Paradoxes

I propose an approach to liar and Curry paradoxes inspired by the work of Roger Swyneshed in his treatise on insolubles (1330-1335). The keystone of the account is the idea that liar sentences and their ilk are false (and only false) and that the so-called ''capture'' direction of the T-schema should be restricted. The proposed account retains what I take to be the attractive features of Swyneshed's approach without leading to some worrying consequences Swyneshed accepts. The approach and the resulting (...) logic (called ''Swynish Logic'') are non-classical, but are consistent and compatible with many elements of the classical picture including modus ponens, modus tollens, and double-negation elimination and introduction. It is also compatible with bivalence and contravalence. My approach to these paradoxes is also immune to an important kind of revenge challenge that plagues some of its rivals. (shrink)

We develop some of Williamson’s ideas regarding how propositional calculus aids in comprehending Aristotelian logic. Specifically, we enhance the utilisation of truth tables to examine the structure of opposition diagrams. Using ‘conditioned truth tables’, we establish logical dependency relationships between the truth values of different propositions. This approach proves effective in interpreting various texts of the Organon concerning the doctrine of opposition.

A famous mathematical theorem says that the sum of an infinite series of numbers can depend on the order in which those numbers occur. Suppose we interpret the numbers in such a series as representing instances of some physical quantity, such as the weights of a collection of items. The mathematics seems to lead to the result that the weight of a collection of items can depend on the order in which those items are weighed. But that is very hard (...) to believe. A puzzle then arises: How do we interpret the metaphysical significance of this mathematical theorem? I first argue that prior solutions to the puzzle lead to implausible consequences. Then I develop my own solution, where the basic idea is that the weight of a collection of items is equal to the limit of the weights of its finite subcollections contained within ever-expanding regions of space. I show how my solution is intuitively plausible and philosophically motivated, how it reveals an underexplored line of metaphysical inquiry about quantities and locations, and how it elucidates some classic puzzles concerning supertasks. (shrink)

Priest's theory of motion is based on Leibniz's Continuity Condition (LCC), which states that any state that exists at each instant in a continuous set of moments also exists at its temporal limit. If we accept the CCL, a free-falling pen would have to be simultaneously in motion and at rest at the instant of change: the critical moment when it hits the ground, thus passing from the state of motion to that of rest. This seems to be a contradictory (...) state of affairs, which is precisely what Priest claims to be the case. In this article, I discuss the logical possibility of empirically testing whether such a contradictory state of affairs occurs. (shrink)

This book will examine paradoxes in diverse areas of thought: philosophy, mathematics, physics, economics, political science, psychology, computer science, logic, statistics, linguistics, law, etc. Though the treatment of each paradox is rigorous, the book will be written accessibly with a lighthearted and humorous tone so as to keep the reader engaged. Each chapter will focus on a single paradox, structured roughly like so: 1. A question is asked in the context of a story. As an answer, the paradox is presented (...) (which often results in an aha moment). The historical background of the paradox is recounted. 2. The dénouement explains how the paradox is resolved or why there is no resolution. 3. The chapter ends with further remarks, usually contemporary real-world examples or applications of said paradox. Some examples of the paradoxes covered are the Axiom of Choice (Mathematics), Monty Hall Problem (Statistics), Morgenbesser's Paradox (Linguistics), Tea Leaves Paradox (Physics), The Ultimatum Game (Economics), and The Chicken or Egg Question (Evolution). (shrink)

David Kaplan famously argued that mainstream semantics for modal logic, which identifies propositions with sets of possible worlds, is affected by a cardinality paradox. Takashi Yagisawa showed that a variant of the same paradox arises when standard possible worlds semantics is extended with impossible worlds to deliver a hyperintensional account of propositions. After introducing the problem, we discuss two general approaches to a possible solution: giving up on sets and giving up on worlds, either in the background semantic framework or (...) in the corresponding conception of propositions. As a result, we conclude that abandoning worlds by embracing a truthmaker-based approach offers a promising way to account for hyperintensional propositions without facing the paradoxical outcome. (shrink)

