Under this head are to be included such problems as the following: Do all judgments contain a reference to reality? In what sense, if any, can all judgments claim to possess universality or necessity? What is the nature of significant denial? Are distinctions of modality subjective or objective?
(2) In the interpretation of propositional forms we have an enquiry of a very different character, an enquiry which relates distinctively to propositions, and not to judgments considered apart from their expression. The problem is indeed to determine what is the precise judgment that a given proposition shall be understood to express; and, in consequence of the uncertainty and ambiguity of ordinary language, the solution of the problem includes an optional or selective element.
71 As a simple illustration of the kind of problem that we here have in view, we may note that in the traditional scheme of propositions, All S is P, No S is P, Some S is P, Some S is not P, the signs of quantity have to be interpreted. The existential and modal import of these propositions is also partly a question of interpretation.
In connexion with the interpretation of propositions, the distinction between meaning and implication has to be considered. What we do in interpreting propositions is to assign to them a meaning; and when the meaning has once been fixed, the implications are determined in accordance with logical principles.
The dividing line between meaning and implication is not in practice always easy to draw, and some writers seek to ignore it by including within the scope of meaning all the implications of a proposition. But this is a fatal error. The assignment of meaning is within certain limits arbitrary and selective. But if element a necessarily involves element b, then a having been assigned as part of the meaning of a given propositional form, it is no question of meaning as to whether the form in question does or does not imply b, and there is nothing arbitrary or selective in the solution of this question.
Sometimes the elements a and b mutually involve one another. It may then be a question of interpretation whether a shall be included in meaning, b thus becoming an implication, or whether b shall be included in meaning, a becoming an implication.
A failure to recognise what is really the point at issue in a case like this has sometimes caused discussions to take a wrong turn. Thus the question is raised whether the import of the proposition All S is P is that the class S is included in the class P, or that the set of attributes S is invariably accompanied by the set of attributes P ; and these are regarded as antagonistic theories. If the implications of a proposition are regarded as part of its import, then the proposition may be said to import both these things. But if by the import of a proposition we intend to signify its meaning only, then we may adopt an interpretation that will make either of them (but not both) part of its import, or our interpretation may be such 72 that the proposition imports neither of them. The question here raised is dealt with in more detail [later on].
(3) A third problem, distinct from both those described above, arises in connexion with the expression of judgments in propositional form.
In ordinary discourse we meet with an infinite variety of forms of statement. To recognise and deal separately with all these forms in our treatment of logical problems would, however, be impracticable. We have, therefore, in some at any rate of our discussions, to limit ourselves to a certain number of selected forms; and in such discussions we have to assume that the judgments with which we are dealing are at the outset expressed in one or other or a combination of these selected forms.
This reduction of a statement to some canonical form has been called by Mr Johnson its formulation.