Popper: Philosophy of Probability

There are (at least) three approaches to quantifying information. The first, algorithmic information or Kolmogorov complexity, takes events as strings and, given a universal Turing machine, quantifies the information content of a string as the length of the shortest program producing it [1]. The second, Shannon information, takes events as belonging to ensembles and quantifies the information resulting from observing the given event in terms of the number of alternate events that have been ruled out [2]. The third, statistical learning (...) theory, has introduced measures of capacity that control (in part) the expected risk of classifiers [3]. These capacities quantify the expectations regarding future data that learning algorithms embed into classifiers. Solomonoff and Hutter have applied algorithmic information to prove remarkable results on universal induction. Shannon information provides the mathematical foundation for communication and coding theory. However, both approaches have shortcomings. Algorithmic information is not computable, severely limiting its practical usefulness. Shannon information refers to ensembles rather than actual events: it makes no sense to compute the Shannon information of a single string – or rather, there are many answers to this question depending on how a related ensemble is constructed. Although there are asymptotic results linking algorithmic and Shannon information, it is unsatisfying that there is such a large gap – a difference in kind – between the two measures. This note describes a new method of quantifying information, effective information, that links algorithmic information to Shannon information, and also links both to capacities arising in statistical learning theory [4, 5]. After introducing the measure, we show that it provides a non-universal analog of Kolmogorov complexity. We then apply it to derive basic capacities in statistical learning theory: empirical VC-entropy and empirical Rademacher complexity. A nice byproduct of our approach is an interpretation of the explanatory power of a learning algorithm in terms of the number of hypotheses it falsifies [6], counted in two different ways for the two capacities. We also discuss how effective information relates to information gain, Shannon and mutual information. (shrink)

This note presents a very brief history of the observation that the probability of the material conditional A⊃B is in general different from, but cannot be less than, the conditional probability of...

David Stove was a philosopher strong on argument and polemic. His work on the logical intepretation of probability led to a defence of induction in The Rationality of Induction (1986). It resulted too in his denunciation of Popper, Kuhn, Lakatos and Feyeraband as irrationalists because of their "deductivism" (the thesis that the only logic is deductive logic). Stove also defended controversial views on the intelligence of women and on Darwinism. The article surveys his life and work.

The question of truth is a broadly broached subject in Philosophy as it features along the entire historical and polemical growth of the discipline right from the time of the Ancients down to our Post-Modern era. Yet, the delimiting realization of being unable to register general success in our dogged attempts at truth and knowledge, mostly stares us blankly in the face, for matters on which philosophy endeavours to speculate on, are beyond the reach of definite knowledge.1 Our theories of (...) the universe open up to modifications, refutations, and further propositions, evidencing a historical development in philosophical inquiry. This generally is the growth of our science, of our knowledge. This paper critically seeks to examine Popper’s notion of verisimilitude. It takes us through the scientist’s journey from ignorance to truth, and the difference between probability and verisimilitude. It addresses the relevance of the theory of content in understanding verisimilitude, under its distinctions as quantitative and qualitative. Finally, it discusses corroboration and the criteria for theory-choice. (shrink)

DNA evidence is one of the most significant modern advances in the search for truth since the cross examination, but its format as a random-match-probability makes it difficult for people to assign an appropriate probative value (Koehler, 2001). While Frequentist theories propose that the presentation of the match as a frequency rather than a probability facilitates more accurate assessment (e.g., Slovic et al., 2000), Exemplar-Cueing Theory predicts that the subjective weight assigned may be affected by the frequency or probability format, (...) and how easily examples of the event, i.e., ‘exemplars’, are generated from linguistic cues that frame the match in light of further evidence (Koehler & Macchi, 2004). This paper presents two juror research studies to examine the difficulties that jurors have in assigning appropriate probative value to DNA evidence when contradictory evidence is presented. Study 1 showed that refuting evidence significantly reduced guilt judgments when exemplars were linguistically cued, even when the probability match and the refuting evidence had the same objective probative value. Moreover, qualitative reason for judgment responses revealed that interpreting refuting evidence was found to be complex and not necessarily reductive; refutation was found indicative of innocence or guilt depending on whether exemplars have been cued or not. Study 2 showed that the introduction of judges’ directions to linguistically cue exemplars, did not increase the impact of refuting evidence beyond its objective probative value, but less guilty verdicts were returned when jurors were instructed to consider all possible explanations of the evidence. The results are discussed in light of contradictory frequentist and exemplar-cueing theoretical positions, and their real-world consequences. (shrink)

