Popper functions allow one to take conditional probabilities as primitive instead of deriving them from unconditional probabilities via the ratio formula P=P/P. A major advantage of this approach is it allows one to condition on events of zero probability. I will show that under plausible symmetry conditions, Popper functions often fail to do what they were supposed to do. For instance, suppose we want to define the Popper function for an isometrically invariant case in two (...) dimensions and hence require the Popper function to be rotationally invariant and defined on pairs of sets from some algebra that contains at least all countable subsets. Then it turns out that the Popper function trivializes for all finite sets: P=1 for all A ) if B is finite. Likewise, Popper functions invariant under all sequence reflections can’t be defined in a way that models a bidirectionally infinite sequence of independent coin tosses. (shrink)

Rational consequence relations and Popper functions provide logics for reasoning under uncertainty, the former purely qualitative, the latter probabilistic. But few researchers seem to be aware of the close connection between these two logics. I’ll show that Popper functions are probabilistic versions of rational consequence relations. I’ll not assume that the reader is familiar with either logic. I present them, and explicate the relationship between them, from the ground up. I’ll also present alternative axiomatizations for each (...) logic, showing them to depend on weaker axioms than usually recognized. (shrink)

A brief historical comment on scientific knowledge as Socratic ignorance -- Some critical comments on the text of this book, particularly on the theory of truth Exposition [1933] -- Problem of Induction (Experience and Hypothesis) -- Two Fundamental Problems of the Theory of Knowledge -- Formulation of the Problem -- The problem of induction and the problem of demarcation -- Deductivtsm and Inductivism -- Comments on how the solutions are reached and preliminary presentation of the solutions -- Rationalism and empiricism-deductivism (...) and inductivism -- The possibility of a deductivist psychology of knowledge -- The Problem of Induction -- The infinite regression (Hume's argument) --The inductivist positions -- The Normal-Statement Positions -- The normal-statement positions: naive inductivism, strict positivism and apriorism -- Critique of strict positivism - twofold transcendence of natural laws -- The transcendental method - presentation of apriorism -- Critique of apriorism -- Kant and Fries -- Supplement to the critique of apriorism. (Psychologism and transcendentalism in Kant and Fries.-On the question of the empirical basis.) -- Probability Positions -- The probability positions - subjective belief in probability -- Statements about the objective probability of events -- Probability as an objective degree of validity of universal empirical statements -- One way of more closely defining the concept of the probability of a hypothesis (primary and secondary probability of hypotheses). The concept of simplicity -- The concept of the corroboration of a hypothesis - positivist, pragmatist and probabilistic interpretations of the concept of corroboration -- The infinite regression of probability statements -- Pseudo-Statement Positions -- The pseudo-statement positions: new formulation of the problem -- Natural laws as "instructions for the formation of statements" -- "True - false" or "useful - useless"? Consistent pragmatism --Difficulties of consistent pragmatism -- Tool and schema as purely pragmatic constructs -- Natural laws as propositional functions -- Conventionalism -- The pseudo-statement positions will temporarily be put away: conventionalism -- The three interpretations of axiomatic systems. (The circle of problems surrounding conventionalism) -- Conventionalist implicit and explicit definitions Propositional function and propositional equation -- Conventionalist propositional equations as tautological general implications -- Can axiomatic-deductive systems also be understood as consequence classes of pure propositional functions (of pseudo-statements)? -- The coordinative definitions of empiricism: synthetic general implications -- Conventionalist and empiricist interpretations, illustrated by the example of applied geometry -- Strictly Universal Statements and Singular Statements -- Implication and general implication -- General implication and the distinction between strictly universal and singular statements -- Universal concept and individual concept-class and element -- Strictly universal statements-the problem of induction and the problem of universals -- Comments on the problem of universals -- Back to the Pseudo-Statement Positions -- Return to the discussion of the pseudo-statement positions -- Symmetry or asymmetry in the evaluation of natural laws? -- The negative evaluation of universal statements. Critique of the strictly symmetrical interpretation of pseudo-statements -- An infinite regression of pseudo-statements -- An apriorist pseudo-statement position -- Interpretation of the critique up to this point; comments on the unity of theory and practice -- A last chance for the pseudo-statement positions -- Pseudo-Statement Positions and the Concept of Meaning -- The concept of meaning and logical positivism -- The concept of meaning and the demarcation problem-the fundamental thesis of inductivism -- Critique of the inductivist dogma of meaning -- Fully decidable and partially decidable empirical statements-the antinomy of the knowability of the world. (Conclusion of the critique of the pseudo-statement positions.) -- The dialectical and the transcendental corroboration of the solution -- Is the problem of induction solved? (shrink)

