Hume and The Traditional Problem of Induction – Bruce Aune

Posted on September 29, 2010 by

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Many secularists or skeptics accuse faith of not being based on reason. Often, religion is contrasted with science which is purported to be objective, rational, etc. However, science has its own philosophical shortcomings to deal with. One of the essential features of modern science is inductive reasoning. Yet, on what grounds is inductive reasoning substantiated? British Empiricist David Hume exposed a fundamental problem with the use of inductive reasoning that has yet to be adequately resolved. The text below is an excerpt from Bruce Aune’s “Rationalism, Empiricism, and Pragmatism: An Introduction”. Aune explains the “Problem of Induction” as stated by Hume while also providing some possible solutions to the problem. He ultimately concludes that inductive reasoning is not justifiable, but we have no choice to use it.

Hume and The Traditional Problem of Induction

By Bruce Aune

“What Hume called “experimental inference” is merely a special case of what is now terms “inductive generalizations.” As the word “generalization” suggests, this form of inference consists in generalizing from a body of data. Omitting certain qualifications, the basic rule involved may be formulated as follows: if n out of m instances of a large, arbitrarily selected class of Ks have been found on examination to possess a property P, then infer that n/m of all Ks probably have P. As an application of this rule, suppose that half of all pigs ever examined (or one out of every two) have been found to have pink eyes. The conclusion to be drawn is that half of all pigs probably have pink eyes. Again, suppose that all crows ever observed (or one out of every one) have been found to be black. The conclusion to be drawn is that all (or one out of every one) crows are probably black.

Although Hume wrote as if the conclusion of an experimental inference must always be a statement of cause and effect, his discussion of causation makes it clear that his method is just a special kind of generalization from experience. As he takes great pains to tell us, what we must establish in showing that A is the cause of B is a constant conjunction between events of kind A and events of kind B. Given this interpretation of general causal claims, the conclusion of an experimental inference can be formulated most perspicuously as “All As (or A-events) are conjoined with Bs.” Since the evidence appropriate to such a conclusion is that all experienced As are (or have been) conjoined with Bs, Hume’s method of experimental inference is nothing but a special kind of inductive generalization. To make it fit the rule given in the last paragraph, we need only take as our property P the complex property of being conjoined to a B. We can then say that since one out of every one (or all) instances of experienced As have been found to have the property of being conjoined to a B, probably all As have this property.

Hume’s argument that experimental inference is incapable of rational justification applies with equal force to the more general method of inductive generalization, and his critique is now regarded as the classical statement of the so-called problem of induction. Most subsequent philosophers have agreed with Hume that inductive inference is indispensable for our reasoning about matters of fact and existence, and countless attempts have been made to show that such inference is justifiable. It cannot be said that any of these attempts is now universally accepted, for, as we shall see in Chapter VI, a number of important contemporary philosophers are convinced that induction is unacceptable. It is inappropriate here to survey the range of solutions that have been suggested, but it will be helpful for the understanding of Hume’s position to consider the main point of one recent and influential attempt to refute him.

According to Hume, inductive or “experimental” inference is not a rational form of reasoning, because it cannot possibly be justified on rational grounds. But this claim, it is urged, is simply false. The definition of a rational being assures us that experimental reasoning is necessarily a rational form of inference. If a man were deliberating about whether to allow his children to play with a neighbor’s pet lion, he would be regarded as irrational if he did not rely on inductive reasoning. In fact, if he did not consider whether in the past the lion had been gentle with children, he would no doubt be regarded as incompetent. This example shows us that it is part of the meaning of “rational” that inductive inference must be considered as a rational form of reasoning. To declare that such inference is not rational because it cannot be justified by some more basic form of inference is simply to create confusion by misusing the word “rational.”

In any case, it is unreasonable to suppose that a form of inference is rationally unjustifiable if it cannot be justified by a more basic form of inference. Obviously, we have basic forms of a priori reasoning as well as a basic form of a posteriori reasoning; if the latter were declared unjustifiable on the mere ground that it is basic, the same would have to be said for our basic forms of a priori reasoning. But this is totally absurd. TO say such a thing is tantamount to saying that all forms of reasoning are unjustifiable – for we certainly cannot hold that an unjustifiable form of reasoning may justify some other form of reasoning. Since at least some form of reasoning is admitted as rational even by Hume, we may conclude that a form of reasoning cannot be regarded as unjustifiable merely because it is basic, merely because there is not a more basic form by which to justify it.

