We group the existing variants of the familiar set-theoretical and truth-theoretical paradoxes into two classes: connective paradoxes, which can in principle be ascribed to the presence of a contracting connective of some sort, and structural paradoxes, where at most the faulty use of a structural inference rule can possibly be blamed. We impute the former to an equivocation over the meaning of logical constants, and the latter to an equivocation over the notion of consequence. Both equivocation sources (...) are tightly related, and can be cleared up by adopting a particular substructural logic in place of classical logic. We then argue that our perspective can be justified via an informational semantics of contraction-free substructural logics. (shrink)

The author believes that large‐scale rationality on the part of the interpretant is essential to his interpretability, and therefore, in his view, to her having a mind. How, then are cases of irrationality, such as akrasia or self‐deception, judged by the interpretant's own standards, possible? He proposes that, in order to resolve the apparent paradoxes, one must distinguish between accepting a contradictory proposition and accepting separately each of two contradictory propositions, which are held apart, which in turn requires to (...) conceive of the mind as containing a number of semi‐independent structures of interlocking beliefs, desires, emotions, memories, and other mental states. (shrink)

All paradoxes of self-reference seem to share some structural features. Russell in 1908 and especially Priest nowadays have advanced structural descriptions that successfully identify necessary conditions for having a paradox of this kind. I examine in this paper Priest’s description of these paradoxes, the Inclosure Scheme (IS), and consider in what sense it may help us understand and solve the problems they pose. However, I also consider the limitations of this kind of structural descriptions and give arguments against (...) Priest’s use of IS in favour of dialetheism. IS fails to identify sufficient conditions for having a paradox of self-reference. That means that, even if we identified a problem common to any reasoning satisfying IS, that problem would not explain why some of those reasonings are paradoxical and some others are not. Therefore IS cannot justify by itself the claim that some particular theory offers the best solution to the paradoxes of self-reference. We still need to consider aspects concerning the content and context of occurrence of every paradox. (shrink)

_Paradoxes of the Infinite_ presents one of the most insightful, yet strangely unacknowledged, mathematical treatises of the 19 th century: Dr Bernard Bolzano’s _Paradoxien_. This volume contains an adept translation of the work itself by Donald A. Steele S.J., and in addition an historical introduction to the masterpiece, which includes a brief biography as well as an evaluation of Bolzano the mathematician, logician and physicist.

Philosophical Issues, Volume 31, Issue 1, Page 355-373, October 2021.

The coordinated attack scenario and the electronic mail game are two paradoxes of common knowledge. In simple mathematical models of these scenarios, the agents represented by the models can coordinate only if they have common knowledge that they will. As a result, the models predict that the agents will not coordinate in situations where it would be rational to coordinate. I argue that we should resolve this conflict between the models and facts about what it would be rational to (...) do by rejecting common knowledge assumptions implicit in the models. I focus on the assumption that the agents have common knowledge that they are rational, and provide models to show that denying this assumption suffices for a resolution of the paradoxes. I describe how my resolution of the paradoxes fits into a general story about the relationship between rationality in situations involving a single agent and rationality in situations involving many agents. (shrink)

Nobody interested in philosophy need be deterred by Kant's reputation for difficulty from familiarizing himself with his ethics. While the Critique of Pure Reason and his other non-ethical works are very hard to follow, the first two chapters of the Fundamental Principles of the Metaphysic of Morals at least are clear and straightforward and presuppose little previous acquaintance with philosophy. The third chapter is not about ethics as such but about the metaphysical problem of freedom and should be omitted by (...) anyone who is not familiar with Kant's general philosophy, but the first two. (shrink)

