Hamilton’s reading breaks down even more completely in the case of particulars and negatives. The attributes constituting the intensions of S and P partly coincide is clearly not equivalent to Some S is P ; for example, the intension (in any sense) of Englishman has something in common with the intension of Frenchman, but it does not follow that Some Englishmen are Frenchmen. Again, from the fact that the intension of S has nothing in common with the intension of P, we cannot infer that No S is P ; suppose, for example, that S stands for “ruminant,” and P for “cloven-hoofed.” Compare Venn, Symbolic Logic, pp. 391–5.
It will be noticed that in the connotative reading we have always to take the attributes which constitute the connotation collectively. In other words, by the attributes constituting the connotation of a term we mean those attributes regarded as a whole. Thus, No S is P does not imply that none of the attributes connoted by S are ever accompanied by any of those connoted by P. This is apparent if we take such a proposition as No oxygen is hydrogen. It follows that when the subject is read in connotation the quantity of the proposition must appear as a separate element, being expressed by the word “always” or “sometimes,” and must not be interpreted as meaning “all” or “some” of the attributes included in the connotation of the subject.
It is argued by those who deny the possibility of the connotative mode of interpreting propositions, that this is not really reading the subject in connotation at all; always and sometimes are said to reduce us to denotation at once. In reply to this, it must of course be allowed that real propositions affirm no relation between attributes independently of the objects to which they belong. The connotative reading implies the denotative, and we must not exaggerate the nature of the distinction between them. Still the connotative reading presents the import of the proposition in a new aspect, and there is at any rate a prima facie difference between regarding one class as included within another, and regarding one attribute as always accompanied by another, even though a little 186 consideration may shew that the two things mutually involve one another.[185]
[185] Mill attaches great importance to the connotative mode of interpreting propositions as compared with the class mode or the predicative mode, on the ground that it carries the analysis a stage further; and this must be granted, at any rate so far as we consider the application of the terms involved to be determined by connotation and not by exemplification. Mill is, however, sometimes open to the charge of exaggerating the difference between the various modes of interpretation. This is apparent, for example, in his rejection of the Dictum de omni et nullo as the axiom of the syllogism, and his acceptance of the Nota notae est nota rei ipsius in its place.
(4) Subject in connotation, predicate in denotation.
Taking the proposition All S is P, and reading the subject in connotation and the predicate in denotation, we have, “The attributes connoted by S are an indication of the presence of an individual belonging to the class P.” This mode of interpretation is always a possible one, but it must be granted that only rarely does the import of a proposition naturally present itself to our minds in this form. There are, however, exceptional cases in which this reading is not unnatural. The proposition No plants with opposite leaves are orchids has already been given as an example. Another example is afforded by the proposition All that glitters is not gold. Taking the subject in connotation and the predicate in denotation we have, The attribute of glitter does not always indicate the presence of a gold object ; and it will be found that this reading of the proverb serves to bring out its meaning really better than any of the three other readings which we have been discussing.
It is worth while noticing here by way of anticipation that on any view of the existential interpretation of propositions, as discussed in [chapter 8], we shall still have a fourfold reading of categorical propositions in connotation and denotation. The universal negative and the particular affirmative may be taken as examples, on the supposition that the former is interpreted as existentially negative and the latter as existentially affirmative. The universal negative yields the following: (1) There is no individual belonging to the class S and possessing the attributes connoted by P ; (2) There is no individual common to the two classes S and P ; (3) The attributes 187 connoted by S and P respectively are never found conjoined; (4) There is no individual possessing the attributes connoted by S and belonging to the class P. Similarly the particular affirmative yields: (1) There are individuals belonging to the class S and possessing the attributes connoted by P ; (2) There are individuals common to the two classes S and P ; (3) The attributes connoted by S and P respectively are sometimes found conjoined; (4) There are individuals possessing the attributes connoted by S and belonging to the class P. We see, therefore, that the question discussed in this section is independent of that which will be raised in [chapter 8]; and that for this reason, if for no other, no solution of the general problem raised in the present chapter can afford a complete solution of the problem of the import of categorical propositions.
136. The Reading of Propositions in Comprehension.—If, in taking the intensional standpoint, we consider comprehension instead of connotation, our problem is to determine what relation is implied in any proposition between the comprehension of the subject and the comprehension of the predicate. This question being asked with reference to the universal affirmative proposition All S is P, the solution clearly is that the comprehension of S includes the comprehension of P. The interpretation in comprehension is thus precisely the reverse of that in denotation (the denotation of S is included in the denotation of P); and we might be led to think that, taking the different propositional forms, we should have a scheme in comprehension, analogous throughout to that in denotation. But this is not the case, for the simple reason that in our ordinary statements we do not distributively quantify comprehension in the way in which we do denotation; in other words, comprehension is always taken in its totality. Thus, reading an I proposition in denotation we have—the classes S and P partly coincide ; and corresponding to this we should have—the comprehensions of S and P partly coincide. But this is clearly not what we express by Some S is P ; for the partial coincidence of the comprehensions of S and P is quite compatible with No S is P, that is to say, the classes S and P may be mutually exclusive, and yet some attributes may be common to the whole of S and 188 also to the whole of P ; for example, No Pembroke undergraduates are also Trinity undergraduates. Again, given an E proposition, we have in denotation—the classes S and P have no part in common ; but for the reason just given, it does not follow that the comprehension of S and the comprehension of P have nothing in common.
It is indeed necessary to obvert I and E in order to obtain a correct reading in comprehension. We then have the following scheme, in which the relation of contradiction between A and O and between E and I is made clearly manifest:
All S is P, The comprehension of S includes the comprehension of P ;
No S is P, The comprehension of S includes the comprehension of not-P;
Some S is P, The comprehension of S does not include the comprehension of not-P;
Some S is not P, The comprehension of S does not include the comprehension of P.