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)

One of the many intriguing features of the axiomatic systems of probability investigated in Popper (1959), appendices _iv, _v, is the different status of the two arguments of the probability functor with regard to the laws of replacement and commutation. The laws for the first argument, (rep1) and (comm1), follow from much simpler axioms, whilst (rep2) and (comm2) are independent of them, and have to be incorporated only when most of the important deductions have been accomplished. It is plain that, (...) in the presence of (comm1), the principle (sub), which says that terms that are intersubstitutable in the first argument are intersubstitutable also in the second argument, implies (comm2), and in Popper’s systems the converse implication obtains. It is naturally asked what is needed in an axiomatic theory of probability in order to enforce this equivalence. Leblanc (1981) offered a rather weak set of axioms, containing (comm1) and (comm2), that suffice for the derivation of (sub). In this paper Leblanc’s result is improved in a number of different ways. Three weaker systems, one of which is incomparable with the other two, are shown to admit the same implication. DOI:10.5007/1808-1711.2011v15n2p271. (shrink)

How are we to understand the use of probability in corroboration functions? Popper says logically, but does not show we could have access to, or even calculate, probability values in a logical sense. This makes the logical interpretation untenable, as Ramsey and van Fraassen have argued. -/- If corroboration functions only make sense when the probabilities employed therein are subjective, however, then what counts as impressive evidence for a theory might be a matter of convention, or even whim. So isn’t (...) so-called ‘corroboration’ just a matter of psychology? -/- In this paper, I argue that we can go some way towards addressing this objection by adopting an intersubjective interpretation, of the form advocated by Gillies, with respect to corroboration. I show why intersubjective probabilities are preferable to subjective ones when it comes to decision making in science: why group decisions are liable to be superior to individual ones, given a number of plausible conditions. I then argue that intersubjective corroboration is preferable to intersubjective confirmation of a Bayesian variety, because there is greater opportunity for principled agreement concerning the factors involved in the former. (shrink)

There are two central questions concerning probability. First, what are its formal features? That is a mathematical question, to which there is a standard, widely (though not universally) agreed upon answer. This answer is reviewed in the next section. Second, what sorts of things are probabilities---what, that is, is the subject matter of probability theory? This is a philosophical question, and while the mathematical theory of probability certainly bears on it, the answer must come from elsewhere. To see why, observe (...) that there are many things in the world that have the mathematical structure of probabilities---the set of measurable regions on the surface of a table, for example---but that one would never mistake for being probabilities. So probability is distinguished by more than just its formal characteristics. The bulk of this essay will be taken up with the central question of what this “more” might be. (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)

