Studies and Exercises in Formal Logic

where N (x₁) means the number of things which are not x₁; N (x₂) means the number of things which are not x₂; &c. [J.]

CHAPTER IX.

THE CHARACTERISTICS OF INFERENCE.

376. The Nature of Logical Inference.—The question as to the nature and characteristics of inference, so far as its solution depends on the more or less arbitrary meaning that we choose to attach to the term “inference,” is a merely verbal question. The controversies to which the question has given rise do not, however, depend mainly on verbal considerations; and the fact that they partly do so has increased rather than diminished the difficulties with which the problem is beset.

It will be generally agreed that inference involves a passage of thought from a given judgment or combination of judgments to some new judgment. This alone, however, is not sufficient to constitute inference in the logical sense. The formation of new judgments by the unconscious association of ideas is a psychological process which might be brought under the above description; but it is not what we mean by logical inference.

(1) It is, in the first place, an essential characteristic of logical inference that the passage of thought should be realised as such. The connexion between the judgment or judgments from which we set out and the new judgment at which we arrive must be one of which we are, at any rate on reflection, explicitly conscious.

In logical inference, then, I do not merely pass from P to Q ; I realise that I am doing so. And I apprehend further that the truth of P being granted, the truth of Q necessarily follows. For logical inference, in short, it is required that there should be a logical relation between a premiss or premisses and a conclusion, not merely a psychological relation between antecedents and consequents in a train of thought.

This distinction between the logical and the psychological may be briefly illustrated by reference to what are known as acquired perceptions. Psychologists are, for example, agreed that our perception of distance through the sense of sight or the sense of sound is not immediate, but acquired in the course of experience. Here then we have a case in which one perception generates another; but there is no conscious passing from premisses to a conclusion, and nothing that can properly be called inference. Hence we must reject Mill’s dictum that “a great part of what seems observation is really inference” (Logic, iv. 1, § 2), so far as the dictum is based—as to a large extent it is—on the position that a great part of our perceptions are acquired, not immediate. Here, as well as in connexion with some of his other and more important logical doctrines, Mill is open to the charge of failing to distinguish between the cause of a belief and its ground or reason.

377. The Paradox of Inference.—The description of logical inference given in the preceding section leads up immediately to the fundamental difficulty which any discussion of the subject must inevitably bring to the forefront. We are in fact face to face with what has aptly been designated the “paradox of inference.” On the one hand, we are to advance to something new; the conclusion of an inference must be different from the premisses, and hence must go beyond the premisses. On the other hand, the truth of the conclusion necessarily follows from the truth of the premisses, and the conclusion must therefore in some sense be contained in the premisses.

On the one hand, by laying stress on the characteristic of novelty, we may be led to doubt whether formal inference of any description can properly be so called. For in all such inference the conclusion is implicitly contained in the premisses, and in uttering the premisses we have virtually committed ourselves to the conclusion. How then can we be said to make any advance to what is really new?

On the other hand, by laying stress on the characteristic of necessity, we may be led to doubt whether any inductive inference can properly be so called. For in such inference the falsity of the conclusion is not demonstrably inconsistent with the truth of the premisses. We may hold that if the premisses are true the conclusion will be true. But can we hold that it must be true, unless we also hold that in affirming the premisses we have virtually affirmed the conclusion too? And then we are back on the other horn of the dilemma.

This is not the place at which to discuss the difficulty from the point of view of inductive logic. We must, however, attempt a solution from the point of view of formal logic.

378. The nature of the difference that there must be between premisses and conclusion in an inference.—In order to find a solution of the difficulty, so far as formal inference is concerned, we must pursue our analysis further. We have said that the conclusion must be different from the premiss or premisses. But we have not yet asked what must be the nature of the difference or wherein it must consist; and it is on the answer to this question that everything turns.

If we consider two sentences we shall find that they may differ from one another from three distinct standpoints, representing three degrees of difference.

This (to take an example from Jevons) applies to two such sentences as Victoria is the Queen of England, Victoria is England’s Queen. It applies also to a statement expressed in a given language and the same statement translated into a second language, assuming that an absolutely literal translation is possible.

