Paradox Research Articles

A normal paradox

Abstract: For the past 40 years, Neil Tennant has defended a proof-theoretic criterion of self-referential paradoxicality. According to this criterion, the defining characteristic of paradoxes is that, when formulated within a natural deduction system, they produce derivations that cannot be normalized. This paper raises doubts about Tennant’s approach. Recently, Tennant has suggested that Russell’s paradox might not truly fit his criterion. I will argue that the reasoning that rules out Russell’s paradox can similarly be applied to some semantic paradoxes. Therefore, if Tennant’s assessment of Russell’s paradox holds, few cases may genuinely qualify as paradoxes by his standards.

Wittgenstein and the liar

Ludwig Wittgenstein’s remarks on contradictions and paradoxes have been met with incomprehension and have fueled the widespread and long-standing prejudice that his later thoughts on the foundations of logic and mathematics are the “surprisingly insignificant product of a sparkling mind” (Kreisel, British Journal for the Philosophy of Science 9:135–158, 1959, p. 158). This paper disagrees; it argues that Wittgenstein’s remarks on semantic paradoxes suggest an account of the Liar and its kin that is not only of historical interest but also represents a hitherto unnoticed paraconsistent alternative to established approaches to the Liar. In what follows, a reading of Wittgenstein’s remarks will be offered according to which Wittgenstein subscribes to a form of dialetheism (that is, the view that there are sentences that are both true and false). In contrast to modern dialetheist approaches to the Liar, however, some of Wittgenstein’s remarks suggest combining a dialetheist position with what is called ‘logical nihilism’ (that is, the view that there are no universally valid inference rules).

Heuristics in philosophy

This article argues that heuristics play a key role in philosophy, in generating both our verdicts on proposed counterexamples to philosophical theories and philosophical paradoxes. Heuristics are efficient ways of answering questions, quick and easy to use, but imperfectly reliable. They have been studied by psychologists and cognitive scientists such as Gigerenzer and Kahneman, but their relevance to philosophical methodology has not been properly recognized. Several heuristics are discussed at length. The persistence heuristic can be summarized in the slogan ‘Small changes don’t matter’. Without it, updating would present an intractable problem for both natural and artificial intelligence. But our reliance on the persistence heuristic also makes us vulnerable to paradoxes of vagueness. Disquotational heuristics of various kinds are considered. They play central roles in our ascriptions of truth, falsity, and belief, but they also generate semantic paradoxes such as the Liar and Frege puzzles about coreference. The use of an additive heuristic for combining reasons is also discussed. Our reliance on fallible heuristics in philosophy does not make philosophical knowledge impossible, just as our reliance on fallible heuristics in perception does not make perceptual knowledge impossible. Nevertheless, it should motivate us to take a more critical attitude to our data. By identifying and analyzing the heuristics on which we rely, we may be able to work out where they make us most vulnerable to error.

Semantic Closure and Classicality

The semantic paradoxes show that semantic theories that internalize their semantic concepts, such as truth and validity, cannot validate all classical logic. That is, it is necessary to weaken some connective of the object language, taken as the guilty of the paradoxes, or give up some property of the consequence relation of the logical theory. Both strategies may distance us from classical logic, the logic commonly used in our current mathematical theories. So, a desirable solution to semantic paradoxes cannot distance us from classical logic. This paper analyzes two interesting proposals that aim to maintain classical logic as most as possible. The first strategy, the Barrio & Pailos & Szmuc-approach (2017) (BPS-approach), proposes the paraconsistent logic MSC that contains in its object language a connective capable of recovering classical inference whenever the sentences at issue are consistent. So, they show that it is possible to build a semantic theory over this logic that is immune to the semantic paradoxes. The second approach is based on the hierarchy STω of non-transitive systems STn, proposed by Pailos (2020a). This hierarchy recovers classical metainferences as many as possible in higher levels of the hierarchy. We argue in favor of the second approach by arguing that the first strategy must adopt weak self-referential procedures to avoid the paradoxes.

