Adopting this interpretation, we may come to the conclusion that the postulate is obscurely expressed, but we can have no hesitation in admitting its validity. It is obviously of importance to the logician to clear up ambiguities and ellipses of language. For this reason it is, amongst other things, desirable that we should avoid condensed and elliptical modes of expression. But whether Hamilton’s postulate, as thus interpreted, supports the doctrine of the quantification of the predicate is another question. This point will be considered in the next two sections.
142. Advantages claimed for the Quantification of the Predicate.—Hamilton maintains that “in thought the predicate is always quantified,” and hence he makes it follow immediately from the postulate discussed in the preceding section that “in logic the quantity of the predicate must be expressed, on demand, in language.” “The quantity of the predicate,” says Dr Baynes in the authorised exposition of Hamilton’s doctrine contained in his New Analytic of Logical Forms, “is not expressed in common language because common language is elliptical. Whatever is not really necessary to the clear comprehension of what is contained in thought, is usually elided in 197 expression. But we must distinguish between the ends which are sought by common language and logic respectively. Whilst the former seeks to exhibit with clearness the matter of thought, the latter seeks to exhibit with exactness the form of thought. Therefore in logic the predicate must always be quantified.” It is further maintained that the quantification of the predicate is necessary for intelligible predication. “Predication is nothing more or less than the expression of the relation of quantity in which a notion stands to an individual, or two notions to each other. If this relation were indeterminate—if we were uncertain whether it was of part, or whole, or none—there could be no predication.”
Amongst the practical advantages said to result from quantifying the predicate are the reduction of all species of the conversion of propositions to one, namely, simple conversion; and the simplification of the laws of syllogism. As regards the first of these points, it may be observed that if the doctrine of the quantification of the predicate is adopted, the distinction between subject and predicate resolves itself into a difference in order of statement alone. Each propositional form can without any alteration in meaning be read either forwards or backwards, and every proposition may, therefore, rightly be said to be simply convertible.
It is further argued that the new propositional forms resulting from the quantification of the predicate are required in order to express relations that cannot otherwise be so simply expressed. Thus, U alone serves to express the fact that two classes are co-extensive; and even ω is said to be needed in logical divisions, since if we divide (say) Europeans into Englishmen, Frenchmen, &c., this requires us to think that some Europeans are not some Europeans (e.g., Englishmen are not Frenchmen).
143. Objections urged against the Quantification of the Predicate.—Those who reject Hamilton’s doctrine of the quantification of the predicate deny at the outset the fundamental premiss upon which it is based, namely, that the predicate of a proposition is always thought of as a determinate quantity. They go further and deny that it is as a rule thought of as a 198 quantity, that is, as an aggregate of objects, at all. We have already in section [135] indicated grounds for the view that, while in the great majority of instances the subject of a proposition is in ordinary thought naturally interpreted in denotation, the predicate is naturally interpreted in connotation. This psychological argument is valid against Hamilton, inasmuch as he really bases his doctrine upon a psychological consideration; and it seems unanswerable.
Mill (in his Examination of Hamilton, pp. 495-7) puts the point as follows: “I repeat the appeal which I have already made to every reader’s consciousness: Does he, when he judges that all oxen ruminate, advert even in the minutest degree to the question, whether there is anything else which ruminates? Is this consideration at all in his thoughts, any more than any other consideration foreign to the immediate subject? One person may know that there are other ruminating animals, another may think that there are none, a third may be without any opinion on the subject: but if they all know what is meant by ruminating, they all, when they judge that every ox ruminates, mean exactly the same thing. The mental process they go through, so far as that one judgment is concerned, is precisely identical; though some of them may go on further, and add other judgments to it. The fact, that the proposition ‘Every A is B’ only means ‘Every A is some B,’ so far from being always present in thought, is not at first seized without some difficulty by the tyro in logic. It requires a certain effort of thought to perceive that when we say, ‘All A’s are B’s,’ we only identify A with a portion of the class B. When the learner is first told that the proposition ‘All A’s are B’s’ can only be converted in the form ‘Some B’s are A’s,’ I apprehend that this strikes him as a new idea; and that the truth of the statement is not quite obvious to him, until verified by a particular example in which he already knows that the simple converse would be false, such as, ‘All men are animals, therefore, all animals are men.’ So far is it from being true that the proposition ‘All A’s are B’s’ is spontaneously quantified in thought as ‘All A is some B.’”
A word may be added in reply to the argument that if the 199 quantity of the predicate were indeterminate—if we were uncertain whether the reference was to the whole or part or none—there could be no predication. This is perfectly true so long as we are left with all three of these alternatives; but we may have predication which involves the elimination of only one of them, so that there is still indeterminateness as regards the other two. To argue that unless we are definitely limited to one of the three we are left with all of them is practically to confuse contradictory with contrary opposition.
A further objection raised to the doctrine of the quantification of the predicate is that some of the quantified forms are composite not simple predications. Thus All S is all P is a condensed mode of expression, which may be analysed into the two propositions All S is P and All P is S. Similarly, if we interpret some as exclusive of all, a point to which we shall presently return, All S is some P is an exponible proposition resolvable into All S is P and Some P is not S. As a rule, however, the use of exponible forms tends to make the detection of fallacy the more difficult, and this general consideration applies with undoubted force to the particular case of the quantification of the predicate. The bearing of the quantification doctrine upon the syllogism will be briefly touched upon subsequently, and it will be found that the problem of discriminating between valid and invalid moods is rendered more complex and difficult. It may indeed be doubted whether any logical problem, with the one exception of conversion, is really simplified by the introduction of quantified predicates.
Even apart from the above objections, the Hamiltonian doctrine of quantification is sufficiently condemned by its want of internal consistency. Its unphilosophical character in this respect will be shewn in the following sections.
144. The meaning to be attached to the word “some” in the eight propositional forms recognised by Sir William Hamilton.—Professor Baynes, in his authorised exposition of Sir William Hamilton’s doctrine, would at the outset lead us to suppose that we have no longer to do with the indeterminate some of the Aristotelian Logic, but that this word is now to be used in the more definite sense of some, but not all. He argues, as we 200 have [seen], that intelligible predication requires an absolutely determinate relation in respect of quantity between subject and predicate, and that this ought to be clearly expressed in language. Thus, “if the objects comprised under the subject be some part, but not the whole, of those comprised under the predicate, we write All X is some P, and similarly with other forms.”