outrageous remarks about contradictions. Perhaps the most striking remark he makes is that they are not false. This claim first appears in his early notebooks (Wittgenstein 1960, p.108). In the Tractatus, Wittgenstein argued that contradictions (like tautologies) are not statements (Sätze) and hence are not false (or true). This is a consequence of his theory that genuine statements are pictures.
Cet article démontre qu'un exemple cité par Ernest Adams pour montrer que l'implication matérielle n'est pas l'interprétation correcte de la sémantique de la conjonction de subordination si, n'est rien d'autre qu'un corollaire d'une observation d'jà faite par Lewis Carroll, il y a cent ans, dans l'exposition de son paradoxe du salon de coiffure.
The Pinocchio paradox poses one dialetheia too many for semantic dialetheists (Eldridge-Smith 2011). However, Beall (2011) thinks that the Pinocchio scenario is merely an impossible story, like that of the village barber who shaves just those villagers who do not shave themselves. Meanwhile, Beall maintains that Liar paradoxes generate dialetheia. The Barber scenario is self-contradictory, yet the Pinocchio scenario requires a principle of truth for a contradiction. In this and other respects the Pinocchio paradox is a version of (...) the Liar, unlike the Barber. One wonders why some Liars would be impossible if others generate dialetheias. (shrink)
It is proposed that subconscious retro-predictions in conjunction with brain state update cycles are instrumental in the physiological generation of conscious sensations and perceptions, and in all abstract thought. In this paper the hypothesis is supported by conducting a detailed a re-evaluation of the self-referential statements in Set Theory and Formal Logic known as antinomies. This study concludes that the recursive behavior exhibited by abstract enigmas such as "Russell’s Paradox" is analogous to the oscillations typical of bistable perceptual phenomena.
As Russell's paradox of "the set of all sets that do not contain themselves" indicated long ago, matters go seriously amiss if one operates an ontology of unrestricted totalization. Some sort of restriction must be placed on such items as "the set of all sets that have the feature F' or "the conjunction of all truths that have the feature G." But generally, logicians here introduce such formalized and complex devices as the theory of types or the doctrine of impredictivity. (...) The present paper argues for the informal and elementary idea that the items invoked in a proper identification have themselves already been identified. Even as an explanation is not satisfactory that proceeds in terms of items that themselves require prior explanation, so the same holds with identification. And heed of this elementary idea suffices to sideline those otherwise paradoxical perplexities. (shrink)
For Quine, a paradox is an apparently successful argument having as its conclusion a statement or proposition that seems obviously false or absurd. That conclusion he calls the proposition of the paradox in question. What is paradoxical is of course that if the argument is indeed successful as it seems to be, its conclusion must be true. On this view, to resolve the paradox is (1) to show either that (and why) despite appearances the conclusion is true after all, or (...) that the argument is fallacious, and (2) if the former, to explain away the deceptive appearances. Quine divides paradoxes into three groups. A veridical paradox is one whose proposition or conclusion is in fact true despite its air of absurdity. We decide that a paradox is veridical when we look carefully at the argument and it convinces us, i.e., it manages to show us how it is that the conclusion is true after all and appearances to the contrary were misleading. Quine’s two main examples of this are the puzzle of Frederic in The Pirates of Penzance (who has reachError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMapError: Illegal entry in bfrange block in ToUnicode CMaped the age of twenty-one after passing only five birthdays), and the Barber Paradox, which Quine considers simply a sound proof that there can be no such barber as is described.1 A falsidical paradox is one whose proposition or conclusion is indeed obviously false or self-contradictory, but which contains a fallacy that is detectably responsible for delivering the absurd conclusion. We decide that a paradox is falsidical when we look carefully at the argument and spot the fallacy. Quine’s leading example here is De Morgan’s trick argument for the proposition that 2 = 1. (shrink)
Copi, Quine and van Heijenoort have each claimed that there are two fundamentally different kinds of logical paradox; namely, genuine paradoxes like Russell's and pseudo-paradoxes like the Barber of Seville. I want to contest this claim and will present my case in three stages. Firstly, I will characterize the logical paradoxes; state standard versions of three of them; and demonstrate that a symbolic formulation of each leads to a formal contradiction. Secondly, I will discuss the reasons Copi, Quine and (...) van Heijenoort have given for the distinction between genuine and pseudo-paradoxes. Thirdly, I will attempt to explain why there is no such class as the class of all and only those classes which are not members of themselves. (shrink)
