (=) is frequently used in logic to express the identity of two classes. For example,
Equilateral triangles = equiangular triangles ;
Exogens = dicotyledons ;
Men = mortal men.
It is, however, important to recognise that in thus borrowing a symbol from mathematics we do not retain its meaning unaltered, and that a so-called logical equation is, therefore, something very different from a mathematical equation. In mathematics the symbol of equality generally means numerical or quantitative equivalence. But clearly we do not mean to express mere numerical equality when we write equilateral triangles = equiangular triangles. Whatever this so-called equation signifies, it is certainly something more than that there are precisely as many triangles with three equal sides as there are triangles with three equal angles. It is further clear that we do not intend to express mere similarity. Our meaning is that the denotations of the terms which are equated are absolutely identical; in other words, that the class of objects denoted by the term equilateral triangle is absolutely identical with the class of objects denoted by the term equiangular triangle.[186] It may, however, be objected that, if this 190 is what our equation comes to, then inasmuch as a statement of mere identity is empty and meaningless, it strictly speaking leaves us with nothing at all; it contains no assertion and can represent no judgment. The answer to this objection is that whilst we have identity in a certain respect, it is erroneous to say that we have mere identity. We have identity of denotation combined with diversity of connotation, and, therefore, with diversity of determination (meaning thereby diversity in the ways in which the application of the two terms identified is determined).[187] The meaning of this will be made clearer by the aid of one or two illustrations. Taking, then, as examples the logical equations already given, we may analyse their meaning as follows. If out of all triangles we select those which possess the property of having three equal sides, and if again out of all triangles we select those which possess the property of having three equal angles, we shall find that in either case we are left with precisely the same set of triangles. Thus, each side of our equation denotes precisely the same class of objects, but the class is determined or arrived at in two different ways. Similarly, if we select all plants that are exogenous and again all plants that are dicotyledonous, our results are precisely the same although our mode of arriving at them has been different. Once more, if we simply take the class of objects which possess the attribute of humanity, and again the class which possess both this attribute and also the attribute of mortality, the objects selected will be the same; none will be excluded by our second method of selection although an additional attribute is taken into account.
[186] It follows that the comprehensions (but of course not the connotations) of the terms will also be identical; this cannot, however, be regarded as the primary signification of the equation.
[187] I have practically borrowed the above mode of expression from Miss Jones, who describes an affirmative categorical proposition as “a proposition which asserts identity of application in diversity of signification” (General Logic, p. 20). Miss Jones’s meaning may, however, be slightly different from that intended in the text, and I am unable to agree with her general treatment of the import of categorical propositions, as she does not appear to allow that before we can regard a proposition as asserting identity of application we must implicitly, if not explicitly, have quantified the predicate.
Since the identity primarily signified by a logical equation is an identity in respect of denotation, any equational mode of reading propositions must be regarded as a modification of the 191 “class” mode. What has been said above, however, will make it clear that here as elsewhere denotation is considered not to the exclusion of connotation but as dependent upon it; and we again see how denotative and connotative readings of propositions are really involved in one another, although one side or the other may be made the more prominent according to the point of view which is taken.
Another point to which attention may be called before we pass on to consider different types of logical equations is that in so far as a proposition is regarded as expressing an identity between its terms the distinction between subject and predicate practically disappears. We have seen that when we have the ordinary logical copula is, propositions cannot always be simply converted, the reason being that the relation of the subject to the predicate is not the same as the relation of the predicate to the subject. But when two terms are connected by the sign of equality, they are similarly, and not diversely, related to each other; in other words, the relation is symmetrical. Such an equation, for example, as S = P can be read either forwards or backwards without any alteration of meaning. There can accordingly be no distinction between subject and predicate except the mere order of statement, and that may be regarded as for most practical purposes immaterial.
138. Types of Logical Equations[188]—Jevons (Principles of Science, chapter 3) recognises three types of logical equations, which he calls respectively simple identities, partial identities, and limited identities.
[188] This section may be omitted on a first reading.
Simple identities are of the form S = P ; for example, Exogens = dicotyledons. Whilst this is the simplest case equationally, the information given by the equation requires two propositions in order that it may be expressed in ordinary predicative form. Thus, All S is P and All P is S ; All exogens are dicotyledons and All dicotyledons are exogens. If, however, we are allowed to quantify the predicate as well as the subject, a single proposition will suffice. Thus, All S is all P, All exogens are all dicotyledons. We shall return presently to a consideration of this type of proposition.