Paradoxes

The paper discusses an original and ingenious puzzle about time and thought due to Saul Kripke. Its connections with related issues and puzzles such as Russell’s paradox and Kaplan’s puzzle are addressed and explored. The problems raised by the puzzle are dealt with briefly, emphasizing the comparative worth of Kripke’s findings in terms of simplicity and modesty of its assumptions. Then the attempted solutions are analyzed and explored, the most important and promising avenue for future work being the extension of (...) the apparatus developed by Kripke in his groundbreaking paper on truth to intensional logic. (shrink)

The Curry–Howard correspondence, according to which propositions are types, suggests that every paradox formulable in natural deduction has a type-theoretical counterpart. I will give a purely type-theoretical formulation of Curry’s paradox. On the basis of the definition of a type, Γ(A), Curry’s reasoning can be adapted to show the existence of an object of the arbitrary type A. This is paradoxical for several reasons, among others that A might be an empty type. The solution to the paradox consists in seeing (...) that Γ(A) is not a well-defined type. (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 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)

This paper clarifies and defends the global approach to defining logical validity for meta- and higher-level inferences. This is contrary to an emerging consensus in favour of local validity. Prevalent recent arguments claim that global validity is either superfluous in virtue of collapsing into local, or else untenable because it overgenerates validities, compromises the formality of logic, or breaks symmetry with regular validity. Accordingly, the literature on higher inferential logic has come to focus almost exclusively on local validity. Many key (...) philosophical takeaways that have since been drawn from the field are effectively based on local monism: the view that local is the One True Validity Criterion. The present paper argues that this is a mistake, resulting largely from an imprecise understanding of global validity. I untangle some common confusions about how global generalizes beyond the metalevel, and submit that its proper formulation is schematic. These clarifications help dispel each of the aforementioned arguments against global, leaving local monism without any justification. Finally, I outline how much of the philosophical literature around higher-level inferences has been built on local monism, and sketch out the sweeping consequences of rejecting it. (shrink)

The concept of truth has many aims but only one source. The article describes the primary concept of truth, here called the synthetic concept of truth, according to which truth does not belong exclusively to us nor exclusively to nature: truth is the objective result of the synthesis of us and nature in the process of rational cognition. It is shown how various aspects of the concept of truth – logical, scientific, and mathematical aspect – arise from the synthetic concept (...) of truth. Related to these aspects, (i) the role of the predicate of truth in the concept of truth is analysed, (ii) Tarski’s definition of truth and its role in the concept of truth are analysed, and (iii) the position of the paradoxes of truth in the concept of truth is analysed. (shrink)

This paper presents the Paradoxical Theorem, which states that no imperfect being can determine the truth or falsity of any statement with absolute perfection. We then apply the theorem to itself for any imperfect being, which leads to a self-referential paradox.