Paradoxes and their Resolutions is a ‘thematic compilation’ by Avi Sion. It collects in one volume the essays that he has written in the past (over a period of some 27 years) on this subject. It comprises expositions and resolutions of many (though not all) ancient and modern paradoxes, including: the Protagoras-Euathlus paradox (Athens, 5th Cent. BCE), the Liar paradox and the Sorites paradox (both attributed to Eubulides of Miletus, 4th Cent. BCE), Russell’s paradox (UK, 1901) and its derivatives the (...)Barber paradox and the Master Catalogue paradox (also by Russell), Grelling’s paradox (Germany, 1908), Hempel's paradox of confirmation (USA, 1940s), and Goodman’s paradox of prediction (USA, 1955). This volume also presents and comments on some of the antinomic discourse found in some Buddhist texts (namely, in Nagarjuna, India, 2nd Cent. CE; and in the Diamond Sutra, date unknown, but probably in an early century CE). (shrink)
Fitch’s Paradox shows that if every truth is knowable, then every truth is known. Standard diagnoses identify the factivity/negative infallibility of the knowledge operator and Moorean contradictions as the root source of the result. This paper generalises Fitch’s result to show that such diagnoses are mistaken. In place of factivity/negative infallibility, the weaker assumption of any ‘level-bridging principle’ suffices. A consequence is that the result holds for some logics in which the “Moorean contradiction” commonly thought to underlie the result is (...) in fact consistent. This generalised result improves on the current understanding of Fitch’s result and widens the range of modalities of philosophical interest to which the result might be fruitfully applied. Along the way, we also consider a semantic explanation for Fitch’s result which answers a challenge raised by Kvanvig. (shrink)
The lottery paradox involves a set of judgments that are individually easy, when we think intuitively, but ultimately hard to reconcile with each other, when we think reflectively. Empirical work on the natural representation of probability shows that a range of interestingly different intuitive and reflective processes are deployed when we think about possible outcomes in different contexts. Understanding the shifts in our natural ways of thinking can reduce the sense that the lottery paradox reveals something problematic about our concept (...) of knowledge. However, examining these shifts also raises interesting questions about how we ought to be thinking about possible outcomes in the first place. (shrink)
In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this paper I show that a number of logics are susceptible to a strengthened version of Curry's paradox. This can be adapted to provide a proof theoretic analysis of the omega-inconsistency in Lukasiewicz's continuum valued logic, allowing us to better evaluate which logics are suitable for a naïve truth theory. On this basis I identify two natural subsystems of Lukasiewicz logic which individually, but (...) not jointly, lack the problematic feature. (shrink)
I present a new argument for the repugnant conclusion. The core of the argument is a risky, intrapersonal analogue of the mere addition paradox. The argument is important for three reasons. First, some solutions to Parfit’s original puzzle do not obviously generalize to the intrapersonal puzzle in a plausible way. Second, it raises independently important questions about how to make decisions under uncertainty for the sake of people whose existence might depend on what we do. And, third, it suggests various (...) difficulties for leading views about the value of a person’s life compared to her nonexistence. (shrink)
When you and I seriously argue over whether a man of seventy is old enough to count as an "old man", it seems that we are appealing neither to our own separate standards of oldness nor to a common standard that is already fixed in the language. Instead, it seems that both of us implicitly invoke an ideal, shared standard that has yet to be agreed upon: the place where we ought to draw the line. As with other normative standards, (...) it is hard to know whether such borderlines exist prior to our coming to agree on where they are. But epistemicists plausibly argue that they must exist whether we ever agree on them or not, as this provides the only logically acceptable response to the sorites paradox. This paper argues that such boundaries do typically exist as hypothetical ideals, but not as determinate features of the present actual world. There is in fact no general solution to the paradox, but attention to practice in resolving vague disagreements shows that its instances can be dealt with separately, as they arise, in many reasonable ways. (shrink)
In this paper, we distinguish two versions of Curry's paradox: c-Curry, the standard conditional-Curry paradox, and v-Curry, a validity-involving version of Curry's paradox that isn’t automatically solved by solving c-curry. A uniﬁed treatment of curry paradox thus calls for a uniﬁed treatment of both c-Curry and v-Curry. If, as is often thought, c-Curry paradox is to be solved via non-classical logic, then v-Curry may require a lesson about the structure—indeed, the substructure—of the validity relation itself.
