Online research in philosophy

The blame for the semantic and set-theoretic paradoxes is often placed on self-reference and circularity. Some years ago, Yablo [1985; 1993] challenged this diagnosis, by producing a paradox that's liar-like but does not seem to involve circularity. But is Yablo's paradox really non-circular? In a recent paper, Beall [2001] has suggested that there are no means available to refer to Yablo's paradox without invoking descriptions, and since Priest [1997] has shown that any such description is circular, Beall concludes that Yablo's paradox itself is circular. In this paper, we argue that Beall's conclusion is unwarranted, given that (i) descriptions are not the only way to refer to Yablo's paradox, and (ii) we have no reason to believe that because the description involves self-reference, the denotation of that description is also circular. As a result, for all that's been said so far, we have no reason to believe that Yablo's paradox is circular.

The Lottery Paradox and the Preface Paradox both involve the thesis that high probability is sufficient for rational acceptability. The standard solution to these paradoxes denies that rational acceptability is deductively closed. This solution has a number of untoward consequences. The present paper suggests that a better solution to the paradoxes is to replace the thesis that high probability suffices for rational acceptability with a somewhat stricter thesis. This avoids the untoward consequences of the standard solution. The new solution will be defended against a seemingly obvious objection. 1 The paradoxes of rational acceptability 2 The standard solution 3 A new solution to the paradoxes 4 Basic assumptions 5 The new solution defended 6 Conclusion 7 Appendix.

The paradoxes of self-reference are genuinely paradoxical. The liar paradox, Russell’s paradox and their cousins pose enormous difficulties to anyone who seeks to give a comprehensive theory of semantics, or of sets, or of any other domain which allows a modicum of self-reference and a modest number of logical principles. One approach to the paradoxes of self-reference takes these paradoxes as motivating a non-classical theory of logical consequence. Similar logical principles are used in each of the paradoxical inferences. If one or other of these problematic inferences are rejected, we may arrive at a consistent (or at least, a coherent) theory. In this paper I will show that such approaches come at a serious cost. The general approach of using the paradoxes to restrict the class of allowable inferences places severe constraints on the domain of possible propositional logics, and on the kind of metatheory that is appropriate in the study of logic itself. Proof-theoretic and model-theoretic analyses of logical consequence make provide different ways for non-classical responses to the paradoxes to be defeated by revenge problems: the redefinition of logical connectives thought to be ruled out on logical grounds. Non-classical solutions are not the “easy way out” of the paradoxes.

In this paper I present two new paradoxes, a definability paradox (related to the paradoxes of Berry, Richard and König), and a paradox about extensions (related to Russell’s paradox). However, unlike the familiar definability paradoxes and Russell’s paradox, these new paradoxes involve no self-reference or circularity.

Badici [2008] criticizes views of Priest [2002] concerning the Inclosure Schema and the paradoxes of self-reference. This article explains why his criticisms are to be rejected.

Graham Priest (1994) has argued that the following paradoxes all have the same structure: Russell’s Paradox, Burali-Forti’s Paradox, Mirimanoff’s Paradox, König’s Paradox, Berry’s Paradox, Richard’s Paradox, the Liar and Liar Chain Paradoxes, the Knower and Knower Chain Paradoxes, and the Heterological Paradox. Their common structure is given by Russell’s Schema: there is a property φ and function δ such that..

In a recent article, Emil Badici contends that the inclosure schema substantially fails as an analysis of the paradoxes of self-reference because it is question-begging. The main purpose of this note is to show that Badici's critique highlights a necessity condition for the success of dialectic about paradoxes. The inclosure argument respects this condition and remains solvent.

that all the paradoxes of set theory and logic fall under one schema; and (2) hence they should be solved by one kind of solution. This reply addresses both claims, and counters that (1) in fact at least one paradox escapes the schema, and also some apparently 'safe' theorems fall within it; and (2) even for the (considerable) range of paradoxes so captured by the schema, the assumption of a common solution is not obvious; each paradox surely depends upon the theory and context in which it arises. Details of Priest's proposed solution are also sought.

This paper builds on work done by Graham Priest (1994, 1995, 1998b, 2000) but does not presuppose knowledge of that work. Priest established that many paradoxes, which had been traditionally divided into different families, have a structure in common – which he calls the Inclosure Schema – and, correlatively, that these paradoxes demand a uniform solution. The uniform solution favoured by Priest is a Dialetheist one. I show that, with minor modification, the Inclosure Schema becomes sufficiently embracing to exhibit the underlying structure not just of the logico-semantical paradoxes discussed by Priest, but of some metaphysical paradoxes too. The uniform solution advocated here is a non-Dialetheist one. Although this is not the concern of the present paper, I am persuaded by some recent work (Bromand 2002; Simmons 1993, pp.80-2) that Dialetheism, whatever its other virtues, does not deliver a solution to the semantical paradoxes.

In Beyond the Limits of Thought [2002], Graham Priest argues that logical and semantic paradoxes have the same underlying structure (which he calls the Inclosure Schema ). He also argues that, in conjunction with the Principle of Uniform Solution (same kind of paradox, same kind of solution), this is sufficient to 'sink virtually all orthodox solutions to the paradoxes', because the orthodox solutions to the paradoxes are not uniform. I argue that Priest fails to provide a non-question-begging method to 'sink virtually all orthodox solutions', and that the Inclosure Schema cannot be the structure that underlies the Liar paradox. Moreover, Ramsey was right in thinking that logical and semantic paradoxes are paradoxes of different kinds.