For a brief formulation of the problem of induction we can turn to Born, who writes: '... no observation or experiment, however extended, can give more than a finite number of repetitions'; therefore, 'the statement of a law — B depends on A — always transcends experience. Yet this kind of statement is made everywhere and all the time, and sometimes from scanty material.'
In other words, the logical problem of induction arises from (1) Hume's discovery (so well expressed by Born) that it is impossible to justify a law by observation or experiment, since it 'transcends experience'; (2) the fact that science proposes and uses laws 'everywhere and all the time'. (Like Hume, Born is struck by the 'scanty material', i.e. the few observed instances upon which the law may be based.) To this we have to add (3) the principle of empiricism which asserts that in science only observation and experiment may decide upon the acceptance or rejection of scientific statements, including laws and theories.
These three principles, (1), (2), and (3), appear at first sight to clash; and this apparent clash constitutes the logical problem of induction.
Faced with this clash, Born gives up (3), the principle of empiricism (as Kant and many others, including Bertrand Russell, have done before him), in favour of what he calls a 'metaphysical principle'; a metaphysical principle which be does not even attempt to formulate; which he vaguely describes as a 'code or rule of craft'; and of which I have never seen any formulation which even looked promising and was not clearly untenable.
But in fact the principles (1) to (3) do not clash. We can see this the moment we realize that the acceptance by science of a law or of a theory is tentative only; which is to say that all laws and theories are conjectures, or tentative hypotheses (a position which I have sometimes called 'hypotheticism'); and that we may reject a law or theory on the basis of new evidence, without necessarily discarding the old evidence which originally led us to accept it. (I do not doubt that Born and many others would agree that theories are accepted only tentatively. But the widespread belief in induction shows that the far-reaching implications of this view are rarely seen.)
The principle of empiricism (3) can be fully preserved, since the fate of a theory, its acceptance or rejection, is decided by observation and experiment — by the results of tests. So long as a theory stands up to the severest tests we can design, it is accepted; if it does not, it is rejected. But it is never inferred, in any sense, from the empirical evidence. There is neither a psychological nor a logical induction. Only the falsity of the theory can be inferred from empirical evidence, and this inference is a purely deductive one.
Hume showed that it is not possible to infer a theory from observation statements; but this does us affect the possibility of refuting a theory by observation statements. The full appreciation of this possibility makes the relation between theories and observations perfectly clear.
This solves the problem of the alleged clash between the principles (1), (2), and (3), and with it Hume's problem of induction.
Hume's problem of induction has almost always been badly formulated by what may be called the philosophical tradition. I will first give a few of these bad formulations, which I shall call the traditional formulations of the problem of induction. I shall replace them, however, by what I regard as better formulations.
Typical examples of formulations of the problem of induction that are both traditional and bad are the following.
What is the justification for the belief that the future will resemble the past? What is the justification of so-called inductive inferences?
By an inductive inference is here meant an inference from repeatedly observed instances to some as yet unobserved instances. It is of comparatively minor significance whether such an inference from the observed to the unobserved is, from the point of view of time, predictive or retrodictive; whether we infer that the sun will rise tomorrow or that it did rise 100,000 years ago. Of course, from a pragmatic point of view, one might say that it is the predictive type of inference which is the more important. No doubt usually it is.
There are various other philosophers who also regard as misconceived this traditional problem of induction. Some say that it is misconceived because no justification is needed for inductive inference; no more in fact than for deductive inference. Inductive inference is inductively valid just as deductive inference is deductively valid. I think it was Professor Strawson who was the first to say this.
I am of a different opinion. I hold with Hume that there simply is no such logical entity as an inductive inference; or, that all so-called inductive inferences are logically invalid — and even inductively invalid, to put it more sharply [see the end of this selection]. We have many examples of deductively valid inferences, and even some partial criteria of deductive validity; but no example of an inductively valid inference exists. And I hold, incidentally, that this result can be found in Hume, even though Hume, at the same time, and in sharp contrast to myself, believed in the psychological power of induction; not as a valid procedure, but as a procedure which animals and men successfully make use of, as a matter of fact and of biological necessity.
I take it as an important task to make clear, even at the cost of some repetition, where I agree and where I disagree with Hume.