A - "All ravens are black." B - "All non-black things are non-ravens." C - "All the elements of the set of ravens are elements of the set of black ravens." D - "All the elements of the set of the things that are not black ravens are elements of the set of the things that are not ravens." -/- The propositions A, B, C, D are logically equivalent: if one proposition is true, the other three propositions are also true; (...) if one proposition is false, the other three propositions are also false. -/- The generalizations A and C have the same instances: a black raven is an instance of A and is an instance of C. -/- The generalizations B and D have some instances in common: a non-black object that is a non-raven is an instance of B and is an instance of D; in addition, the generalization D also has as its instances black objects that are non-ravens (for example, a black shoe is an instance of D but is not an instance of B). -/- E - "All ravens are non-black." F - "All black things are non-ravens." G - "All the elements of the set of ravens are elements of the set of the things that are not black ravens". H - "All the elements of the set of black ravens are elements of the set of the things that are not ravens." -/- The generalizations E and G have the same instances: a non-black raven is an instance of E and is an instance of G. -/- By contrast, the generalizations F and H do not have the same instances: a black thing that is a non-raven is an instance of F; by contrast, H is a generalization that cannot have instances because a black raven that is a non-raven is a contradiction in terms. -/- To summarize: A and C have the same instances; B and D have some instances in common; E and G have the same instances; by contrast, F and H do not have instances in common because H has no instances, H cannot have instances. -/- To state that all black ravens are not ravens (as proposition H does) implies that the set of black ravens is not a subset of the set of ravens: hence proposition H is self-contradictory. Affirming that the set of black ravens is not a subset of the set of ravens contradicts the very definition of what a black raven is. Thus, even though the proposition H is the contrapositive of the proposition G, the propositions G and H are not logically equivalent since the proposition G is coherent and is a contingent proposition; the proposition H is incoherent and is a necessary falsehood. -/- Furthermore: Peter Godfrey-Smith writes in the section 3.3 “The Ravens Problem” of his book “Theory and Reality” [chapter “Induction and Confirmation”]: -/- “First, the logical empiricists were concerned to deal with the case where generalizations cover an infinite number of instances. In that case, as we see each raven we are not reducing the number of ways in which the hypothesis might fail”. -/- In the raven paradox it is assumed that the set of ravens has infinite elements, therefore it is assumed that the set of ravens is not empty. -/- The proposition G implies that the set of ravens contains an empty subset of black ravens: the proposition G is true in case there are not black ravens. The proposition H implies that the set of black ravens contains an empty subset of ravens: but in the raven paradox it is assumed that the set of ravens is not empty; hence the proposition H is false. The proposition H is not logically equivalent to the proposition G: in case there are ravens (as assumed in the raven paradox) but there are not black ravens, the proposition H is false but the proposition G is true. Thus, even though the proposition H is the contrapositive of the proposition G, the propositions G and H are not logically equivalent. -/- . (shrink)

En este breve ensayo me interesa analizar el concepto de lenguaje privado desarrollado por Ludwig Wittgenstein. La importancia radica en el desafío que plantea esta noción cuando nos ponemos a pensar sobre las posibilidades reales de un lenguaje puramente privado. En este sentido, el argumento del enfoque wittgensteiniano sostiene que el significado de las palabras no se origina en experiencias subjetivas individuales, sino en la interacción social dentro de una comunidad de hablantes. De modo que, si cada persona tuviera un (...) lenguaje exclusivamente privado, la comunicación sería imposible. Por ello, el objetivo de este escrito se centra en debatir si efectivamente el lenguaje es un fenómeno colectivo, arraigado en prácticas compartidas y acuerdos tácitos, o bien, puede tratarse de una construcción personal subjetiva. En síntesis, para examinar este planteo me centraré en el argumento propuesto por Kripke contra el lenguaje privado, y desde allí, analizaré la paradoja escéptica de Wittgenstein. (shrink)

This manuscript presents an English translation of the work titled “Begründung Einer Strengen Implikation” by the German logician and mathematician Wilhelm Ackermann (1896-1962), first published in June 1956.

The present paper investigates whether strictly classical inferences contribute to the formalization of (genuine) paradoxes within natural deduction. Tennant's criterion for paradoxicality relies on the generation of an infinite reduction sequence, which distinguishes genuine paradoxes from mere inconsistencies. His methodological conjecture posits that genuine paradoxes are never strictly classical and can be derived without classical inferences such as the Law of Excluded Middle, Dilemma, Classical Reductio, and Double Negation Elimination. -/- It appears that there were two reasons for Tennant's proposal (...) of the methodological conjecture. The one is that strictly classical inferences hinder the generation of an infinite reduction sequence and the other is that strictly classical inferences have no role in the formalization of genuine paradoxes. -/- This paper raises questions about these two reasons. Focusing on the liar paradox, it will be argued that strictly classical inferences do not interfere with the generation of an infinite reduction sequence and that the liar sentence may implicitly entail strictly classical inferences. Should this analysis hold, it would call into question not only Tennant’s motivation for advancing the methodological conjecture, but also challenge his contention that genuine paradoxes—exemplified by the liar paradox—are never constructed with reliance on strictly classical inferences. (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)