Many ecological economists have argued that some natural capital should be preserved for posterity. Yet, among environmental philosophers, the preservation paradox entails that preserving parts of nature, including those denoted by natural capital, is impossible. The paradox claims that nature is a realm of phenomena independent of intentional human agency, that preserving and restoring nature require intentional human agency, and, therefore, no one can preserve or restore nature (without making it artificial). While this article argues that the preservation paradox is (...) more difficult to resolve than ordinarily recognized, it also concludes by sketching a positive way to understand what it means to preserve natural capital during the Anthropocene. (shrink)
One of the hardest problems in philosophy, one that has been around for over two thousand years without generating any significant consensus on its solution, involves the concept of vagueness: a word or concept that doesn't have a perfectly precise meaning. There is an argument that seems to show that the word or concept simply must have a perfectly precise meaning, as violently counterintuitive as that is. Unfortunately, the argument is usually so compressed that it is difficult to see why (...) exactly the problem is so hard to solve. In this article I attempt to explain just why it is that the problem – the sorites paradox – is so intractable.Export citation. (shrink)
This paper is a response to David Oderberg's discussion of the Tristram Shandy paradox. I defend the claim that the Tristram Shandy paradox does not support the claim that it is impossible that the past is infinite.
The principle of universal instantiation plays a pivotal role both in the derivation of intensional paradoxes such as Prior’s paradox and Kaplan’s paradox and the debate between necessitism and contingentism. We outline a distinctively free logical approach to the intensional paradoxes and note how the free logical outlook allows one to distinguish two different, though allied themes in higher-order necessitism. We examine the costs of this solution and compare it with the more familiar ramificationist approaches to higher-order logic. Our assessment (...) of both approaches is largely pessimistic, and we remain reluctantly inclined to take Prior’s and Kaplan’s derivations at face value. (shrink)
Thomas Kroedel argues that the lottery paradox can be solved by identifying epistemic justification with epistemic permissibility rather than epistemic obligation. According to his permissibility solution, we are permitted to believe of each lottery ticket that it will lose, but since permissions do not agglomerate, it does not follow that we are permitted to have all of these beliefs together, and therefore it also does not follow that we are permitted to believe that all tickets will lose. I present two (...) objections to this solution. First, even if justification itself amounts to no more than epistemic permissibility, the lottery paradox recurs at the level of doxastic obligations unless one adopts an extremely permissive view about suspension of belief that is in tension with our practice of doxastic criticism. Second, even if there are no obligations to believe lottery propositions, the permissibility solution fails because epistemic permissions typically agglomerate, and the lottery case provides no exception to this rule. (shrink)
ABSTRACTMany discussions of the ‘preface paradox’ assume that it is more troubling for deductive closure constraints on rational belief if outright belief is reducible to credence. I show that this is an error: we can generate the problem without assuming such reducibility. All that we need are some very weak normative assumptions about rational relationships between belief and credence. The only view that escapes my way of formulating the problem for the deductive closure constraint is in fact itself a reductive (...) view: namely, the view that outright belief is credence 1. However, I argue that this view is unsustainable. Moreover, my version of the problem turns on no particular theory of evidence or evidential probability, and so cannot be avoided by adopting some revisionary such theory. In sum, deductive closure is in more serious, and more general, trouble than some have thought. (shrink)
The canonical Bayesian solution to the ravens paradox faces a problem: it entails that black non-ravens disconfirm the hypothesis that all ravens are black. I provide a new solution that avoids this problem. On my solution, black ravens confirm that all ravens are black, while non-black non-ravens and black non-ravens are neutral. My approach is grounded in certain relations of epistemic dependence, which, in turn, are grounded in the fact that the kind raven is more natural than the kind black. (...) The solution applies to any generalization “All F’s are G” in which F is more natural than G. (shrink)
The lottery paradox can be solved if epistemic justification is assumed to be a species of permissibility. Given this assumption, the starting point of the paradox can be formulated as the claim that, for each lottery ticket, I am permitted to believe that it will lose. This claim is ambiguous between two readings, depending on the scope of ‘permitted’. On one reading, the claim is false; on another, it is true, but, owing to the general failure of permissibility to agglomerate, (...) does not generate the paradox. The solution generalizes to formulations of the paradox in terms of rational acceptability and doxastic rationality. (shrink)