The relation between body and mind is one of the oldest riddles that has puzzled mankind. That material and mental events may interact is accepted even by the law: our mental capacity to concentrate on the task can be seriously reduced by drugs. Physical and chemical processes may act upon the mind; and when we are writing a difficult letter, our mind acts upon our body and, through a chain of physical events, upon the mind of the recipient of the (...) letter. This is what the authors of this book call the 'interaction of mental and physical events'. We know very little about this interaction; and according to recent philosophical fashions this is explained by the alleged fact that we have brains but no thoughts. The authors of this book stress that they cannot solve the body mind problem; but they hope that they have been able to shed new light on it. Eccles especially with his theory that the brain is a detector and amplifier; a theory that has given rise to important new developments, including new and exciting experiments; and Popper with his highly controversial theory of 'World 3'. They show that certain fashionable solutions which have been offered fail to understand the seriousness of the problems of the emergence of life, or consciousness and of the creativity of our minds. In Part I, Popper discusses the philosophical issue between dualist or even pluralist interaction on the one side, and materialism and parallelism on the other. There is also a historical review of these issues. In Part II, Eccles examines the mind from the neurological standpoint: the structure of the brain and its functional performance under normal as well as abnormal circumstances. The result is a radical and intriguing hypothesis on the interaction between mental events and detailed neurological occurrences in the cerebral cortex. Part III, based on twelve recorded conversations, reflects the exciting exchange between the authors as they attempt to come to terms with their opinions. (shrink)

This paper uses Popper's treatment of probability and an epistemic constraint on probability assignments to conditionals to extend the Bayesian representation of rational belief so that revision of previously accepted evidence is allowed for. Results of this extension include an epistemic semantics for Lewis' theory of counterfactual conditionals and a representation for one kind of conceptual change.

Despite the results of David Lewis, Peter Gärdenfors, and others, showing that imaging and classical conditionalization coincide only in the most trivial probabilistic models of belief revision, it turns out that imaging on a proposition A can always be described via Popper function conditionalization on a proposition that entails A. This result generalizes to any method of belief revision meeting certain minimal requirements. The proof is illustrated by an application of imaging in the context of the Monty Hall Problem.

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 is famous for favoring an open society, one in which the individual is treated as an end in himself and social arrangements are subjected to critical evaluation, which he defends largely by appeal to a Kantian ethic of respecting the dignity of rational beings. In this essay, I consider for the first time what the implications of a characteristically African ethic, instead prescribing respect for our capacity to relate communally, are for how the state should operate in (...) an open society. I argue that, while an Afro-communal moral foundation does not prescribe a closed society, it supports an open society politics of a sort different from the one that Popper specifies. For Popper, the state in an open society should improve social arrangements albeit without seeking to promote objective goods, should protect rights that merely serve the function of facilitating individual choice, and should employ majoritarian democracy to be able to avoid unwelcome rulers and policies. On all three counts, I show that a relational ethic typical of the African philosophical tradition entails different, intuitively attractive approaches to politics. This chapter is reprinted from an article that initially appeared in Jus Cogens (2020). (shrink)

This article shows that Popper’s measure of corroboration is inapplicable if, as Popper argued, the logical probability of synthetic universal statements is zero relative to any evidence that we might possess. It goes on to show that Popper’s definition of degree of testability, in terms of degree of logical content, suffers from a similar problem. 1 The Corroboration Function and P(h|b) 2 Degrees of Testability and P(h|b).