I will now describe several applications of the theory of n-universes across different probabilistic scenarios. First, I will explain how n-universes can extend the probability spaces used in traditional probability theory. These extended probability spaces enable more refined modelling of complex probabilistic situations and align more intuitively with our perceptions of the physical universe. Next, I will demonstrate the use of n-universes as a methodological tool through two thought experiments described by John Leslie. Finally, I will model Goodman's paradox within (...) the framework of n-universes, showing how they closely resemble Goodmanian worlds. (shrink)

Much of the debate about the mathematical refutation of Zeno’s paradoxes surrounds the logical possibility of completing supertasks—tasks made up of an infinite number of subtasks. Max Black and J.F. Thomson attempt to show that supertasks entail logical contradictions, but their arguments come up short. In this paper, I take a different approach to the mathematical refutations. I argue that even if supertasks are possible, we do not have a non-question-begging reason to think that Achilles’ supertask is possible. The justification (...) for the possibility of Achilles’ supertask lies in the possibility of him completing other supertasks of the same kind, and the justification for the possibility of him completing these other supertasks lies in the possibility of him completing yet more supertasks ad infinitum. (shrink)

I argue for a transformative revival or actualization of the very core of an integrative, methodologically secured form of intellect‑mystical asceticism. This approach draws on traditional sources that are re‑examined from a systematic—synthetic and transcultural—philosophical perspective and in light of the multi‑civilizational global environment of the 21st century. The main traditional points of reference in this paper are provided by Nicolaus de Cusa and Ibn Sīnā, and I refer toa few others, such as Attar of Nishapur, in passing. I begin (...) by developing a basic concept of intellect‑mystical asceticism. It is distinguished from mystification, science, scientism, and modes of everyday communication and cognition. Then, I make the case for an updated, transcultural approach to intellect‑mysticism that can foster the internal (social) and external (environmental) reintegration of the human noosphere and technosphere in future planetary development. In this context, a modern intellect‑mystical philosophical notion of “knowing non‑knowing” (wissendes Nichtwissen, docta ignorantia) is developed. It is inspired by Nicolaus de Cusa and contextualized from a systematic transcultural angle at the same time. Finally, I discuss the problem of the practical, or rather ascetic, realization of the related possibilities of intellect‑mystical self‑enfolding. Here, the preceding steps of the reflection are mapped onto an outline regarding distinct developmental stages of such a transformative intellect‑mystical practice in Ibn Sīnā’s Remarks and Admonitions (al‑Ishārāt wat‑Tanbīhāt). (shrink)

English translation of an article originally published in French in Liber Amicorum Pascal Engel, J. Dutant, G. Fassio & A. Meylan (eds.), University of Geneva, 2014, pp. 581-608. In what follows, I strive to present the elements of a philosophical doctrine, which can be defined as dialectical contextualism. I proceed first to define the elements of this doctrine: dualities and polar contraries, the principle of dialectical indifference and the one-sidedness bias. I emphasize then the special importance of this doctrine in (...) one specific field of meta-philosophy: the methodology for solving philosophical paradoxes. Finally, I describe several applications of this methodology on the following paradoxes: Hempel's paradox, the surprise examination paradox and the Doomsday Argument. (shrink)

Dans ce qui suit, je m'attache à présenter les éléments d'une doctrine philosophique, qui peut être définie comme un contextualisme dialectique. Je m'efforce tout d'abord de définir les éléments constitutifs de cette doctrine, à travers les dualités et pôles duaux, le principe d'indifférence dialectique et le biais d'uni-polarisation. Je m'attache ensuite à souligner l'intérêt spécifique de cette doctrine au sein d'un domaine particulier de la méta-philosophie : la méthodologie utilisée pour la résolution des paradoxes philosophiques. Je décris enfin une application (...) de cette dernière aux paradoxes suivants : le paradoxe de Hempel, le paradoxe de l'examen-surprise et l'argument de l'Apocalypse. (shrink)

