Paradoxes Of Self-reference Research Articles

A Comprehensive Look at Self-Referential Paradoxes and Their Evolution

A Comprehensive Look at Self-Referential Paradoxes and Their Evolution

SettingThe Waste Landin Order

Are T. S. Eliot’s notes on The Waste Land a scholarly resource or a literary hoax? This oft-repeated question gets to the heart of the poem, which thrives on its allusions, whether seriously or cynically. However, scholars have largely passed over the notes’ (and the poem’s) numberings, despite their complexity and superabundance—a panoply of quantitative relations running alongside the qualitative references. These numberings, with startling frequency, do not compute, which poses a philosophical dilemma greater than arithmetical errors would seem to imply. As a graduate student at Harvard, Eliot took course notes on mathematical logic and number theory that show him grappling again and again with a concept of numerical irrationality, a dilemma that, for him, seems to threaten the coherence of the world itself, the failures of enumeration auguring broader pandemonium. Under the tutelage of Bertrand Russell, Eliot turns to logic in an attempt to discern a coherent system for numbers (and therefore life), but he grows disenchanted with how logic’s paradoxes of self-reference undermines that very possibility. In turn, these paradoxes inform The Waste Land as an irrational subtext, as small miscalculations in the poem and the notes herald impending physical disasters, psychological hazards, and metaphysical perils. In the end, how we count its numbers turns out to have important implications for how we account for The Waste Land’s puzzling and even deadly subjects.

Towards a Non-classical Meta-theory for Substructural Approaches to Paradox

In the literature on self-referential paradoxes one of the hardest and most challenging problems is that of revenge. This problem can take many shapes, but, typically, it besets non-classical accounts of some semantic notion, such as truth, that depend on a set of classically defined meta-theoretic concepts, like validity, consistency, and so on. A particularly troubling form of revenge that has received a lot of attention lately involves the concept of validity. The difficulty lies in that the non-classical logician cannot accept her own definition of validity because it is given in a classical meta-theory. It is often suggested that this mismatch between the consequence relation of the account being espoused and the consequence relation of the meta-theory is a serious embarrassment. The main goal of the paper is to explore whether certain substructural accounts of the paradoxes can avoid this sort of embarrassment. Typically, these accounts are expressively incomplete, since they cannot assert in the object language that certain invalid arguments are in fact invalid. To overcome this difficulty I develop a novel type of hybrid proof-procedure, one that takes invalidities to be just as fundamental as validities. I prove that this proof-procedure enjoys a number of interesting properties and I analyze the prospects of applying it to languages capable of expressing self-referential statements.

When Curry met Abel

AbstractBased on his Inclosure Schema and the Principle of Uniform Solution (PUS), Priest has argued that Curry’s paradox belongs to a different family of paradoxes than the Liar. Pleitz (2015, The Logica Yearbook 2014, pp. 233–248) argued that Curry’s paradox shares the same structure as the other paradoxes and proposed a scheme of which the Inclosure Schema is a particular case and he criticizes Priest’s position by pointing out that applying the PUS implies the use of a paraconsistent logic that does not validate Contraction, but that this can hardly seen as uniform. In this paper, we will develop some further reasons to defend Pleitz’ thesis that Curry’s paradox belongs to the same family as the rest of the self-referential paradoxes & using the idea that conditionals are generalized negations. However, we will not follow Pleitz in considering doubtful that there is a uniform solution for the paradoxes in a paraconsistent spirit. We will argue that the paraconsistent strategies can be seen as special cases of the strategy of restricting Detachment and that the latter uniformly blocks all the connective-involving self-referential paradoxes, including Curry’s.

Elementary cellular automata and self-referential paradoxes

AbstractWe associate an elementary cellular automaton with a set of self-referential sentences, whose revision process is exactly the evolution process of that automaton. A simple but useful result of this connection is that a set of self-referential sentences is paradoxical, iff (the evolution process for) the cellular automaton in question has no fixed points. We sort out several distinct kinds of paradoxes by the existence and features of the fixed points of their corresponding automata. They are finite homogeneous paradoxes and infinite homogeneous paradoxes. In some weaker sense, we will also introduce no-no-sort paradoxes and virtual paradoxes. The introduction of these paradoxes, in turn, leads to a new classification of the cellular automata.

