Popper: Induction

Karl Popper (1902-1994) was one of the most influential philosophers of science of the 20th century. He made significant contributions to debates concerning general scientific methodology and theory choice, the demarcation of science from non-science, the nature of probability and quantum mechanics, and the methodology of the social sciences. His work is notable for its wide influence both within the philosophy of science, within science itself, and within a broader social context. Popper’s early work attempts to solve the problem of (...) demarcation and offer a clear criterion that distinguishes scientific theories from metaphysical or mythological claims. Popper’s falsificationist methodology holds that scientific theories are characterized by entailing predictions that future observations might reveal to be false. When theories are falsified by such observations, scientists can respond by revising the theory, or by rejecting the theory in favor of a rival or by maintaining the theory as is and changing an auxiliary hypothesis. In either case, however, this process must aim at the production of new, falsifiable predictions. While Popper recognizes that scientists can and do hold onto theories in the face of failed predictions when there are no predictively superior rivals to turn to. He holds that scientific practice is characterized by its continual effort to test theories against experience and make revisions based on the outcomes of these tests. By contrast, theories that are permanently immunized from falsification by the introduction of untestable ad hoc hypotheses can no longer be classified as scientific. Among other things, Popper argues that his falsificationist proposal allows for a solution of the problem of induction, since inductive reasoning plays no role in his account of theory choice. Along with his general proposals regarding falsification and scientific methodology, Popper is notable for his work on probability and quantum mechanics and on the methodology of the social sciences. Popper defends a propensity theory of probability, according to which probabilities are interpreted as objective, mind-independent properties of experimental setups. Popper then uses this theory to provide a realist interpretation of quantum mechanics, though its applicability goes beyond this specific case. With respect to the social sciences, Popper argued against the historicist attempt to formulate universal laws covering the whole of human history and instead argued in favor of methodological individualism and situational logic. Table of Contents 1. Background 2. Falsification and the Criterion of Demarcation a. Popper on Physics and Psychoanalysis b. Auxiliary and Ad Hoc Hypotheses c. Basic Sentences and the Role of Convention d. Induction, Corroboration, and Verisimilitude 3. Criticisms of Falsificationism 4. Realism, Quantum Mechanics, and Probability 5. Methodology in the Social Sciences 6. Popper’s Legacy 7. References and Further Reading a. Primary Sources b. Secondary Sources -/- . (shrink)

The problem of rational prediction, launched by Wesley Salmon, is without doubt the Achilles heel of the critical method defended by Popper. In this paper, I assess the response given both by Popper and by the popperian Alan Musgrave to this problem. Both responses are inadequate and thus the conclusion of Salmon is reinforced: without appeal to induction, there is no way to make of the practical prediction a rational action. Furthermore, the critical method needs to be vindicated if one (...) pretends that its application is suitable for the preference of a hypothesis. I argue that the nature of this vindication is such that it may be applied also to induction. Thus, to be a popperian is a good reason also to be an inductivist. (shrink)

It is generally accepted that Popper‘s degree of corroboration, though “inductivist” in a very general and weak sense, is not inductivist in a strong sense, i.e. when by ‘inductivism’ we mean the thesis that the right measure of evidential support has a probabilistic character. The aim of this paper is to challenge this common view by arguing that Popper can be regarded as an inductivist, not only in the weak broad sense but also in a narrower, probabilistic sense. In section (...) 2, first, I begin by briefly characterizing the relevant notion of inductivism that is at stake here; second, I present and discuss the main Popperian argument against it and show that in the only reading in which the argument is formally it is restricted to cases of predicted evidence, and that even if restricted in this way the argument is formally valid it is nevertheless materially unsound. In section 3, I analyze the desiderata that, according to Popper, any acceptable measure for evidential support must satisfy, I clean away its ad-hoc components and show that all the remaining desiderata are satisfied by inductuvist-in-strict-sense measures. In section 4 I demonstrate that two of these desiderata, accepted by Popper, imply that in cases of predicted evidence any measure that satisfies them is qualitatively indistinguishable from conditional probability. Finally I defend that this amounts to a kind of strong inductivism that enters into conflict with Popper’s anti-inductivist argument and declarations, and that this conflict does not depend on the incremental versus non-incremental distinction for evidential-support measures, making Popper’s position inconsistent in any reading.Keywords: Popper; Inductivism; Confirmation; Corroboration. (shrink)

We compare Jan ?ukasiewicz's and Karl Popper's views on induction. The English translation of the two ?ukasiewicz's papers is included in the Appendix.