In the Western philosophical tradition, the earliest recognized paradoxes are attributed to Zeno of Elea (ca. 490–430 B.C.E.) and to Eubulides of Miletus (fl. 4th century B.C.E.). In the Chinese tradition, the earliest and most well-known paradoxes are ascribed to figures associated with the “School of Names” (ming jia 名家), a diverse group of Warring States (479–221 B.C.E.) thinkers who shared an interest in language, logic, and metaphysics. Their investigations led some of these thinkers to propound puzzling, paradoxical (...) statements such as “Today go to Yue but arrive yesterday,” “White horses are not horses,” and “Mountains and gorges are level.” Such paradoxes seem to have been intended to highlight fundamental features of reality or subtleties in semantic relations between words and things. Why were thinkers who advanced paradoxes categorized as a school of “names”? In ancient China, philosophical inquiry concerning language and logic focused on the use of “names” (ming 名, also terms, labels, or reputation) and their semantic relations to “stuff” (shi 實, also objects, features, events, or situations). Hence for classificatory purposes, second-century B.C.E. Han-dynasty archivists grouped together assorted pre-Han figures whose most prominent ideas seemed to concern the relation between names and stuff—or at least strange, unorthodox uses of names—and dubbed them a school or lineage (jia 家) devoted to the study of names. Unfortunately, both the label and the grouping are misleading. Historically, the school was a retrospective, taxonomical fiction. The figures classified under the “School of Names” never formed a distinct circle, movement, tradition, or line of influence devoted to any particular doctrine, theme, method, or way of life. Their intellectual interests overlapped at most only partly, while also overlapping extensively with those of texts associated with other schools or traditions, such as the Mohist “Dialectics” 墨辯, the Zhuangzi 莊子, the Xunzi 荀子, and the Annals of Lü Buwei 呂氏春秋. What perhaps does set the School of Names apart is that some (though not all) of the figures associated with it apparently delighted in propounding paradoxical or preposterous sayings, while the other texts just mentioned generally (though not exclusively) seek to explain and debunk such utterances. (shrink)

According to the revision theory of truth, the binary sequences generated by the paradoxical sentences in revision sequence are always unstable. In this paper, we work backwards, trying to reconstruct the paradoxical sentences from some of their binary sequences. We give a general procedure of constructing paradoxes with specific binary sequences through some typical examples. Particularly, we construct what Herzberger called “unstable statements with unpredictably complicated variations in truth value.” Besides, we also construct those paradoxes with infinitely many (...) finite primary periods but without any infinite primary period, those with an infinite critical point but without any finite primary period, and so on. This is the first formal appearance of these paradoxes. Our construction demonstrates that the binary sequences generated by a paradoxical sentence are something like genes from which we can even rebuild the original sentence itself. (shrink)

1 Background for the Uninitiated RICHMOND CAMPBELL Paradoxes are intrinsically fascinating. They are also distinctively ...

Smith argues that, unlike other forms of evidence, naked statistical evidence fails to satisfy normic support. This is his solution to the puzzles of statistical evidence in legal proof. This paper focuses on Smith’s claim that DNA evidence in cold-hit cases does not satisfy normic support. I argue that if this claim is correct, virtually no other form of evidence used at trial can satisfy normic support. This is troublesome. I discuss a few ways in which Smith can respond.

Heidegger tries to explain our emotional life applying three schemes: causal explanation, mental internalisation of emotions and metaphorical expression. None of the three schemes explains emotion though. Either because the causal nexus does not always occur or because objects and people in the external world are carriers of emotional agents or because language is already on a metaphorical level. Moreover, how is it possible that there are presently emotions constituting our life without our being aware of their existence? From the (...) analysis of boredom in its three varieties (“bored by X”, “get oneself bored”, and “it is boring”) we will get to the depth where emotions lie, trying to rouse them and to keep them awake. Although it surfaces with the force and energy of the present, every emotion has its past and future constitution. How can we understand the future of a present emotion along with its past? (shrink)

We identify a class of paradoxes that is neither set-theoretical nor semantical, but that seems to depend on intensionality. In particular, these paradoxes arise out of plausible properties of propositional attitudes and their objects. We try to explain why logicians have neglected these paradoxes, and to show that, like the Russell Paradox and the direct discourse Liar Paradox, these intensional paradoxes are recalcitrant and challenge logical analysis. Indeed, when we take these paradoxes seriously, we may (...) need to rethink the commonly accepted methods for dealing with the logical paradoxes. (shrink)

How can we experience real emotions when viewing a movie or reading a novel or watching a play when we know the characters whose actions have this effect on us do not exist? This is a conundrum that has puzzled philosophers for a long time, and in this book Robert Yanal both canvasses previously proposed solutions to it and offers one of his own. First formulated by Samuel Johnson, the paradox received its most famous answer from Samuel Taylor Coleridge, who (...) advised his readers to engage in a "willing suspension of disbelief." More recently, philosophers have argued that we are irrational in emoting toward fiction, or that we do not emote toward fiction but rather toward factual counterparts, or that we do not have real but only quasi-emotion toward fiction, generated by our playing games of make-believe. All of these proposed solutions are critically reviewed. Finding these answers unsatisfactory, Yanal offers an alternative, providing a new version of what has been dubbed "thought theory." On this theory, mere thoughts not believed true are seen as the functional equivalent of belief at least insofar as stimulating emotion is concerned. The emoter's disbelief in the actuality of components of the thoughts must be rendered relatively inactive. Such emotion is real and typically has the character of being richly generated yet unconsummated. The book extends this theory also to resolving other paradoxes arising from emotional response to fiction: how we feel suspense over what comes next in a story even when we are re-reading it for a second or third time; and how we take pleasure in narratives, such as tragedy, that excite unpleasant emotions such as fear, pity, or horror. (shrink)