Following the pioneer work of Bruno De Finetti [12], conditional probability spaces (allowing for conditioning with events of measure zero) have been studied since (at least) the 1950's. Perhaps the most salient axiomatizations are Karl Popper's in [31], and Alfred Renyi's in [33]. Nonstandard probability spaces [34] are a well know alternative to this approach. Vann McGee proposed in [30] a result relating both approaches by showing that the standard values of infinitesimal probability functions are representable as Popper functions, and (...) that every Popper function is representable in terms of the standard real values of some infinitesimal measure. Our main goal in this article is to study the constraints on (qualitative and probabilistic) change imposed by an extended version of McGee's result. We focus on an extension capable of allowing for iterated changes of view. Such extension, we argue, seems to be needed in almost all considered applications. Since most of the available axiomatizations stipulate (definitionally) important constraints on iterated change, we propose a non-questionbegging framework, Iterative Probability Systems (IPS) and we show that every Popper function can be regarded as a Bayesian IPS. A generalized version of McGee's result is then proved and several of its consequences considered. In particular we note that our proof requires the imposition of Cumulativity, i.e. the principle that a proposition that is accepted at any stage of an iterative process of acceptance will continue to be accepted at any later stage. The plausibility and range of applicability of Cumulativity is then studied. In particular we appeal to a method for defining belief from conditional probability (first proposed in [42] and then slightly modified in [6] and [3]) in order to characterize the notion of qualitative change induced by Cumulative models of probability kinematics. The resulting cumulative notion is then compared with existing axiomatizations of belief change and probabilistic supposition. We also consider applications in the probabilistic accounts of conditionals [1] and [30]. (shrink)

Sir Karl Popper is well known for explicating science in falsificationist terms, for which his degree of corroboration formalism, C(h,e,b), has become little more than a symbol. For example, de Queiroz and Poe in this issue argue that C(h,e,b) reduces to a single relative (conditional) probability, p(e,hb), the likelihood of evidence e, given both hypothesis h and background knowledge b, and in reaching that conclusion, without stating or expressing it, they render Popper a verificationist. The contradiction they impose is easily (...) explained--de Queiroz and Poe fail to take account of the fact that Popper derived C(h,e,b) from absolute (logical) probability and severity of test, S(e,h,b), where critical evidence, p(e,b), is fundamental. Thus, de Queiroz and Poe's conjecture that p(e,hb) = C(h,e,b) is refuted. Falsificationism, not verificationism, remains a fair description of the parsimony method of inference used in phylogenetic systematics, not withstanding de Queiroz and Poe's mistaken understanding that "statistical" probability justifies that method. Although de Queiroz and Poe assert that maximum likelihood has the power "to explain data", they do not successfully demonstrate how causal explanation is achieved or what it is that is being explained. This is not surprising, bearing in mind that what is assumed about character evolution in the accompanying likelihood model M cannot then be explained by the results of a maximum likelihood analysis. (shrink)

The propensity interpretation of probability was introduced by Popper ([1957]), but has subsequently been developed in different ways by quite a number of philosophers of science. This paper does not attempt a complete survey, but discusses a number of different versions of the theory, thereby giving some idea of the varieties of propensity. Propensity theories are classified into (i) long-run and (ii) single-case. The paper argues for a long-run version of the propensity theory, but this is contrasted with two single-case (...) propensity theories, one due to Miller and the later Popper, and the other to Fetzer. The three approaches are compared by examining how they deal with a key problem for the propensity approach, namely the relationship between propensity and causality and Humphreys' paradox. (shrink)

This paper is a re-examination of Popper’s propensity interpretation of probability in respect of its potential methodological value in social science. A long standing problem for the frequency interpretation of probability is that whilst it is able to treat both aggregate and individual phenomena as having measurable properties, it cannot explain the ontological relationship between such concrete individual cases and aggregates. Popper’s interpretation treats single cases as both real, but also as realisations of a propensity to occur. The frequency and (...) propensity interpretations are compared and whilst some common objections raised must be upheld, they do not devalue the importance of the propensity interpretation as at least a metaphysical basis for probabilistic claims in social science. However the value of the approach may also lie in its methodological potential. Here I sugest that single case probabilities must be analysed in terms of the anterior probabilities of prior constituent events. In this I move beyond Popper’s own programme, but suggest that such a move is theoretically compatible with recent complexity approaches in social science and goes some way toward meeting anti-naturalist concerns about intentionality. (shrink)