This paper constitutes one extended argument, which touches on various topics of Critical Rationalism as it was initiated by Karl Popper and further developed in his aftermath. The result of the argument will be that critical rationalism either offers no solution to the problem of induction at all, or that it amounts, in the last resort, to a kind of Critical Rationalist Inductivism as it were, a version of what I call Good Old Induction. One may think of David Miller (...) as a contemporary representative of what I consider as the ‘no solution’ version of critical rationalism, while Alan Musgrave stands for the version of ‘critical rationalist induction’. Popper’s own writings admit of either interpretation. (shrink)

Popper uses the "Humean challenge" as a justification for his falsificationism. It is claimed that in his basic argument he confuses two different doubts: (a) the Humean doubt (Popper's problem of induction), and (b) the "Popperean" doubt whether - presupposing that there are laws of nature - the laws we accept are in fact valid. Popper's alleged solution of the problem of induction does not solve the problem in a straightforward way (as Levison and Salmon have remarked before). But if (...) Popper's solution of the Humean challenge is re-interpreted as being close to Kant's it makes sense. Even though Popper explicitly rejects Kant's synthetic judgements a priori, it is claimed here that this is so because he misinterprets Kant's argument. If he had understood Kant correctly he should have been a modern "Kantianer"! (shrink)

In section I the notions of logical and inductive probability will be discussed as well as two explicanda, viz. degree of confirmation, the base for inductive probability, and degree of evidential support, Popper's favourite explicandum. In section II it will be argued that Popper's paradox of ideal evidence is no paradox at all; however, it will also be shown that Popper's way out has its own merits.

Despite his well‐known deductivism, in his early (unpublished) writings, Popper held an inductivist position. Up to 1929 epistemology entered Popper's reflections only as far as the problem was that of the justification of the scientific character of these fields of research. However, in that year, while surveying the history of non‐Euclidean geometries, Popper explicitly discussed the cognitive status of geometry without referring to psycho‐pedagogical aspects, thus turning from cognitive psychology to the logic and methodology of science. As a consequence of (...) his reflections on the problematic relationship between geometrical‐mathematical constructions and physical reality Popper was able to get over a too direct notion of such a relationship, cast doubts on inductive inference and started conceiving in a new (strictly non‐inductivist) manner the relationship between theoretical and observational propositions. (shrink)

Unlike almost all other philosophers of science, Karl Popper sought to contribute to natural philosophy or cosmology – a synthesis of science and philosophy. I consider his contributions to the philosophy of science and quantum theory in this light. There is, however, a paradox. Popper’s most famous contribution – his principle of demarcation – in driving a wedge between science and metaphysics, serves to undermine the very thing he professes to love: natural philosophy. I argue that Popper’s philosophy of science (...) is, in this respect, defective. Science cannot proceed without making highly problematic metaphysical assumptions concerning the comprehensibility and knowability of the universe. Precisely because these assumptions are problematic, rigour requires that they be subjected to sustained critical scrutiny, as an integral part of science itself. Popper’s principle of demarcation must be rejected. Metaphysics and philosophy of science become a vital part of science. Natural philosophy is reborn. A solution to the problem of what it means to say a theory is unified is proposed, a problem Popper failed to solve. In The Logic of Scientific Discovery, Popper made important contributions to the interpretation of quantum theory, especially in connection with Heisenberg's uncertainty relations. Popper's advocacy of natural philosophy has important implications for education. (shrink)

