Lecture Ix Inference and the Syllogistic Forms

Inference in general

1. The Problem of Inference is something of a paradox. Inference consists in asserting as fact or truth, on the ground of certain given facts or truths, something which is not included in those data. We have not got inference unless the conclusion, (i.) is necessary from the premisses, and (ii.) goes beyond the premisses. To put the paradox quite roughly--we have not got inference unless the conclusion is (i.) in the premisses, and (ii.) outside the premisses. This is the problem which exercises Mill so much in the chapter, “Function and Value of the Syllogism.” We should notice especially his § 7, “the universal type of the reasoning process.” The point of it is to make the justice of inference depend upon relations of content, which are judged of by what he calls induction. That is quite right, but the question still returns upon us, “What kind of relations of content must we have, in order to realise the paradox of Inference?” This the “type of inference” rather shirks. See Mill’s remarks when he is brought face to face with {138} Induction, Bk. III. ch. f. § 2. An Inference, as he there recognises, either does not hold at all, or it holds “in all cases of a certain description,” i.e., it depends on universals.

Read for Lectures IX. and X., Mill, Bk. II ch. i., ii., iii.; Bk. III. ch. i. and ii. at least; Venn, ch. xiv., xv.; Jevons, Lessons xv. and xxiv.; De Morgan’s Budget of Paradoxes.

I ought to warn you at once that though we may have novelty in the conclusion of Inference (as in multiplication of large numbers), the necessity is more essential than the novelty. In fact, much of Inference consists in demonstrating the connection of matters that as facts are pretty familiar. Of course, however, they are always modified in the process, and in that sense there is always novelty. You obtain the most vital idea of Inference by starting from the conclusion as a suggestion, or even as an observation, and asking yourself how it is proved, or explained, and treating the whole process as a single mediate judgment, i.e. a reasoned affirmation. Take the observation, “The tide at new and full moon is exceptionally high.” In scientific inference this is filled out by a middle term. We may profitably think of the “middle term,” as the copula or grip which holds the conclusion together, made explicit and definitely stated. Thus the judgment pulls out like a telescope, exhibiting fresh parts within it, as it passes into inference. “The tide at new and full moon, being at these times the lunar tide plus the solar tide, is exceptionally high.” This is the sort of inference which is really commonest in science. Such an inference would no doubt give us the conclusion if we did not know it by observation, but it happens in many cases that we do know it by observation, and what the inference gives us is the connection, which of course may enable us to correct the observation.

{139} Conditions of the possibility of Inference

2. In the strictest formal sense there can be no inference from particulars to particulars. When there seems to be such inference, it is merely that the ground of inference is not mentioned, sometimes because it is obvious, sometimes because it is not clearly specified in the mind. Suppose we say, “Morley and Harcourt will go for Disestablishment, and I think, therefore, that Gladstone will.” I do not express any connecting link, merely because every one sees at once that I am inferring from the intentions of some Liberal leaders to those of another. If the terms are really particulars, “X is A, Y is B, Z is C,” one is helpless; they do not point to anything further at all; there is no bridge from one to the other.

Inference cannot possibly take place except through the medium of an identity or universal which acts as a bridge from one case or relation to another. If each particular was shut up within itself as in the letters taken as an instance just now, you could never get from one which is given to another which is not given, or to a connection not given between two which are given.

Take the simplest conceivable case, which hardly amounts to Inference, that of producing a given straight line. How is it that this is possible? Because the direction of the straight line is universal and self-identical as against possible directions in space, and it acts as a rule which carries you beyond the given portion of it. This might fairly be called an “immediate inference.” So I presume that any curve can be constructed out of a sufficient portion of the curve, although, except with a circle, this is more than repeating the same line over again. The content has a nature which {140} is capable of prescribing its own continuation. A curve is not a direction; a truth which is a puzzle to the non-mathematician--it is a law of continuous change of direction.