This work is in two parts. The main aim of part 1 is a systematic examination of deductive, probabilistic, inductive and purely inductive dependence relations within the framework of Kolmogorov probability semantics. The main aim of part 2 is a systematic comparison of (in all) 20 different relations of probabilistic (in)dependence within the framework of Popper probability semantics (for Kolmogorov probability semantics does not allow such a comparison). Added to this comparison is an examination of (in all) 15 purely inductive (...) dependence relations. ————Part 1 leads in an axiomatic step-by-step development from the elementary classical truth value semantics of a sentential-logical language, called ‘L’, (chapter 1) to the elementary Kolmogorov probability semantics of L (chapter 2), which is then extended to four axiomatic semantical theories of dependence relations between the formulae of L. First the elementary Kolmogorov probability semantics of L is extended to a theory, called ‘Kdd’, of the relations of deductive dependence and deductive independence between formulae of L (chapter 3). Then Kdd is extended to a theory, called ‘Kpd1’, of the degree to which formulae of L probabilistically depend on each other in regard to a given probability distribution on the set of all formulae of L (chapter 4). Kpd1, in its turn, gets extended to a theory, called ‘Kpd2’, of the relations of probabilistic dependence and independence, relativized to unary Kolmogorov probability functions defined on L (chapter 5). Then Kpd2 is extended to a theory, called ‘Kid’, of the relations of inductive dependence and inductive independence, again relativized to unary Kolmogorov probability functions defined on L (chapter 6). Finally, Kid is extended to a theory, called ‘Kpid’, of the relations of purely inductive positive and negative dependence, relativized to unary Kolmogorov probability functions defined on L (chapter 7). ——Chapter 1, which deals with the familiar notions of truth value functions, tautologies, consequence relations and relations of logical opposition, is naturally the shortest chapter of part 1.——In chapter 2, the elementary classical semantics of L is extended to the elementary Kolmogorov probability semantics of L, i.e. to an axiomatic theory of unary and of binary Kolmogorov probability functions defined on the set of formulae of L. Because of the elementary character of this theory, chapter 2 is also rather short.——Chapter 3 introduces the first theory on dependence relations, to wit: Kdd, the theory of deductive (in)dependence between formulae of L. I follow here the well-known idea of Popper and Miller, who have used it in a famous discussion on the nature of probabilistic support for their arguments that probabilistic support is deductive, not inductive. I develop Kdd in the form of about 100 theorems, making ample use of the fact that deductive independence is nothing but subcontrary opposition, and close with a remark on the fundamental difference between deductive and logical dependence—two relations the ideas of which are all too easily mixed up.——Chapters 4 and 5 deal extensively with the traditional ideas of probabilistic (in)dependence, applied to formulae rather than to events. As always, I proceed axiomatically in a step-by-step process under systematic viewpoints and obtain about 300 theorems in this way. In the formulation of the theorems, I took special care to state clearly and expressly so-called tacit assumptions, especially those concerning the probability values of the formulae said to be dependent on each other. These assumptions are usually missing in the literature, due either to economy of writing or to sloppiness of thinking. Presumably, both chapters contain little that is new, their value lying more in the systematic grouping and organic development of the theorems than in the newness of these.——In chapter 6, I extend the axiomatic theory about probabilistic (in)dependence which has been elaborated in chapter 5, to an axiomatic theory of inductive (in)dependence by requiring of the relation of inductive (in)dependence that it be probabilistic (in)dependence, but not also logical implication or logical opposition. I point out the differences between probabilistic and inductive (in)dependence by means of some 60 theorems and close my examination of inductive (in)dependence by considering its relationship to the notion of support in the philosophy of science.