Supererogatory acts—good deeds “beyond the call of duty”—are a part of moral common sense, but conceptually puzzling. I propose a unified solution to three of the most infamous puzzles: the classic Paradox of Supererogation (if it’s so good, why isn’t it just obligatory?), Horton’s All or Nothing Problem, and Kamm’s Intransitivity Paradox. I conclude that supererogation makes sense if, and only if, the grounds of rightness are multi-dimensional and comparative.

Recently, several authors have pointed out that substructural logics are adequate for developing naive theories that represent semantic concepts such as truth. Among them, three proposals have been explored: dropping cut, dropping contraction and dropping reflexivity. However, nowhere in the substructural literature has anyone proposed rejecting the structural rule of weakening, while accepting the other rules. Some theorists have even argued that this task was not possible, since weakening plays no role in the derivation of semantic paradoxes. In this (...) article, I introduce a theory for naive truth based on the logic resulting from dropping the rule of weakening from classical logic, and maintaining the other structural rules. (shrink)

Critical theory must add to its agenda “disrupt[ing] the easy passage from critique [to] its neoliberal double”, Nancy Fraser recently argued. Emancipatory movements have not only been transformed by neoliberalism. They have, “unwittingly”, provided powerful “ingredients” for the transition to neoliberalism. This essay examines Axel Honneth and Nancy Fraser’s assessment of and normative proposal for addressing the paradoxes of neoliberalism. The constraints of neoliberalism, I argue, bring into focus the structural challenge of immanent critique as understood within second and (...) third generation Critical Theory. Normative ambivalence within neoliberalism, I maintain, calls for a shift in the way that critical and justificatory practices are understood within the Frankfurt School. (shrink)

In The Enterprise of Knowledge, I proposed a general theory of rational choice which I intended as a characterization of a prescriptive theory of ideal rationality. A cardinal tenet of this theory is that assessments of expected value or expected utility in the Bayesian sense may not be representable by a numerical indicator or indeed induce an ordering of feasible options in a context of deliberation. My reasons for taking this position are related to my commitment to the inquiry-oriented approach (...) to human knowledge and valuation favored by the American pragmatists, Charles Peirce and John Dewey. A feature of any acceptable view of inquiry ought to be that during an inquiry points under dispute ought to be kept in suspense pending resolution through inquiry. (shrink)

To solve the highly counterintuitive paradox of confirmation represented by the statement, “A pair of red shoes confirms that all ravens are black,” Hempel employed a strategy that retained the equivalence condition but abandoned Nicod’s irrelevance condition. However, his use of the equivalence condition is fairly ad hoc, raising doubts about its applicability to this problem. Furthermore, applying the irrelevance condition from Nicod’s criterion does not necessarily lead to paradoxes, nor does discarding it prevent the emergence of paradoxes. (...) Hempel’s approach fails to adequately resolve the paradox. (shrink)

Winner of the 2014 Edward Goodwin Ballard Award for an Outstanding Book in Phenomenology, awarded by the Center for Advance Research in Phenomenology. -/- Merleau-Ponty and the Paradoxes of Expression offers a comprehensive reading of the philosophical work of Maurice Merleau-Ponty, a central figure in 20th-century continental philosophy. -/- By establishing that the paradoxical logic of expression is Merleau-Ponty's fundamental philosophical gesture, this book ties together his diverse work on perception, language, aesthetics, politics and history in order to establish (...) the ontological position he was developing at the time of his sudden death in 1961. Donald A. Landes explores the paradoxical logic of expression as it appears in both Merleau-Ponty’s explicit reflections on expression and his non-explicit uses of this logic in his philosophical reflection on other topics, and thus establishes a continuity and a trajectory of his thought that allows for his work to be placed into conversation with contemporary developments in continental philosophy. The book offers the reader a key to understanding Merleau-Ponty's subtle methodology and highlights the urgency and relevance of his research into the ontological significance of expression for today's work in art and cultural theory. (shrink)