Harman’s lottery paradox, generalized by Vogel to a number of other cases, involves a curious pattern of intuitive knowledge ascriptions: certain propositions seem easier to know than various higher-probability propositions that are recognized to follow from them. For example, it seems easier to judge that someone knows his car is now on Avenue A, where he parked it an hour ago, than to judge that he knows that it is not the case that his car has been stolen and driven (...) away in the last hour. Contextualists have taken this pattern of intuitions as evidence that ‘knows’ does not always denote the same relationship; subject-sensitive invariantists have taken this pattern of intuitions as evidence that non-traditional factors such as practical interests figure in knowledge; still others have argued that the Harman Vogel pattern gives us a reason to abandon the principle that knowledge is closed under known entailment. This paper argues that there is a psychological explanation of the strange pattern of intuitions, grounded in the manner in which we shift between an automatic or heuristic mode of judgment and a controlled or systematic mode. Understanding the psychology behind the pattern of intuitions enables us to see that the pattern gives us no reason to abandon traditional intellectualist invariantism. The psychological account of the paradox also yields new resources for clarifying and defending the single premise closure principle for knowledge ascriptions. (shrink)
This paper presents an approach to truth and the Liar paradox which combines elements of context dependence and hierarchy. This approach is developed formally, using the techniques of model theory in admissible sets. Special attention is paid to showing how starting with some ideas about context drawn from linguistics and philosophy of language, we can see the Liar sentence to be context dependent. Once this context dependence is properly understood, it is argued, a hierarchical structure emerges which is neither ad (...) hoc nor unnatural. (shrink)
Judgment aggregation theory, or rather, as we conceive of it here, logical aggregation theory generalizes social choice theory by having the aggregation rule bear on judgments of all kinds instead of merely preference judgments. It derives from Kornhauser and Sager’s doctrinal paradox and List and Pettit’s discursive dilemma, two problems that we distinguish emphatically here. The current theory has developed from the discursive dilemma, rather than the doctrinal paradox, and the final objective of the paper is to give the latter (...) its own theoretical development along the line of recent work by Dietrich and Mongin. However, the paper also aims at reviewing logical aggregation theory as such, and it covers impossibility theorems by Dietrich, Dietrich and List, Dokow and Holzman, List and Pettit, Mongin, Nehring and Puppe, Pauly and van Hees, providing a uniform logical framework in which they can be compared with each other. The review goes through three historical stages: the initial paradox and dilemma, the scattered early results on the independence axiom, and the so-called canonical theorem, a collective achievement that provided the theory with its specific method of analysis. The paper goes some way towards philosophical logic, first by briefly connecting the aggregative framework of judgment with the modern philosophy of judgment, and second by thoroughly discussing and axiomatizing the ‘general logic’ built in this framework. (shrink)
Any theory of truth must find a way around Curry’s paradox, and there are well-known ways to do so. This paper concerns an apparently analogous paradox, about validity rather than truth, which JC Beall and Julien Murzi call the v-Curry. They argue that there are reasons to want a common solution to it and the standard Curry paradox, and that this rules out the solutions to the latter offered by most “naive truth theorists.” To this end they recommend a radical (...) solution to both paradoxes, involving a substructural logic, in particular, one without structural contraction. In this paper I argue that substructuralism is unnecessary. Diagnosing the “v-Curry” is complicated because of a multiplicity of readings of the principles it relies on. But these principles are not analogous to the principles of naive truth, and taken together, there is no reading of them that should have much appeal to anyone who has absorbed the morals of both the ordinary Curry paradox and the second incompleteness theorem. (shrink)