This open access book is a superb collection of some fifteen chapters inspired by Schroeder-Heister's groundbreaking work, written by leading experts in the field, plus an extensive autobiography and comments on the various contributions by Schroeder-Heister himself. For several decades, Peter Schroeder-Heister has been a central figure in proof-theoretic semantics, a field of study situated at the interface of logic, theoretical computer science, natural-language semantics, and the philosophy of language. -/- The chapters of which this book is composed discuss the (...) subject from a rich variety of angles, including the history of logic, the proper interpretation of logical validity, natural deduction rules, the notions of harmony and of synonymy, the structure of proofs, the logical status of equality, intentional phenomena, and the proof theory of second-order arithmetic. All chapters relate directly to questions that have driven Schroeder-Heister's own research agenda and to which he has made seminal contributions. The extensive autobiographical chapter not only provides a fascinating overview of Schroeder-Heister's career and the evolution of his academic interests but also constitutes a contribution to the recent history of logic in its own right, painting an intriguing picture of the philosophical, logical, and mathematical institutional landscape in Germany and elsewhere since the early 1970s. The papers collected in this book are illuminatingly put into a unified perspective by Schroeder-Heister's comments at the end of the book. Both graduate students and established researchers in the field will find this book an excellent resource for future work in proof-theoretic semantics and related areas. (shrink)

Paradoxes evoke astonishment, confusion, and delight in the extraordinary. But that is not all: They point to fundamental problems of philosophy, mathematics, and the natural sciences. This volume presents a number of the most important paradoxes from an analytical-philosophical perspective. -/- German abstract: Paradoxien rufen Staunen, Verwirrung und die Lust am Außergewöhnlichen hervor. Aber nicht nur das: Es sind Paradoxien, die bis heute auf Grundprobleme der Philosophie, der Mathematik sowie der Naturwissenschaften hinweisen und uns zu revolutionären Lösungsvorschlägen herausfordern. -/- Einige (...) Paradoxien markieren dabei vielleicht sogar unüberwindbare Grenzen unseres Wissens. Dieser Band stellt eine Reihe der wichtigsten Paradoxien – Paradoxien der Wahrheit, des Infiniten, der Bestätigung, der Vagheit, der Quantenmechanik, der Zeit, des Visuellen und des Auditiven – sowie Überlegungen zu allgemeinen Lösungswegen aus einer analytisch-philosophischen Perspektive vor. Dabei richtet er sich an interessierte Einsteiger in die Thematik, ohne den Gegenstand dabei zu sehr zu verkürzen. (shrink)

Emile Borel regards the Banach-Tarski Paradox as a reductio ad absurdum of the Axiom of Choice. Peter Forrest instead blames the assumption that physical space has a similar structure as the real numbers. This paper argues that Banach and Tarski's result is not paradoxical and that it merely illustrates a surprising feature of the continuum: dividing a spatial region into disjoint pieces need not preserve volume.

Recall that B is PA relative to A if B computes a member of every nonempty $\Pi ^0_1(A)$ class. This two-place relation is invariant under Turing equivalence and so can be thought of as a binary relation on Turing degrees. Miller and Soskova [23] introduced the notion of a $\Pi ^0_1$ class relative to an enumeration oracle A, which they called a $\Pi ^0_1{\left \langle {A}\right \rangle }$ class. We study the induced extension of the relation B is PA relative (...) to A to enumeration oracles and hence enumeration degrees. We isolate several classes of enumeration degrees based on their behavior with respect to this relation: the PA bounded degrees, the degrees that have a universal class, the low for PA degrees, and the ${\left \langle {\text {self}\kern1pt}\right \rangle }$ -PA degrees. We study the relationship between these classes and other known classes of enumeration degrees. We also investigate a group of classes of enumeration degrees that were introduced by Kalimullin and Puzarenko [14] based on properties that are commonly studied in descriptive set theory. As part of this investigation, we give characterizations of three of their classes in terms of a special sub-collection of relativized $\Pi ^0_1$ classes—the separating classes. These three can then be seen to be direct analogues of three of our classes. We completely determine the relative position of all classes in question. (shrink)

The Russell's paradox concerns the foundations of naive set theory. This short short paper is about how it can be interpreted in other contexts and has significance in the world of commands. Understanding the paper assumes that the reader is broadly familiar with the foundations of set theory and its history. The text contains many formulas and therefore the reader should be comfortable in the world of logical formulas. My example is somewhat similar to the barber paradox. There, too, we (...) are puzzled by the feasibility of a task. In the case of the barber paradox there is a solution: the barber is a woman, in my example there is no such escape. (shrink)

