The formulae advanced by Popper to calculate the degree of corroboration C of a scientific hypothesis are unsatisfactory in that the probability values required in the computation are often not available. An attempt is made to define a quantitative measure B* in the place of C in which only countable empirical values would be used. This condition is fulfilled in two basic formulae and eq. ), which could be applied to calculate the degree of corroboration. When m successful falsifications of (...) the hypothesis h appear after a sequence of n unsuccessful ones, the pragmatic degree of corroboration B* results from the difference between B and the analogue value of the falsification F* and eq. ). In the event that the hypothesis can be "rescued" through a correction, increased experimental effort must be made with the revision. This would be taken into account through the correction equation ). When the hypothesis cannot be corrected, however, it can no longer exceed the value of 1-F*, even after numerous verification experiments. In the process of deciding whether a hypothesis is "adequately" verified or falsified, a lower threshold ${\rm B}_{{\rm u}}$ und Bₒ must be selected. These values have to be chosen on an intuitive subjective basis. The same applies to the strength of unsuccessful falsifications . This evaluation is expressed by a multiplicative constant a̱ ) or an additive constant kₐ in eq. . Despite the subjectivity of the values a̱, kₐ necessary for the calculation of B*, and of the values ${\rm B}_{{\rm u}}$ , Bₒ for the decision to discontiue the examination of the hypothesis, I feel that amongst the scientists within a particular field of research, a consensus for these values should be possible. Through a quantitative determination of B* the procedure for testing a hypothesis, considered to be a closed loop process, becomes clearer. To estimate the values of B* could not only have an influence upon the decisions of the respective researchers or research groups examining a hypothesis, but also be of assistance in decisions regarding research grants when financial resources are limited. (shrink)

This article revisits Karl Popper’s falsificationist methodology with respect to three tasks. The first is to illuminate and systematize Popper’s methodological views in light of his core epistemological commitments. A second and related objective is to gauge which aspects of falsificationism should be identified as “conventionalist”—a label that Popper himself uses (albeit with qualifications) but that is compromised by and, thus, stands in need of elucidation because of Popper’s idiosyncratic understanding of conventionalism. Third, by elaborating Popper’s virtue-epistemological, dialogical model of (...) rationality, I show how Popper’s conventionalism, fallibilism, and critical rationalism form a coherent system. This system allows Popper to accord science the status of a privileged source of knowledge—without naïve appeal to authority. My systematization of Popper’s views yields an intrinsically and exegetically more satisfactory reading of Popper’s falsificationism than usual presentations. Thanks to its marked flexibility and methodological liberalism, it bypasses many standard objections but still offers sound and relevant methodological advice. With its “virtue-methodological” thrust, Popper sketches an original and promising approach to methodology, the fertility of which deserves further exploration for contemporary debates. (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 B given A. The difference between the two probabilities is significant for the interpretation of conditionals and for the possibility of inductive probability. It can be quantitatively specified in so-called ‘excess laws’ for which Popper appears to have claimed priority. I argue that such a priority claim should be (...) rejected and credit should instead be given, if to anyone, to Boole and Reichenbach instead. (shrink)

A prized property of theories of all kinds is that of generality, of applicability or least relevance to a wide range of circumstances and situations. The purpose of this article is to present a pair of distinctions that suggest that three kinds of generality are to be found in mathematics and logics, not only at some particular period but especially in developments that take place over time: ‘omnipresent’ and ‘multipresent’ theories, and ‘ubiquitous’ notions that form dependent parts, or moments, of (...) theories. The category of ‘facets’ is also introduced, primarily to assess the roles of diagrams and notations in these two disciplines. Various consequences are explored, starting with means of developing applied mathematics, and then reconsidering several established ways of elaborating or appraising theories, such as analogising, revolutions, abstraction, unification, reduction and axiomatisation. The influence of theories already in place upon theory-building is emphasised. The roles in both mathematics and logics of set theory, abstract algebras, metamathematics, and model theory are assessed, along with the different relationships between the two disciplines adopted in algebraic logic and in mathematical logic. Finally, the issue of monism versus pluralism in these two disciplines is rehearsed, and some suggestions are made about the special character of mathematical and logical knowledge, and also the differences between them. Since the article is basically an exercise in historiography, historical examples and case studies are described or noted throughout. (shrink)