perfectly well 'predicate of the extension of one term the extension of another.' What we have already said may perhaps be sufficient to establish the point. But it may be useful, further, to draw attention to the ambiguity of the verb 'to predicate.' It is possible to use this verb in such a way as to presuppose the predicative view of import, and so exclude ab initio the possibility of adopting an extensive reading. In the proposition 'All S's are P's' the predicate is not predicated of the subject as its attribute. P cannot be a differentia or a proprium of S. But it may still be a predicable, for it may stand to S as genus to species, or as a class-term in one division may stand to a class-term in another division. What is predicated of the extension-of-S is that it is related in the way of at least partial coincidence to the extension of a certain correlate P. Thus P is predicated of S, not as its attribute, but as its extensional correlate. Our conclusion, then, is that we are justified in retaining the fourfold scheme in its extensional form, and that we may posit the following equivalences :* All S's are P's Each of all S's is a P. Each of some S's is a P. Each of all S's is-not any P. Each of some S's is-not any P. The Quantification of the Predicate. Sir William Hamilton proposed to develop the fourfold scheme of Categorical Propositions by adding the quantity-mark 'all' or 'some' to the predicate of A and I, and 'any' or 'some' to that of E and O, thus obtaining an eight-fold scheme. The interpretation and discussion of this scheme still occupies a considerable place in modern treatises on Logic. The reader will find excellent critical appreciations in Professor Welton's 'Manual of Logic' (vol. i., bk. ii., ch. ii., pp. 200 ff.) and Mr. Joseph's 'Introduction to Logic' (ch. ix., pp. 198, f.). The ambiguities of the * For the relation of this 'coincidence' or 'identity' form of extensional import to the class-inclusion view, vide p. 239. words 'All' and 'Some' and the confusion between the distributive and collective uses of these marks supply ample opportunity for criticism and reconstruction. Into the complexities of this discussion we do not propose to enter. We content ourselves with connecting the doctrine of a quantified predicate with the fourfold scheme as we have adopted it, and considering the logical significance and importance of the eightfold scheme from this single point of view. In the first place, the conception of a quantified predicate appears to us to be perfectly reasonable. The quantified predicate is indeed already present in the fourfold scheme under the guise of the distributed and undistributed predicate-term. Hence, assuming as we do the undistributed meaning of the word 'Some,' and reading both subject and predicate terms in extension, we hold that there is obviously no difference between the A, I, E, and O of the quantified scheme and the corresponding propositions of the fourfold scheme. Our sole criticism of the Hamiltonian scheme when interpreted in this way is that the four additional propositions, U, Y, η, ω are superfluous. The U proposition, as we have already seen, is equivalent to the compound proposition 'All Sis Pand All P is S,' of which the elements are A propositions. Again, the Y proposition is equivalent to the A proposition 'All P is S.' The η proposition may be disposed of in a similar way. We may diagrammatically represent the η proposition - No S is some P'as follows: But if we exchange the S and P in these diagrams, we obtain the diagrams of the O proposition : 'No S is some P' is, in fact, identical with 'Some P is no S.' As for the w proposition, it cannot be denied whatever we intend to state. Even if we intend to state that the extension of the class S exactly coincides with that of P, still we evidently cannot deny the statement that a certain number of S's are not coincident with a certain other number of P's. Thus, of the four propositions U, Y, η, ω, U is reducible to two A propositions, Y is reducible to an A proposition, ω is truistic, and therefore useless.* CHAPTER XVIII V. (iii.) THE REDUCTION OF CATEGORICAL PROPOSITIONS TO STRICT LOGICAL FORM. IN reducing a proposition to strict logical form, our first care must be to interpret the given statement as an 'extensive' proposition. Thus, if 'S is P' is naturally read on the predicative view, to the effect that the objects indicated by S possess the attribute P, it must be reinterpreted, so as to read as follows: 'The objects indicated by S are objects which possess the attribute P.' In the case of abstract propositions (statements, that is, with an abstract term as subject) the object indicated by the subject-term will be abstract, and the correct interpretation will be somewhat as follows: 'The abstract object indicated by the term S is identified with an abstract object indicated by the term P.' Thus, 'Mercy is twice blest' may be interpreted as 'Mercy is a twice-blest virtue.' It will then rank as a universal affirmative with distributed subject and undistributed predicate. The singular proposition need cause no difficulty from the point of view we are here considering. For, as we have seen, it is but the limiting form of the universal proposition. 