In this paper I show that one of the most fruitful ways of employing paradoxes has been as a philosophical method that forces us to reconsider basic assumptions. After a brief discussion of recent understandings of the notion of paradoxes, I show that Zeno of Elea was the inventor of paradoxes in this sense, against the background of Heraclitus’ and Parmenides’ way of argumentation: in contrast to Heraclitus, Zeno’s paradoxes do not ask us to embrace a paradoxical reality; and in (...) contrast to Parmenides, Zeno shows common assumptions to be internally problematic, not just in light of Eleatic positions. (shrink)

Concerning Nicolaus Cusanus’ (Nicholas of Cusa, 1401–1464) mysticism of the intellect, his approach to the problem of ineffability deserves the special attention of researchers. Preceded by a general exposition on the topic of the inconceivability of the experience of the foundational autopoietic self-reference of thinking and speaking, this article shows how Nicolaus Cusanus has developed a complex approach to the problem of an “ineffable way of speaking” (ineffable fari). Cusanus developed a set of approaches to non-negatable cataphatic “pointing rods” (Max (...) Scheler) and apophatic ways of thinking about what is to be understood as ineffable in the sense of a philosophical form of mysticism. Both are still inspiring and highly relevant for the discussion today. In terms of the overall interior development of his philosophical way of “eloquent silence” (German: beredtes Schweigen), it is notable that Cusanus eventually referred to both ways of affirmative and negative theology in their dialectic interdependence. Eventually, he found increasingly simple ways to point the way towards the “likeness of the path along which the seeker must walk.” In his later works, Cusanus developed a unique understanding of the problem of ineffability about philosophical mysticism, the potential of which remains to be explored further in the future. (shrink)

I defend the hypothesis that the semantic paradoxes, the paradoxes about collections, and the sorites paradoxes, are all paradoxes of reference fixing: they show that certain conventionally adopted and otherwise functional reference-fixing principles cannot provide consistent assignments of reference to certain relevant expressions in paradoxical cases. I note that the hypothesis has interesting implications concerning the idea of a unified account of the semantic, collection and sorites paradoxes, as well as about the explanation of their “recalcitrance”. I also note that (...) it does not necessarily imply that one should not expect the sometimes hoped for “unique” solution to a paradox of these kinds. (shrink)

Causal finitism, the view that the causal history of any event must be finite, has garnered much philosophical interest recently—especially because of its applicability to the Kalām cosmological argument. The most prominent argument for causal finitism is the Grim Reaper argument, which attempts to show that, if infinite causal histories are possible, then other paradoxical states of affairs must also be possible. However, this style of argument has been criticized on the grounds of (i) relying on controversial modal principles, and (...) (ii) providing a false diagnosis of the paradoxes involved. In this paper, I develop a new kind of Grim Reaper argument immune to these criticisms. I show that, by using insights from the literature on time travel, causal finitists should instead argue that infinite causal histories are problematically inexplicable, as they entail the possibility of unexplained foiling mechanisms. The fruits of this paper are that (i) a novel supporting argument for the Kalām is developed, and (ii) along the way of building this argument, it is shown that the literatures on time travel and causal finitism are deeply and intimately connected. (shrink)

This essay evaluates Hegel's claim that the phenomenon of time exhibits a quantitative logic in the context of a paradox concerning temporal presence. On the one hand, in time, the present always is. It seems that the very nature of time, assuming that it is really passing, requires us to assent to the continuous being of the present. If time is always passing, there must always be a present when the passing actually occurs and thus when beings actually exist. On (...) the other hand, any particular moment of presence, as a point or an interval, immediately ceases to be or has not yet come to be. And, because of this, no delineated moment can be purely self-present. Conceived as an unextended point, presence would be nothing enduring of its own against time's passing, while, conceived as an interval, presence contains before and after within itself, meaning as an interval that is not actually present at once. The paradox is therefore that time's passing demands we think being present and presence as being, while being present, strictly speaking, seems impossible due precisely to that passing. Hegel claims to reconcile the self-same form of presence, a presence that always is, with continuous change under the category of quantity. However, I argue that the non-identity between the logical category and the phenomenon of time renders this reconciliation ineffective against the paradox, breaking down, more specifically, as it concerns the formation of a temporal magnitude. I evaluate alternative Hegelian interpretations for determining whether the irresolvability of the paradox proves problematic after all, arguing that the paradox in fact presents a significant problem for the conceivability of temporal existence. (shrink)