Identifying logical evidence

Given the plethora of competing logical theories of validity available, it’s understandable that there has been a marked increase in interest in logical epistemology within the literature. If we are to choose between these logical theories, we require a good understanding of the suitable criteria we ought to judge according to. However, so far there’s been a lack of appreciation of how logical practice could support an epistemology of logic. This paper aims to correct that error, by arguing for a practice-based approach to logical epistemology. By looking at the types of evidence logicians actually appeal to in attempting to support their theories, we can provide a more detailed and realistic picture of logical epistemology. To demonstrate the fruitfulness of a practice-based approach, we look to a particular case of logical argumentation—the dialetheist’s arguments based upon the self-referential paradoxes—and show that the evidence appealed to support a particular theory of logical epistemology, logical abductivism.

A short remark on Godel incompleteness theorem and its self-referential paradox from Neutrosophic Logic perspective

A short remark on Godel incompleteness theorem and its self-referential paradox from Neutrosophic Logic perspective

Noncontractive Classical Logic

One of the most fruitful applications of substructural logics stems from their capacity to deal with self-referential paradoxes, especially truth-theoretic paradoxes. Both the structural rules of contraction and the rule of cut play a crucial role in typical paradoxical arguments. In this paper I address a number of difficulties affecting noncontractive approaches to paradox that have been discussed in the recent literature. The situation was roughly this: if you decide to go substructural, the nontransitive approach to truth offers a lot of benefits that are not available in the noncontractive account. I sketch a noncontractive theory of truth that has these benefits. In particular, it has both a proof- and a model-theoretic presentation, it can be extended to a first-order language, and it retains every classically valid inference.

A short remark on Gödel incompleteness theorem and its self-referential paradox from Neutrosophic Logic perspective

It is known from history of mathematics, that Gödel submitted his two incompleteness theorems, which can be considered as one of hallmarks of modern mathematics in 20th century. Here we argue that Gödel incompleteness theorem and its self-referential paradox have not only put Hilbert’s axiomatic program into question, but he also opened up the problem deep inside the then popular Aristotelian Logic. Although there were some attempts to go beyond Aristotelian binary logic, including by Lukasiewicz’s three-valued logic, here we argue that the problem of self-referential paradox can be seen as reconcilable and solvable from Neutrosophic Logic perspective. Motivation of this paper: These authors are motivated to re-describe the self-referential paradox inherent in Godel incompleteness theorem. Contribution: This paper will show how Neutrosophic Logic offers a unique perspective and solution to Godel incompleteness theorem.

Dialetheism and the Problem of the Missing Difference

Abstract During the past few decades, Graham Priest has advocated for Dialetheism, the controversial position that some contradictions are true. Dialetheism entails that the Law of Non-Contradiction fails. In recent decades the philosophical community has begun to recognize the significant challenge posed by Priest’s arguments. Priest has primarily appealed to paradoxes of self-reference, such as the Liar Paradox, to support his position. Following Priest’s approach, I offer another argument for Dialetheism, which appeals to a self-referential paradox that has been more or less ignored in the philosophical literature on the subject: the paradox of the missing difference. When we reflect on the question ‘what is a concept?’ from the perspective of a classical model of conceptual analysis, we arrive at the paradox of the missing difference. Although contradictions may be improbable, when we reflect on the question ‘how is the domain of concepts possible?’ we are led to a startling principle: without dialetheia any theory concerning concept formation (from a classical perspective on concepts) would be impossible. Dialetheism is a necessary condition for the existence of a domain of concepts in general. As a result, Dialetheism may even be more central to philosophical reflection than even dialetheists themselves have recognized.

Tigres azules de J. L. Borges. Lógica, verdad y mundos imposibles

El propósito de este artículo es mostrar el modo en que Borges, en su relato Tigres azules, desarrolla varias ideas interesantes en torno a la lógica, el lenguaje y las matemáticas. El punto de partida es el escepticismo borgeano sobre la capacidad de la lógica clásica para abordar el problema de la verdad y los mundos imposibles. Se defiende que objetos inconsistentes e, incluso, autocontradictorios no necesariamente dan lugar a mundos física o lógicamente imposibles. Se sugiere que las paradojas que Borges construye en el cuento no son paradojas de autorreferencia, sino variantes de la paradoja de Moore y de las paradojas de implicación material, por lo que para superarlas se requiere un enfoque no-clásico que neutralice ECQ, esto es, un sistema de lógica paraconsistente.