Consider the following argument written on the board in room 227: 1 = 1. So, the argument on the board in room 227 is not valid. This argument generates a paradox. The aim of this paper is to present a resolution of this paradox and related paradoxes of validity, including a version of the Curry paradox. The proposal stresses the close connections between these validity paradoxes and paradoxes of truth and paradoxes of denotation. So a more (...) general aim is to provide a unified response to semantic paradox. The positive proposal is in part inspired by a brief, tantalizing suggestion of Gödel’s, that the paradoxes might appear “as something analogous to dividing by zero”—so that the concept of validity, for example, is everywhere applicable except for certain singular points or singularities. A second central claim is that ‘valid’ is a context-sensitive predicate. The key notions of this contextual-singularity theory are presented and applied to a variety of cases. Any purported solution to paradox must deal with the phenomenon of revenge, and a response to revenge is outlined. The paper closes with remarks about the accommodation of Tarski’s claim that natural languages are universal. (shrink)

Aharon Aviram; The Paradoxes of Education for Democracy, or the Tragic Dilemmas of the Modern Liberal Educator, Journal of Philosophy of Education, Volume 20, I.

Though we may be competent at using many technologies, most of what we think we know about technology in general is false. Our error stems from the everyday conception of things as separate from each other and from us. In reality technologies belong to an interconnected network the nodes of which cannot exist independently qua technologies. What is more we tend to see technologies as quasi-natural objects, but they are just as much social as natural, just as much determined by (...) the meanings we give them as by the causal laws that rule over their powers. The errors of common sense have political consequences in domains such as, development, medicine and environmental policy. In this paper I summarize many of the conclusions philosophy of technology has reached reflecting on the reality of our technological world. These conclusions appear as paradoxes judged from our everyday perspective.This paper presents a philosophy of technology. It draws on what we have learnt in the last 30 years as we abandoned old Heideggerian and positivist notions and faced the real world of technology. It turns out that most of our common sense ideas about technology are wrong. This is why I have put my ten propositions in the form of paradoxes, although I use the word loosely here to refer to the counter-intuitive nature of much of what we know about technology. (shrink)

Recently, it has been observed that the usual type-theoretic restrictions are not enough to block certain paradoxes involving two or more predicates. In particular, when we have a self-referential language containing modal predicates, new paradoxes might appear even if there are type restrictions for the principles governing those predicates. In this article we consider two type-theoretic solutions to multimodal paradoxes. The first one adds types for each of the modal predicates. We argue that there are a number (...) of problems with most versions of this approach. The second one, which we favour, represents modal notions by using the truth predicate together with the corresponding modal operator. This way of doing things is not only useful because it avoids multimodal paradoxes, but also because it preserves the expressive capacity of the language. As an example of the sort of theory we have in mind, we provide a type-theoretic axiomatization that combines truth with necessity and knowledge. (shrink)

My paper focuses on the theme of visibility by teasing out some paradoxes of invisibility. In the ordinary social world, what is said to be invisible is generally what is here, right before our eyes, but to which we pay no attention. Care is invisible because it goes on without us seeing it. By suddenly making visible what is ordinarily invisible, the COVID pandemic has been a strange pedagogical moment, making visible the people who take care of “us”, and (...) revealing our entire society’s ignorance of what allows it to live—whether in the context of everyday life or in the urgency of the risk of death. The grammar of care has thus imposed itself on everyone, because care is never so visible as in those situations where a form of life is shaken. Care work has been revealed as invisible work that keeps everyone going. “Invisible” does not refer to a difficulty in perceiving but rather a refusal to see. A refusal to see something that is not hidden, but which we do not see precisely because it is right before our eyes. Invisibility is thus denial, in both the social and the theoretical realms, especially when care work is envisioned in the terms of the further invisibilization of care work when it is done for the benefit of women as in the “care drain” from poor to rich countries. The asymmetry in the relations between North and South is part of the invisibility of what sustains societies. The invisible chains of care reveal the extent to which the question of service is the fundamental question of social invisibility. (shrink)

Christopher Cowie argues that moral error theory is uniquely placed to avoid embracing any of the troubling and counter-intuitive horns of moral paradoxes. Contra Cowie, I argue that moral non-cognitivists can also avoid embracing any such troubling and counter-intuitive horns. Rather than supporting moral error theory, I argue that moral paradoxes more precisely support moral nihilism—a first-order view that is consistent with moral error theory and with moral non-cognitivism. Moreover, I argue that reconstructing the argument from moral (...) class='Hi'>paradoxes as an argument for moral nihilism enhances its philosophical significance in two ways. First, the argument from moral paradoxes is the first unified argument in support of moral nihilism. Second, given that the logical space of moral nihilism is greater than the logical space of moral error theory, the threat that the argument from moral paradoxes poses to moral philosophy is more serious than Cowie acknowledges. (shrink)