——Finally, in chapter 7, the last of part 1, I take the step from inductive dependence to what I call ‘purely inductive dependence’ by combining the idea of inductive dependence with that of deductive independence in a way which is suggested by writings of Popper and Miller. I arrive at two noteworthy theorems. Firstly, there is indeed no purely inductive support. But secondly, and perhaps amazingly, countersupport is purely inductive.————Whereas the probabilistic framework of part 1 of the present work is Kolmogorov probability semantics, the framework of part 2 is Popper probability semantics, which is not only worth examining as a fascinating alternative to orthodox Kolmogorov probability semantics, but also allows us to examine dependence relations more deeply, than Kolmogorov probability semantics does. Part 2 leads—again in an axiomatic step-by-step development—from the basic Popper probability semantics of L, called ‘Pb’, (chapter 8) via a probabilistic theory of logical attributes, called ‘Ps’, (chapter 9) to four axiomatic semantical theories of dependence relations between the formulae of L. First, Ps is extended to a theory, called ‘Pdd’, of the relations of deductive dependence and deductive independence between formulae of L (chapter 10). Then Pdd is extended to a theory, called ‘Ppd’, of (in all) 20 relations of probabilistic (in)dependence, relativized to binary Popper probability functions defined on L (chapter 11). Ppd, in its turn, is extended to a theory, called ‘Pid’, of (in all) 10 relations of inductive dependence, again relativized to binary Popper probability functions defined on L (first part of chapter 12). Finally, Pid is extended to a theory, called ‘Ppid’, of (in all) 15 relations of purely inductive positive and negative dependence, relativized to binary Popper probability functions defined on L (second part of chapter 12).——Chapter 8, the first chapter of part 2 of the present work, is entirely preparatory. It introduces the axioms and about 180 theorems (150 of them together with their proofs) of basic Popper probability semantics in order to set this kind of semantics under way.——Then, in chapter 9, basic Popper probability semantics is extended to a probabilistic theory of logical properties of and relations between the formulae of L. Although I think that the way I did this extension is of some interest in itself, the main task of chapter 9 is again a preparatory one: to yield the indispensable lemmata (about 90 in number) for the theorems concerning probabilistic dependence relations in chapter 11 and concerning inductive dependence relations in chapter 12.——Chapter 10 brings the extension of Ps to the theory Pdd of deductive (in)dependence. Only half a dozen theorems are noted here for later use in the Pdd-extensions Ppd and Ppid. In view of the over 100 theorems already gained on this topic in the Kolmogorovian framework (cf. chapter 3), a similar extensive elaboration of Pdd would have been superfluous.——Chapter 11 is the most important one of part 2. It consists of a systematic comparison of 20 probabilistic (in)dependence concepts by means of about 230 theorems, obtained within the axiomatic theory Ppd, which is built up as an extension of Pdd. The main points of comparison were: differences in logical strength; reflexivity and symmetry; behaviour under the condition that the probability values of the formulae in question are extreme. It turned out that each of the examined concepts violates a strong and straightforward version of the intuitive requirement that probabilistic dependence should go with logical dependence. Whereas the corresponding chapter 5 in part 1 of the present work may not have led to new theorems, chapter 11 yields dozens of them in the process of comparison of concepts of dependence and independence which had—as far as I know—never before been treated in a single theoretical framework. With Popper probability semantics, this framework has become available, and here I have simply made full use of it.——In chapter 12, I extend the theory Ppd of probabilistic (in)dependence to the theories Pid and Ppid of inductive and purely inductive dependence, in a way very similar to that in which I have extended the theory Kpd2 to the theories Kid and Kpid in chapters 6 and 7. The first main result of Kpid (roughly: there is no purely inductive support) could be repeated for four of the five purely inductive positive dependence relations considered in chapter 12, whereas the second main result of Kpid (roughly: there is purely inductive countersupport ) could be repeated for each of the five examined purely inductive negative dependence relations. Chapter 12 closes with a brief recapitulation and critical discussion of the main results. (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)

