On the consistency of a three-valued logical calculus

[This résumé was published in English in Matematicheskii Sbornik along with the article.]The present paper contains an investigation of a three-valued logical calculus (the system) previously described by the author [Recueil Mathématique 4 (46), 2 (1938)].A constructive consistence proof is given for a part of this calculus rendering the results previously published concerning the Russell paradox. A method for a non-constructive completeness proof for the complete calculus is briefly indicated.     

Paradoxical hypodoxes

Synthese - Most paradoxes of self-reference have a dual or ‘hypodox’. The Liar paradox (Lr = ‘Lr is false’) has the Truth-Teller (Tt = ‘Tt is true’)....     

On a medieval solution to the liar paradox

In this paper, I examine a solution to the Liar paradox found in the work of Ockham, Burley, and Pseudo-Sherwood. I reject the accounts of this solution offered by modern commentators. I argue that this medieval line suggests a non-hierarchical solution to the Liar, according to which ‘true’ is analysed as an indexical term, and paradox is avoided by minimal restrictions on tokens of ‘true’. In certain respects, this solution resembles the recent approaches of Charles Parsons and Tyler Burge; in other respects, it is related to a suggestion of Gödel. But, as a whole, it suggests an original solution to the Liar paradox, quite unlike any current proposals.     

Indefinite Divisibility

Some hold that the lesson of Russell’s paradox and its relatives is that mathematical reality does not form a ‘definite totality’ but rather is ‘indefinitely extensible’. There can always be more sets than there ever are. I argue that certain contact puzzles are analogous to Russell’s paradox this way: they similarly motivate a vision of physical reality as iteratively generated. In this picture, the divisions of the continuum into smaller parts are ‘potential’ rather than ‘actual’. Besides the intrinsic interest of this metaphysical picture, it has important consequences for the debate over absolute generality. It is often thought that ‘indefinite extensibility’ arguments at best make trouble for mathematical platonists; but the contact arguments show that nominalists face the same kind of difficulty, if they recognize even the metaphysical possibility of the picture I sketch.     

In what sense is the no-no paradox a paradox?

Philosophical Studies - Cook regards Sorenson’s so-called ‘the no-no paradox’ as only a kind of ‘meta-paradox’ or ‘quasi-paradox’ because the symmetry...     

Fiction,Pleasurable Tragedy,and the HOT Theory of Consciousness

Abstract

The ‘feeling fiction problem’ asks: is it rational to be moved by what happens to fictional characters? The so-called ‘paradox of tragedy’ is embodied in the question: Why or how is it that we take pleasure in artworks which are clearly designed to cause in us such feelings as sadness and fear? My focus in this paper is to examine these problems from the point of view of the so-called ‘higher-order thought theory of consciousness’ (HOT theory) which says that the best explanation for what makes a mental state conscious is that it is accompanied by a thought that one is in that state. I examine the feeling fiction problem in light of the HOT theory and through a critique of Colin Radford's view. For example, I argue that Radford equivocates in his use of the term ‘aware’ in his response to some of the proposed solutions to the feeling fiction problem. Finally, I show how Susan Feagin's approach to the paradox of tragedy can be analysed and supported by the HOT theory.     

The Mere Addition Paradox,Parity and Vagueness*

Derek Parfit’s mere addition paradox has generated a large literature. This paper articulates one response to this paradox—which Parfit himself suggested—in terms of a formal account of the relation of parity. I term this response the ‘parity view’. It is consistent with transitivity of ‘at least as good as’, but implies incompleteness of this relation. The parity view is compatible with critical‐band utilitarianism if this is adjusted to allow for vagueness. John Broome argues against accounts which involve incompleteness. He thinks they are based on an intuition of ‘neutrality’, which is most naturally understood in terms of equality. There is no rationale, on Broome’s view, for seeing it as ‘incommensurateness’ which leads to incompleteness. Parity provides one. Broome’s worries that ‘incommensurateness’ makes neutrality implausibly ‘greedy’, and that ‘incommensurateness’ and vagueness are incompatible do not constitute a knock‐down case against the parity view. Similar worries arise for his preferred vagueness view.     