In recent years, substructural approaches to paradoxes have become quite popular. But whatever restrictions on structural rules we may want to enforce, it is highly desirable that such restrictions be accompanied by independent philosophical motivation, not directly related to paradoxes. Indeed, while these recent developments have shed new light on a number of issues pertaining to paradoxes, it seems that we now have even more open questions than before, in particular two very pressing ones: what (independent) motivations (...) do we have (if any) for restrictions on structural rules, and what to make of the plurality of new logics emerging from these restrictions, i.e. how to ‘choose’ among the different options. In this paper, we address these two questions from the perspective of a dialogical conception of logic that we've been advocating in recent years. We will argue that dialogical interpretations of structural rules, that is, as rules determining specific properties of the dialogues in question, provide a conveniently neutral framework to adjudicate between the different substructural proposals that have been made in the literature on paradoxes. (shrink)

Ryan Wasserman explores a range of fascinating puzzles raised by the possibility of time travel, with entertaining examples from physics, science fiction, and popular culture, and he draws out their implications for our understanding of time, tense, freedom, fatalism, causation, counterfactuals, laws of nature, persistence, change, and mereology.

"The book is divided into 8-10 chapters that are each only 2 or 3 pages long... this feels like a nice feature of the book, since you can dip in and just read a short bite before moving on. The author clearly has some interesting ideas and at times I found his writing to be quite engaging." MAA Review "I did enjoy reading (and re-reading) this book very much. Reading it deserves a warm recommendation not only for mathematicians but for (...) anybody... this book makes you think about how and what you think you are thinking." European Mathematical Society "Circularity" is the story of a Janus-faced conceptual structure, that on the one hand led to deep scientific discoveries, and on the other hand is used to trick the mind into believing the impossible. Alongside mathematical revolutions that eventually led to the invention of the computer, the book describes ancient paradoxes that arise from circular thinking. Another aspect of circularity, its ability to entertain, leads to a surprising insight on the time old question "What is humor." The book presents the ubiquity of circularity in many fields, and its power to confuse and to instruct. (shrink)

This paper presents and motivates a new philosophical and logical approach to truth and semantic paradox. It begins from an inferentialist, and particularly bilateralist, theory of meaning---one which takes meaning to be constituted by assertibility and deniability conditions---and shows how the usual multiple-conclusion sequent calculus for classical logic can be given an inferentialist motivation, leaving classical model theory as of only derivative importance. The paper then uses this theory of meaning to present and motivate a logical system---ST---that conservatively extends classical (...) logic with a fully transparent truth predicate. This system is shown to allow for classical reasoning over the full (truth-involving) vocabulary, but to be non-transitive. Some special cases where transitivity does hold are outlined. ST is also shown to give rise to a familiar sort of model for non-classical logics: Kripke fixed points on the Strong Kleene valuation scheme. Finally, to give a theory of paradoxical sentences, a distinction is drawn between two varieties of assertion and two varieties of denial. On one variety, paradoxical sentences cannot be either asserted or denied; on the other, they must be both asserted and denied. The target theory is compared favourably to more familiar related systems, and some objections are considered. (shrink)

The paradoxes act as restraints on undisciplined or erroneous reasoning and so perform a valuable role. The fact that they have resisted solution for so long suggests that the current systems of logic are defective. The paradoxes of set theory, in my opinion, completely condemn all the current forms of set theory and their associated definitions of the natural numbers. Nothing short of a complete review of the whole area seems capable of remedying the situation.

One of the world's pre-eminent Max Weber scholars here presents a comprehensive analysis of Weber's ambiguous stance toward modernity considered from a normative, theoretical, and historical point of view. The book is in two parts. Part I scrutinises Weber's world view. On the basis of his thinking about the meaning and inter-relationships of science, politics, and ethics in the modern era, Weber is seen as the embodiment of a social scientist and political thinker who exposes himself to intellectual risks and (...) existential tensions while resisting final solutions. It includes a masterly analysis of Weber's two famous speeches 'Science as a Vocation' and 'Politics as a Vocation'. Part II considers Weber's unfinished project on the sociology of religion. Piecing together planned and partially completed works on Islam and Western Christianity, Schluchter locates them in the history and theory of Weber's overall work. This reconstruction of Weber's work on religion emphasises its interplay between religion, economy, politics, and law. (shrink)