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)

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)

Accounting for the epistemic contribution of deduction has been a pervasive problem for logicians interested in deduction, such as, among others, Jakko Hintikka. The problem arises because the conclusion validly deduced from a set of premises is said to be “contained” in that set; because of this containment relation, the conclusion would be known from the moment the premises are known. Assuming this, it is problematic to explain how we can gain knowledge by deducing a logical consequence implied by a (...) set of known premises. To address this problem, we offer an alternative account of the epistemic contribution of deduction as the process required to deduce a conclusion or a theorem, understanding such a process not only in terms of the number of steps in the derivation but also, more importantly, in terms of the reason for or justification for every step. That is, we do not know a proposition unless we have a justification or proof of that proposition. With this goal in mind, we develop a justification logic system which exhibits the epistemic contribution of a deductive derivation as the resulting justified formula. (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.

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)

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)

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.

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)

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)

We can illustrate the phenomenon of model-induced escape by examining the phenomenon of spam filters. Spam filter A is, we can assume, very effective at blocking spam. Indeed it is so effective that it motivates the authors of spam to invent new types of spam that will beat the filters of spam filter A. -/- An example of this phenomenon in the realm of philosophy is illustrated in the work of Nyíri on Wittgenstein's political beliefs. Nyíri writes a paper demonstrating (...) convincingly that there are strong signals of a conservative strain of thought in the writings of Wittgenstein. This has initially a tiny effect; but then a more significant effect sets in as the authors of Wittgenstein secondary literature, consciously or unconsciously, draw attention to features of Wittgenstein which cast the conservatism thesis in a negative light. This is an example of what I call model-induced escape: the Nyíri model of Wittgenstein initiates a process which undermines the Nyíri model of Wittgenstein. -/- Wittgenstein, as obiectum philosophiae, is reduced to a constantly mutating set of interpretations of a certain body of work. -/- Perhaps this is what makes philosophy so problematic when viewed from the perspective of results, or in other words of signs of progress commonly accepted across the discipline. There are just too many ways of inducing escape from any given putative discovery; too many dimensions along which an interpretative or definitional arms race can be triggered. (shrink)

In our research about Fuzzy Time and modeling time, "Unexpected Hanging Paradox" plays a major role. Here, we compare this paradox to the Zeno Paradox and the relations of them with our standard models of continuum and Fuzzy numbers. To do this, we review the project "Fuzzy Time and Possible Impacts of It on Science" and introduce a new way in order to approach the solutions for these paradoxes. Additionally, we have a more general discussion about paradoxes, as Philosophical back (...) ground of the subject and in this way, we introduce the concepts of General View, the first picture, BBF-General View. (shrink)

In this article, some classical paradoxes of infinity such as Galileo’s paradox, Hilbert’s paradox of the Grand Hotel, Thomson’s lamp paradox, and the rectangle paradox of Torricelli are considered. In addition, three paradoxes regarding divergent series and a new paradox dealing with multiplication of elements of an infinite set are also described. It is shown that the surprising counting system of an Amazonian tribe, Pirah ̃a, working with only three numerals (one, two, many) can help us to change our perception (...) of these paradoxes. A recently introduced methodology allowing one to work with finite, infinite, and infinitesimal numbers in a unique computational framework not only theoretically but also numerically is briefly described. This methodology is actively used nowadays in numerous applications in pure and applied mathematics and computer science as well as in teaching. It is shown in the article that this methodology also allows one to consider the paradoxes listed above in a new constructive light. (shrink)