In these prefatory remarks, which are designed to locate my topic within the complex and wide-stretching field of Popper's thought and writings, I shall not say anything that those familiar with his work will not already know. Moreover, what I do say will take as understood many of the problems and theories, not to mention the terminology, that I shall later be doing my best to make understandable. My apologies are therefore due equally to those who know something about Popper's (...) discussions of indeterminism and of the propensity interpretation of probability, and to those who know nothing. (shrink)

Popper's writings cover a remarkably wide range of subjects. The spectrum runs from Plato's theory of politics to the foundations of quantum mechanics. Yet even amidst this variety the philosophy of probability occupies a prominent place. David Miller once pointed out to me that more than half of Popper's The Logic of Scientific Discovery is taken up with discussions of probability. I checked this claim using the 1972 6th revised impression of The Logic of Scientific Discovery , and found that (...) of the approximately 450 pages of text, approximately 250 are to do with probability. Thus Miller's claim is amply justified. It seems indeed that the philosophy of probability was one of Popper's favourite subjects, and, as we shall see, Popper certainly enriched the field with several striking innovations. In this area, as in others, Popper held very definite views, and criticized his opponents in no uncertain terms. Popper was an objectivist and anti-Bayesian, and his criticisms were directed against subjectivism and Bayesianism. (shrink)

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)

Shown here is that a constraint used by Popper in The Logic of Scientific Discovery (1959) for calculating the absolute probability of a universal quantification, and one introduced by Stalnaker in "Probability and Conditionals" (1970, 70) for calculating the relative probability of a negation, are too weak for the job. The constraint wanted in the first case is in Bendall (1979) and that wanted in the second case is in Popper (1959).

Karl Popper discovered in 1938 that the unconditional probability of a conditional of the form ‘If A, then B’ normally exceeds the conditional probability of B given A, provided that ‘If A, then B’ is taken to mean the same as ‘Not (A and not B)’. So it was clear (but presumably only to him at that time) that the conditional probability of B given A cannot be reduced to the unconditional probability of the material conditional ‘If A, then B’. (...) I describe how this insight was developed in Popper’s writings and I add to this historical study a logical one, in which I compare laws of excess in Kolmogorov probability theory with laws of excess in Popper probability theory. (shrink)

In the first part of this work(1) we have explicated the spacetime structure of the probabilistic organization of quantum mechanics. We have shown that each quantum mechanical state, in consequence of the spacetime characteristics of the epistemic operations by which the observer produces the state to be studied and the processes of qualification of these, brings in a tree-like spacetime structure, a “quantum mechanical probability tree,” thattransgresses the theory of probabilities as it now stands. In this second part we develop (...) the general implications of these results.Starting from the lowest level of cognitive action and creating an appropriate symbolism, we construct a “relativizing epistemic syntax,” a “general method of relativized conceptualization” where—systematically—each description is explicitly referred to the epistemic operations by which the observer produces the entity to be described and obtains qualifications of it. The method generates a typology of increasingly complex relativized descriptions where the question of realism admits of a particularly clear pronouncement. Inside this typology the epistemic processes that lie—UNIVERSALLY—at thebasis ofany conceptualization, reveal a tree-like spacetime structure. It appears in particular that the spacetime structure of the relativized representation of aprobabilistic description, which transgresses the nowadays theory of probabilities,is the general mould of which the quantum mechanical probability trees are only particular realizations. This entails a clear definition of the descriptional status of quantum mechanics. While the recognition of theuniversal cognitive content of the quantum mechanical formalism opens up vistas towardmathematical developments of the relativizing epistemic syntax.The relativized representation of a probabilistic description leads with inner necessity to a “morphic” interpretation of probabilities thatcan be regarded as a formalized and deepening elaboration of Sir Karl Popper's “propensity” interpretation. A functional is then constructed, the “opacity functional,” that associatesa mathematical expression to the Popperian “propensities”. Furthermore the opacity functional produces adeductive definition of Shannon's “informational entropy.” Thereby there appears an explicitly unified relativized probabilistic-informational approach. This sketches out a second branch of a future mathematical epistemic syntax, to be connected with the branch stemming from quantum mechanics.The problem of the objectivity of probabilistic descriptions acquires certain precise rephrasings and—in a sense—solution. (shrink)