SummaryAfter presumably cleaning science of induction, Karl Popper claims to offer a purely noninductivist theory of science. In critically evaluating this theory, I focus on the allegedly noninductive character of this theory. First, I defend and expand Wesley Salmon's charge that Popper's dismissal of induction renders science useless for practical purposes. Without induction practitioners have no grounds for believing that the predicted event will actually take place. Second, despite Popper's demands to the contrary, his theory of science (...) is shown to rest on induction. In particular, the function he attributes to background knowledge in testing a scientific hypothesis requires induction. (shrink)

Popper’s theory of World 3 is often regarded as incongruent with his defense of methodological individualism. This article criticizes this widespread view. Methodological individualism is said to be at odds with three crucial assumptions of the theory of World 3: the impossibility of reducing World 3 to subjective mental states because it exists objectively, the view that the mental functions cannot be explained by assuming that individuals are isolated atoms, and the idea that World 3 has causal power (...) and influences both individual minds and actions. This article demonstrates that the inconsistency thesis stems from a confusion between methodological individualism as understood by Popper and reductionism. The reasons for this confusion are analyzed and clarified. It is argued that two variants of methodological individualism can be distinguished, and that unlike psychologistic individualism, Popper’s nonatomistic individualism is fully consistent with his theory of World 3. (shrink)

In the present paper I shall first summarize Popper's criticism of the traditional method of definition, and then go on to comment critically on his own views on the form and function of so-called nominalist definitions.

Karl Popper is famous for favoring an open society, one in which the individual is treated as an end in himself and social arrangements are subjected to critical evaluation, which he defends largely by appeal to a Kantian ethic of respecting the dignity of rational beings. In this essay, I consider for the first time what the implications of a characteristically African ethic, instead prescribing respect for our capacity to relate communally, are for how the state should operate in (...) an open society. I argue that while an under-appreciated Afro-communal moral foundation does not prescribe a closed society, it supports an open society politics and law of a sort different from the one that Popper specifies. For Popper, the state in an open society should improve social arrangements albeit without seeking to promote a particular conception of the good life, should protect rights that merely serve the function of facilitating individual choice, and should employ majoritarian democracy to be able to avoid unwelcome rulers and policies. On all three counts, I show that a relational ethic typical of the African philosophical tradition, but having broad intuitive appeal, entails different, intuitively attractive approaches to politics and law. (shrink)

Popper repeatedly emphasised the significance of a critical attitude, and a related critical method, for scientists. Kuhn, however, thought that unquestioning adherence to the theories of the day is proper; at least for ‘normal scientists’. In short, the former thought that dominant theories should be attacked, whereas the latter thought that they should be developed and defended (for the vast majority of the time). -/- Both seem to have missed a trick, however, due to their apparent insistence that each (...) individual scientist should fulfil similar functions (at any given point in time). The trick is to consider science at the group level; and doing so shows how puzzle solving and ‘offensive’ critical activity can simultaneously have a legitimate place in science. This analysis shifts the focus of the debate. The crucial question becomes ‘How should the balance between functions be struck?’. (shrink)

The functional approach to scientific progress has been mainly developed by Kuhn, Lakatos, Popper, Laudan, and more recently by Shan. The basic idea is that science progresses if key functions of science are fulfilled in a better way. This chapter defends the function approach. It begins with an overview of the two old versions of the functional approach by examining the work of Kuhn, Laudan, Popper, and Lakatos. It then argues for Shan’s new functional approach, in which (...) scientific progress is defined as an increase of usefulness of exemplary practices. (shrink)

ZusammenfassungIn diesem Aufsatz vertrete ich die These, dass Poppers Konversion zum naiven Realismus seine eigene Theorie der Falsifikation aushöhlt. Tarskis Definition der Wahrheit ist eine semantische; wir dürfen sie nicht so interpretieren, als ob sie uns ein erkenntnistheoretisches Kriterium fur die Wahrheit lieferte. Poppers Wahrheitsbegriff hat keine semantische Funktion und bietet uns kein brauchbares Kriterium für die Beurteilung von Theorien. Seine Einführung markiert meines Erachtens nur einen Rückzug in die Metaphysik.SummaryIn this paper I have tried to show that Popper's (...) conversion to naive realism undermines his own idea of falsification. Tarski's definition of truth is a semantical notion and therefore should not be interpreted as having something to do with the epistemological question regarding the criteria of truth. Since Poppers notion of truth has no semantical function and does not provide us with a criterion for judging the truth of theories, I think it fosters a retreat into metaphysics.RésuméJ'ai essayé de montrer, dans cet article, que la conversion de Popper au réalisme naïf vide de sa substance sa propre théorie de la falsification. La définition tarskienne de la vérité est une définition sémantique; on ne peut pas ľinterpréter comme fournissant un critère épistémologique de vérité Le concept de vérité de Popper n'a pas de fonction sémantique et n'apporte pas de critère utilisable pour juger les théories. Son introduction ne représente, à mon avis, qu'une régression vers la métaphysique. (shrink)