Las máquinas aceleradas de Turing (ATMs) son dispositivos capaces de ejecutar súper-tareas. Sin embargo, el simple ejercicio de definirlas ha generado varias paradojas. En el presente artículo se definirán las nociones de súper-tarea y ATM de manera exhaustiva y se aclarará qué debe entenderse en un contexto lógico-formal cuando se pregunta por la existencia de un objeto. A partir de la distinción entre posibilidades lógicas y físicas se disolverán las paradojas y se concluirá que las ATMs son posibles y existen (...) como objetos abstractos. (shrink)

Das Metameta-Paradox ist darauf zurückzuführen, daß man die Eigenständigkeit jeder Realität annimmt, während es in Wirklichkeit nur eine Realität gibt, zu deren Beschreibung eine Sprache genutzt wird, die jedoch keine Metasprache im Verhältnis zu sich selbst sein kann, weil sie aus Elementen besteht, die zu gleicher Klasse angehören. Daher ist jede Beschreibung der Realität der Realität untergeordnet, und ihre Beschreibung sowie die Beschreibung ihrer Beschreibung u.s.w. dürfen nicht als eigenständige Klassen im Sinne der Gruppentheorie eingestuft werden.

In this paper, we examine a fundamental problem that appears in Greek philosophy: the paradoxes of self-reference of the type of “Third Man” that appears first in Plato’s 'Parmenides', and is further discussed in Aristotle and the Peripatetic commentators and Proclus. We show that the various versions are analysed using different language, reflecting different understandings by Plato and the Platonists, such as Proclus, on the one hand, and the Peripatetics (Aristotle, Alexander, Eudemus), on the other hand. We show that the (...) Peripatetic commentators do not focus on Plato’s solution but primarily on the formulation of the “Third Man” paradox. On the contrary, Proclus seems to be convinced that Plato suggests a sound solution to the paradox by defining the predicate of similarity (homogeneity) that demarcates two types of homogeneous entities – the eide and the participants in them in a way that their confusion would be inadmissible. We claim that Plato’s solution follows a sound line of reasoning that is formalisable in a language of Frege-Russell type; hence there exists a model in which Plato’s reasoning is valid. Furthermore, we notice that Plato’s definition of the second-order predicate of similarity is attained by resorting to first-order entities. In this sense, Plato’s definition is comparable to Eudoxus’ definition of ratio, which is also attained by resorting to first-order objects. Consequently, Plato seems to follow a logical practice established by the mathematicians of the 5th century, notably Eudoxus, in his solution to the paradox. (shrink)

Substructural solutions to the semantic paradoxes have been broadly discussed in recent years. In particular, according to the non-transitive solution, we have to give up the metarule of Cut, whose role is to guarantee that the consequence relation is transitive. This concession—giving up a meta rule—allows us to maintain the entire consequence relation of classical logic. The non-transitive solution has been generalized in recent works into a hierarchy of logics where classicality is maintained at more and more metainferential levels. All (...) the logics in this hierarchy can accommodate a truth predicate, including the logic at the top of the hierarchy—known as CMω—which presumably maintains classicality at all levels. CMω has so far been accounted for exclusively in model-theoretic terms. Therefore, there remains an open question: how do we account for this logic in proof-theoretic terms? Can there be found a proof system that admits each and every classical principle—at all inferential levels—but nevertheless blocks the derivation of the liar? In the present paper, I solve this problem by providing such a proof system and establishing soundness and completeness results. Yet, I also argue that the outcome is philosophically unsatisfactory. In fact, I’m afraid that in light of my results this metainferential solution to the paradoxes can hardly be called a “solution,” let alone a good one. (shrink)