It has indeed been maintained by some writers that a difference of expression necessarily involves some difference of thought. But this at any rate appears not to be the case where one single word is substituted for another completely coincident with it both in denotation and in connotation (as thought by the speaker). Where one complex term is substituted for another (for example, England’s Queen for Queen of England) there may no doubt be involved some change in the order of thought; but this does not necessitate any change of meaning in the thought considered as a whole. Again we ought perhaps not to say that the same proposition expressed in two different languages has absolutely the same mental equivalent, since a consciousness of the actual words of which a proposition consists may constitute part of its mental equivalent. But, as before, this makes no difference in the meaning that the proposition is intended to convey.

To this distinction we shall return shortly from a more controversial point of view.

(3) In the third place, our sentences may differ not merely from the verbal and subjective standpoints, but also from the objective standpoint; they may affirm distinct matters of fact. As, for example, if one of them states that all potassium with which we have experimented takes fire when thrown on water, and the other that a piece of potassium with which we have not yet experimented will do the same.

Now in all three of these cases we have novelty, and the question to be decided is which of the three kinds of novelty is requisite in order that we may have inference. I hold the right answer to be that, for inference, subjective novelty is necessary and sufficient.

Here then we have one form in which the point mainly at issue in regard to the nature of inference presents itself. Is it possible for two judgments to be different quâ judgments, although from the objective standpoint one of them states nothing that is not also stated by the other? Or, to put the question differently, can two judgments (or sets of judgments) be distinct as judgments although they are not logically independent, that is, although self-evident relations exist between them such that the truth of one of them involves the truth of the other?

I am ready to admit that it is no easy matter to draw a hard and fast line determining where mere verbal novelty ends and subjective novelty begins. Before attempting to deal with this difficulty, however, I will endeavour to shew that there undoubtedly are cases in which we have progress in thought without reaching anything that is objectively new.

We have cited cases in which it appears clear that we have inference and not mere verbal transformation. But in most of these cases intermediate steps may be inserted; and if this is done to the fullest possible extent, the progress at each step may be so slight that it may not be at all easy to say wherein precisely the inference is to be found.

We must then proceed to consider the limiting cases in which there may be legitimate doubt as to whether we have inference or not. One of these cases is that of conversion. The question whether there is inference in conversion may be in itself, as Mr Bosanquet puts it, “a point of little interest” (Essentials of Logic, p. 141). Nevertheless, as a limiting case, it is not lightly to be put on one side when we are attempting to decide what fundamentally constitutes inference.

It appears to me that conversion is a process of inference if we are dealing with a schedule of propositions in which the predicative reading is adopted. In such a schedule the primary import of the various propositions involves a differentiation between subject and predicate, and to predicate P of S or to deny that P can be predicated of S is a different thing from predicating S of P or denying that S can be predicated of P. Moreover we may grasp the one relation without necessarily realising whether it does or does not involve the other. But in an equational system it is different. If two classes are affirmed to be identical it is merely a verbal question which is mentioned first, and we cannot consider that we have made any progress in thought when we merely alter the order in which they are named. It follows that we must consider that we have inference when we reduce a proposition expressed predicatively to the equational form.

In either schedule, contraposition (or a process analogous to contraposition) presents itself as an inference. In the one case, we have All S is P, therefore, Anything that is not P is not S ; in the other case, S = SP, therefore, Pʹ = PʹSʹ.

The given syllogism might also be reduced as follows:
 From (1) it follows that Everything is m or P ; (4)
   and from (2) we get Everything is s or M. (5)
 Combining (4) and (5), Everything is (s or M) and (m or P);
   therefore, Everything is sm or sP or MP ;
   therefore, Every S is P.

We may note in passing that if elimination is regarded as constituting the essence of inference, then in each of the above resolutions of the syllogism all the inference is concentrated in the last step, and this again seems paradoxical.

In his systematic treatment of fallacies, Mill classifies petitio principii amongst fallacies of confusion, and quotes with approval Whately’s definition: it is the fallacy “in which one of the premisses either is manifestly the same in sense with the conclusion, or is actually proved from it, or is such as the persons you are addressing are not likely to know, or to admit, except as an inference from the conclusion” (Logic, v. 7, § 2 n.). This fallacy has been described as being a fallacy of proof rather than a fallacy of inference ; that is to say, it arises when we ask how a given thesis is to be established, rather than when we ask what follows from a given hypothesis. We have to enquire whether every syllogism is open to the charge of petitio principii in this sense.

It is obvious that the answer to the question in the case of any particular syllogism depends upon the grounds on which the premisses are themselves affirmed; and we may begin by calling attention to certain cases in which the justice of the charge must be admitted, the conclusion of the syllogism being regarded as a thesis to be proved.