Frege's theorem and his logicism

As is well known, Frege gave an explicit definition of number (belonging to some concept) in §68 of his Die Grundlagen der Arithmetik     

On Russell's vulnerability to Russell's paradox

Influenced by G. E. Moore, Russell broke with Idealism towards the end of 1898; but in later years he characterized his meeting Peano in August 1900 as ‘the most important event’ in ‘the most important year in my intellectual life’. While Russell discovered his paradox during his post-Peano period, the question arises whether he was already committed, during his pre-Peano Moorean period, to assumptions from which his paradox may be derived. Peter Hylton has argued that the pre-Peano Russell was thus vulnerable to (at least one version of) Russell's paradox and hence that the paradox exposes a pre-existing difficulty in Russell's Moorean philosophy. Contrary to Hylton, I argue that the Moorean Russell adhered to views which insulated him against the paradox. Further, I argue that Russell became vulnerable to his paradox as a result of changes in his Moorean position occasioned, first, by his acceptance of Cantor's theory of the transfinite, and, second, by his correspondence with Frege. I conclude with some general comments regarding Russell's acceptance of naïve set theory.     

Hilbert,logicism, and mathematical existence

David Hilbert’s early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind’s footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the evolution of Hilbert’s foundational ideas, including his early contributions to the foundations of geometry and the real number system. The context of Dedekind-style logicism makes it possible to offer a new analysis of the emergence of Hilbert’s famous ideas on mathematical existence, now seen as a revision of basic principles of the “naive logic” of sets. At the same time, careful scrutiny of his published and unpublished work around the turn of the century uncovers deep differences between his ideas about consistency proofs before and after 1904. Along the way, we cover topics such as the role of sets and of the dichotomic conception of set theory in Hilbert’s early axiomatics, and offer detailed analyses of Hilbert’s paradox and of his completeness axiom (Vollständigkeitsaxiom).     

Russell's 1925 logic

In a recent series of articles, J. P. Moreland has attempted to revive the idea that bare particulars are indispensable for individuating concrete particulars. The success of the project turns on Moreland's proposal that while bare particulars are indeed 'partially clad'--that is, exemplify at least some properties--they are nevertheless 'bare' in that they lack internal constituents. I argue that 'partially clad' bare particulars (PCBPs) are impervious not only to traditional objections, but also those recently urged in this journal by D. W. Mertz. The real problem with Moreland's view, I contend, is that together with his containment model of predication, it leads to the unwanted conclusion that PCBPs actually contain themselves as constituents, thereby ensnaring them in a vicious (individuative) circularity.     

Epistemic Principles and Epistemic Circularity

Can we show that our senses are reliable sources of information about the world? To show this, we need to establish that most of our perceptual judgments have been true. But we cannot determine these inductive instances without relying upon sense perception. Thus, it seems, we cannot establish the reliability of sense perception by means of an argument without falling into epistemic circularity. In this paper, I argue that this consequence is not an epistemological disaster. For this purpose, I defend a normative claim that it is reasonable to accept the general reliability of our perceptual judgments, instead of a factual claim that our perceptual judgments are generally reliable. More specifically, I offer a normative practical argument which explains why it is reasonable to accept the general reliability of our perceptual judgments, even though we cannot establish the general reliability of our perceptual judgments by means of theoretical reasoning.     

Epistemic Intuitions and Epistemic Contextualism

In this paper I examine the way appeals to pretheoretic intuition are used to support epistemological theses in general and the thesis of epistemic contextualism in particular. After outlining the sceptical puzzle and the contextualist's resolution of that puzzle, I explore the question of whether this solution fits better with our intuitive take on the puzzle than its invariantist rivals. I distinguish two kinds of fit a theory might have with pretheoretic intuitions–accommodation and explanation, and consider whether achieving either kind of fit would be a virtue for a theory. I then examine how contextualism could best claim to accommodate and explain our intuitions, building the best case that 1 can for contextualism, but concluding that there is no reason to accept contextualism either in the way it accommodates nor the way it explains our intuitions about the sceptical puzzle.     

Epistemic Pluralism

In this paper epistemic pluralism concerning knowledge is taken to be the claim that very different facts may constitute knowledge. The paper argues for pluralism by arguing that very different facts can constitute the knowledge-making links between beliefs and facts. If pluralism is right, we need not anxiously seek a unified account of the links between beliefs and facts that partly constitute knowledge in different cases of knowledge. The paper argues that no good reasons have been put forward in favour of believing in a unified maker of knowledge. It then appeals to the role of knowledge in order to argue that we have positive reason to embrace pluralism.     

Epistemic Contextualism

Any contextualist approach to knowledge has to provide a plausible definition of the concept of context and spell out the mechanisms of context changes. Since it is the dynamics of context change that carry the main weight of the contextualist position, not every mechanism will be capable of filling that role. In particular, I argue that one class of mechanisms that is most popularly held to account for context changes, namely those that arise out of shifts of conversational parameters in discourses involving knowledge claims, are not suited to the job because they cannot account for the genuinely epistemic nature of the context shift. A form of epistemic contextualism that defines the context through the structure of our epistemic projects is suggested. Context changes in this account are linked to changes in the background assumptions operative in our epistemic projects and the methods used to carry out our inquiries.