I discuss a neglected solution to the skeptical problem introduced by Lewis Carroll’s “What the Tortoise Said to Achilles” (1895) in terms of a self-citational inferential license. I then consider some responses to this solution. The most significant response on behalf of the skeptic utilizes the familiar distinction between two ways of accepting a rule: as action-guiding and as a mere truth. I argue that this is ultimately unsatisfactory and conclude by opting for an alternative conception of rules as representations (...) of behavior deployed for various purposes, some theoretical and others practical. This alternative conception does not allow the skeptical problem to get off the ground. (shrink)

In the first of the Insolubles in Chapter 8 of his Sophismata, Buridan contends that the inference Omnis propositio est affirmativa; ergo, nulla propositio est negativa (PS) is valid, even though it appeals to the self-reference in the conclusion to show that what we (following Read 2001) call the classical conception of validity (CCV) fails. This requires that we accept that there are good inferences in which a false conclusion follows from true premises. Partially following Hughes’ proposal (1982), we argue (...) that the First Sophism (PS) involves three different notions of validity. Two of them correspond to the ones described by Hughes (1982, 80–86), who calls them Theory A and Theory B. The third one—that will we call Theory C—is not mentioned by Hughes; instead, it is suggested by Buridan himself in the first three arguments in favor of the validity of PS. We show that: a) from what Buridan says in his Theory C it follows that PS is a formal and material consequence, and hence, a valid one. Then we show that: b) the rejection of CCV and the acceptance of Nulla propositio est negativa (NPN) as a (formal) consequence of Omnis propositio est affirmativa (OPA) leads to a paradox that bears similarities with the one put forward by Pseudo Scotus—which has been studied by Read (2001) and is related to Curry’s paradox. However, there are enough differences to merit considering this paradox separately, especially in relation to the so-called validity paradoxes. Interestingly, our work suggests that Buridan was aware of these problems, which explains why he introduced a new criterion for validity, one that is not based on truth-preservation but on what Spade (1988) calls firmness, and Klima (2016) correspondence. (shrink)

Classical theists hold that God is omnipotent. But now suppose a critical atheologian were to ask: Can God create a stone so heavy that even he cannot lift it? This is the dilemma of the stone paradox. God either can or cannot create such a stone. Suppose that God can create it. Then there's something he cannot do – namely, lift the stone. Suppose that God cannot create the stone. Then, again, there's something he cannot do – namely, create it. (...) Either way, God cannot be omnipotent. Among the variety of known theological paradoxes, the paradox of the stone is especially troubling because of its logical purity. It purports to show that one cannot believe in both God and the laws of logic. In the face of the stone paradox, how should the contemporary analytic theist respond? Ought they to revise their belief in theology or their belief in logic? Ought they to lose their religion or lose their mind? (shrink)