System the ultimate condition of Inference

3. Ultimately the condition of inference is always a system. And it will help us in getting a vital notion of inference if we think, to begin with, of the interdependence of relations in space--in geometrical figures, or, to take a commonplace example, in the adjustment of a Chinese puzzle or a dissected map. Or any of the propositions about the properties of triangles are a good example. How can one property or attribute determine another, so that you can say, “Given this, there must be that”? This can only be answered by pointing to the nature of a whole with parts, or a system, which just means this, a group of relations or properties or things so held together by a common nature that you can judge from some of them what the others must be. Not all systems admit of precise calculation and demonstration, but wherever there is inference at all there is at least an identity of content which may be more or less developed into a precise relation between parts. For example, we cannot construct geometrically the life and character of an individual man; we can argue from his character to some extent, but the connection of facts in his personal identity is all that we can infer for certain; and even this involves a certain context of facts, as in circumstantial evidence. Yet this simplest linking together of occurrences by personal identity is enough to give very startling inferences. Thackeray’s story of the priest is a good instance of inference from mere identity. “An old abbé, talking among a party of intimate friends, happened {141} to say, ‘A priest has strange experiences; why, ladies, my first penitent was a murderer.’ Upon this, the principal nobleman of the neighbourhood enters the room. ‘Ah, Abbé, here you are; do you know, ladies, I was the Abbé’s first penitent, and I promise you my confession astonished him!’” Here the inference depends solely on individual identity, which is, as we saw, a kind of universal.

But in this case was there really an inference? Does not the conclusion fall inside the premisses? It must in one sense fall inside the premisses, or it is not true. But it does not fall inside them until we have brought them into contact by their point of identity and melted them down into the same judgment. I admit that these inferences from individual identity, assuming the terms not to be ambiguous, are only just within the line of rational inference, but, as we see in this case, they bring together the parts of a very extended universal. What is the lower limit of inference?

Immediate Inference

4. In the doctrine of immediate Inference common Logic treats of Conversion and the Opposition of judgments.

Is a mere transposition of Subject and Predicate, where the truth of the new judgment follows from that of the old, an inference? It is a matter of degree. Does it give anything new? “The Queen is a woman.” “A woman is the Queen.” If we make a real difference between the implications of a Subject and a Predicate, we seem to get something new; but it is a point of little interest. {142} Comparison or Recognition are more like immediate inferences. Comparison means that we do not let ourselves perceive freely, but take a particular content as the means of apperception of another content, i.e. as the medium through which we look at it. I do not merely look at the second, but I look at it with the first in my mind. And so far I may be said to infer, without the form of proof, from data of perception to a relation between them. “You are taller than me,” is a result obtained by considering your height from the point of view of mine, or vice versâ. Recognition is somewhat similar. It is more than a mere perception, because it implies reproduction of elements not given, and an identification with them. I recognise this man as so-and-so, i.e. I see he is identical with the person who did so-and-so. It is a judgment, but it goes beyond the primary judgment, “He is such and such,” and is really inferred from it. It is a matter of degree. Almost every Judgment can be broken up into elements, and recognition fades gradually into cognition--we “recognise” an example of a law, a right, a duty, an authority; not that we knew it, the special case, before, but that in analysing it we find a principle which commands our assent, and with which we identify the particular instance before us.

The collective or general judgment, as commonly explained, cannot be converted “simply,” because the predicate is “wider” than the subject, and the same rule is accepted for the relation of consequent to antecedent. The aim of science, it might almost be said, is to get beyond the kind of judgment to which this rule applies.

Number of Instances

5. The difference between guess-work and demonstration rests on the difference between a detached quality or relation striking enough to suggest something to us, and a system thoroughly known in its parts as depending on one another. This is so even in recognising an individual person; it is necessary to know that the quality by which you recognise him is one that no one else possesses, or else {143} it is guess-work. Still more is this the case in attempting a scientific connection. All scientific connection is really by system as between the parts of the content. A quality is often forced on our attention by being repeated a great many times in some particular kind of occurrence, but as long as we do not know its causal connection with the properties and relations involved in the occurrence it is only guess-work to treat them as essentially connected. This is a matter very easy to confuse, and very important. It is easy to confuse, because a number of instances does help us really in inference, as it always insensibly gives us an immense command of content; that is to say, without knowing it we correct and enlarge our idea of the probable connection a little with every instance. So the connection between the properties that strike us becomes much larger and also more correct than it is to people who have only seen a few instances. But this is because the instances are all a little different, and so correct each other, and show transitions from more obvious forms to less obvious forms of the properties in question which lead us up to a true understanding of them. If the instances were all exactly the same they would not help us in this way, but our guess would still be a guess, however many instances might have suggested it.