Review of True Competition: A Guide to Pursuing Excellence in Sport and Society

Stephens and Feezell argue, in ‘The Ideal of the Stoic Sportsman’ (2004), that ‘one need not be a scholar of ancient Greek philosophy to refer to “stoic” conduct or a “stoic” approach to certain matters, because the vocabulary related to this apparently antiquarian view of life has seeped into our common language’. Nonetheless, Stephens and Feezell go on to give a scholarly account of Stoicism as it relates to athletic participation. Their account, in part, takes the form of a distinction between ‘simple Stoicism’ and ‘sophisticated Stoicism’– the former being a common, contemporary grasp of Stoic moral psychology; the latter being a more sophisticated and historically accurate grasp of Stoic moral psychology. In fleshing out their more sophisticated account, they disclose a paradox. Given the Stoic sufficiency thesis – i.e., that the sole (Stoic) good is virtue – the Stoic sportsman must be indifferent to failure or winning. Yet the Stoic sportsman must be sufficiently attached to the athletic experience to use it as a means of developing virtuous states of character. That they dub the paradox the ‘paradox of Stoic detachment’. ‘Curiosity? Paradox? Or psychological incoherence?’ they ask. The aim of the present undertaking is a ‘soft’ critique of Stephens and Feezell – soft, because the critique is not so much a critical rejection of the authors' view tout court. Instead, I aim to point out deficiencies with their account and expand on other points not fully elucidated in it. The most salient point I make is that what they deem paradoxical is not really paradoxical, once there is a more thorough account and clearer grasp of Stoic ‘detachment’.     

Gödelizing the Yablo Sequence

We investigate what happens when ‘truth’ is replaced with ‘provability’ in Yablo’s paradox. By diagonalization, appropriate sequences of sentences can be constructed. Such sequences contain no sentence decided by the background consistent and sufficiently strong arithmetical theory. If the provability predicate satisfies the derivability conditions, each such sentence is provably equivalent to the consistency statement and to the Gödel sentence. Thus each two such sentences are provably equivalent to each other. The same holds for the arithmetization of the existential Yablo paradox. We also look at a formulation which employs Rosser’s provability predicate.     

Embracing Ambiguity: Transpersonal Development and the Phenomenological Tradition

This article offers a perspective on transpersonal development that has been inspired by the phenomenological tradition. This philosophical movement as exemplified by Husserl, Heidegger, and Merleau-Ponty may provide a non-dualistic vision in which human beings participate in both development and no development. Some implications of this paradox are taken forward to indicate a basic open and non-deterministic dimension of our depths which enters ‘nature’ and ‘time’ in unknown ways. In this view, the tension between the ‘personal’ and the ‘transpersonal’ functions in any moment and forms a deep motivation and creative tension in the human heart. How is this tension resolved?     

Telling lies? An exploration of self-deception and bad faith

If self deception is a lie that one tells to oneself, then paradoxically the deceived is also simultaneously the deceiver. This paper considers two contrasting conceptualizations of self-deception: A.R. Mele's () account of cognitive functioning and J.P Sartre's () existential exploration of ‘bad faith’. Both writers seek to resolve the ‘paradox’ of self-deception or bad faith without recourse to the positing of mental partitioning or the Freudian unconscious. Mele uncovers and lays bare the underlying structure of self-deception, revealing it to consist in desire-driven perceptual and interpretational shortcomings. Sartre's analysis of bad faith emerges from his ontology and is bound up with his notions of ‘being-for’ and the tension between personal ‘transcendence’ and ‘facticity’. Finally, the implications for psychotherapy of Mele's and Sartre's accounts are considered.     

Rose's Prevention Paradox

Geoffrey Rose's ‘prevention paradox’ occurs when a population‐based preventative health measure that brings large benefits to the community – such as compulsory seatbelts, a ‘fat tax’, or mass immunisation – offers little to each participating individual. Although the prevention paradox is not obviously a paradox in the sense in which philosophers understand the term, it does raise important normative questions. In particular, should we implement population‐based preventative health measures when the typical individual is not expected to gain from them? After canvassing other attempts to address the paradox, I argue that what is significant about the prevention paradox is that it involves intra‐personal trade‐offs; the costs and benefits of the choice to implement or not implement a preventative health measure fall on the same individuals. The intra‐personal nature of these trade‐offs has two implications. First, the solutions to the paradox proposed by other authors are deficient. Second, the policy choice to not implement some preventative health measures can be normatively justified.     

The Liar Paradox and the Inclosure Schema

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.     