A Mug's Game? Solving the Problem of Induction with Metaphysical Presuppositions Nicholas Maxwell Emeritus Reader in Philosophy of Science at University College London Email: [email protected] Website: www.ucl.ac.uk/from-knowledge-to-wisdom . Abstract This paper argues that a view of science, expounded and defended elsewhere, solves the problem of induction. The view holds that we need to see science as accepting a hierarchy of metaphysical theses concerning the comprehensibility and knowability of the universe, these theses asserting less and less as we go up the (...) hierarchy. It may seem that this view must suffer from vicious circularity, in so far as accepting physical theories is justified by an appeal to metaphysical theses in turn justified by the success of science. But this is rebutted. A thesis high up in the hierarchy asserts that the universe is such that the element of circularity, just indicated, is legitimate and justified, and not vicious. Acceptance of the thesis is in turn justified without appeal to the success of science. It may seem that the practical problem of induction can only be solved along these lines if there is a justification of the truth of the metaphysical theses in question. It is argued that this demand must be rejected as it stems from an irrational conception of science. (shrink)

Popper famously claimed that he had solved the problem of induction, but few agree. This paper explains what Popper's solution was, and defends it. The problem is posed by Hume's argument that any evidence-transcending belief is unreasonable because (1) induction is invalid and (2) it is only reasonable to believe what you can justify. Popper avoids Hume's shocking conclusion by rejecting (2), while accepting (1). The most common objection is that Popper must smuggle in induction somewhere. But this objection smuggles (...) in precisely the justificationist assumption (2) that Popper, as here undestood, rejects. Footnotes1 Invited address at the Karl Popper 2002 Centenary Conference, Vienna, 3–7 July 2002. (shrink)

In 1983, in an open letter to the journal Nature, Karl Popper and David Miller set forth a particularly strong critical argument which sought to demonstrate the impossibility of inductive probability. Since its publication the argument has faced many criticisms and we argue in this article that they do not reach their objectives. We will first reconstruct the demonstration made by Popper and Miller in their initial article and then try to evaluate the main arguments against it. Although it is (...) possible to conceptualize logically the idea of induction, it is shown that it is not possible on traditional Bayesian grounds. (shrink)

I examine Popper’s claims about Newton’s use of induction in Principia with the actual contents of Principia and draw two conclusions. Firstly, in common with most other philosophers of his generation, it appears that Popper had very little acquaintance with the contents and methodological complexities of Principia beyond what was in the famous General Scholium. Secondly Popper’s ideas about induction were less sophisticated than those of Newton, who recognised that it did not provide logical proofs of the results obtained using (...) it, because of the possibilities of later, contrary evidence. I also trace the historical background to commonplace misconceptions about Newton’s method.Author Keywords: Newton; Popper; Induction; Principia; Kepler’s laws. (shrink)

On the basis of his unpublished thesis 'Gewohnheit und Gesetzerlebnis in der Erziehung' (1926-7) a historical reconstruction is given of the genesis of Popper's ideas on induction and demarcation which differs radically from his own account in Unended quest. It is shown not only that he wholeheartedly endorses inductive epistemology and psychology but also that his 'demarcation' criterion is inductivistic. Moreover it is shown that his later demarcation thesis arises not from his worries about, on the one hand, Marxism and (...) psychoanalysis and, on the other hand, Einstein's physics, but rather from his urgent preoccupation with providing pedagogy with a psychological foundation, which has its sources in Karl Buhler's cognitive psychology as well as, surprisingly, Adler's Characterology. Aside from Adler some lesser known psychologists, such as Karl Groos, will also be seen to have played a formative role on Popper's early thinking. (shrink)

A reprint of Popper and After: Four Modern Irrationalists.

Reprint of Popper and After: Four Modern Irrationalists. In an afterword, James Franklin discusses reactions to Stove's work.

