On the other hand, the so-called problematic judgment, interpreted as expressing mere uncertainty,[85] cannot be regarded as in itself expressing a judgment at all. It may imply a judgment in regard to the validity of arguments brought forward in support or in disproof of a given thesis; and it implies also a judgment as to the state of mind of the person who is in a state of uncertainty; but it is in itself a mere suspension of judgment.
[85] The problematic judgment as interpreted in the following section does more than express mere uncertainty. The form of proposition S may be P is no doubt ambiguous.
59. Objective Distinctions of Modality.—We have next to consider whether, having regard not to the judgment as a 88 subjective product, but to the objective fact expressed in a judgment, any valid distinction can be drawn between the necessary, the actual (or contingent), and the possible ; and our answer must be in the affirmative, provided that we are prepared to admit the conception of the operation of law.
Thus the judgment Planets move in elliptic orbits is in this sense a judgment of necessity. It expresses something which we regard as the manifestation of a law, and it has an indefinitely wide application. For we believe it to hold good not only of the planets with which we are acquainted, but also of other planets (if such there be) which have not yet been discovered.
Now take the judgment, All the kings who ruled in France in the eighteenth century were named Louis. This is a statement of fact, but clearly is not the expression of any law. The proposition relates to a limited number of individuals who happened to have the same name given to them; but we recognise that their names might have been different, and that their being kings of France was not dependent on their possessing the name in question. This then we may call a judgment of actuality.
We have a judgment of possibility when we make such a statement as that a seedling rose may be produced different in colour from any roses with which we are at present acquainted, meaning that there is nothing in the inherent nature of roses (or in the laws regulating the production of roses) to render this impossible.
We have then a judgment of necessity (an apodeictic judgment) when the intention is to give expression to some law relating to the class of objects denoted by the subject-term; we have a judgment of actuality (an assertoric judgment) when the intention is to state a fact, as distinguished from the affirmation or denial of a law; we have a judgment of possibility (a problematic judgment) when the intention is to deny the operation of any law rendering some complex of properties impossible.[86]
[86] The case of a proposition which may be regarded as expressing a particular instance of the operation of a law needs to be specially considered. Granting, for instance, that the proposition Every triangle has its angles equal to two right angles is apodeictic, are we to describe the proposition This triangle has its angles equal to two right angles as apodeictic or as assertoric? The right answer seems to be that, as thus barely stated, the proposition may be merely assertoric; for it may do no more than express a fact that has been ascertained by measurement. If, however, the proposition is interpreted as meaning This figure, being a triangle, has its angles equal to two right angles, then it is apodeictic.
89 I shall not attempt to give here any adequate philosophic analysis of the conception of objective necessity. It must suffice to say that we all have the conception of the operation of law, and that for our present purpose the validity of this conception is assumed.
With regard to this treatment of modality the objection may perhaps be raised that, whatever their value in themselves, the distinctions involved are not of a kind with which formal logic has any concern. It is true that, in a sense, judgments of necessity are the peculiar concern of inductive, as distinguished from formal, logic. The main function of inductive logic is indeed to determine how apodeictic judgments (as above defined) are to be established on the basis of individual observations; for what we mean by induction is the process of passing from particulars to the laws by which they are governed. Granting this, however, there are also many problems, with which logic in its more formal aspects has to deal, in the solution of which some recognition of the distinctions under discussion is desirable, if not essential.