Engaging marginal stakeholders on social networking sites. A cross‐country exploratory analysis among Generation Z consumers

AbstractThis research explores the marginal stakeholder engagement and propensity to value cocreation in the fast‐fashion industry by taking Generation Z consumers (GZCs) as observation unit and social networking sites (SNSs) as context of investigation. By undertaking 24 in‐depth interviews with US and Italian GZCs, the study uncovers the main elements that influence their engagement generation on SNSs and highlights that at least four main paradoxes (PXs) exist in this scenario. Specifically, the interviewees reported that they do not trust those brands that make them feel as marginal consumers (PX1) and that they are likely to be extremely loyal to those brands that actively take them into consideration in their activities on SNSs (PX2). In addition, although GZCs concerned about the environment, in most cases, they will buy repeatedly from fast‐fashion brands for convenience reasons (PX3). Finally, social dimension of GZCs engagement appears to be influenced by their peers' preferences (PX4). This research is the first that explored GZCs by considering them under the lenses of stakeholder engagement and offering novel insights about common believes on this generation of consumers. Therefore, theoretical and managerial implications are discussed to advance current knowledge on GZCs and to help the development of new strategies to engage them on SNSs.

Systems for Non-Reflexive Consequence

Substructural logics and their application to logical and semantic paradoxes have been extensively studied. In the paper, we study theories of naïve consequence and truth based on a non-reflexive logic. We start by investigating the semantics and the proof-theory of a system based on schematic rules for object-linguistic consequence. We then develop a fully compositional theory of truth and consequence in our non-reflexive framework.

Procedural Semantics and its Relevance to Paradox

Two semantic paradoxes, the Liar and Curry’s paradox, are analysed using a newly developed conception of procedural semantics (semantics according to which the truth of propositions is determined algorithmically), whose main characteristic is its departure from methodological realism. Rather than determining pre-existing facts, procedures are constitutive of them. Of this semantics, two versions are considered: closed (where the halting of procedures is presumed) and open (without this presumption). To this end, a procedural approach to deductive reasoning is developed, based on the idea of simulation. As is shown, closed semantics supports classical logic, but cannot in any straightforward way accommodate the concept of truth. In open semantics, where paradoxical propositions naturally ‘belong’, they cease to be paradoxical; yet, it is concluded that the natural choice—for logicians and common people alike—is to stick to closed semantics, pragmatically circumventing problematic utterances.

Outline of a Dialogical Theory of Truth

In this essay I propose two theories of truth and show how they deal with semantic paradoxes. Their most salient feature is that they are based on a gametheoretic understanding of logic and meaning. “Truth”, therefore, is understood dialogically, as agreement between parts. I compare this proposal with a similar one already existing in the literature —Dutilh Novaes and French 2018—, and highlight the advantages of mine. The theories of truth I present are non-trivial, substructural (in a sense to be clarified) and capture strong intuitions about truth. Some philosophical recommendations in favor of our understanding of Dialogics are delivered along the way.

NON-CONTRACTIVE LOGICS, PARADOXES, AND MULTIPLICATIVE QUANTIFIERS

Abstract The paper investigates from a proof-theoretic perspective various non-contractive logical systems, which circumvent logical and semantic paradoxes. Until recently, such systems only displayed additive quantifiers (Grišin and Cantini). Systems with multiplicative quantifiers were proposed in the 2010s (Zardini), but they turned out to be inconsistent with the naive rules for truth or comprehension. We start by presenting a first-order system for disquotational truth with additive quantifiers and compare it with Grišin set theory. We then analyze the reasons behind the inconsistency phenomenon affecting multiplicative quantifiers. After interpreting the exponentials in affine logic as vacuous quantifiers, we show how such a logic can be simulated within a truth-free fragment of a system with multiplicative quantifiers. Finally, we establish that the logic for these multiplicative quantifiers (but without disquotational truth) is consistent, by proving that an infinitary version of the cut rule can be eliminated. This paves the way to a syntactic approach to the proof theory of infinitary logic with infinite sequents.