The present paper explores three interrelated topics in Popper's theory of science: (1) his view of conjecture, (2) the aim of science, and (3) his (never fully articulated) theory of meaning. Central to Popper's theory of science is the notion of conjecture. Popper writes as if scientists faced with a problem proceed to tackle it by conjecture, that is, by guesses uninformed by inferential considerations. This paper develops a contrast between guesses and educated guesses in an attempt to show that (...) there is more to scientific conjecture than conjecture. The suggestion is made that some inductive considerations enter into the process of educated guessing or scientific conjecture in such a way that the ‘context of discovery’ cannot be sharply separated from the ‘context of justification’. This discussion leads to a tension between Popper's negative method of conjecture and his realism. Given Popper's (implicit) theory of meaning it seems Popper's epistemology (the conjecture and refutation method) is incompatible with his metaphysical realism. (shrink)

1. Introduction. Attempts to utilize the probability calculus to prove or disprove various inductive or inductive skeptical theses are, I believe, highly problematic. Inductivism and inductive skepticism are substantive philosophical positions that do not allow of merely formal proofs or disproofs. Often the problems with particular alleged formal proofs of inductive or inductive sceptical theses turn on subtle technical considerations. In the following I highlight such considerations in pointing out the flaws of two proofs, one an alleged proof of an (...) inductive sceptical conclusion due to Karl Popper, the other an alleged proof of an inductivist thesis originally due to Harold Jeffreys and later advocated by John Earman. Surprisingly, in examining Popper's argument it is shown that certain apparently weak premises, often embraced by both inductivists and deductivists, lend themselves to inductive conclusions. However it is argued that those premises are still philosophically substantive and not amenable to a purely formal demonstration. The lesson to be learnt here is twofold. First, we need to be very careful in determining which formal theses entail, and which are entailed by, inductive skepticism and inductivism. Second, we need to take great care in laying out and examining the assumptions presumed in formal arguments directed for and against such formal theses. In a follow up article I will consider various attempts by David Stove and Karl Popper and David Miller to identify the exact content of inductive skepticism and propose a new identification based on the theory of content developed in Gemes 1994. Finally I will compare this new version with that proposed recently in Alberto Mura 1990. (shrink)

1. Introduction. Attempts to utilize the probability calculus to prove or disprove various inductive or inductive skeptical theses are, I believe, highly problematic. Inductivism and inductive skepticism are substantive philosophical positions that do not allow of merely formal proofs or disproofs. Often the problems with particular alleged formal proofs of inductive or inductive sceptical theses turn on subtle technical considerations. In the following I highlight such considerations in pointing out the flaws of two proofs, one an alleged proof of an (...) inductive sceptical conclusion due to Karl Popper, the other an alleged proof of an inductivist thesis originally due to Harold Jeffreys and later advocated by John Earman. Surprisingly, in examining Popper's argument it is shown that certain apparently weak premises, often embraced by both inductivists and deductivists, lend themselves to inductive conclusions. However it is argued that those premises are still philosophically substantive and not amenable to a purely formal demonstration. The lesson to be learnt here is twofold. First, we need to be very careful in determining which formal theses entail, and which are entailed by, inductive skepticism and inductivism. Second, we need to take great care in laying out and examining the assumptions presumed in formal arguments directed for and against such formal theses. In a follow up article I will consider various attempts by David Stove and Karl Popper and David Miller to identify the exact content of inductive skepticism and propose a new identification based on the theory of content developed in Gemes 1994. Finally I will compare this new version with that proposed recently in Alberto Mura 1990. (shrink)

Elementary results concerning the connections between deductive relations and probabilistic support are given. These are used to show that Popper-Miller's result is a special case of a more general result, and that their result is not very unexpected as claimed. According to Popper-Miller, a purely inductively supports b only if they are deductively independent — but this means that a b. Hence, it is argued that viewing induction as occurring only in the absence of deductive relations, as Popper-Miller sometimes do, (...) is untenable. Finally, it is shown that Popper-Miller's claim that deductive relations determine probabilistic support is untrue. In general, probabilistic support can vary greatly with fixed deductive relations as determined by the relevant Lindenbaum algebra. (shrink)

Neopositivistic philosophers held that Popper's destructive criticism to inductive methods is wrong. The legend according to which Popper's criticism, in the final analysis, is inconsistent is greatly widespread also amongst neopositivistic Italian scholars. I argue that they are wrong, and that, in general, Popper's view about induction is true. According to Popper all scientific concepts are theoretical, for every assertion not only entails hypotheses but it is also hypothetical, that is not sure and always falsifiable. I argue that the validity (...) and the strength of Popper's criticism to induction is independent from the view according to which all scientific concepts are theoretical. (shrink)

The basic idea by means of which Popper and Miller proved the non-existence of inductive probabilistic support in 1983/1985/1987, is used to prove that inductive probabilistic countersupport does exist. So it seems that after falsification has won over verification on the deductive side of science, countersupport wins over support on the inductive side.