The black hole information loss paradox is a catch-all term for a family of puzzles related to black hole evaporation. For almost 50 years, the quest to elucidate the implications of black hole evaporation has not only sustained momentum, but has also become increasingly populated with proposals that seem to generate more questions than they purport to answer. Scholars often neglect to acknowledge ongoing discussions within black hole thermodynamics and statistical mechanics when analyzing the paradox, including the interpretation of Bekenstein-Hawking (...) entropy, which is far from settled. To remedy the dialectical gridlock, I have formulated an overarching, unified framework, which I call ``Black Hole Paradoxes'', that integrates the debates and taxonomizes the relevant `camps' or philosophical positions. -/- I demonstrate that black hole evaporation within Hawking's semi-classical framework insinuates how late-time Hawking radiation is an entangled global system, a contradiction in terms. The relevant forms of information loss are associated with a decrease in maximal Boltzmann entropy and an increase in global von Neumann entropy respectively, which engender what I've branded the ``paradox of phantom entanglement''. Prospective solutions are then tasked with demonstrating how late-time Hawking radiation is either exclusively an entangled subsystem, in which a black hole remnant lingers as an information safehouse, or exclusively an unentangled global system, in which information is evacuated to the exterior. -/- The disagreement between safehouse and evacuation solutions boils down to the statistical interpretation of thermodynamic black hole entropy, i.e., Bekenstein-Hawking entropy. Safehouse solutions attribute Bekenstein-Hawking entropy to a minority of black hole degrees of freedom, those that are associated with the horizon. Evacuation solutions, in contrast, attribute Bekenstein-Hawking entropy to all black hole degrees of freedom. I argue that the interpretation of Bekenstein-Hawking entropy is the litmus test to vet the overpopulated proposal space. So long as any proposal rejecting Hawking's original calculation independently derives black hole evaporation, globally conserves degrees of freedom and entanglement, preserves a version of semi-classical gravity at sub-Planckian scales, and describes black hole thermodynamics in statistical terms, then it counts as a genuine solution to the paradox. (shrink)

In this new kind of introduction to ethical theory, Daniel Muñoz and Sarah Stroud present 50 of the field's most exciting puzzles, paradoxes, and thought experiments. Over the course of 11 chapters, the authors cover a huge variety of topics, starting with the classic debate between utilitarians and deontologists and ending on existential questions about the future of humanity. Every chapter begins with a helpful introduction, and each of the 50 entries includes references for further reading and questions for (...) reflection. Among the entries are such classics as the Ring of Gyges, Jim and the Villagers, the Repugnant Conclusion, JoJo, "One Thought Too Many," the Miners Puzzle, the Gentle Murder Paradox, Nowheresville, the Experience Machine, and the Trolley Problem. The book also explores several more recent topics of interest, such as doxastic wronging and the ethics of AI, so that even advanced students are likely to discover something new. Each entry can be read on its own, and the writing is accessible and conversational throughout, making this an ideal resource for undergraduate teaching. Readers at any level can pick up this book and see for themselves how fascinating-and puzzling-ethical theory can be. (shrink)

We will formulate some analogous higher-order versions of Skolem’s paradox and assess the generalizability of two solutions for Skolem’s paradox to these paradoxes: the textbook approach and that of Bays (2000). We argue that the textbook approach to handle Skolem’s paradox cannot be generalized to solve the parallel higher-order paradoxes, unless it is augmented by the claim that there is no unique language within which the practice of mathematics can be formalized. Then, we argue that Bays’ solution to (...) the original Skolem’s paradox, unlike the textbook solution, can be generalized to solve the higher-order paradoxes without any implication about the possibility or order of a language in which mathematical practice is to be formalized. (shrink)

Both in dealing with the semantic paradoxes and in dealing with vagueness and indeterminacy, there is some temptation to weaken classical logic: in particular, to restrict the law of excluded middle. The reasons for doing this are somewhat different in the two cases. In the case of the semantic paradoxes, a weakening of classical logic (presumably involving a restriction of excluded middle) is required if we are to preserve the naive theory of truth without inconsistency. In the case (...) of vagueness and indeterminacy, there is no worry about inconsistency; but a central intuition is that we must reject the factual status of certain sentences, and it hard to see how we can do that while claiming that the law of excluded middle applies to those sentences. So despite the different routes, we have a similar conclusion in the two cases. (shrink)