While discussing his notion of objective knowledge Popper introduces the idea of dispensability of knowing subjects, the autonomy of knowledge and the argumentative function of language. The main claim of our paper is that, on our view, objective knowledge produced by argumentative interaction is not achieved by dispensing knowing subjects but by differentiating between the play and the strategic levels of argumentation, where a distinction should be drawn between a particular concrete knowing subject engaged in that interaction, say Karl, (...) and an arbitrary one. Moreover, the theory of meaning deployed at the level of plays, the so-called local meaning, provides already some kind of objective knowledge, since it is player-independent and it does not reduce to the purely logical one of the strategy level. In fact, despite our criticism we think that Popper's discussion of the notion objective knowledge still deserves to be reflected on, particularly so in view of new results concerning the formalization of Brouwer’s Creating Subject and the dialogical interpretation of the notion of epistemic assumption. (shrink)

How are we to understand the use of probability in Popper’s corroboration function? Popper says logically, but this raises a problem that becomes apparent when his views on logical probability are compared with those of Keynes. Specifically, Popper does not make it clear how we could have access to, or even calculate, probability values in a logical sense. For first, he would likely want to deny the Keynesian distinction between primary and secondary propositions, and the underlying notion (...) of knowledge-by-acquaintance with Neo-Platonic entities. Second, he would presumably reject not only the notion of non-numerical probabilities, but also the claim that the principle of indifference is aught other than a heuristic, given paradoxes such as Bertrand’s. An attempt might be made to solve this problem by appeal to semantic or possible world analyses over the relative truth-values of statements, but this seems to fail due to concerns relating to infinity: if the set of possible worlds in which p is true is infinite, and the set of possible worlds in which q is true is infinite, then no numerical value for P can be reasonably determined by a comparison between those worlds. As such, an all-too-familiar sort of criticism of Popper’s view of science seems to loom large: if the corroboration function only makes sense when the probabilities employed therein are subjective, then what counts as impressive evidence for a theory might seem to be a matter of convention, or even whim. However, we now have a new interpretation of probability at our disposal: the intersubjective interpretation, recently championed by Gillies, according to which probabilities can lie on a spectrum between the aleatory and the epistemic. Might we employ this to preserve the corroboration function, and deliver ourselves from Bayesianism, without succumbing to radical subjectivism? (shrink)

This paper is a supplement to, and provides a proof of principle of, Kuhn vs. Popper on Criticism and Dogmatism in Science: A Resolution at the Group Level. It illustrates how calculations may be performed in order to determine how the balance between different functions in science—such as imaginative, critical, and dogmatic—should be struck, with respect to confirmation (or corroboration) functions and rules of scientific method.

In other words, classically, probabilities add; quantum mechanically, the probability amplitudes add, leading to the presence of the extra product terms in the quantum case. What this means is that in quantum theory, even though always only one of the various outcomes is obtained in any given observation, some aspect of the non -occurring events, represented by the corresponding complex-valued quantum amplitudes, plays a role in determining the overall probabilities. Indeed, the observed quantum interference effects are correctly captured by the (...) quantum statistical description only because of the presence of these product terms. Therefore, in a realistic construal of quantum theory, if we treat the superposed Ψ function as representing the real state of an individual quantum system, these quantum amplitudes need to be given an ontological status. (shrink)

For many, Rorty was a postmodern relativist and Popper was a positivist and Cold War liberal ideologue. The argument developed here rejects such views and explores how Rorty?s work is best understood from a Popperian problem-solving perspective. It is argued that Rorty erred in seeking justification for beliefs, unlike Popper who replaced the search for justification with criticism. Nonetheless, Rorty?s arguments about post-Nietzschean theory and reformism function as important updates to Popper?s arguments about methodological essentialism and piecemeal (...) social engineering, respectively. (shrink)