How Braess’ paradox solves Newcomb's problem*

Newcomb's problem is regularly described as a problem arising from equally defensible yet contradictory models of rationality. Braess’ paradox is regularly described as nothing more than the existence of non‐intuitive (but ultimately non‐contradictory) equilibrium points within physical networks of various kinds. Yet it can be shown that Newcomb's problem is structurally identical to Braess’ paradox. Both are instances of a well‐known result in game theory, namely that equilibria of non‐cooperative games are generally Pareto‐inefficient. Newcomb's problem is simply a limiting case in which the number of players equals one. Braess’ paradox is another limiting case in which the ‘players’ need not be assumed to be discrete individuals. The result is that Newcomb's problem is no more difficult to solve than (the easy to solve) Braess’ paradox.     

The lesson of Newcomb’s paradox

In Newcomb’s paradox you can choose to receive either the contents of a particular closed box, or the contents of both that closed box and another one. Before you choose though, an antagonist uses a prediction algorithm to accurately deduce your choice, and uses that deduction to fill the two boxes. The way they do this guarantees that you made the wrong choice. Newcomb’s paradox is that game theory’s expected utility and dominance principles appear to provide conflicting recommendations for what you should choose. Here we show that the conflicting recommendations assume different probabilistic structures relating your choice and the algorithm’s prediction. This resolves the paradox: the reason there appears to be two conflicting recommendations is that the probabilistic structure relating the problem’s random variables is open to two, conflicting interpretations. We then show that the accuracy of the prediction algorithm in Newcomb’s paradox, the focus of much previous work, is irrelevant. We end by showing that Newcomb’s paradox is time-reversal invariant; both the paradox and its resolution are unchanged if the algorithm makes its ‘prediction’ after you make your choice rather than before.     

FINE'S MCTAGGART,TEMPORAL PASSAGE,AND THE A VERSUS B‐DEBATE

I offer an interpretation and a partial defense of Kit Fine's ‘Argument from Passage’, which is situated within his reconstruction of McTaggart's paradox. Fine argues that existing A‐theoretic approaches to passage are no more dynamic, i.e. capture passage no better, than the B‐theory. I argue that this comparative claim is correct. Our intuitive picture of passage, which inclines us towards A‐theories, suggests more than coherent A‐theories can deliver. In Finean terms, the picture requires not only Realism about tensed facts, but also Neutrality, i.e. the tensed facts not being ‘oriented towards’ one privileged time. However unlike Fine, and unlike others who advance McTaggartian arguments, I take McTaggart's paradox to indicate neither the need for a more dynamic theory of passage nor that time does not pass. A more dynamic theory is not to be had: Fine's ‘non‐standard realism’ amounts to no more than a conceptual gesture. But instead of concluding that time does not pass, we should conclude that theories of passage cannot deliver the dynamicity of our intuitive picture. For this reason, a B‐theoretic account of passage that simply identifies passage with the succession of times is a serious contender.     

Idealism and Williams's Semantic Paradox

Bernard Williams's essay ‘Wittgenstein and Idealism’ argues that that the conventionality of language entails the dependence of the truth of sentences and ultimately of corresponding states of affairs as truth‐makers on the existence of thinking subjects. Peter Winch and Colin Lyas try to avoid William's paradox by distinguishing between the existence conditions of a sentence and its assertion. The Winch‐Lyas solution is criticized and a stronger Winch‐Lays resistant version of Williams's paradox is proposed. A more satisfactory countercriticism is given, involving an ineliminable modal fallacy occurring in the paradox inference, that arises because of the argument's invalid combination of categorical and counterfactual assumptions and conclusions.     

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.     

Two-state solution to the lottery paradox

This paper elaborates a new solution to the lottery paradox, according to which the paradox arises only when we lump together two distinct states of being confident that p under one general label of ‘belief that p’. The two-state conjecture is defended on the basis of some recent work on gradable adjectives. The conjecture is supported by independent considerations from the impossibility of constructing the lottery paradox both for risk-tolerating states such as being afraid, hoping or hypothesizing, and for risk-averse, certainty-like states. The new proposal is compared to views within the increasingly popular debate opposing dualists to reductionists with respect to the relation between belief and degrees of belief.

     

Semantic Knowledge,Semantic Guidance,and Kripke's Wittgenstein

Saul Kripke's influential ‘sceptical paradox’ of semantic rule‐following alleges that speakers cannot have any justification for using a word one way rather than another. If it is correct, there can be no such thing as meaning anything by a word. I argue that the paradox fails to undermine meaning. Kripke never adequately motivates its excessively strict standard for the justified use of words. The paradox lacks the resources to show that its standard is truly mandatory or that speakers do not frequently satisfy the well‐motivated competitor I offer. So the paradox fails.