Bicontextualism

Can one quantify over absolutely everything? Absolutists answer positively, while relativists answer negatively. Here, I focus on the absolutism versus relativism debate in the framework of theories of truth, where relativism becomes a form of contextualism about truth predications. Contextualist theories of truth provide elegant and uniform solutions to the semantic paradoxes while preserving classical logic. However, they interpret harmless generalizations (such as “everything is self-identical”) in less than absolutely comprehensive domains, thus systematically misconstruing them. In this article, I show that contextualism is broadly compatible with absolute generality. More specifically, I develop a bipartite contextualist semantics, or “bicontextualism,” on which sentences are split in two groups: the unproblematic sentences, which are compatible with absolute generality, and the problematic ones, which are given a relativist semantics. I then argue that bicontextualism retains the advantages of (orthodox) contextualism and does not give rise to new revenge paradoxes.

A Recipe for Paradox

In this paper, we provide a recipe that not only captures the common structure of semantic paradoxes but also captures our intuitions regarding the relations between these paradoxes. Before we unveil our recipe, we first talk about a well-known schema introduced by Graham Priest, namely, the Inclosure Schema. Without rehashing previous arguments against the Inclosure Schema, we contribute different arguments for the same concern that the Inclosure Schema bundles together the wrong paradoxes. That is, we will provide further arguments on why the Inclosure Schema is both too narrow and too broad.
 We then spell out our recipe. The recipe shows that all of the following paradoxes share the same structure: The Liar, Curry's paradox, Validity Curry, Provability Liar, Provability Curry, Knower's paradox, Knower's Curry, Grelling-Nelson's paradox, Russell's paradox in terms of extensions, alternative Liar and alternative Curry, and hitherto unexplored paradoxes.
 We conclude the paper by stating the lessons that we can learn from the recipe, and what kind of solutions the recipe suggests if we want to adhere to the Principle of Uniform Solution.

The Final Cut

In a series of works, Pablo Cobreros, Paul Égré, David Ripley and Robert van Rooij have proposed a nontransitive system (call it ‘K3LP’) as a basis for a solution to the semantic paradoxes. I critically consider that proposal at three levels. At the level of the background logic, I present a conception of classical logic on which K3LP fails to vindicate classical logic not only in terms of structural principles, but also in terms of operational ones. At the level of the theory of truth, I raise a cluster of philosophical difficulties for a K3LP-based system of naive truth, all variously related to the fact that such a system proves things that would seem already by themselves repugnant, even in the absence of transitivity. At the level of the theory of validity, I consider an extension of the K3LP-based system of naive validity that is supposed to certify that validity in that system does not fall short of naive validity, argue that such an extension is untenable in that its nontriviality depends on the inadmissibility of a certain irresistible instance of transitivity (whence the advertised “final cut”) and conclude on this basis that the K3LP-based system of naive validity cannot coherently be adopted either. At all these levels, a crucial role is played by certain metaentailments and by the extra strength they afford over the corresponding entailments: on the one hand, such strength derives from considerations that would seem just as compelling in a general nontransitive framework, but, on the other hand, such strength wreaks havoc in the particular setting of K3LP.

On the Metainferential Solution to the Semantic Paradoxes

On the Metainferential Solution to the Semantic Paradoxes

Dissolving the Paradoxicality Paradox

Non-classical solutions to semantic paradox can be associated with conceptions of paradoxicality understood in terms of entailment facts. In a K3-based theory of truth, for example, it is prima facie natural to say that a sentence φ is paradoxical iff φ ∨ ¬φ entails an absurdity. In a recent paper, Julien Murzi and Lorenzo Rossi exploit this idea to introduce revenge paradoxes for a number of non-classical approaches, including K3. In this paper, I show that on no understanding of ‘is paradoxical’ (for K3) should both rules needed for their paradox be expected to hold unrestrictedly. Just which rule fails, however, depends on various factors, including whether the derivability relation of a target system of reasoning is arithmetically definable.