This paper studies the extent to which probability functions are recursively definable. It proves, in particular, that the (absolute) probability of a statement A is recursively definable from a certain point on, to wit: from the (absolute) probabilities of certain atomic components and conjunctions of atomic components of A on, but to no further extent. And it proves that, generally, the probability of a statement A relative to a statement B is recursively definable from a certain point on, to (...) wit: from the probabilities relative to that very B of certain atomic components and conjunctions of atomic components of A, but again to no further extent. These and other results are extended to the less studied case where A and B are compounded from atomic statements by means of `` ∀ '' as well as `` ∼ '' and "&". The absolute probability functions considered are those of Kolmogorov and Carnap, and the relative ones are those of Kolmogorov, Carnap, Renyi, and Popper. (shrink)

In his article, Jakub Lampart addresses the social, cultural, and historical functions of various forms of knowledge (and of science in particular) as they can be reconstructed on the basis of the few descriptive remarks found in Paul Feyerabend’s works in the three periods of his scholarly career: moderate, transitional, and radical. Lampart interprets Feyerabend’s views on the relationship between knowledge and society as influenced by the following: the early concepts of Karl Popper (in the moderate period), some (...) of the theses of Benjamin L. Whorf and the late Ludwig Wittgenstein (in the radical period). The article also contains: a) a juxtaposition of Feyerabend’s views with the theories of these thinkers; (b) an attempt to explain Feyerabend’s use of the term “ideology”; (c) a description of two trends characterizing different systems of knowledge: isolation and interaction; d) a description of the two types of ideal members of a given tradition: rationalists and pragmatists. (shrink)

I give a probabilistic semantics for modal logic in which modal operators function as quantifiers over Popper functions in probabilistic model sets, thereby generalizing Kripke's semantics for modal logic.

It is natural and important to have a formal representation of plain belief, according to which propositions are held true, or held false, or neither. (In the paper this is called a deterministic representation of epistemic states). And it is of great philosophical importance to have a dynamic account of plain belief. AGM belief revision theory seems to provide such an account, but it founders at the problem of iterated belief revision, since it can generally account only for one step (...) of revision. The paper discusses and rejects two solutions within the confines of AGM theory. It then introduces ranking functions (as I prefer to call them now; in the paper they are still called ordinal conditional functions) as the proper (and, I find, still the best) solution of the problem, proves that conditional independence w.r.t. ranking functions satisfies the so-called graphoid axioms, and proposes general rules of belief change (in close analogy to Jeffrey's generalized probabilistic conditionalization) that encompass revision and contraction as conceived in AGM theory. Indeed, the parallel to probability theory is amazing. Probability theory can profit from ranking theory as well since it is also plagued by the problem of iterated belief revision even if probability measures are conceived as Popper measures (see No. 11). Finally, the theory is compared with predecessors which are numerous and impressive, but somehow failed to explain the all-important conditional ranks in the appropriate way. (shrink)

It has already been argued by Frazer and Boland (1983) that, interpreted in an instrumentalist fashion, Milton Friedman’s well known and much criticized 1953 paper on “The Methodology of Positive Economics”1 proved to be convergent with Popper's falsificationist philosophy of science2. I think that this comparison is flawed. For one can assuredly contest this interpretation in view of the fact that Popper always opposed any kind of instrumentalist philosophy of science3. It is not even clear that what Friedman (...) has to say on the intricate question of the status of theories and on the function of tests has anything to do with what Popper criticizes under the general category of “instrumentalism”. What I will do here is not so much to directly challenge the Frazer-Boland interpretation as to replace it with a completely new comparison of both methodologies in order to get a new look at Friedman's methodological stance. Of course, one has to focus only on the arguments that really are comparable and leave aside those that could easily be proved to be fundamentally different, if not plainly incompatible. But, as I will try to show, the exercise is worthwhile, for this new comparison will force us to reinterpret, at least partially, what seem to me to be the two central tenets of Friedman's methodological doctrine. Of course, in order to find convergence between Friedman's and Popper's views on economic methodology, we have to read.. (shrink)

