Studies and Exercises in Formal Logic

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.

It is to be added that a statement which does not admit of being translated into any one of the simple forms included in a given scheme may still be capable of being expressed by a conjunctive or disjunctive combination of such simple forms. Thus, if the statement Some S is P is made with an emphasis on some, implying not all, then the statement cannot be expressed in any one of the forms of the traditional schedule of propositions, but it is equivalent to Some S is P and some S is not P.

(b) In the reduction of a statement to a form in which it belongs to a given schedule there may be involved what must be admitted to be inference. As, for instance, if statements are given in the ordinary predicative form and have to be expressed in an equational scheme.

It may perhaps be urged that this is legitimate, simply on the ground that one of the postulates of logic is that we be allowed to substitute for any given form of words the technical form (and in an equational system this will be an equation) which is equivalent to it. Have we not, however, in reality a vicious circle if a process which involves inference is to be regarded as a postulate of logic?

The difficulty here raised is a serious one only if we suppose ourselves rigidly limited in logic to a single scheme of formulation; and the solution is to be found in our not confining ourselves to any one scheme, but in our recognising several and investigating the logical relations between them. We can then refuse to regard any substitution of one set of words for another as pre-logical except in so far as it consists of a merely verbal transformation: and our postulate will merely be that we are free to make verbal changes as we please; it will not by itself authorise any change of an inferential character. For a change of this kind, appeal must be made to logical principles.

These problems are inter-related and do not admit of being discussed in complete isolation. It is clear, for instance, that the drawing up of a schedule of propositions needs to be supplemented by the exact interpretation of the different forms which it is proposed to recognise; and both in the drawing up of the schedule and in the interpretation we shall be guided and controlled by a consideration of fundamental distinctions between judgments.

The problems are, however, in themselves distinct; and some misunderstanding may be avoided if we can make it clear what is the actual problem that we are discussing at any given point.

In particular, it is important to recognise that in the formulation and interpretation of propositions there is an arbitrary and selective element which is absent from the more fundamental problem. Systems of formulation and interpretation, therefore, if only they are intelligible and self-consistent, can hardly be condemned as radically wrong, though they may be rejected as inconvenient or unsuitable. When, however, we are dealing with the fundamental import of judgments, the questions raised do become questions of absolute right or wrong.

It should be added that in the present treatise, since it is concerned with logic in its more formal aspects, questions of interpretation and formulation occupy a position of greater relative importance than they would in a treatment of the science more fully developed on the philosophical side.

This claim to be true implies an objective reference. For a merely subjective state is not, as such, either true or false; it is simply an occurrence. Thus, the distinction between truth and falsity is inapplicable to an emotion or a volition. An emotion may be pleasurable or painful; it may be strong or weak; it may or may not impel to action; but we cannot describe it as true or false.

And the same applies to a judgment regarded as no more than a subjective connexion of ideas. The claim to truth necessarily involves more than this, namely, a reference to something external to the psychical occurrence involved in the formation of the judgment. Every judgment implies, therefore, on the part of the judging mind, the recognition of an objective system of reality of some sort. The validity that is claimed for judgment is an objective validity.

The word “objective” is always a dangerous word to use, and some further explanation may be given of the meaning to be attached to it here. When we say that a judgment refers to an objective system, we mean a system that subsists independently of the act of judgment itself, and that is not dependent on the passing fancy of the person who forms the judgment. An objective system of reality in this sense may, however, include subjective states, that is, states of consciousness. A body of psychological doctrine consists of judgments relating to states of mind. But such judgments have an external reference (that is, external to the judgments themselves) just as much as a body of judgments relating to material phenomena. Indeed the doctrine of judgment here laid down is not inconsistent with the theory of subjective idealism that resolves all phenomena into states of consciousness.

Even when a judgment relates to purely fictitious objects there is still an external reference,—in this case, to the world of convention.

50. The Universality of Judgments.—The fundamental characteristic then of judgments is their objective reference, their claim to objective validity. It follows that all judgments claim universality, that is to say, they claim to be acknowledged as true not for a given person only, or for a limited number of persons, but for everyone; and again, not for a given time only, or for a limited time, but for all time. In other words, the import of a judgment is not merely to express some connexion of ideas in my own mind; but to express something that claims to be true. And truth is not relative to the individual, nor is it when fully set forth limited by considerations of time.

We shall have subsequently to deal with the ordinary distinction between universal and particular propositions; but it will be clear that the claim to universality which we are now considering is one that must be made on behalf of so-called particular, as well as of so-called universal, propositions. The judgment that some men are six feet in height claims universal acceptance just as much as the judgment that all men are mortal.

In any discussion of the modality of judgments, other senses in which the term “necessary” may be applied to judgments have to be considered. In here affirming necessity as a characteristic of all judgments, we are merely declaring over again in another aspect their objective character. The merely subjective sequence of ideas in our minds is more or less under our own control. At any rate we can at will bring given ideas together in our mind. But a judgment is more than a relation between ideas. It claims to be true of some system of reality; and hence it is not so much determined by us, as for us by the knowledge which we have come to possess or think we have come to possess about that system of reality.