The American rock band KISS is notorious. Its notoriety derives not only from the band’s otherworldly costumes (except for of course during the unmasked period), the fact that they were inducted into the Rock and Roll Hall of Fame, their numerous hit records or the amazing stage theatrics and pyrotechnics of their live shows. It’s also related to the band’s constantly changing makeup (and I don’t mean the kind on their faces!). Of the four members, only Paul Stanley and Gene (...) Simmons were fixtures. With so many changes to the band’s composition, has KISS always remained the same band? Some see this head-scratcher as roughly similar to a conundrum in philosophical metaphysics (that’s the area of philosophy addressing problems of existence): the puzzle of Theseus’s ship. (shrink)

Hofstadter [1979, 2007] offered a novel Gödelian proposal which purported to reconcile the apparently contradictory theses that (1) we can talk, in a non-trivial way, of mental causation being a real phenomenon and that (2) mental activity is ultimately grounded in low-level rule-governed neural processes. In this paper, we critically investigate Hofstadter’s analogical appeals to Gödel’s [1931] First Incompleteness Theorem, whose “diagonal” proof supposedly contains the key ideas required for understanding both consciousness and mental causation. We maintain that bringing sophisticated (...) results from Mathematical Logic into play cannot furnish insights which would otherwise be unavailable. Lastly, we conclude that there are simply too many weighty details left unfilled in Hofstadter’s proposal. These really need to be fleshed out before we can even hope to say that our understanding of classical mind-body problems has been advanced through metamathematical parallels with Gödel’s work. (shrink)

This article proposes a general strategy to overcome normative conflicts, namely, paradoxes represented in Standard Deontic Logic. This solution is based on preference relations between norms that circumvent situations of conflict. Pragmatic justifications of the proposed method are also given.

Nosso objetivo neste trabalho é apresentar uma proposta de solução a paradoxos relacionados à lógica deôntica presentes na literatura, reunidos sob o que é chamado de dilemas deônticos - situações nas quais duas obrigações conflitantes estão presentes num mesmo sistema normativo. Situações deste tipo, quando formalizadas (em SDL - standard deontic logic - ou em outras lógicas relacionadas), levam a uma inconsistência. Nossa proposta baseia-se em relações de preferência que geram uma ferramenta de escolha dentre as duas soluções normativas conflitantes, (...) o que evita a inconsistência e permite o pleno cumprimento do sistema. Justificativas filosóficas são fornecidas as ferramentas lógicas, bem como as suas implicações. (shrink)

O propósito deste trabalho é analisar a estrutura e discutir a aplicação da(s) lógica(s) deôntica(s) na representação de normas jurídicas. Após uma apresentação desta(s) lógic(s) e, em particular, do sistema de von Wright e da SDL (Standard Deontic Logic), veremos as dificuldades de se aplicar uma lógica à análise do direito positivo: as regras da lógica devem permitir realizar sobre as normas formalizadas somente os tipos de inferência e operaçôes lógicas intuitivamente feitas pelos juristas, ou seja, a reconstrução lógica do (...) direito positivo não se trata de uma tarefa de criação do direito, mas uma tarefa de representação daquilo que o conceito de direito usual reconhece como tal. Para ilustrar estas dificuldades, abordaremos as propostas de Carlos Alchourrón em artigo de 1975, “Condicionalidad y la representación de las normas jurídicas”. (shrink)