According to Popper and Miller [1983 and 1987], the part of a hypothesis that transcends the evidence is probablistically countersupported by the evidence. Therefore, inductive support is not probabilistic support. Their argument hinges on imposing the following necessary condition on ‘the part of a hypothesis h that goes beyond the evidence e’: that transcendent part, called k, must share no nontrivial consequences with e. I propose a new condition on k that is incompatible with Popper and Miller's condition. I then (...) show why the new condition is a viable alternative to Popper and Miller's. By doing so, I refute their argument that probabilistic support cannot be inductive. *I'd like to thank Michael Redhead. Jeremy Butterfield. and an anonymous referee for comments on earlier drafts. Thanks especially to David Miller for insightful criticisms. Adopted from work done in the Department of History and Philosophy of Science. University of Cambridge. 1989. (shrink)

I set up two axiomatic theories of inductive support within the framework of Kolmogorovian probability theory. I call these theories ‘Popperian theories of inductive support’ because I think that their specific axioms express the core meaning of the word ‘inductive support’ as used by Popper (and, presumably, by many others, including some inductivists). As is to be expected from Popperian theories of inductive support, the main theorem of each of them is an anti-induction theorem, the stronger one of them saying, (...) in fact, that the relation of inductive support is identical with the empty relation. It seems to me that an axiomatic treatment of the idea(s) of inductive support within orthodox probability theory could be worthwhile for at least three reasons. Firstly, an axiomatic treatment demands from the builder of a theory of inductive support to state clearly in the form of specific axioms what he means by ‘inductive support’. Perhaps the discussion of the new anti-induction proofs of Karl Popper and David Miller would have been more fruitful if they had given an explicit definition of what inductive support is or should be. Secondly, an axiomatic treatment of the idea(s) of inductive support within Kolmogorovian probability theory might be accommodating to those philosophers who do not completely trust Popperian probability theory for having theorems which orthodox Kolmogorovian probability theory lacks; a transparent derivation of anti-induction theorems within a Kolmogorovian frame might bring additional persuasive power to the original anti-induction proofs of Popper and Miller, developed within the framework of Popperian probability theory. Thirdly, one of the main advantages of the axiomatic method is that it facilitates criticism of its products: the axiomatic theories. On the one hand, it is much easier than usual to check whether those statements which have been distinguished as theorems really are theorems of the theory under examination. On the other hand, after we have convinced ourselves that these statements are indeed theorems, we can take a critical look at the axioms—especially if we have a negative attitude towards one of the theorems. Since anti-induction theorems are not popular at all, the adequacy of some of the axioms they are derived from will certainly be doubted. If doubt should lead to a search for alternative axioms, sheer negative attitudes might develop into constructive criticism and even lead to new discoveries. -/- I proceed as follows. In section 1, I start with a small but sufficiently strong axiomatic theory of deductive dependence, closely following Popper and Miller (1987). In section 2, I extend that starting theory to an elementary Kolmogorovian theory of unconditional probability, which I extend, in section 3, to an elementary Kolmogorovian theory of conditional probability, which in its turn gets extended, in section 4, to a standard theory of probabilistic dependence, which also gets extended, in section 5, to a standard theory of probabilistic support, the main theorem of which will be a theorem about the incompatibility of probabilistic support and deductive independence. In section 6, I extend the theory of probabilistic support to a weak Popperian theory of inductive support, which I extend, in section 7, to a strong Popperian theory of inductive support. In section 8, I reconsider Popper's anti-inductivist theses in the light of the anti-induction theorems. I conclude the paper with a short discussion of possible objections to our anti-induction theorems, paying special attention to the topic of deductive relevance, which has so far been neglected in the discussion of the anti-induction proofs of Popper and Miller. (shrink)

The form of argument used by Popper and Miller to attack the concept of probabilistic induction is applied to the slightly different situation in which some evidence undermines a hypothesis. The result is seemingly absurd, thus bringing the form of argument under suspicion.