I agree with Hume's opinion that induction is invalid and in no sense justified. Consequently neither Hume nor I can accept the traditional formulations which uncritically ask for the justification of induction; such a request is uncritical because it is blind to the possibility that induction is invalid in every sense, and therefore unjustifiable.
I disagree with Hume's opinion (the opinion incidentally of almost all philosophers) that induction is a fact and in any case needed. I hold that neither animals nor men use any procedure like induction, or any argument based on the repetition of instances. The belief that we use induction is simply a mistake. It is a kind of optical illusion.
What we do use is a method of trial and the elimination of error; however misleadingly this method may look like induction, its logical structure, if we examine it closely, totally differs from that of induction. Moreover, it is a method which does not give rise to any of the difficulties connected with the problem of induction.
Thus it is not because induction can manage without justification that I am opposed to the traditional problem; on the contrary, it would urgently need justification. But the need cannot be satisfied. Induction simply does not exist, and the opposite view is a straightforward mistake.
There are many ways to present my own non-inductivist point of view. Perhaps the simplest is this. I will try to show that the whole apparatus of induction becomes unnecessary once we admit the general fallibility of human knowledge or, as I like to call it, the conjectural character of human knowledge.
Let me point this out first for the best kind of human knowledge we have; that is, for scientific knowledge. I assert that scientific knowledge is essentially conjectural or hypothetical.
Take as an example classical Newtonian mechanics. There never was a more successful theory. If repeated observational success could establish a theory, it would have established Newton's theory. Yet Newton's theory was superseded in the field of astronomy by Einstein's theory, and in the atomic field by quantum theory. And almost all physicists think now that Newtonian classical mechanic is no more than a marvellous conjecture, a strangely successful hypothesis, and a staggeringly good approximation to the truth.
I can now formulate my central thesis, which is this. Once we fully realize the implications of the conjectural character of human knowledge, then the problem of induction changes its character completely: there is no need any longer to be disturbed by Hume's negative results, since there is no need any longer to ascribe to human knowledge a validity derived from repeated observations. Human knowledge possesses no such validity. On the other hand, we can explain all our achievements in terms of the method of trial and the elimination of error. To put it in a nutshell, our conjectures are our trial balloons, and we test them by criticizing them and by trying to replace them — by trying to show that there can be better or worse conjectures, and that they can be improved upon. The place of the problem of induction is usurped by the problem of the comparative goodness or badness of the rival conjectures or theories that have been proposed.
The main barrier to accepting the conjectural character of human knowledge, and to accepting that it contains the solution of the problem of induction, is a doctrine which may be called the commonsense theory of human knowledge or the bucket theory of the human mind.
I think very highly of common sense. In fact, I think that all philosophy must start from commonsense views and from their critical examination.
For our purposes here I want to distinguish two parts of the commonsense view of the world and draw attention to the fact that they clash with one another.
The first is commonsense realism; this is the view that there is a real world, with real people, animals and plants, cars and stars in it. I think that this view is true and immensely important, and I believe that no valid criticism of it has ever been proposed.
A very different part of the commonsense view of the world is the commonsense theory of knowledge. The problem is the problem of how we get knowledge about the world. The commonsense solution is: by opening our eyes and ears. Our senses are the main if not the only sources of our knowledge of the world.
This second view I regard as thoroughly mistaken, and as
insufficiently criticized (in spite of Leibniz and Kant). I call it the
bucket theory of the mind, because it can be summed up by this
diagram:
Both halves of commonsense philosophy, commonsense realism and the commonsense theory of knowledge, were held by Hume; he found, as did Berkeley before him, that there is a clash between them. For the commonsense theory of knowledge is liable to lead to a kind of anti-realism. If knowledge results from sensations, then sensations are the only certain elements of knowledge, and we can have no good reason to believe that anything but sensation exists.
Hume, Berkeley, and Leibniz were all believers in a principle of sufficient reason. For Berkeley and Hume the principle took the form: if you do not have sufficient reasons for holding a belief, then this fact is itself a sufficient reason for abandoning this belief. Genuine knowledge consisted for both Berkeley and Hume essentially of belief, backed by sufficient reasons: but this led them to the position that knowledge consists, more or less, of sensations on their own.
Thus for these philosophers the real world of common sense does not really exist; according to Hume, even we ourselves do not fully exist. All that exist are sensations, impressions, and memory images.