I develop an original view of the structure of space---called "infinitesimal atomism"---as a reply to Zeno's paradox of measure. According to this view, space is composed of ultimate parts with infinitesimal size, where infinitesimals are understood within the framework of Robinson's (1966) nonstandard analysis. Notably, this view satisfies a version of additivity: for every region that has a size, its size is the sum of the sizes of its disjoint parts. In particular, the size of a finite region is the (...) sum of the sizes of its infinitesimal parts. Although this view is a coherent approach to Zeno's paradox and is preferable to Skyrms's (1983) infinitesimal approach, it faces both the main problem for the standard view (the problem of unmeasurable regions) and the main problem for finite atomism (Weyl's tile argument), leaving it with no clear advantage over these familiar alternatives. (shrink)
Expressivists explain the expression relation which obtains between sincere moral assertion and the conative or affective attitude thereby expressed by appeal to the relation which obtains between sincere assertion and belief. In fact, they often explicitly take the relation between moral assertion and their favored conative or affective attitude to be exactly the same as the relation between assertion and the belief thereby expressed. If this is correct, then we can use the identity of the expression relation in the two (...) cases to test the expressivist account as a descriptive or hermeneutic account of moral discourse. I formulate one such test, drawing on a standard explanation of Moore's paradox. I show that if expressivism is correct as a descriptive account of moral discourse, then we should expect versions of Moore's paradox where we explicitly deny that we possess certain affective or conative attitudes. I then argue that the constructions that mirror Moore's paradox are not incoherent. It follows that expressivism is either incorrect as a hermeneutic account of moral discourse or that the expression relation which holds between sincere moral assertion and affective or conative attitudes is not identical to the relation which holds between sincere non-moral assertion and belief. A number of objections are canvassed and rejected. (shrink)
The Borel–Kolmogorov Paradox is typically taken to highlight a tension between our intuition that certain conditional probabilities with respect to probability zero conditioning events are well defined and the mathematical definition of conditional probability by Bayes’ formula, which loses its meaning when the conditioning event has probability zero. We argue in this paper that the theory of conditional expectations is the proper mathematical device to conditionalize and that this theory allows conditionalization with respect to probability zero events. The conditional probabilities (...) on probability zero events in the Borel–Kolmogorov Paradox also can be calculated using conditional expectations. The alleged clash arising from the fact that one obtains different values for the conditional probabilities on probability zero events depending on what conditional expectation one uses to calculate them is resolved by showing that the different conditional probabilities obtained using different conditional expectations cannot be interpreted as calculating in different parametrizations of the conditional probabilities of the same event with respect to the same conditioning conditions. We conclude that there is no clash between the correct intuition about what the conditional probabilities with respect to probability zero events are and the technically proper concept of conditionalization via conditional expectations—the Borel–Kolmogorov Paradox is just a pseudo-paradox. (shrink)
We provide a 'verisimilitudinarian' analysis of the well-known Linda paradox or conjunction fallacy, i.e., the fact that most people judge the probability of the conjunctive statement "Linda is a bank teller and is active in the feminist movement" (B & F) as more probable than the isolated statement "Linda is a bank teller" (B), contrary to an uncontroversial principle of probability theory. The basic idea is that experimental participants may judge B & F a better hypothesis about Linda as compared (...) to B because they evaluate B & F as more verisimilar than B. In fact, the hypothesis "feminist bank teller", while less likely to be true than "bank teller", may well be a better approximation to the truth about Linda. (shrink)
The Liar paradox raises foundational questions about logic, language, and truth (and semantic notions in general). A simple Liar sentence like 'This sentence is false' appears to be both true and false if it is either true or false. For if the sentence is true, then what it says is the case; but what it says is that it is false, hence it must be false. On the other hand, if the statement is false, then it is true, since it (...) says (only) that it is false. -/- How, then, should we classify Liar sentences? Are they true or false? A natural suggestion would be that Liars are neither true nor false; that is, they fall into a category beyond truth and falsity. This solution might resolve the initial problem, but it beckons the Liar's revenge. A sentence that says of itself only that it is false or beyond truth and falsity will, in effect, bring back the initial problem. The Liar's revenge is a witness to the hydra-like nature of Liars: in dealing with one Liar you often bring about another. -/- JC Beall presents fourteen new essays and an extensive introduction, which examine the nature of the Liar paradox and its resistance to any attempt to solve it. Written by some of the world's leading experts in the field, the papers in this volume will be an important resource for those working in truth studies, philosophical logic, and philosophy of language, as well as those with an interest in formal semantics and metaphysics. (shrink)