This note makes a contribution to the issue raised in a paper by Popper and Miller (1983) in which it was claimed that probabilistic support is purely deductive. Developing R. C. Jeffrey's remarks, a new general approach to the crucial concept of "going beyond" is here proposed. By means of it a quantitative measure of the inductive component of a probabilistic inference is reached. This proposal leads to vindicating the view that typical predictive probabilistic inferences by enumeration and analogy are (...) purely inductive. (shrink)

Watkins proposes a neo-Popperian solution to the pragmatic problem of induction. He asserts that evidence can be used non-inductively to prefer the principle that corroboration is more successful over all human history than that, say, counter-corroboration is more successful either over this same period or in the future. Watkins's argument for rejecting the first counter-corroborationist alternative is beside the point. However, as whatever is the best strategy over all human history is irrelevant to the pragmatic problem of induction since we (...) are not required to act in the past, and his argument for rejecting the second presupposes induction. (shrink)

Dunn and Hellman's objection to Popper and Miller's alleged disproof of inductive probability is considered and rejected. Dunn and Hellman base their objection on a decomposition of the incremental support P(h/e)-P(h) of h by e dual to that of Popper and Miller, and argue, dually to Popper and Miller, to a conclusion contrary to the latters' that all support is deductive in character. I contend that Dunn and Hellman's dualizing argument fails because the elements of their decomposition are not supports (...) of parts of h. I conclude by reinforcing a different line of criticism of Popper and Miller due to Redhead. (shrink)

In this paper I distinguish two problems of induction: a problem of the uniformity of nature and a problem of the variety of nature. I argue that the traditional problem of induction that Popper poses—the problem of uniformity—is not that which is relevant to science. The problem relevant to science is that of the variety of nature. *I would like to thank Bob Hale, Russell Keat and the Journal's referee for their comments on earlier drafts of this paper.

Popper and Miller (1983) have presented an argument purporting to establish the impossibility of inductive probability. Here I discuss critically their characterization of a deductive part of nondeductive support, a point that has not figured centrally in previous responses.

Popper and Miller argued, in a 1983 paper, that there is no such thing as 'probabilistic inductive support' of hypotheses. They show how to divide a hypothesis into two "parts," where evidence only 'probabilistically supports' the "part" that the evidence 'deductively' implies, and 'probabilistically countersupports' the "rest" of the hypothesis. I argue that by distinguishing between 'support that is purely deductive in nature' and 'support of a deductively implied hypothesis', we can see that their argument fails to establish (in any (...) important way of interpreting it) their conclusion that "all probabilistic support is purely deductive." Their argument is 'not' "completely devastating to the inductive interpretation of the calculus of probability," as claimed. (shrink)

The controversy surrounding Popper's proposed solution to the problem of induction is beginning to display many of the symptoms of being interminable. For decades the discussion has continued, apparently without any progress being made. Again and again, Popperians and their critics have accused each other of ‘missing the point’. The essay attempts to explain what exactly is ‘the point’ of the problem of induction, and asks whether Popper does indeed miss it. An answer is proposed, and on this basis an (...) explanation for the puzzling interminability and emptiness of the above dialogue is put forward. (shrink)

In their 1983 article, Popper and Miller present an argument against inductive probability. This argument is criticized by Redhead in his 1985 article. The aim of the present note is to state one form of the Popper-Miller argument, and defend it against Redhead's criticisms.

In section I the notions of logical and inductive probability will be discussed as well as two explicanda, viz. degree of confirmation, the base for inductive probability, and degree of evidential support, Popper's favourite explicandum. In section II it will be argued that Popper's paradox of ideal evidence is no paradox at all; however, it will also be shown that Popper's way out has its own merits.