I remember that a great many years ago I hardly believed in the stone-age tools being really tools made by men. I had only seen a few bad specimens, one or two of which I still think were just accidentally broken flints which an old country clergyman took for stone-age tools. This was to me then a mere guess, viz. that the cutting shape proved {144} the flints to have been made by men. And obviously, if I had seen hundreds of specimens no better than these, I should have treated it as a mere guess all the same. But I happened to go to Salisbury, and there I saw the famous Blackmore Museum, where there are not only hundreds of specimens, but specimens arranged in series from the most beautiful knives and arrow-heads to the rudest. There one’s eye caught the common look of them at once, the better specimens helping one to interpret the worse, and the guess was almost turned into a demonstration, because one’s eyes were opened to the sort of handwork which these things exhibit, and to the way in which they are chipped and flaked.

Now this very important operation of number of examples, in helping the mind to an explanation, is always being confused with the effect of mere repetition of examples, which does not help you to an explanation, i.e. a repetition in which one tells you no more than another. But these mere repetitions operate prima facie in a different way, viz. by making you think there is an unknown cause in favour of the combination of properties which recurs, and lead up to the old-fashioned perfect Induction and the doctrine of chances, and not to demonstration.

Ultimately the calculus of chances may be said to rest on the same principle as Induction, in so far as the repetition of examples derives its force from the (unspecified) variety of contexts through which this repetition shows a certain result to be persistent. But in such a calculus the presumption from recurrence in such a variety of contexts is only estimated, and not analysed.

On the road from guess-work to demonstration, and generally assisted by great experience, we have skilful {145} guess-work, the first stage of discovery. This depends on the capacity for hitting upon qualities which are connected by causation, though the connection remains to be proved. So a countryman or a sailor gets to judge of the weather; it is not merely that he has seen so many instances, but he has been taught by a great variety of instances to recognise the essential points, and has formed probably a much more complex judgment than he can put into words. So again a doctor or a nurse can see how ill a patient is, though it does not follow that they could always say why this appearance goes with this degree of illness. In proportion as you merely presume a causal connection, it is guess-work or pure discovery. In as far as you can analyse a causal connection it is demonstration or proof; and for Logic, discovery cannot be treated apart from proof, except as skilful guess-work. In as far as there is ground for the guess, so far it approaches to proof; in as far as there is no ground, it gives nothing for Logic to get hold of--is mere caprice. A good scientific guess really depends on a shrewd eye for the essential points. I am not mathematician enough to give the history of the discovery of Neptune by Leverrier and Adams, “calculating a planet into existence by enormous heaps of algebra,” but it must have begun as a guess, I should suppose it was suggested before Adams and Leverrier took it up, on the ground of the anomalous movements of Uranus indicating an attraction unaccounted for by the known solar system. And I suppose that this guess would gradually grow into demonstration as it became clear that nothing but a new planet would explain the anomalies of {146} the orbit of Uranus. And at last the calculators were able to tell the telescopist almost exactly where to look for the unknown planet. The proof in this case preceded the observation or discovery by perception, and this makes it a very dramatic example; but if the observation had come earlier, it would not I suppose have dispensed with the precise proof of Neptune’s effect on Uranus, though it might have made it easier.

De Morgan, Budget of Paradoxes, p. 53.

Figures of Syllogism

6. In illustration of this progress from guess-work to science, I will give an example of the three Aristotelian figures of the Syllogism. I omit the fourth. I assume that the heavier term, or the term most like a “thing,” is fitted to be the Subject, and the term more like an attribute to be the Predicate. The syllogistic rules depend practically on the fact that common Logic, following common speech and thought, treats the Predicate as wider than the Subject, which corresponds to Mill’s view (also the common scientific view), that the same effect may have several alternative causes (not a compound cause, but different possible causes), and that consequent is wider than antecedent. It is this assumption that prevents affirmative propositions from being simply convertible, i.e. prevents “All men are mortal” from being identical with “All mortals are men,” and but for it there would be no difference of figure at all, as there is not for inference by equation.

Cf. Plato’s Republic, Bk. VI., end. See p. 141, note.

This progression is here merely meant to illustrate the universal or systematic connection of particulars in process of disengaging itself. But I do not say that the first {147} figure with a major premise is a natural form for all arguments.