This anti-realistic view can be characterized by various names, but the most usual name seems to be 'idealism'. Hume's idealism appeared to him to be a strict refutation of commonsense realism. But though he felt rationally obliged to regard commonsense realism as a mistake, he himself admitted that he was in practice quite unable to disbelieve in realism for more than an hour.
Thus Hume experienced very strongly the clash between the two parts of commonsense philosophy: realism, and the commonsense theory of knowledge. And although he was aware that emotionally he was unable to give up realism, he looked on this fact as a mere consequence of irrational custom or habit; he was convinced that a consistent adherence to the more critical results of the theory of knowledge ought to make us abandon realism. Fundamentally, Hume's idealism has remained the mainstream of British empiricism.
Hume's two problems of induction — the logical problem and the psychological problem — can best be presented, I think, against the background of the commonsense theory of induction. This theory is very simple. Since all knowledge is supposed to be the result of past observation, so especially is all expectational knowledge such as that the sun will rise tomorrow, or that all men are bound to die, or that bread nourishes. All this has to be the result of past observation.
It is to Hume's undying credit that he dared to challenge the commonsense view of induction, even though he never doubted that it must be largely true. He believed that induction by repetition was logically untenable — that rationally, or logically, no amount of observed instances can have the slightest bearing upon unobserved instances. This is Hume's negative solution of the problem of induction, a solution which I fully endorse.
But Hume held, at the same time, that although induction was rationally invalid, it was a psychological fact, and that we all rely on it.
Thus Hume's two problems of induction were:
(1) The logical problem: Are we rationally justified in reasoning from repeated instances of which we have had experience to instances of which we have had no experience?
Hume's unrelenting answer was: No, we are not justified, however great the number of repetitions may be. And he added that it did not make the slightest difference if, in this problem, we ask for the justification not of certain belief, but of probable belief. Instances of which we have had experience do not allow us to reason or argue about the probability of instances of which we have had no experience, any more than to the certainty of such instances.
(2) The following psychological question: How is it that nevertheless all reasonable people expect and believe that instances of which they have had no experience will conform to those of which they have had experience? Or in other words, why do we all have expectations, and why do we hold on to them with such great confidence, or such strong belief?
Hume's answer to this psychological problem of induction was: Because of 'custom or habit'; or in other words, because of the irrational but irresistible power of the law of association. We are conditioned by repetition; a conditioning mechanism without which, Hume says, we could hardly survive.
My own view is that Hume's answer to the logical problem is right and that his answer to the psychological problem is, in spite of its persuasiveness, quite mistaken.
The answers given by Hume to the logical and psychological problems of induction lead immediately to an irrationalist conclusion. According to Hume, all our knowledge, especially all our scientific knowledge, is just irrational habit or custom, and it is rationally totally indefensible.
Hume himself thought of this as a form of scepticism; but it was rather, as Bertrand Russell pointed out, an unintended surrender to irrationalism. It is an amazing fact that a peerless critical genius, one of the most rational minds of all ages, not only came to disbelieve in reason, but became a champion of unreason, of irrationalism.
Nobody has felt this paradox more strongly than Bertrand Russell, an admirer and, in many respects, even a late disciple of Hume. Thus in the Hume chapter in A History of Western Philosophy, published in 1946, Russell says about Hume's treatment of induction: 'Hume's philosophy ... represents the bankruptcy of eighteenth-century reasonableness' and, 'It is therefore important to discover whether there is any answer to Hume within a philosophy that is wholly or mainly empirical. If not, there is no intellectual difference begins sanity and insanity. The lunatic who believes that he is a poached egg is to be condemned solely on the ground that he is in a minority....'
Russell goes on to assert that if induction (or the principle of induction) is rejected, 'every attempt to arrive at general scientific laws from particular observations is fallacious, and Hume's scepticism is inescapable for an empiricist.'
And Russell sums up his view of the situation created by the clash between Hume's two answers, by the following dramatic remark:
'The growth of unreason throughout the nineteenth century and what has passed of the twentieth is a natural sequel to Hume's destruction of empiricism.'
This last quotation of Russell's goes perhaps too far. I do not wish to overdramatize the situation; and although I sometimes feel that Russell is right in his emphasis, at other moments I doubt it.
