(2) Subject in denotation, predicate in denotation.
If we read both the subject and the predicate of a proposition in denotation, we have a relation between two classes, and hence this is called the class mode of interpreting the proposition. It must be particularly observed that the relation between the subject and the predicate is now one of inclusion in or exclusion from, not one of possession. It may at once be admitted that the class mode of interpreting the categorical proposition is neither the most ultimate, nor—generally speaking—that which we naturally or spontaneously adopt. It is, however, extremely convenient for manipulative purposes, and hence is the mode of interpretation usually selected, either explicitly or implicitly, by the formal logician. Attention may be specially called to the following points:
(a) When subject and predicate are both read in denotation, they are homogeneous.
(b) In the diagrammatic illustration of propositions both subject and predicate are necessarily read in denotation, since it is the denotation—not the connotation—of a term that we represent by means of a diagram.
(c) The predicate of a proposition must be read in denotation in order to give a meaning to the question whether it is or is not distributed.
(d) The predicate as well as the subject must be read in denotation before such a process as conversion is possible.
(e) In the treatment of the syllogism both subject and predicate must be read in denotation (or else both in connotation), since either the middle term (first and fourth figures) or the major term (second and fourth figures) or the minor term (third and fourth figures) is subject in one of the propositions in which it occurs and predicate in the other.
The class mode of interpreting categorical propositions is nevertheless treated by some writers as being positively 182 erroneous. But the arguments used in support of this view will not bear examination.
(i) It is said that to read both subject and predicate in denotation is psychologically false. It has indeed been pointed out already that the class mode of interpretation is not that which as a rule first presents itself to our mind when a proposition is given us; but we have also seen that there are exceptions to this, as, for example, in the propositions All daisies are compositae. All Hindoos are Aryan, All Tartars are Turanians. It is, therefore, clearly wrong to describe the reading in question as in all cases psychologically false. On the same shewing, any other reading would equally be psychologically false, for what is immediately present to the mind in judgment varies very much in different cases. Undoubtedly there are many judgments in regard to which we do not spontaneously adopt the class reading. Still, analysis shews that in these judgments, as in others, inclusion in or exclusion from a class is really implicated along with other things, although this relation may be neither that which first impresses itself upon us nor that which is most important or characteristic.
(ii) It is asked what we mean by a class, by the class of birds, for example, when we say All owls are birds. “It is nothing existing in space; the birds of the world are nowhere collected together so that we can go and pick out the owls from amongst them. The classification is a mental abstraction of our own, founded upon the possession of certain definite attributes. The class is not definite and fixed, and we do not find out whether any individual belongs to it by going over a list of its members, but by enquiring whether it possesses the necessary attributes.”[182] In so far as this argument applies against reading the predicate in denotation, it applies equally against reading the subject in denotation. It is in effect the argument used by Mill (Logic, i. 5, § 3) in order to lead up to his position that the ultimate interpretation of the categorical proposition requires us to read both subject and predicate in connotation, since denotation is determined by connotation. But if this be granted, it does not prove the class reading of the 183 proposition erroneous; it only proves that in the class reading, the analysis of the import of the proposition has not been carried as far as it admits of being carried.
[182] Welton, Logic, p. 218.
(iii) It is argued that when we regard a proposition as expressing the inclusion of one class within another, even then the predicate is only apparently read in denotation. “On this view, we do not really assert P but ‘inclusion in P,’ and this is therefore the true predicate. For example, in the proposition ‘All owls are birds,’ the real predicate is, on this view, not ‘birds’ but ‘included in the class birds.’ But this inclusion is an attribute of the subject, and the real predicate, therefore, asserts an attribute. It is meaningless to say ‘Every owl is the class birds,’ and it is false to say ‘The class owls is the class birds.’”[183] This argument simply begs the question in favour of the predicative mode of interpretation. It overlooks the fact that the precise kind of relation brought out in the analysis of a proposition will vary according to the way in which we read the subject and the predicate. An analogous argument might also be used against the predicative reading itself. Take the proposition, “All men are mortal.” It is absurd to say that “Every man is the attribute mortality,” or that “The class men is the attribute mortality.”
[183] Welton, Logic, p. 218.
(iv) It is said that a class interpretation of both S and P would lead properly to a fivefold, not a fourfold, scheme of propositions, since there are just five relations possible between any two classes, as is shewn by the Eulerian diagrams. This contention has force, however, only upon the assumption that we must have quite determinate knowledge of the class relation between S and P before being able to make any statement on the subject; and this assumption is neither justifiable in itself nor necessarily involved in the interpretation in question. It may be added that if a similar view were taken on the adoption of the predicative mode of interpretation, we should have a threefold, not a fourfold scheme. For then the quantity of our subject at any rate would have to be perfectly determinate, and with S and P for subject and predicate, the three possible statements would be—All S is P, Some S is P and 184 some is not, No S is P. The point here raised will presently be considered further in connexion with the quantification of the predicate.
(3) Subject in connotation, predicate in connotation.
If we read both the subject and the predicate of a proposition in connotation, we have what may be called the connotative mode of interpreting the proposition. In the proposition All S is P, the relation expressed between the attributes connoted by S and those connoted by P is one of concomitance—“the attributes which constitute the connotation of S are always found accompanied by those which constitute the connotation of P.”[184] Similarly, in the case of Some S is P,—“the attributes 185 which constitute the connotation of S are sometimes found accompanied by those which constitute the connotation of P”; No S is P,—“the attributes which constitute the connotation of S are never found along with those which constitute the connotation of P”; Some S is not P,—“the attributes which constitute the connotation of S are sometimes found unaccompanied by those which constitute the connotation of P.”
[184] This is the only possible reading in connotation, so far as real propositions are concerned, if the term connotation is used in the strict sense as distinguished both from subjective intension and from comprehension. Unfortunately confusion is apt to be introduced into discussions concerning the intensive rendering of propositions simply because no clear distinction is drawn between the different points of view which may be taken when terms are regarded from the intensive side. Hamilton distinguished between judgments in extension and judgments in intension, the relation between the subject and the predicate in the one case being just the reverse of the relation between them in the other. Thus, taking the proposition All S is P, we have in extension S is contained under P, and in intension S comprehends P. On this view the intensive reading of All men are mortal is “mortality is part of humanity” (the extensive reading being “the class man is part of the class mortal”). This reading may be accepted if the term intension is used in the objective sense which we have given to comprehension, so that by humanity is meant the totality of attributes common to all men, and by mortality the totality of attributes common to all mortals. To this point of view we shall return in the next section. Leaving it for the present on one side, it is clear that if by humanity we mean only what may be called the distinctive or essential attributes of man, then in order that the above reading may be correct, the given proposition must be regarded as analytical. In other words, if humanity signifies only those attributes which are included in the connotation of man, then, if mortality is included in humanity, we shall merely have to analyse the connotation of the name man, in order to obtain our proposition. Hence on this view it must either be maintained that all universal affirmative propositions are analytical, or else that some universal affirmatives cannot be read in intension. But obviously the first of these alternatives must be rejected, and the second practically means that the reading in question breaks down so far as universal affirmatives are concerned.