I take the scheme of the first three figures from Jevons, and suggest their meaning as follows:--

X denotes the major term. Y denotes the middle term. Z denotes the minor term.

1st Fig. 2nd Fig. 3rd Fig. Major Premise YX XY YX Minor Premise ZY ZY YZ Conclusion ZX ZX ZX

Fig. 3. An observation and a guess.

Yesterday it rained in the evening. All yesterday the smoke tended to sink. ∴ The smoke sinking ( may be ) a sign of rain. ( is sometimes )

The conclusion cannot be general in this figure, because nothing general has been said in the premisses about the subject of the conclusion. So it is very suitable for a mere suggested connection given in a single content--that of the time “yesterday,” implying moreover that both the points in question have something to do with the state of the atmosphere on that single day.

Fig. 2. A tentative justification.

Smoke that goes downwards is heavier than air Particles of moisture are heavier than air. ∴ Particles of moisture may be in the descending smoke.

A universal conclusion in this figure would be formally bad. But we do not care for that, because we only mean it to be tentative, and we do not draw a universal affirmative {148} conclusion. We express its badness by querying it, or by saying “may be.” The reason why it is formally bad is that nothing general has been said in the premisses about the middle term or reason, so that it is possible that the two Subjects do not touch each other within it, i.e. that the suggested special cause, moisture, is not connected with the special effect, the sinking of the smoke. The general reason “heavier than air” may include both special suggested cause and special suggested effect without their touching. Smoke and moisture may both sink in air, but for different and unconnected reasons. Still, when a special cause is suggested which is probably present in part, and which would act in the way required by the general character of the effect, there is a certain probability that it is the operative cause, subject to further analysis; and the argument has substantive value, though bad in form. The only good arguments in this figure have negative conclusions, e.g.--

Smoke that is heavier than air goes downwards. Smoke on dry days does not go downwards. ∴ Smoke on dry days is not heavier than air.

This conclusion is formal, because the negative throws the second Subject altogether outside the Predicate, and so outside the first Subject. The one content always has a characteristic which can never attach to the other, and consequently it is clear that some genuine underlying difference keeps them apart. Such an inference would corroborate the suggestion previously obtained that the presence of moisture was the active cause of the descending smoke on days when rain was coming.

Fig. 1. A completely reasoned judgment. {149} All particles that sink in the air in damp weather more than in dry, are loaded with moisture when they sink.

Smoke that descends before rain is an example of particles that sink in the air in damp weather more than in dry.

∴ Smoke that descends before rain is loaded with moisture when it descends (and therefore its sinking is not accidentally a sign of rain, but is really connected with the cause of rain).

The major premise belongs only to this figure. In the other it is mere tradition to call it so, and their two premisses are the same in kind, and contribute equally to the conclusion, and for that reason the affirmative conclusion was not general or not formal. If your general conclusion is to follow by mere form, you must show your principle as explicitly covering your conclusion. But if you do this, then of course you are charged with begging the question. And, in a sense, that is what you mean to do, when you set out to make your argument complete by its mere form. If you have bonâ fide to construct a combination of your data, you cannot predict whether the conclusion will take this form or that form. Using a major premise meant, “We have got a principle that covers the conclusion, and so explains the case before us.” Granting that the major premise involves the minor premise and conclusion, that is just the reason why it is imperative to express them. The meaning of the Syllogism is that it analyses the whole actual thought; the fault is to suppose that novelty is the point of inference. The Syllogism shows you how you must understand either premise in order that it may cover {150} the conclusion. Or, starting from the conclusion as a current popular belief, or as an isolated observation or suggestion by an individual observer (and this is practically the way in which our science on any subject as a rule takes its rise), the characteristic process through the three stages described above consists in first noting the given circumstances under which, according to the prima facie belief or observation, the conjunction in question takes place (“yesterday,” i.e. “in the state of the atmosphere yesterday”); secondly in analysing or considering those given circumstances, to find within them something which looks like a general property, a law, or causal operation, which may attach the conjunction in question to the systematic whole of our experience (the presence of something heavier than air in the atmosphere); and thirdly, in the exhibition of this ground or reason as a principle, in the light of which the primary belief or observation (probably a good deal modified) becomes a part of our systematic intelligible world.

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