What we must recognize here is that there is more than one way in which a form of reasoning may be justified. Sometimes this may be accomplished by referring to a more basic form of reasoning, and sometimes it may be accomplished by referring to our accepted standards of rationality. A basic form of inference obviously cannot be justified in the first way, but it can be justified in the second way. This is true of inductive reasoning. Since it is our basic form of a posteriori reasoning, there is no question of justifying it by reference to some other form of reasoning. But this does not mean that we cannot justify it by reference to our standards of rationality. This justification is, in fact, easy to give. Since a man who does not follow the accepted canons of inductive reasoning in thinking about matters of fact is, by definition, irrational, it immediately follows that we are entitled to regard this form of reasoning as eminently rational and therefore justifiable.

The reply to Hume just sketched brings out some important points, but it does not really refute the position he is concerned with defending. Hume agrees that we do employ experimental reasoning in our everyday life; in fact, he insists that we cannot avoid using it. Although he did not state the point explicitly, he would also grant that our ordinary standards of rationality are such that the man who did not employ inductive reasoning in his deliberations about the lion would be declared irrational. But these standards, Hume would say, are merely matters of custom, and they cannot themselves guarantee that the conclusions we draw when using inductive inference are even likely to be true. The latter point is, for him, crucial; his concern is not with the fact of our customary standards but with the credentials they possess.

Hume’s view on this matter emerges more clearly when we consider the charge that if inductive inference is unjustifiable merely because it cannot be justified, then any form of inference must be unjustifiable. To evaluate this charge we must first note that the general purpose of inference is, at least for empiricists, to draw true or at least probably true conclusions from true premises. If a form of inference is to be rationally justifiable, we must therefore have some assurance, Hume would say, that any conclusions it allows us to draw from true premises will be true at least more often than note. For any main-line empiricist, we have this assurance for basic forms of a priori (or deductive) inference, but we do not have it for inductive inference. And this is why we have a special problem of induction.

Take, for example, the basic rule of deductive inference, “From the premises P and Q one may infer P.” We know, according to the empiricist, that this form of inference will always yield true conclusions from true premises because the corresponding hypothetical statement, “If P and Q, then P,” is analytically true and empty of content. For any main-line empiricist, and we include Hume here, the same holds for all forms of deductive inference: A rule of the form “From A one may infer B” is deductively valid when and only when the corresponding conditional statement “If A, then B” is analytically true.

But now consider the rule “From the premise that all observed As have been conjoined with Bs one may infer that all As, whether observed or not, are conjoined with Bs.” What assurance do we have that the conclusions this rule allows us to draw from true premises will be true or even likely to be true? Obviously, it is not even analytically true that if all observed As have been conjoined with Bs, then most As are conjoined with Bs; the unobserved As may greatly outnumber the observed ones. Since the conditional statement corresponding to the rule is not analytic, our belief in its truth or even its high probability cannot be defended a priori. Is there any other way to defend it? Hume’s answer is, of course, “No,” and the line of objection considered above does not show that there is anything wrong with this answer.

As mentioned earlier, our conception of inductive inference has undergone some refinement since Hume’s day. This refinement has not made the solution of his problem any easier, but it has improved our understanding of what inductive generalization (assuming that it is acceptable) accomplishes. A major lesson, emphasized in recent discussion, is that we cannot really suppose that this form of inference will ever lead us to the truth in individual cases or even to a close approximation of the truth. The most we can assume is that the method is self-correcting in the sense that its continued use will permit us to eliminate erroneous conclusions in favor of others that are, we hope, progressively more accurate. Our acceptance of the method cannot, in other words, be based on the belief that individual inductive inferences are likely to yield true conclusions; it can, at best, reflect our confidence that we are thereby provided with a general form of a posteriori reasoning which, if used consistently and systematically, is self-correcting and capable of bringing us increasingly closer to the truth.