Yet the following quotation from Professor Strawson seems to me to support Russell's grave opinion: '[If] ... there is a problem of induction, and ... Hume posed it, it must be added that he solved it... [;] our acceptance of the "basic canons" [of induction] ... is forced upon us by Nature.... Reason is, and ought to be the slave of the passions.'
However this may be, I assert that I have an answer to Hume's psychological problem which completely removes the clash between the logic and the psychology of knowledge; and with it, it removes all of Hume's and Strawson's reasoning against reason.
My own way of avoiding Hunt's irrationalist consequences is very simple. I solve the psychological problem of induction (and also such formulations as the pragmatic problem) in a manner which satisfies the following 'principle of the primacy of the logical solution', or, more briefly, the 'principle of transference'. The principle runs like this: the solution of the logical problem of induction, far from clashing with those of the psychological or pragmatic problems, can, with some care, be directly transferred to them. As a result, there is no clash, and there are no irrationalist consequences.
The logical problem of induction itself needs some reformulation to start with.
First, it must be formulated in terms not only of 'instances' (as by Hume) but of universal regularities or laws. Regularities or laws are presupposed by Hume's own term 'instance'; for an instance is an instance of something — of a regularity or of a law. (Or, rather, it is an instance of many regularities or many laws.)
Secondly, we must widen the scope of reasoning from instances to laws so that we can take heed also of counterinstances.
In this way, we arrive at a reformulation of Hume's logical problem of induction along the following lines: -
Are we rationally justified in reasoning from instances or from counterinstances of which we have had experience to the truth or falsity of the corresponding laws, or to instances of which we have had no experience?
This is a purely logical problem. It is essentially merely a slight extension of Hume's logical problem of induction formulated here, earlier, in section V.
The answer to this problem is: as implied by Hume, we certainly are not justified in reasoning from an instance to the truth of the corresponding law. But to this negative result a second result, equally negative, may be added: we are justified in reasoning from a counterinstance to the falsity of the corresponding universal law (that is, of any law of which it is a counterinstance). Or in other words, from a purely logical point of view, the acceptance of one counterinstance to 'All swans are white' implies the falsity of the law 'All swans are white' — that law, that is, whose counterinstance we accepted. Induction is logically invalid; but refutation or falsification is a logically valid way of arguing from a single counterinstance to — or, rather, against — the corresponding law.
This shows that I continue to agree with Hume's negative logical result; but I extend it.
This logical situation is completely independent of any question of whether we would, in practice, accept a single counterinstance — for example, a solitary black swan — in refutation of a so far highly successful law. I do not suggest that we would necessarily be so easily satisfied; we might well suspect that the black specimen before us was not a swan. And in practice, anyway, we would be most reluctant to accept an isolated counterinstance. But this is a different question. Logic forces us to reject even the most successful law the moment we accept one single counterinstance.
Thus we can say: Hume was right in his negative result that there can be no logically valid positive argument leading in the inductive direction. But there is a further negative result; there are logically valid negative arguments leading in the inductive direction: a counterinstance may disprove a law.
Hume's negative result establishes for good that all our universal laws or theories remain for ever guesses, conjectures, hypotheses. But the second negative result concerning the force of counterinstances by no mean rules out the possibility of a positive theory of how, by purely rational arguments, we can prefer some competing conjectures to others.
In fact, we can erect a fairly elaborate logical theory of preference — preference from the point of view of the search for truth.
To put it in a nutshell, Russell's desperate remark that if with Hume we reject all positive induction, 'there is no intellectual difference between sanity and insanity' is mistaken. For the rejection of induction does not prevent us from preferring, say, Newton's theory to Kepler's, or Einstein's theory to Newton's: during our rational critical discussion of these theories we may have accepted the existence of counterexamples to Kepler's theory which do not refute Newton's, and of counterexamples to Newton's which do not refute Einstein's. Given the acceptance of these counterexamples we can say that Kepler's and Newton's theories are certainly false; whilst Einstein's may be true or it may be false: that we don't know. Thus there may exist purely intellectual preferences for one or the other of these theories; and we are very far from having to say with Russell that all the difference between science and lunacy disappears. Admittedly, Hume's argument still stands, and therefore the difference between a scientist and a lunatic is not that the first bases his theories securely upon observation while the second does not, or anything like that. Nevertheless we may now see that there may be a difference: it may be that the lunatic's theory is easily refutable by observation, while the scientist's theory has withstood severe tests.