Philosophers often explain what could be the case in terms of what is, in fact, the case at one possible world or another. They may differ in what they take possible worlds to be or in their gloss of what is for something to be the case at a possible world. Still, they stand united by the threat of paradox. A family of paradoxes akin to the set-theoretic antinomies seem to allow one to derive a contradiction from apparently plausible principles. (...) Some of them concern the interaction between propositions and worlds, and they appear to afford the means to map classes of propositions into propositions – or, likewise, classes of worlds into worlds – in a one-to-one fashion that leads to contradiction. Yet another family of paradoxes threaten the view that whatever could exist does, in fact, exist, which is in line with modal realism, for example. This article aims to survey and identify the source of each family of paradoxes as well as to outline some responses to them. (shrink)
I argue that Meno’s Paradox targets the type of knowledge that Socrates has been looking for earlier in the dialogue: knowledge grounded in explanatory definitions. Socrates places strict requirements on definitions and thinks we need these definitions to acquire knowledge. Meno’s challenge uses Socrates’ constraints to argue that we can neither propose definitions nor recognize them. To understand Socrates’ response to the challenge, we need to view Meno’s challenge and Socrates’ response as part of a larger disagreement about the value (...) of inquiry. (shrink)
I give an interpretation according to which Meno’s paradox is an epistemic regress problem. The paradox is an argument for skepticism assuming that acquired knowledge about an object X requires prior knowledge about what X is and any knowledge must be acquired. is a principle about having reasons for knowledge and about the epistemic priority of knowledge about what X is. and jointly imply a regress-generating principle which implies that knowledge always requires an infinite sequence of known reasons. Plato’s response (...) to the problem is to accept but reject : some knowledge is innate. He argues from this to the conclusion that the soul is immortal. This argument can be understood as a response to an Eleatic problem about the possibility of coming into being that turns on a regress-generating causal principle analogous to the regress-generating principle presupposed by Meno’s paradox. (shrink)
We shall evaluate two strategies for motivating the view that knowledge is the norm of belief. The first draws on observations concerning belief's aim and the parallels between belief and assertion. The second appeals to observations concerning Moore's Paradox. Neither of these strategies gives us good reason to accept the knowledge account. The considerations offered in support of this account motivate only the weaker account on which truth is the fundamental norm of belief.
McTaggart’s argument for the unreality of time, first published in 1908, set the agenda for 20th-century philosophy of time. Yet there is very little agreement on what it actually says—nobody agrees with the conclusion, but still everybody finds something important in it. This book presents the first critical overview of the last century of debate on what is popularly called "McTaggart’s Paradox". Scholars have long assumed that McTaggart’s argument stands alone and does not rely on any contentious ontological principles. The (...) author demonstrates that these assumptions are incorrect—McTaggart himself explicitly claimed his argument to be dependent on the ontological principles that form the basis of his idealist metaphysics. The result is that scholars have proceeded to understand the argument on the basis of their own metaphysical assumptions, duly arriving at very different interpretations. This book offers an alternative reading of McTaggart’s argument, and at the same time explains why other commentators arrive at their mutually incompatible interpretations. It will be of interest to students and scholars with an interest in the philosophy of time and other areas of contemporary metaphysics. (shrink)
If I were to say, “Agnes does not know that it is raining, but it is,” this seems like a perfectly coherent way of describing Agnes’s epistemic position. If I were to add, “And I don’t know if it is, either,” this seems quite strange. In this chapter, we shall look at some statements that seem, in some sense, contradictory, even though it seems that these statements can express propositions that are contingently true or false. Moore thought it was paradoxical (...) that statements that can express true propositions or contingently false propositions should nevertheless seem absurd like this. If we can account for the absurdity, we shall solve Moore’s Paradox. In this chapter, we shall look at Moore’s proposals and more recent discussions of Moorean absurd thought and speech. (shrink)