To see the point of this lesson, consider the problem of estimating the relative frequency with which a certain coin turns up heads when thrown on a table. If four throws are made and heads appears only twice, we may generalize that the relative frequency of getting heads with this coin is, in general, 1/2. This conclusion is, however, subject to reappraisal by further inductions. If we observe a total of eight throws, three of which are heads, we shall then infer that the general frequency is 3/8. Since the evidence for the second estimate is based on a greater class of throws, we may regard it as correcting the previous one. Further throws will provide further evidence, and if we continue the process we shall hopefully move closer and closer to a correct estimate. Given certain reasonable assumptions, it can be proved mathematically that if this process were continued indefinitely, we should reach an estimate that differs from the truth by no more than arbitrary fraction e, where e is as small as we may wish to specify.

This recent conception of induction as yielding more or less accurate estimates rather than true or probably true conclusions would not really surprise Hume, nor would he find it destructive of his basic point of view. In numerous passages of the Enquiry, particularly in the chapter “Of Probability,” Hume acknowledged that our inductive conclusions are constantly modified by our experience of further cases, both positive and negative. Instead of discussing the relative acceptability of various estimates or the degree to which a given estimate might approximate the truth, however, he turned his attention to the purely psychological question “How is the strength of a man’s belief in a certain conclusion affected by his experience of positive and negative cases?” This attention to psychological matters at the expense of a more refined theory of induction is wholly understandable, given Hume’s fundamental convictions about induction. If experimental inference is not “based on reason or any operation of the understanding,” he could scarcely be required to discuss how close to the truth a given estimate might be or whether one estimate is more acceptable than another.

The mathematical fact, noted above, that in the long run continued used of the inductive method would result in extremely accurate estimates cannot by itself solve Hume’s problem, because the long run in question is an infinite run that can never be completed. In practice, therefore, we are always faced with the stubborn question whether our most recent estimate, made on admittedly limited evidence, is a decent approximation to the truth – and we have no a priori means of obtaining a definitive answer to this question. If our evidence is extensive and carefully assembled, we may adopt the so-called straight rule of inductive logic and conclude that our estimate is the same as what would be attained in the long run mentioned above. But we have no way of proving that our estimate is this accurate or that additional evidence will not yield an estimate that diverges significantly from our present one. Hume’s fundamental critique of induction is therefore still applicable to current conceptions. Our inductive procedures are much more sophisticated, mathematically, than his were, but we still cannot prove that the conclusions or estimates we obtain are true, likely, or even close approximations to the truth.

In concluding this section we might mention that some influential philosophers accepting Hume’s skeptical attitude have attempted to vindicate, as they call it, our practice of using induction while admitting that any validation of our inductive method is impossible. Their fundamental idea is that we have no alternative to using some form of a posterior inference and that the method we do use is at least preferable to any other method we can think of. Our method is said to be preferable to others in the sense that if empirical truth is attainable by any method at all, it will be attained more readily by the method we have than by any other. Since unlike stones and carrots we are forced by nature to draw conclusions from the character of our experience, our only reasonable course is to adopt the forms of inference that are preferable to others. Since the inductive method can be shown to be preferable to all known alternatives, we are therefore completely justified in using it even though we cannot prove that it will bring us to the truth or to a close approximation of it.

Instead of trying to prove that the inductive method is preferable to any conceivable alternative, some philosophers have worked to establish the weaker conclusion that our accepted method is at least as good as any alternative. Whichever of these approaches is taken, however, the arguments employed are generally very technical, and they are impossible to survey in a book of this kind. We may note, however, that even these so-called pragmatic approaches to justifying induction are highly controversial. As we shall see in Chapter V, some philosophers insist that such strategies cannot possibly succeed because inductive inference, at least as commonly understood, is demonstrably untenable. The last allegation is not widely accepted – least of all by empiricists, who are virtually unanimous in accepting the inductive method as fundamental to all empirical thought. For them, as for Hume, we might not be able to prove that induction is justifiable, but we have no real alternative to using it in our reasoning about the world.”

(p57 – 64)

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