EXERCISE.

CHAPTER II.

KINDS OF JUDGMENTS AND PROPOSITIONS.

53. The Classification of Judgments.—It is customary for logicians to offer a classification of judgments or propositions. There is, however, so much variation in the objects they have in view in drawing up their classifications, that very often their results are not really comparable.

(1) Our object in classifying propositions may, in the first place, be to produce a working scheme for the formulation of judgments. An illustration of this is afforded by the traditional scheme of propositions (All S is P, No S is P, etc.), or by the Hamiltonian scheme based upon the quantification of the predicate. A classification of this kind is essentially formal. The different propositional forms that are recognised must receive clearly defined interpretations; and the resulting scheme, if it is worth anything at all, will be orderly and compact. On the other hand, it is not likely to be comprehensive or exhaustive; for many natural modes of judgment will not find a place in it, at any rate until they have been expressed in a modified, though as nearly as possible equivalent, form.

There are many ways of formulating judgments, each of which has its special merits and is from some particular point of view specially appropriate. We must, however, give up the idea that any one of these ways can hold the field as a fundamental and essentially suitable classification of judgments looked at from the psychological point of view.

In a classification of this kind the dividing lines are not so clear and sharply defined as in a scheme framed for the logical formulation of judgments. The different types, moreover, do not stand out in marked distinction from one another, and it is difficult to arrange the different classes in due subordination, and with complete avoidance of cross divisions. The underlying plan is indeed apt to be obscured by details, so that the whole discussion tends to become somewhat cumbrous.

(3) A classification of propositions of still another kind is given by Mill in the later part of his chapter on the Import of Propositions. The conclusion at which he arrives is that every proposition affirms, or denies, either simple existence, or else some sequence, coexistence, causation, or resemblance. This classification is certainly not a formal one; it is not a scheme for the logical formulation of judgments. Nor, on the other hand, can it be regarded as a psychological classification of types of judgment, designed to illustrate the nature and growth of thought. Mill’s point of view is objective and material. In one place he describes his scheme as a classification of matters of fact, of all things that can be believed; and the main use that he subsequently makes of it is in connexion with the enquiry as to the methods of proof that are appropriate according to the nature of the matter of fact that is asserted.

In the pages that follow various schemes for formulating judgments will be considered. For reasons already stated, however, no scheme of this kind can be regarded as constituting an exhaustive classification of judgments. The traditional scheme, for example, is ludicrously unsatisfactory and incomplete if put forward as affording such a classification.

Again, such a classification as Mill’s involves material considerations that are outside the scope of this treatise.

Without, however, professing to give any complete scheme of classification, we shall endeavour to touch upon the most fundamental differences that may exist between judgments.

54. Kant’s Classification of Judgments.—Kant classified judgments according to four different principles (Quantity, Quality, Relation, and Modality) each yielding three subdivisions, as follows:

This arrangement is open to criticism from several points of view; and its symmetry, although attractive, is not really defensible. At the same time it has the great merit of making prominent what really are the fundamental distinctions between judgments.

55. Simple Judgments and Compound Judgments.—Under the head of relation, Kant gave the well-known threefold division of judgments into categorical, where the affirmation or denial is absolute (S is P); hypothetical (or conditional), where the affirmation or denial is made under a condition (If A is B then S is P); and disjunctive, where the affirmation or denial is made with an alternative (Either S is P or Q is R).

These three kinds of judgment cannot, however, properly be co-ordinated as on an equality with one another in a threefold division. For the categorical judgment appears as an element in both the others, and hence the distinction between the categorical, on the one hand, and the hypothetical and the disjunctive, on the other, appears to be on a different level from that between the two latter. Moreover, the hypothetical and the disjunctive do not exhaust the modes in which categorical judgments may be combined so as to form further judgments. It is, therefore, better not to start from the above threefold division, but from a twofold, namely, into simple and compound.

The main point at issue is whether distinctions of modality are subjective or objective. In attempting to decide this question it will be convenient to deal separately with simple judgments and compound judgments.

Another suggested ground of distinction is that between immediate knowledge and knowledge that is based on inference, the former being expressed by the assertoric judgment, and the latter by the apodeictic.

There is no doubt that we often say a thing is so and so when this is a matter of direct perception, while we say it must be so and so when we cannot otherwise account for certain perceived facts. Thus, if I have been out in the rain, I say it has rained ; if, without having observed any rain fall, I notice that the roads and roofs are wet, I say it must have rained.

It is obvious, however, that this distinction is quite inconsistent with the ascription of any superior certainty to the apodeictic judgment. For that which we know mediately must always be based on that which we know immediately; and, since in the process of inference error may be committed, it follows that that which we know mediately must have inferior certainty to that of which we have immediate knowledge. Accordingly in ordinary discourse the statement that anything must be so and so would generally be understood as expressing a certain degree of doubt.

We cannot then justify the recognition of the apodeictic judgment as expressing a higher degree of certainty than the merely assertoric.

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.