Popper's paradox of ideal evidence has long been viewed as a telling criticism of Keynes's logical theory of probability and its associated concept of the weight of argument. This paper shows that a simple addition to Keynes's definitions of irrelevance enables his theory to elude the paradox with ease. The modified definition draws on ideas already present in Keynes's Treatise on Probability (1973). As a consequence, relevant evidence and the weight of argument may increase, even when new evidence leaves the (...) probability unaltered. (shrink)

The propensity interpretation of probability, bred by Popper in 1957(K. R. Popper, in Observation and Interpretation in the Philosophy of Physics,S. Körner, ed. (Butterworth, London, 1957, and Dover, New York, 1962), p. 65; reprinted in Popper Selections,D. W. Miller, ed. (Princeton University Press, Princeton, 1985), p. 199) from pure frequency stock, is the only extant objectivist account that provides any proper understanding of single-case probabilities as well as of probabilities in ensembles and in the long run. In Sec. 1 of (...) this paper I recall salient points of the frequency interpretations of von Mises and of Popper himself, and in Sec. 2 I filter out from Popper's numerous expositions of the propensity interpretation its most interesting and fertile strain. I then go on to assess it. First I defend it, in Sec. 3, against recent criticisms(P. Humphreys, Philos. Rev.94,557 (1985); P. Milne, Erkenntnis25,129 (1986)) to the effect that conditional [or relative] probabilities, unlike absolute probabilities, can only rarely be made sense of as propensities. I then challenge its predominance, in Sec. 4, by outlining a rival theory: an irreproachably objectivist theory of probability, fully applicable to the single case, that interprets physical probabilities asinstantaneous frequencies. (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)

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)

Kolmogorov's account in his [1933] of an absolute probability space presupposes given a Boolean algebra, and so does Rényi's account in his [1955] and [1964] of a relative probability space. Anxious to prove probability theory ‘autonomous’. Popper supplied in his [1955] and [1957] accounts of probability spaces of which Boolean algebras are not and [1957] accounts of probability spaces of which fields are not prerequisites but byproducts instead.1 I review the accounts in question, showing how Popper's issue from and how (...) they differ from Kolmogorov's and Rényi's, and I examine on closing Popper's notion of ‘autonomous independence’. So as not to interrupt the exposition, I allow myself in the main text but a few proofs, relegating others to the Appendix and indicating as I go along where in the literature the rest can be found. (shrink)

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)

I survey the difficulties of several probabilistic views of non-deductive argument and of inductive probability and propose to explicate non-deductive reasoning in terms of rational preference. Following a critical examination of Popper’s allegedly deductive theory of rational preference, I draw upon the work of Popper and Rescher to present my view which includes: (i) the conjecturing of a set of alternative answers to or theories or hypotheses about the questions prompting the inquiry and (ii) the “reduction” of this set via (...) plausibilistic principles of rational preference. (shrink)

The concepts of supportive evidence and of relevant evidence seem very closely related to each other. Supportive evidence is clearly always relevant as well. But must relevant evidence be defined as evidence which is either supportive or weakeking? In an explicit or implicit manner, this is indeed the position of many philosophers. The paradox of ideal evidence, however, shows us that this is to restrictive. Besides increasing or decreasing the probability attached to some hypothesis, evidence can alter or interact with (...) the background assumptions underlying the hypothesis.In most circumstances, the (post hoc) relevance of evidence can indeed be judged by its effect on the confidence one attaches to hypotheses. Occasionally, as in the circumstances described by Example I, and more generally called “the Paradox of Ideal Evidence”, the relevance of evidence to an hypothesis can only be understood by appeal to a broader sense. (shrink)

Stove argues that Popper and his successors in the philosophy of science, Kuhn, Lakatos and Feyerabend, were irrationalists because they were deductivists. That is, they believed all logic is deductive, and thus denied that experimental evidence could make scientific theories logically more probable. The book was reprinted as Anything Goes (1998) and Scientific Irrationalism: Origins of a Postmodern Cult (1998).