Lewis Carroll, both logician and writer, suggested a logical paradox containing furthermore two connotations (connotations or metaphors are inherent in literature rather than in mathematics or logics). The paradox itself refers to implication demonstrating that an intermediate implication can be always inserted in an implication therefore postponing its ultimate conclusion for the next step and those insertions can be iteratively and indefinitely added ad lib, as if ad infinitum. Both connotations clear up links due to the shared formal structure with (...) other well-known mathematical observations: (1) the paradox of Achilles and the Turtle; (2) the transitivity of the relation of equality. Analogically to (1), one can juxtapose the paradox of the Liar (for Lewis Carroll’s paradox) and that of the arrow (for “Achilles and the Turtle”), i.e. a logical paradox, on the one hand, and an aporia of motion, on the other hand, suggesting a shared formal structure of both, which can be called “ontological”, on which basis “motion” studied by physics and “conclusion” studied by logic can be unified being able to bridge logic and physics philosophically in a Hegelian manner: even more, the bridge can be continued to mathematics in virtue of (2), which forces the equality (for its property of transitivity) of any two quantities to be postponed analogically ad lib and ad infinitum. The paper shows that Hilbert arithmetic underlies naturally Lewis Carroll’s paradox admitting at least three interpretations linked to each other by it: mathematical, physical and logical. Thus, it can be considered as both generalization and solution of his paradox therefore naturally unifying the completeness of quantum mechanics (i.e. the absence of hidden variables) and eventual completeness of mathematics as the same and isomorphic to the completeness of propositional logic in relation to set theory as a first-order logic (in the sense of Gödel (1930)’s completeness theorems). (shrink)

The present work outlines a logical and philosophical conception of propositions in relation to a group of puzzles that arise by quantifying over them: the Russell-Myhill paradox, the Prior-Kaplan paradox, and Prior's Theorem. I begin by motivating an interpretation of Russell-Myhill as depending on aboutness, which constrains the notion of propositional identity. I discuss two formalizations of of the paradox, showing that it does not depend on the syntax of propositional variables. I then extend to propositions a modal predicative response (...) to the paradoxes articulated by an abstraction principle for propositions. On this conception, propositions are “shadows” of the sentences that express them. Modal operators are used to uncover the implicit relation of dependence that characterizes propositions that are about propositions. The benefits of this approach are shown by application to other intensional puzzles. The resulting view is an alternative to the plenitudinous metaphysics of impredicative comprehension principles. (shrink)

There are three questions associated with Simpson’s paradox (SP): (i) Why is SP paradoxical? (ii) What conditions generate SP? and (iii) How to proceed when confronted with SP? An adequate analysis of the paradox starts by distinguishing these three questions. Then, by developing a formal account of SP, and substantiating it with a counterexample to causal accounts, we argue that there are no causal factors at play in answering questions (i) and (ii). Causality enters only in connection with action.

We address the need for a model by considering two competing theories regarding the origin of life: (i) the Metabolism First theory, and (ii) the RNA World theory. We discuss two interrelated points, namely: (i) Models are valuable tools for understanding both the processes and intricacies of origin-of-life issues, and (ii) Insights from models also help us to evaluate the core objection to origin-of-life theories, called “the inefficiency objection”, which is commonly raised by proponents of both the Metabolism First theory (...) and the RNA World theory against each other. We use Simpson’s Paradox (SP) as a tool for challenging this objection. We will use models in various senses, ranging from taking them as representations of reality to treating them as theories/accounts that provide heuristics for probing reality. In this paper, we will frequently use models and theories interchangeably. Additionally, we investigate Conway’s Game of Life and contrast it with our SP-based approach to emergence-of-life issues. Finally, we discuss some of the consequences of our view. A scientific model is testable in three senses: (i) a logical sense, (ii) a nomological sense, and (iii) a current technological sense. The SP-based model is testable in the first two senses but it is not feasible to test it using current technology. (shrink)

