An early, very preliminary edition of this book was circulated in 1962 under the title Set-theoretical Structures in Science. There are many reasons for maintaining that such structures play a role in the philosophy of science. Perhaps the best is that they provide the right setting for investigating problems of representation and invariance in any systematic part of science, past or present. Examples are easy to cite. Sophisticated analysis of the nature of representation in perception is to be found already (...) in Plato and Aristotle. One of the great intellectual triumphs of the nineteenth century was the mechanical explanation of such familiar concepts as temperature and pressure by their representation in terms of the motion of particles. A more disturbing change of viewpoint was the realization at the beginning of the twentieth century that the separate invariant properties of space and time must be replaced by the space-time invariants of Einstein's special relativity. Another example, the focus of the longest chapter in this book, is controversy extending over several centuries on the proper representation of probability. The six major positions on this question are critically examined. Topics covered in other chapters include an unusually detailed treatment of theoretical and experimental work on visual space, the two senses of invariance represented by weak and strong reversibility of causal processes, and the representation of hidden variables in quantum mechanics. The final chapter concentrates on different kinds of representations of language, concluding with some empirical results on brain-wave representations of words and sentences. (shrink)

Baker claims to provide an example of mathematical explanation of an empirical phenomenon which leads to ontological commitment to mathematical objects. This is meant to show that the positing of mathematical entities is necessary for satisfactory scientific explanations and thus that the application of mathematics to science can be used, at least in some cases, to support mathematical realism. In this paper I show that the example of explanation Baker considers can actually be given without postulating mathematical objects and thus (...) cannot be used by the mathematical realist. I also show that, despite this, mathematics keeps playing an important methodological role in the explanation and does not reduce to a merely computational or descriptive framework. (shrink)

Part II: Suppes and the mature theory. Representation and uniqueness.

The purpose of this paper is to improve on the logical and measure-theoretic foundations for the notion of probability in the law of evidence, which were given in my contributions Åqvist [ (1990) Logical analysis of epistemic modality: an explication of the Bolding–Ekelöf degrees of evidential strength. In: Klami HT (ed) Rätt och Sanning (Law and Truth. A symposium on legal proof-theory in Uppsala May 1989). Iustus Förlag, Uppsala, pp 43–54; (1992) Towards a logical theory of legal evidence: semantic analysis (...) of the Bolding–Ekelöf degrees of evidential strength. In: Martino AA (ed) Expert systems in law. Elsevier Science Publishers BV, Amsterdam, North-Holland, pp 67–86]. The present approach agrees with the one adopted in those contributions in taking its main task to be that of providing a semantic analysis, or explication, of the so called Bolding–Ekelöf degrees of evidential strength (“proof-strength”) as applied to the establishment of matters of fact in law-courts. However, it differs from the one advocated in our earlier work on the subject in explicitly appealing to what is known as “Pro-et-Contra Argumentation”, after the famous Norwegian philosopher Arne Naess. It tries to bring out the logical form of that interesting kind of reasoning, at least in the context of the law of evidence. The formal techniques used here will be seen to be largely inspired by the important work done by Patrick Suppes, notably Suppes [(1957) Introduction to logic. van Nostrand, Princeton and (1972) Finite equal-interval measurement structures. Theoria 38:45–63]. (shrink)

The paper presents an infinite hierarchy PR m [ m = 1, 2, . . . ] of sound and complete axiomatic systems for modal logic with graded probabilistic modalities , which are to reflect what I have elsewhere called the Bolding-Ekelöf degrees of evidential strength as applied to the establishment of matters of fact in law-courts. Our present approach is seen to differ from earlier work by the author in that it treats the logic of these graded modalities not (...) only from a semantical or model-theoretic viewpoint but from a prooftheoretical and axiomatic stance as well. A paramount feature of the approach is its use of so-called systematic frame constants as labels of diverse grades of probability. Apart from this novel feature our approach can be seen to go back to pioneering work by Lou Goble in 1970. (shrink)

Theoria , the international Swedish philosophy journal, was founded in 1935. Its contributors in the first 75 years include the major Swedish philosophers from this period and in addition a long list of international philosophers, including A. J. Ayer, C. D. Broad, Ernst Cassirer, Hector Neri Castañeda, Arthur C. Danto, Donald Davidson, Nelson Goodman, R. M. Hare, Carl G. Hempel, Jaakko Hintikka, Saul Kripke, Henry E. Kyburg, Keith Lehrer, Isaac Levi, David Lewis, Gerald MacCallum, Richard Montague, Otto Neurath, Arthur N. (...) Prior, W. V. Quine, Nicholas Rescher, Ernest Sosa, Robert C. Stalnaker, P. F. Strawson, Patrick Suppes, Johan van Benthem, Georg Henrik von Wright and many others. Hempel's confirmation paradoxes, Ross's deontic paradox, Montague's universal grammar and Lindström's theorem are among the contributions to philosophy that were first published in Theoria. (shrink)

It has been argued that the attempt to meet indispensability arguments for realism in mathematics, by appeal to counterfactual statements, presupposes a view of mathematical modality according to which even though mathematical entities do not exist, they might have existed. But I have sought to defend this controversial view of mathematical modality from various objections derived from the fact that the existence or nonexistence of mathematical objects makes no difference to the arrangement of concrete objects. This defense of the controversial (...) view of mathematical modality obviously falls far short of a full endorsement of the counterfactual approach, but I hope my remarks may serve to help keep such an approach a live option. (shrink)