The acidity function is a thermodynamic quantitative measure of acid strength for non-aqueous and concentrated aqueous Brønsted acids, with acid strength being defined as the extent to which the acid protonates a base of known basicity. The acidity function, which was developed, both theoretically and experimentally, by Louis P. Hammett of Columbia University during the 1930s, has proven useful in the area of physical organic chemistry where it has been used to correlate rates of acid-catalyzed reactions and to quantitate the (...) acidity of superacids, acids with protonating abilities greater than pure sulfuric acid. All Brønsted acids can now be compared using a common measure. Karl Popper’s seminal idea of theory falsification does not apply here because of the many successful applications of the acidity function. Likewise, Thomas Kuhn’s idea of a paradigm shift does not apply here, even though the acidity function concept was revolutionary, because the acidity function is commensurate with classical concepts of acidity. (shrink)

In his magnum opus, The Logic of Scientific Discovery (first published in German in 1934, English translation, 1959), Karl Popper make two fundamental philosophical moves. First, he relocates the center of gravity of the philosophical treatment of science around what he calls the problem of demarcation. This is the problem of distinguishing between science, on the one hand, and everything else on the other. (By contrast, his contemporaries of the Vienna Circle, whose positivism would prove the most influential brand (...) of empiricism of the day, located the center of gravity around the problem of linguistic meaning, and use a criterion according to which a statement is meaningful to the extent that one can identify verification conditions for it.) Popper excludes from science such things as logic, metaphysics, Freudian psychoanalysis, and Marx’s theory of history. Second, Popper propounds the doctrine of falsification, which handles the problem of demarcation, as well as answers David Hume’s shattering attack on science as the premier form of knowledge centuries before. The arguments he mounts for falsification would function also as an attack on any account of the scientific enterprise that, like positivism, adheres to the idea that science progresses logically from instances (given in observation or experience) to the high-order generalizations characteristic of mature scientific theory. In this small space I shall undertake neither to illuminate the nuances of Popper’s position, nor to trace the (numerous) lines of criticism that have accumulated against it some seventy years later. I shall busy myself instead with tracing a trajectory of thought on the subject of scientific reasoning and its relation to individual decision-making, reflecting on Popper’s contribution and on how his legacy might be further enlarged. (shrink)

Whether defended or criticized, Darwinism is generally considered a scientific theory. However, Popper contests this point. According to him, Darwinism (defined as the explanation of evolution by natural selection) is less a scientific theory than a metaphysical research program: a theoretical framework that has a heuristic function for science, but that does not fulfill the necessary condition for any science, namely testability. In terms of deconstructing Darwinism, Popper’s position is therefore interesting. It raises the question of the scientificity (...) of Darwinism. More generally, it raises the question of the scientificity of the debate on the mechanisms of evolution: what type(s) of argument(s) does this debate bring into play? Can this debate be reduced to an opposition between a scientifically founded theory (Darwinism) and other theories of evolution that are purely philosophical, or even ideological? First, I analyze Popper’s position. According to him, Darwinism is rationally defensible, but difficult to test because it is not capable of making precise and sufficiently specific predictions. Darwinism is therefore more metaphysical than scientific, even if it has a heuristic function for science. Moreover, Popper proposes an improvement of Darwinism by asserting that it must give a more important place to the activity of organisms in evolution. In a second step, I discuss Popper’s position, taking into account the criticisms that this position has generated. As regards the non-testability of Darwinism, it appears that the tests proposed in the literature do not meet the defects pointed out by Popper. On the metaphysical character of Darwinism, I discuss Popper’s assertion on two levels: on the possibility of a clear demarcation between science and non-science, and on the value of the demarcation criterion proposed by Popper. Finally, on the improvement of Darwinism, I attempt to underline the originality of the Popperian proposal. (shrink)