W. V. Quine famously defended two theses that have fallen rather dramatically out of fashion. The first is that intensions are “creatures of darkness” that ultimately have no place in respectable philosophical circles, owing primarily to their lack of rigorous identity conditions. However, although he was thoroughly familiar with Carnap’s foundational studies in what would become known as possible world semantics, it likely wouldn’t yet have been apparent to Quine that he was fighting a losing battle against intensions, due in (...) large measure to developments stemming from Carnap’s studies and culminating in the work of Kripke, Hintikka, and Bayart. These developments undermined Quine’s crusade against intensions on two fronts. First, in the context of possible world semantics, intensions could after all be given rigorous identity conditions by defining them (in the simplest case) as functions from worlds to appropriate extensions, a fact exploited to powerful and influential effect in logic and linguistics by the likes of Kaplan, Montague, Lewis, and Cresswell. Second, the rise of possible world semantics fueled a strong resurgence of metaphysics in contemporary analytic philosophy that saw properties and propositions widely, fruitfully, and unabashedly adopted as ontological primitives in their own right. This resurgence — happily, in my view — continues into the present day. -/- For a time, at any rate, Quine experienced somewhat better success with his second thesis: that higher-order logic is, at worst, confused and, at best, a quirky notational alternative to standard first-order logic. However, Quine notwithstanding, a great deal of recent work in formal metaphysics transpires in a higher-order logical framework in which properties and propositions fall into an infinite hierarchy of types of (at least) every finite order. Initially, the most philosophically compelling reason for embracing such a framework since Russell first proposed his simple theory of types was simply that it provides a relatively natural explanation of the paradoxes. However, since the seminal work of Prior there has been a growing trend to consider higher-order logic to be the most philosophically natural framework for metaphysical inquiry, many of the contributors to this volume being among the most important and influential advocates of this view. Indeed, this is now quite arguably the dominant view among formal metaphysicians. -/- In this paper, and against the current tide, I will argue in §1 that there are still good reasons to think that Quine’s second battle is not yet lost and that the correct framework for logic is first-order and type-free — properties and propositions, logically speaking, are just individuals among others in a single domain of quantification — and that it arises naturally out of our most basic logical and semantical intuitions. The data I will draw upon are not new and are well-known to contemporary higher-order metaphysicians. However, I will try to defend my thesis in what I believe is a novel way by suggesting that these basic intuitions ground a reasonable distinction between “pure” logic and non-logical theory, and that Russell-style semantic paradoxes of truth and exemplification arise only when we move beyond the purely logical and, hence, do not of themselves provide any strong objection to a type-free conception of properties and propositions. -/- Most of my arguments in §1 are largely independent of any specific account of the nature of properties and propositions beyond their type-freedom. However, I will in addition argue that there are good reasons to take propositions, at least, to be very fine-grained. My arguments are thus bolstered significantly if it can be shown that there are in fact well-defined examples of logics that are not only type-free but which comport with such a conception of propositions. It is the purpose of §2 to lay out a logic of this sort in some detail, drawing especially upon work by George Bealer and related work of my own. With the logic in place, it will be possible to generalize the line of argument noted above regarding Russell-style paradoxes and, in §3, apply it to two propositional paradoxes — the Prior-Kaplan paradox and the Russell-Myhill paradox — that are often taken to threaten the sort of account developed here. (shrink)

En The paradox of Addition and its dissolution (1969), Mario Bunge presenta algunos argumentos para mostrar que la Regla de Adición puede ocasionar paradojas o problemas semánticos. Posteriormente, Margáin (1972) y Robles (1976) mostraron que las afirmaciones de Bunge son insostenibles, al menos desde el punto de vista de la lógica clásica. Aunque estamos de acuerdo con las críticas de Margáin y Robles, no estamos de acuerdo en el diagnóstico del origen del problema y tampoco con la manera en la (...) que se proponen solucionarlo. En esta medida, en este texto mostraremos cómo la Regla de Adición sí trae problemas semánticos que pueden ser planteados desde diferentes perspectivas contemporáneas, como la extensionalidad del condicional, los principios proscriptivos y la validez conexiva. (shrink)

Although Benjamin Schnieder’s theory of the “ordinary conception” of properties successfully handles paradoxical properties—particularly, the property of non-self-instantiation—it fails to account for ordinary, non-pathological cases. The theory allows the inference of ‘a has the property of being F’ only given F(a) and the prior assertibility of ‘the property of being F can exist’. While this allows us to block an inference to a contradiction, it also blocks all of the non-pathological instances of the inference from ‘a is F’ to ‘a (...) has the property of being F’. It thereby fails both as a theory of the ordinary conception and as a replacement of the ordinary notion, assuming the latter is defective. (shrink)

This paper investigates a particular philosophical puzzle via an examination of its status in the writings of Wittgenstein. The puzzle concerns negation and can take on three interrelated guises. The first puzzle is how not-p can so much as negate p at all – for if p is not the case, then nothing corresponds to p. The second puzzle is how not-p can so much as negate p at all when not-p rejects p not as false but as unintelligible – (...) for if p is unintelligible, then p is nothing but scratches and sounds and does not seem apt for negation. And the third puzzle is how “not” could be anything but hopelessly equivocal if it sometimes (per the first puzzle) requires, and sometimes (per the second puzzle) precludes the intelligibility of p. The paper investigates these three puzzles, their respective structures, and their relations to each other. The second puzzle is expounded as the centre of gravity, and in countering two objections to the threefold puzzle, a special predicament is expounded with regard to the second puzzle’s concern with unipolar propositions – propositions that do not admit of an intelligible negation. The text concludes by indicating the first steps that could potentially lead us out of the threefold puzzle. (shrink)