Dogma, Assertive grounds and forms of Truth-assignment failure

This short paper focuses on Kripke's paper on truth from 1975. It is 1. a historiographical commentary, 2. an argument about the advantages of the theory, and 3. an interpretation of its philosophical meaning. 1. Kripke presents a diagnosis of semantic paradoxes based on their similarity with ungrounded sentences. Based on Kleene's three-value logic, he then shows that it is possible to find fixed points in which the assertion of an unsubstantiated (non-paradoxical) sentence can sustain a cumulative distance with its anti-extension. 2. We argue that Kripke's paper has the advantage of explaining risk in truth assessments. It provides a framework to solve problems of languages that have their truth predicate. Although compatible with Tarski's, this solution more faithfully paints the speculative and revisionist representation of assignments of truth. It exhibits the conditions of stable risk assertions (whose fixed point accumulates semantic value in a single direction) and distinguishes it from irrational assertions, which, as dogmas, base their risk on arbitrary points and provide an unstable basis for truth assertions.

"The White Hair with the Black Thought". A Study on Baratynskij’s Poem "There Were Storms, Bad Weather…..."

This study is dedicated to the poem There Were Storms, Bad Weather… (Byli buri, nepogody…) (1839) by Еvgenij Baratynskij. The study analyzes in detail the semantic development of the text, leading to a strong and somewhat paradoxical ending that causes a particular emotional experience for the reader. To understand how the meanings of the poem are shaped in its semantic development is one of the aims of our study. Additionally, we analyze the folkloric features of the poem: repetitions, a parallelism, and the use of paremia. A deep analysis of the text, considering both its semantic movement and lexical nuances, allows us to discuss the semantic paradox of Baratynskij’s poem and affirm that There were storms, bad weather… is part of a group of texts that by the very use of poetic speech overcomes the impossibility of the utterance.

Anti-exceptionalism and methodological pluralism in logic

According to methodological anti-exceptionalism, logic follows a scientific methodology. There has been some discussion about which methodology logic has. Authors such as Priest, Hjortland and Williamson have argued that logic can be characterized by an abductive methodology. We choose the logical theory that behaves better under a set of epistemic criteria (such as fit to data, simplicity, fruitfulness, or consistency). In this paper, I analyze some important discussions in the philosophy of logic (intuitionism versus classical logic, semantic paradoxes, and the meaning of conditionals), and I show that they presuppose different methodologies, involving different notions of evidence and different epistemic values. I argue that, rather than having a specific methodology such as abductivism, logic can be characterized by methodological pluralism. This position can also be seen as the application of scientific pluralism to the realm of logic.

Is the HYPE about strength warranted?

In comparing classical and non-classical solutions to the semantic paradoxes arguments relying on strength have been influential. In this paper I argue that non-classical solutions should preserve the proof-theoretic strength of classical solutions. Leitgeb’s logic of HYPE is then presented as an interesting possibility to strengthen FDE with a suitable conditional. It is shown that HYPE allows for a non-classical Kripkean theory of truth, called KFL, that is strong enough for the relevant purposes and has additional attractive properties.

Expressing logical disagreement from within

Against the backdrop of the frequent comparison of theories of truth in the literature on semantic paradoxes with regard to which inferences and metainferences are deemed valid, this paper develops a novel approach to defining a binary predicate for representing the valid inferences and metainferences of a theory within the theory itself under the assumption that the theory is defined with a classical meta-theory. The aim with the approach is to obtain a tool which facilitates the comparison between a theory and its competitors within the theory itself, thereby expressing the disagreement between the theories within the theories. After discussing what we can and should require of an object-linguistic representation of a theory for that purpose, this paper proposes to restrict the representation of valid metainferences to locally valid metainferences, a requirement which turns out to be omega -consistent and conservative over classical first-order arithmetic. This approach is then applied to four theories definable on strong Kleene models using a labelled nested sequent calculus.

Paradoxicality in Kripke’s theory of truth

A lot has been written on solutions to the semantic paradoxes, but very little on the topic of general theories of paradoxicality. The reason for this, we believe, is that it is not easy to disentangle a solution to the paradoxes from a specific conception of what those paradoxes consist in. This paper goes some way towards remedying this situation. We first address the question of what one should expect from an account of paradoxicality. We then present one conception of paradoxicality that has been offered in the literature: the fixed-point conception. According to this conception, a statement is paradoxical if it cannot obtain a classical truth-value at any fixed-point model. In order to assess this proposal rigorously we provide a non-metalinguistic characterization of paradoxicality and we evaluate whether the resulting account satisfies a number of reasonable desiderata.