Biological self reference idioms in brain-centered or nervous-system-centered self determination of the consious Self reveal an interesting contrast with biological self-determination by immunological self/non-self discrimination. This contrast is both biological and epistemological. In contrast to the consciousness conscious of itself, the immunological self-determination imposes a protective mechanism against self-recognition (Coutinho et al. 1984), which adds to a largely unconscious achievement of the biological Self (Popper 1977; Medawar 1959). The latter viewpoint is in contrast with the immunological Self-determination as an (...) essentially cognitive process as expressed in the analysis of Tauber (1994). Comparison of the immune system in vertebrates and invertebrates, according to new biological insights, has contributed to a better understanding of the relative role of innate (or inherited) immunity versus immunity acquired during each individual life. Also in this respect, immunological self-achievement shows both a striking analogy and a fundamental discrepancy with the activity of the nervous system. The analysis of immunological Self/non-self discrimination versus brain-centered self-determination adds to the understanding of the function paradigm in biological self-reference idioms, especially when regarding the importance of the connectivity notion in both systems. Adopting functional explanatory schemes for understanding immunological self-non-self discrimination, as well as for the understanding of functional mapping of the brain at (conscious) activity (Friston et al. 1993; Frith et al. 1995), forwards the notion of effective/functional connectivity. Network connectivity not only is a primary question in solving the dimensionality question for immunological ‘idiotypic networks’ (Jerne 1974a, b; 1984), it may also have an important value in describing phase transitions in the development of both immune and nervous systems. (shrink)

Peter Munz, a former student of both Popper and Wittgenstein, begins his comparison of the two great twentieth-century philosophers, by explaining that since the demise of positivism there have emerged, broadly speaking, two philosophical options: Wittgenstein, with the absolute relativism of his theory that meaning is a function of language games and that social configurations are determinants of knowledge; and Popper’s evolutionary epistemology – conscious knowledge is a special case of the relationship which exists between all living beings (...) and their environments. Professor Munz examines and rejects the Wittgensteinian position. Instead, _Our Knowledge of the Growth of Knowledge_, first published in 1985, elaborates the potentially fruitful link between Popper’s critical rationalism and Neo-Darwinism. Read in the light of the latter, Popper’s philosophy leads to the transformation of Kant’s Transcendental Idealism into ‘Hypothetical Realism’, whilst the emphasis on the biological orientation of Popper’s thought helps to illumine some difficulties in Popper’s ‘falsificationism’. (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 (...) class='Hi'>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)

One of the most enduring contributions of Sir Karl Popper to the philosophy of science was his deductive approach to the scientific method, as opposed to Hilary Putnam’s absolute faith in science as an inductive process. Popper’s logic of discovery counters the whole inductive procedure that modern science is so often identified with. While the inductive method has generally characterized how scientists commence their work in laboratories, for Popper scientific theories actually start with generalizations inside our mind (...) whose validity the scientific method must test until those come to be falsified. A step further in the scientific method is the function of paradigms that Thomas Kuhn’s revolutionary science has developed. Kuhn’s community and consensus-based approach and Popper’s hypothesis-based approach are both important in the development of science as it is. This paper seeks to show how models of development may be integrated in the above debate in order to derive insightful implications that are crucial to the understanding of economic progress and human development. (shrink)

We relate Popper functions to regular and perfectly additive such non-Archimedean probability functions by means of a representation theorem: every such non-Archimedean probability function is infinitesimally close to some Popper function, and vice versa. We also show that regular and perfectly additive non-Archimedean probability functions can be given a lexicographic representation. Thus Popper functions, a specific kind of non-Archimedean probability functions, and lexicographic probability functions triangulate to the same place: they are (...) in a good sense interchangeable. (shrink)

In this paper we describe two approaches to the revision of probability functions. We assume that a probabilistic state of belief is captured by a counterfactual probability or Popper function, the revision of which determines a new Popper function. We describe methods whereby the original function determines the nature of the revised function. The first is based on a probabilistic extension of Spohn's OCFs, whereas the second exploits the structure implicit in the Popper function itself. This (...) stands in contrast with previous approaches that associate a unique Popper function with each absolute (classical) probability function. We also describe iterated revision using these models. Finally, we consider the point of view that Popper functions may be abstract representations of certain types of absolute probability functions, but we show that our revision methods cannot be naturally interpreted as conditionalization on these functions. (shrink)

