But it will be said that the distinctions cannot be applied formally: that, for example, given a proposition in the bare form S is P, or given an ordinary universal affirmative proposition All S is P, it cannot be determined, apart from the matter of the proposition, whether it is apodeictic (in the sense in which that term is used in this section) or merely assertoric. This is true if we are limited to the traditional schedule of propositions. But it is to be remembered that the formulation and the interpretation of propositions are within certain limits under our own control, and that it is within our power so to interpret propositional forms for logical purposes as to bring out distinctions that are not made clear in ordinary discourse or in the traditional logic. Thus, the form S as such is P might be used for giving formal expression to the apodeictic judgment, S is P being interpreted as merely assertoric.
90 Another solution, however, and one that may be made to yield a symmetrical scheme, is to utilise the conditional (as distinguished from the true hypothetical,[87]) proposition, and to differentiate it from the categorical, by interpreting it as modal,[88] while the categorical remains merely assertoric.
[88] Here and elsewhere in speaking of a proposition as modal (in contradistinction to assertoric) we mean a proposition that is either apodeictic or problematic.
Thus, we should have,—
If anything is S it is P,—apodeictic;
All S is P,—assertoric;
If anything is S it may be P,—problematic.[89]
[89] It will be observed that in this scheme (leaving on one side the question of existential import) the categorical proposition All S is P is inferable from the conditional If anything is S it is P, but not vice versâ.
It is of course not pretended that the differentiation here proposed is adopted in the ordinary use of the propositional forms in question; we shall, for example, have presently to point out that in the customary usage of categoricals the universal affirmative has frequently an apodeictic force. We shall return to a discussion of the suggested scheme [later on].
60. Modality in relation to Compound Judgments.—We may now consider the application of distinctions of modality to compound judgments, that is, to judgments which express a relation in which simple judgments stand one to another. It is one thing to say that as a matter of fact two judgments are not both true; it is another thing to say that two judgments are so related to one another that they cannot both be true. We may describe the one statement as assertoric, the other as apodeictic. An apodeictic judgment thus conceived expresses a relation of ground and consequence; an obligation, therefore, to affirm the truth of a certain proposition when the truth of a certain other proposition or combination of propositions is admitted. The obligation may sometimes depend upon the assistance of certain other propositions which are left unexpressed.[90]
[90] In an apodeictic compound judgment, the necessity may (at any rate in certain cases) be described as subjective. This is so in the case of a formal hypothetical; as, for example, in the proposition If all S is P then all not-P is not-S, or in the proposition If all S is M and all M is P then all S is P.
91 In section [55] a threefold classification of compound judgments was given; the distinction now under consideration points, however, to a more fundamental twofold classification. From this point of view a scheme may be suggested in which conjunctives (P and Q) and so-called disjunctives (P or Q) would be regarded as assertoric, while hypotheticals (If P then Q) would be regarded as modal. The enquiry as to how far this is in accordance with the ordinary usage of the propositional forms in question must be deferred. It may, however, be desirable to point out at once that, if this scheme is adopted, certain ordinarily recognised logical relations are not valid. For the hypothetical If P then Q is ordinarily regarded as equivalent to the disjunctive Either not-P or Q, and this as equivalent to the denial of the conjunctive Both P and not-Q. If, however, the conjunctive (and, therefore, its denial) and also the disjunctive are merely assertoric, while the hypothetical is apodeictic, it is clear that this equivalence no longer holds good. The disjunctive can indeed still be inferred from the hypothetical, but not the hypothetical from the disjunctive. This result will be considered further at a [later] stage.