What the scientist's and the lunatic's theories have in common is that both belong to conjectural knowledge. But some conjectures are much better than others; and this is a sufficient answer to Russell, and it is sufficient to avoid radical scepticism. For since it is possible for some conjectures to be preferable to others, it is also possible for our conjectural knowledge to improve, and to grow. (Of course, it is possible that a theory that is preferred to another at one time may fall out of favour at a later time so the other is now preferred to it. But, on the other hand, this may not happen.)
We may prefer some competing theories to others on purely rational grounds. It is important that we are clear what the principles of preference or selection are.
In the first place they are governed by the idea of truth. We want, if at all possible, theories which are true, and for this reason, we try to eliminate the false ones.
But we want more than this. We want new and interesting truth. We are thus led to the idea of the growth of informative content, and especially of truth content. That is, we are led to the following principle of preference: a theory with a great informative content is on the whole more interesting, even before it has been tested, than a theory with little content. Admittedly, we may have to abandon the theory with the greater content, or as I also call it, the bolder theory, if it does not stand up to tests. But even in this case we may have learned more from it than from a theory with little content, for falsifying tests can sometimes reveal new and unexpected facts and problems.
Thus our logical analysis leads us direct to a theory of method, and especially to the following methodological rule: try out, and aim at, bold theories, with great informative content; and then let these bold theories compete, by discussing them critically and by testing them severely.
My solution of the logical problem of induction was that we may have preferences for certain of the competing conjectures; that is, for those which are highly informative and which so far have stood up to eliminative criticism. These preferred conjectures are the result of selection, of the struggle for survival of the hypotheses under the strain of criticism, which is artificially intensified selection pressure.
The same holds for the psychological problem of induction. Here too we are faced with competing hypotheses, which may perhaps be called beliefs, and some of them are eliminated, while others survive, anyway for the time being. Animals are often eliminated along with their beliefs; or else they survive with them. Men frequently outlive their beliefs; but for as long as the beliefs survive (often a very short time), they form the (momentary or lasting) basis of action.
My thesis is that this Darwinian procedure of the selection of beliefs and actions can in no sense be described as irrational. In no way does it clash with the rational solution of the logical problem of induction. Rather, it is just the transference of the logical solution to the psychological field. (This does not mean, of course, that we never suffer from what are called 'irrational beliefs'.)
Thus with an application of the principle of transference to Hume's psychological problem Hume's irrationalist conclusions disappear.
In talking of preference I have so far discussed only the theoretician's preference — if he has any; and why it will be for the 'better', that is, more testable, theory, and for the better tested one. Of course, the theoretician may not have any preference: he may be discouraged by Hume's, and my, 'sceptical' solution to Hume's logical problem; he may say that, if he cannot make sure of finding the true theory among the competing theories, he is not interested in any method like the one described — not even if the method makes it reasonably certain that, if a true theory should be among the theories proposed, it will be among the surviving, the preferred, the corroborated ones. Yet a more sanguine or more dedicated or more curious 'pure' theoretician may well be encouraged, by our analysis, to propose again and again new competing theories in the hope that one of them may be true — even if we shall never be able to make sure of any one that it is true.
Thus the pure theoretician has more than one way of action open to him; and he will choose a method such as the method of trial and the elimination of error only if his curiosity exceeds his disappointment at the unavoidable uncertainty and incompleteness of all our endeavours.
It is different with him qua man of practical action. For a man of practical action has always to choose between some more or less definite alternatives, since even inaction is a kind of action.
But every action presupposes a set of expectations, that is, of theories about the world. Which theory shall the man of action choose? Is there such a thing as a rational choice?
This leads us to the pragmatic problems of induction, which to start with, we might formulate thus:
(1) Upon which theory should we rely for practical action, from a rational point of view?
(2) Which theory should we prefer for practical action, from a rational point of view?
My answer to (1) is: from a rational point of view, we should not 'rely' on any theory, for no theory has been shown to be true, or can be shown to be true (or 'reliable').
My answer to (2) is: we should prefer the best tested theory as a basis for action.
In other words, there is no 'absolute reliance'; but since we have to choose, it will be 'rational' to choose the best tested theory. This will be 'rational' in the most obvious sense of the word known to me: the best tested theory is the one which, in the light of our critical discussion, appears to be the best so far; and I do not know of anything more rational than a well-conducted critical discussion.