We propose a new account of indicative conditionals, giving acceptability and logical closure conditions for them. We start from Adams’ Thesis: the claim that the acceptability of a simple indicative equals the corresponding conditional probability. The Thesis is widely endorsed, but arguably false and refuted by empirical research. To fix it, we submit, we need a relevance constraint: we accept a simple conditional 'If φ, then ψ' to the extent that (i) the conditional probability p(ψ|φ) is high, provided that (ii) (...) φ is relevant for ψ. How (i) should work is well-understood. It is (ii) that holds the key to improve our understanding of conditionals. Our account has (i) a probabilistic component, using Popper functions; (ii) a relevance component, given via an algebraic structure of topics or subject matters. We present a probabilistic logic for simple indicatives, and argue that its (in)validities are both theoretically desirable and in line with empirical results on how people reason with conditionals. (shrink)

The orthodoxy that conditional probabilities reflect what are for a subject evidential bearings is seconded. This significance suggests that there should be principles equating rationally revised probabilities on new information with probabilities reached by conditionalizing on this information. Several principles, two of which are endorsed, are considered. A book is made against a violator of these, and it is argued that there must be something wrong with a person against whom such books can be made. Appendices comment on Popper- (...) class='Hi'>functions, elaborate on bets and odds, and relate dutch books and strategies to conditions of inconsistency (Ramsey's idea) and imperfection. (shrink)

We investigate the discrete (finite) case of the Popper–Renyi theory of conditional probability, introducing discrete conditional probabilistic models for knowledge and conditional belief, and comparing them with the more standard plausibility models. We also consider a related notion, that of safe belief, which is a weak (non-negatively introspective) type of “knowledge”. We develop a probabilistic version of this concept (“degree of safety”) and we analyze its role in games. We completely axiomatize the logic of conditional belief, knowledge and safe (...) belief over conditional probabilistic models. We develop a theory of probabilistic dynamic belief revision, introducing probabilistic “action models” and proposing a notion of probabilistic update product, that comes together with appropriate reduction laws. (shrink)

Some propositions add more information to bodies of propositions than do others. We start with intuitive considerations on qualitative comparisons of information added . Central to these are considerations bearing on conjunctions and on negations. We find that we can discern two distinct, incompatible, notions of information added. From the comparative notions we pass to quantitative measurement of information added. In this we borrow heavily from the literature on quantitative representations of qualitative, comparative conditional probability. We look at two ways (...) to obtain a quantitative conception of information added. One, the most direct, mirrors Bernard Koopman’s construction of conditional probability: by making a strong structural assumption, it leads to a measure that is, transparently, some function of a function P which is, formally, an assignment of conditional probability (in fact, a Popper function). P reverses the information added order and mislocates the natural zero of the scale so some transformation of this scale is needed but the derivation of P falls out so readily that no particular transformation suggests itself. The Cox–Good–Aczél method assumes the existence of a quantitative measure matching the qualitative relation, and builds on the structural constraints to obtain a measure of information that can be rescaled as, formally, an assignment of conditional probability. A classical result of Cantor’s, subsequently strengthened by Debreu, goes some way towards justifying the assumption of the existence of a quantitative scale. What the two approaches give us is a pointer towards a novel interpretation of probability as a rescaling of a measure of information added. (shrink)

A metric approach to Popper's verisimilitude question is proposed which is related to point-free geometry. Indeed, we define the theory of approximate metric spaces whose primitive notions are regions, inclusion relation, minimum distance, and maximum distance between regions. Then, we show that the class of possible scientific theories has the structure of an approximate metric space. So, we can define the verisimilitude of a theory as a function of its (approximate) distance from the truth. This avoids some of the (...) difficulties arising from the known definitions of verisimilitude. (shrink)

In a previous paper I described a range of nonmonotonic conditionals that behave like conditional probability functions at various levels of probabilistic support. These conditionals were defined as semantic relations on an object language for sentential logic. In this paper I extend the most prominent family of these conditionals to a language for predicate logic. My approach to quantifiers is closely related to Hartry Field's probabilistic semantics. Along the way I will show how Field's semantics differs from a substitutional (...) interpretation of quantifiers in crucial ways, and show that Field's approach is closely related to the usual objectual semantics. One of Field's quantifier rules, however, must be significantly modified to be adapted to nonmonotonic conditional semantics. And this modification suggests, in turn, an alternative quantifier rule for probabilistic semantics. (shrink)