65. Universal Propositions.—In discussing the import of the universal proposition All S is P, attention must first be called to a certain ambiguity resulting from the fact that the word all may be used either distributively or collectively. In the proposition, All the angles of a triangle are less than two right angles, it is used distributively, the predicate applying to each and every angle of a triangle taken separately. In the proposition. All the angles of a triangle are equal to two right angles, it is used collectively, the predicate applying to all the 98 angles taken together, and not to each separately. This ambiguity attaches to the symbolic form All S is P, but not to the form All S’s are P’s. Ambiguity may also be avoided by using every instead of all, as the sign of quantity. In any case the ambiguity is not of a dangerous character, and it may be assumed that all is to be interpreted distributively, unless by the context or in some other way an indication is given to the contrary.
A more important distinction between propositions expressed in the form All S is P remains to be considered. For such propositions may be merely assertoric or they may be apodeictic, in the sense given to these terms in section [59].
It will be convenient here to commence with a threefold distinction.
(1) The proposition All S is P may, in the first place, make a predication of a limited number of particular objects which admit of being enumerated: e.g., All the books on that shelf are novels, All my sons are in the army, All the men in this year’s eleven were at public schools. A proposition of this kind may be called distinctively an enumerative universal. It is clear that such a proposition cannot claim to be apodeictic.
(2) The proposition All S is P may, in the second place, express what is usually described as an empirical law or uniformity: e.g., All lions are tawny, All scarlet flowers are without sweet scent, All violets are white or yellow or have a tinge of blue in them. Many propositions relating to the use of drugs, to the succession of certain kinds of weather to certain appearances of sky, and so on, fall into this class. A proposition of this kind expresses a uniformity which has been found to hold good within the range of our experience, but which we should hesitate to extend much beyond that range either in space or in time. The predication which it makes is not limited to a definite number of objects which can be enumerated, but at the same time it cannot be regarded as expressing a necessary relation between subject and predicate. Such a proposition is, therefore, assertoric, not apodeictic.
(3) The proposition All S is P may, in the third place, express a law in the strict sense, that is to say, a uniformity 99 that we believe to hold good universally and unconditionally: e.g., All equilateral triangles are equiangular, All bodies have weight, All arsenic is poisonous. A proposition of this kind is to be regarded as expressing a necessary relation between subject and predicate, and it is, therefore, apodeictic.
Propositions falling under the first two of the above categories may be described as empirically universal, and those falling under the third as unconditionally universal.[97]
[97] I have borrowed these terms from Sigwart, Logic, § 27; but I cannot be sure that my usage of them corresponds exactly with his. In section 27 he appears to include under empirically universal judgments only such judgments as belong to the first of the three classes distinguished from one another above. At the same time, his description of the unconditionally universal judgment applies to the third class only: such a judgment, he says, expresses a necessary connexion between the predicate P and the subject S ; it means, If anything is S it must also be P. And it seems clear from his subsequent treatment (in § 96) of judgments belonging to the second class that he does not regard them as unconditionally universal.
Lotze (Logic, § 68) indicates the distinction we are discussing by the terms universal and general. But again there seems some uncertainty as to which term he would apply to judgments belonging to our second class. In the universal judgment, he says, we have merely a summation of what is found to be true in every individual instance of the subject; in the general judgment the predication is of the whole of an indefinite class, including both examined and unexamined cases. From this it would appear that the universal judgment corresponds to (1) only, while the general judgment includes both (2) and (3). Lotze, however, continues, “The universal judgment is only a collection of many singular judgments, the sum of whose subjects does as a matter of fact fill up the whole extent of the universal concept; … the universal proposition, All men are mortal, leaves it still an open question whether, strictly speaking, they might not all live for ever, and whether it is not merely a remarkable concatenation of circumstances, different in every different case, which finally results in the fact that no one remains alive. The general judgment, on the other hand, Man is mortal, asserts by its form that it lies in the character of mankind that mortality is inseparable from everyone who partakes in it.” The illustration here given seems to imply that a judgment may be regarded as universal, though it relates to a class of objects, not all of which can be enumerated.
If this distinction is regarded merely as a distinction between different ways in which judgments may be obtained (for example, by enumeration or empirical generalisation on the one hand, or by abstract reasoning or the aid of the principle of causality on the other hand), without any real difference of content, it becomes merely genetic and can hardly be retained as a 100 distinction between judgments considered in and by themselves. If we are so to retain it, it must be as a distinction between the merely assertoric and the apodeictic in the sense already explained. In order to be able to deal with it as a formal distinction, we must further be prepared to assign distinctive forms of expression to the two kinds of universal judgments respectively. Lotze appears to regard the forms All S is P and S is P as sufficiently serving this purpose. But this is hardly borne out by the current usage of these forms. All the S’s are P might serve for the enumerative universal and S as such is P for the unconditionally universal. These forms do not, however, fit into any generally recognised schedules; and our second class of universal would be left out. Another solution, which has been already indicated in section [59], would be to use the categorical form for the empirically universal judgment only, adopting the conditional form for the unconditionally universal judgment.