In this paper a theory of finitistic and frequentistic approximations — in short: f-approximations — of probability measures P over a countably infinite outcome space N is developed. The family of subsets of N for which f-approximations converge to a frequency limit forms a pre-Dynkin system $${{D\subseteq\wp(N)}}$$. The limiting probability measure over D can always be extended to a probability measure over $${{\wp(N)}}$$, but this measure is not always σ-additive. We conclude that probability measures can be regarded as idealizations of (...) limiting frequencies if and only if σ-additivity is not assumed as a necessary axiom for probabilities. We prove that σ-additive probability measures can be characterized in terms of so-called canonical and in terms of so-called full f-approximations. We also show that every non-σ-additive probability measure is f-approximable, though neither canonically nor fully f-approximable. Finally, we transfer our results to probability measures on open or closed formulas of first-order languages. (shrink)

_Zettel, _ an en face bilingual edition, collects fragments from Wittgenstein's work between 1929 and 1948 on issues of the mind, mathematics, and language.

G. H. Bird; Discussions: Kantian Myths, Proceedings of the Aristotelian Society, Volume 96, Issue 1, 1 June 1996, Pages 245–252, https://doi.org/10.1093/aristot.

In "Realism and Reason" Hilary Putnam has offered an apparently strong argument that the position of metaphysical realism provides an incoherent model of the relation of a correct scientific theory to the world. However, although Putnam's attack upon the notion of the "intended" interpretation of a scientific theory is sound, it is shown here that realism may be formulated in such a way that the realist need make no appeal to any "intended" interpretation of such a theory. Consequently, it can (...) be shown that realism is immune to Putnam's criticism and that attempts at reformulating this criticism are not likely to meet with success. (shrink)

It is argued that Hempel's original rejection of the prediction criterion of confirmation in [8] (on the grounds that it leads to a circular definition of confirmation) was ill-conceived, and that his own approach exhibits undesirable consequences to the degree that it deviates from this criterion. A version of the prediction criterion is formulated which, in addition to being-non circular, escapes the criticisms advanced against Hempel's satisfaction criterion, offers certain clear advantages over alternative approaches, and may serve as the basis (...) for a theory of qualitative confirmation. The definition of confirmation developed here violates two of Hempel's three criteria of adequacy, and in showing why it should do so some light is shed on various issues in the debate concerning the acceptability of these criteria. (shrink)

Given Conciliar Christology and a compositionalist metaphysics of the incarnation, I explore whether ‘qua-propositions’ are capable of solving the coherence problem in Christology. I do this by probing the metaphysical aspect of qua-propositions, since ‘semantics presupposes metaphysics’. My proposal focuses on the fact that the Word accidentally owns an individual human nature. Due to that individuality, the human properties first characterise the individual human nature and, in a ‘next step’, this individual human nature characterises the Word. I call this ‘stepped (...) characterisation’. Subsequently, I show that stepped characterisation validates the use of qua-propositions in Conciliar Christology. Hence, qua-propositions are not merely ‘muddling the waters of logic’. (shrink)

At first sight, the philosophy of Spinoza may seem wholly alien to what is now generally regarded as philosophy in the English-speaking world. For some decades, the dominant trend in that philosophy has been linguistic and anti-metaphysical; the philosopher is held to be concerned with the analysis of language, and not with speculative system-building. Spinoza, on the other hand, is very much a system-builder; as to the analysis of language, he says explicitly that this is of no interest to him. (...) ‘It is not my intention’, he says, ‘to explain the meanings of words; it is my intention to explain the nature of things’. However, the paper which follows will attempt to show that Spinoza’s philosophy is not wholly without relevance today. It will try to do this by placing one of Spinoza’s most important doctrines, his theory of human freedom, within the context of recent discussions. (shrink)

Early paper on the nature of the a priori in Kant and its relationship to modality.