Since this point appears not to have got home I shall try to restate it here in a slightly new way, suggested to me by David Miller. Let us forget momentarily about what theories we 'use' or 'choose' or 'base our practical actions on', and consider only the resulting proposal or decision (to do X; not to do X; to do nothing; or so on). Such a proposal can, we hope, be rationally criticized; and if we are rational agents we will want it to survive, if possible, the most testing criticism we can muster. But such criticism will freely make use of the best tested scientific theories in our possession. Consequently any proposal that ignores these theories (where they are relevant, I need hardly add) will collapse under criticism. Should any proposal remain, it will be rational to adopt it.
This seems to me all far from tautological. Indeed, it might well be challenged by challenging the italicized sentence in the last paragraph. Why, it might be asked, does rational criticism make use of the best tested although highly unreliable theories? The answer, however, is exactly the same as before. Deciding to criticize a practical proposal from the standpoint of modern medicine (rather than, say, in phrenological terms) is itself a kind of 'practical' decision (anyway it may have practical consequences). Thus the rational decision is always: adopt critical methods that have themselves withstood severe criticism.
There is, of course, an infinite regress here. But it is transparently harmless.
Now I do not particularly want to deny (or, for that matter, assert) that, in choosing the best tested theory as a basis for action, we 'rely' on it, in some sense of the word. It may therefore even be described as the most 'reliable' theory available, in some sense of this term. Yet this is not to say that it is 'reliable'. It is 'unreliable' at least in the sense that we shall always do well, even in practical action, to foresee the possibility that something may go wrong with it and with our expectations.
But it is not merely this trivial caution which we must derive from our negative reply to the pragmatic problem (1). Rather, it is of the utmost importance for the understanding of the whole problem, and especially of what I have called the traditional problem, that in spite of the 'rationality' of choosing the best tested theory as a basis of action, this choice is not 'rational' in the sense that it is based upon good reasons in favour of the expectation that it will in practice be a successful choice: there can be no good reasons in this sense, and this is precisely Hume's result. On the contrary, even if our physical theories should be true, it is perfectly possible that the world as we know it, with all its pragmatically relevant regularities, may completely disintegrate in the next second. This should be obvious to anybody today; but I said so before Hiroshima: there are infinitely many possible causes of local, partial, or total disaster.
From a pragmatic point of view, however, most of these possibilities are obviously not worth bothering about because we cannot do anything about them: they are beyond the realm of action. (I do not, of course, include atomic war among those disasters which are beyond the realm of human action, although most of us think in just this way since we cannot do more about it than about an act of God.)
All this would hold even if we could be certain that our physical and biological theories were true. But we do not know it. On the contrary, we have very good reason to suspect even the best of them; and this adds, of course, further infinities to the infinite possibilities of catastrophe.
It is this kind of consideration which makes Hume's and my own negative reply so important. For we can now see very clearly why we must beware lest our theory of knowledge proves too much. More precisely, no theory of knowledge should attempt to explain why we are successful in our attempts to explain things.
Even if we assume that we have been successful — that our physical theories are true — we can learn from our cosmology how infinitely improbable this success is: our theories tell us that the world is almost completely empty, and that empty space is filled with chaotic radiation. And almost all places which are not empty are occupied either by chaotic dust, or by gases, or by very hot stars — all in conditions which seem to make the application of any physical method of acquiring knowledge impossible.
There are many worlds, possible and actual worlds, in which a search for knowledge and for regularities would fail. And even in the world as we actually know it from the sciences, the occurrence of conditions under which life, and a search for knowledge, could arise — and succeed — seems to be almost infinitely improbable. Moreover, it seems that if ever such conditions should appear, they would be bound to disappear again, after a time which, cosmologically speaking, is very short.
It is in this sense that induction is inductively invalid, as I said above. That is to say, any strong positive reply to Hume's logical problem (say, the thesis that induction is valid) would be paradoxical. For, on the one hand, if induction is the method of science, then modern cosmology is at least roughly correct (I do not dispute this); and on the other, modern cosmology teaches us that to generalize from observations taken, for the most part, in our incredibly idiosyncratic region of the universe would almost always be quite invalid. Thus if induction is 'inductively valid' it will almost always lead to false conclusions; and therein it is inductively invalid.
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