A resolution to the Russell Paradox is presented that is similar to Russell's “theory of types” method but is instead based on the definition of why a thing exists as described in previous work by this author. In that work, it was proposed that a thing exists if it is a grouping tying "stuff" together into a new unit whole. In tying stuff together, this grouping defines what is contained within the new existent entity. A corollary is that a thing, (...) such as a set, does not exist until after the stuff is tied together, or said another way, until what is contained within is completely defined. A second corollary is that after a grouping defining what is contained within is present and the thing exists, if one then alters what is tied together (e.g., alters what is contained within), the first existent entity is destroyed and a different existent entity is created. A third corollary is that a thing exists only where and when its grouping exists. Based on this, the Russell Paradox's set R of all sets that aren't members of themselves does not even exist until after the list of the elements it contains (e.g. the list of all sets that aren't members of themselves) is defined. Once this list of elements is completely defined, R then springs into existence. Therefore, because it doesn't exist until after its list of elements is defined, R obviously can't be in this list of elements and, thus, cannot be a member of itself; so, the paradox is resolved. This same type of reasoning is then applied to Godel's first Incompleteness Theorem. Briefly, while writing a Godel Sentence, one makes reference to a future, not yet completed and not yet existent sentence, G, that claims its unprovability. However, only once the sentence is finished does it become a new unit whole and existent entity called sentence G. If one then goes back in and replaces the reference to the future sentence with the future sentence itself, a totally different sentence, G1, is created. This new sentence G1 does not assert its unprovability. An objection might be that all the possibly infinite number of possible G-type sentences or their corresponding Godel numbers already exist somehow, so one doesn't have to worry about references to future sentences and springing into existence. But, if so, where do they exist? If they exist in a Platonic realm, where is this realm? If they exist pre-formed in the mind, this would seem to require a possibly infinite-sized brain to hold all these sentences. This is not the case. What does exist in the mind is the system for creating G-type sentences and their corresponding numbers. This mental system for making a G-type sentence is not the same as the G-type sentence itself just as an assembly line is not the same as a finished car. In conclusion, a new resolution of the Russell Paradox and some issues with proofs of Godel's First Incompleteness Theorem are described. (shrink)

Many philosophers think that common sense knowledge survives sophisticated philosophical proofs against it. Recently, however, Bryan Frances (forthcoming) has advanced a philosophical proof that he thinks common sense can’t survive. Exploiting philosophical paradoxes like the Sorites, Frances attempts to show how common sense leads to paradox and therefore that common sense methodology is unstable. In this paper, we show how Frances’s proof fails and then present Frances with a dilemma.

Counterfactuals are somewhat tolerant. Had Socrates been at least six feet tall, he need not have been exactly six feet tall. He might have been a little taller—he might have been six one or six two. But while he might have been a little taller, there are limits to how tall he would have been. Had he been at least six feet tall, he would not have been more than a hundred feet tall, for example. Counterfactuals are not just tolerant, (...) then, but bounded. This paper presents a surprising paradox: If counterfactuals are tolerant and bounded, then we can prove a flat contradiction using natural rules of inference. Something has to go then. But what? (shrink)

As a response to the semantic and logical paradoxes, theorists often reject some principles of classical logic. However, classical logic is entangled with mathematics, and giving up mathematics is too high a price to pay, even for nonclassical theorists. The so-called recapture theorems come to the rescue. When reasoning with concepts such as truth/class membership/property instantiation, if ones is interested in consequences of the theory that only contain mathematical vocabulary, nothing is lost by reasoning in the nonclassical framework. It is (...) shown that this claim is highly misleading, if not simply false. Under natural assumptions, recapture claims are incorrect. (shrink)

When are two formal theories of broadly logical concepts, such as truth, equivalent? The paper investigates a case study, involving two well-known variants Kripke-Feferman truth. The first, KF+CONS, features a consistent but partial truth predicate. The second, KF+COMP, an inconsistent but complete truth predicate. It is well-known that the two truth predicates are dual to each other. We show that this duality reveals a much stricter correspondence between the two theories: they are intertraslatable. Intertranslatability under natural assumptions coincides with definitional (...) equivalence, and is arguably the strictest notion of theoretical equivalence different from logical equivalence. The case of KF+CONS and KF+COMP raises a puzzle: the two theories can be proved to be strictly related, yet they appear to embody remarkably different conceptions of truth. We discuss the significance of the result for the broader debate on formal criteria of conceptual reducibility for theories of truth. (shrink)