'This Sis P' means 'All objects indicated by the term "this S" are objects indicated by the term P.' As a matter of fact, this S,' like a proper name, restricts the extension to one object, so that, while the proposition is singular, its subject-term is distributed. The singular proposition ranks, therefore, as a universal. It could not rank as a particular proposition for another reason. For 'some' means one at least,' and 'one at least' is not the same as 'one.' Hence 'This Sis P' could not be brought under the form 'Some S's are P's,' for ، * In the one case in which it might seem possible not to accept w (namely, where S and P extensionally coincide, and both refer to one and the same individual) the use of the proposition would be inappropriate. If we wished to state such coincidence we should employ a singular proposition. the latter form implies that one S at least is P, and this rendering goes beyond the meaning of the singular proposition. Once a proposition is understood extensively, the main rules for reduction to strict logical form may be briefly formulated. Putting a categorical proposition into strict logical form means, we may say, expressing it in one of the four typical forms, A, E, I, O. This involves : 1. Finding which is the true logical subject and which the true logical predicate. 2. Giving the subject its correct quantity-mark, either 'All ’ or 'Some.' 3. Giving the proposition its correct quality-mark, either 'is' or 'is-not.' The second of these requisites, however, is subject to an important modification. We have seen that, for the purposes of a Formal treatment, the Singular Proposition ranks as a universal. Consequently we have a 'singular' form of the A proposition, the form typically represented by 'This S is a P.' We have, then, as the two recognized forms of the A proposition 1. 'All the S's are P's,' where 'All' is understood distributively. 2. 'S (singular) is a P.' A difficulty frequently arises from the fact that propositions collective in meaning are presented in the ordinary distributive form. Thus, a sentence given to us in the form 'All the S's are P's' may be incorrectly expressed. A proposition, we know, is distributive if, on putting each of' in the place of 'all,' we find that the sense remains unaffected. The distributive use of 'all' being accepted as its correct Formal use, the word 'all' should be retained, in the logically stated proposition, whenever this substitution does not alter the sense. But if the substitution changes the sense, then 'all' is used collectively, and therefore must not be retained in the proposition when this is stated in logical form. In its place we must use a collective expression with a singular import,* so that the elaborated proposition will take the form 'S (singular) is a P'instead of 'All the S's are P's'; the essential defect of the latter expression in such a case being that it has * Collective terms, like 'family' or 'regiment,' which refer to a collection of objects qua aggregate, may be either singular or general. Such terms as 'This regiment' or 'The Light Brigade' are singular collectives. 'Regiment' is a general collective. Collective terms, whether singular or general, are always used collectively with regard to the individuals of the group or kind of group specified. But general collectives are used disjunctively with regard to the various kinds or classes which constitute their denotation. 'Family' indicates the members collectively, but it denotes the various kinds of family disjunctively. A family is large or small, rich or poor. The term 'family' may correctly be predicated of each of these types taken apart from the rest. distributive form but collective meaning. This distinction applies also to some propositions which are apparently particular. 'Some' may be used in a collective sense, as meaning, for instance, 'a handful of'; and, when so used, it should not appear as mark of quantity in the proposition logically expressed, but should be superseded by a collective expression. Thus, 'All these weeds choke the flower-bed' should be transformed into 'This mass of weeds is a mass that chokes the flower-bed.' So also 'Some soothing words appeased him' is only apparently an I proposition, for the 'some' is here used collectively. Reduced to proper logical form, the proposition would run : 'A string of soothing words is a thing that appeased him.' Further difficulty is caused by propositions which refer to a single individual, but do not specify that individual, and therefore are not singular in the sense required to justify the use of the singular form. Examples of such propositions are : ' A friend of mine has gone abroad.' 'An earthquake had occurred in Jamaica.' A legitimate method of reducing sentences of this type would be to utilize x, the usual symbol for the unknown quantity, as follows : 'The friend whom I call X is a person who has gone abroad.' In arguments this substitution would perhaps be convenient. The third of the requisites of strict reduction also needs some words of comment. The quality-mark may take any one of several forms. If affirmative, it may be 'am,' 'art,' 'is,' or 'are.' If negative, 'am-not,' 'art-not,' 'is-not,' or 'are-not.' Thus, 'I am a man,' 'Thou art a woman,' 'We are human beings,' may be regarded as reduced propositions. They are all universal affirmatives. It is important that the quality-mark should not be confused with the tense-mark. The quality-mark is the copula-mark, and has a strictly logical significance. Distinctions between present, past, and future belong to the predicate. Thus 'were' and 'were not,' 'will be' and 'will not be' are not strictly permissible as copula-marks. Example.The Drake was in harbour. She is a splendid vessel, and will be the Admiral's flagship.' |