Davidson and a Twist of Wittgenstein: Metaontology, Self-Canceling Paradox, and Settled Insight

The paper proposes with Davidson that the talk of metaontology is literally meaningless, but with Wittgenstein that it is so in a way that grants a unique type of insight. More specifically, it argues both that Davidson’s arguments have a cogency that is hard to dismiss, and also that, since his own arguments are metaontological, they are self-referential, and consequently in turn undermine their own meaning as well. The paper argues further that metaontological statements cannot be avoided. Consequently, this kind of statement constitutes an unavoidable self-referential paradox that means what it also excludes as capable of meaning. The result is a reinstatement of the meaning of ontological insight and in fact, the paper argues, a deep enrichment and also a particularly cogent justification of it. In addition, the logical peculiarity of the paradox involved has further useful consequences for the outcome of this justification, including a mutually illuminating commonality with some versions of metaethics.

Axiomatic Theories of Partial Ground II

This is part two of a two-part paper in which we develop an axiomatic theory of the relation of partial ground. The main novelty of the paper is the of use of a binary ground predicate rather than an operator to formalize ground. In this part of the paper, we extend the base theory of the first part of the paper with hierarchically typed truth-predicates and principles about the interaction of partial ground and truth. We show that our theory is a proof-theoretically conservative extension of the ramified theory of positive truth up to \U0001d7160 and thus is consistent. We argue that this theory provides a natural solution to Fine’s “puzzle of ground” about the interaction of truth and ground. Finally, we show that if we apply the truth-predicate to sentences involving our ground-predicate, we run into paradoxes similar to the semantic paradoxes: we get ground-theoretical paradoxes of self-reference.

The Problem of Kierkegaard's Socrates

This essay re-examines Kierkegaard's view of Socrates. I consider the problem that arises from Kierkegaard's appeal to Socrates as an exemplar for irony. The problem is that he also appears to think that, as an exemplar for irony, Socrates cannot be represented. And part of the problem is the paradox of self-reference that immediately arises from trying to represent x as unrepresentable. On the solution I propose, Kierkegaard does not hold that, as an exemplar for irony, Socrates is in no way representable. Rather, he holds that, as an exemplar for irony, Socrates cannot be represented in a purely disinterested way. I show how, in The Concept of Irony, Kierkegaard makes use of 'limiting cases' of representation in order to bring Socrates into view as one who defies purely disinterested representation. I also show how this approach to Socrates connects up with Kierkegaard's more general interest in the problem of ethical exemplarity, where the problem is how ethical exemplars can be given as such, that is, in such a way that purely disinterested contemplation is not the appropriate response to them.

A Yabloesque paradox in epistemic game theory

The Brandenburger–Keisler paradox is a self-referential paradox in epistemic game theory which can be viewed as a two-person version of Russell’s Paradox. Yablo’s Paradox, according to its author, is a non-self referential paradox, which created a significant impact. This paper gives a Yabloesque, non-self-referential paradox for infinitary players within the context of epistemic game theory. The new paradox advances both the Brandenburger–Keisler and Yablo results. Additionally, the paper constructs a paraconsistent model satisfying the paradoxical statement.

How Pure Should Justice Be? Reflections on G. A. Cohen's Rhetorical Rescue

AbstractIn this article I argue for two closely related conclusions: one concerned more narrowly with the internal consistency of G. A. Cohen's theorizing about justice and the unique rhetoric in which it is couched, the other connected to a more sweeping set of recommendations about how theorizing on justice is most promisingly undertaken. First, drawing on a famous insight of G. E. Moore, I argue that although the (Platonic) purity of Cohenian justice provides Cohen a platform from which to put some extremely challenging criticisms to Rawls and Rawlsian liberals, at the same time it generates a sort of self-referential paradox for many of the theses about the concept of justice to which Cohen himself is committed. I go on to conclude, using Rawls's theory of justice as a model, that it would serve political philosophy well to conceive of justice with less purity than Cohen conceives of it.

Structural Reflexivity and the Paradoxes of Self-Reference

Structural Reflexivity and the Paradoxes of Self-Reference

On Priest on Nonmonotonic and Inductive Logic

Graham Priest defends the use of a nonmonotonic logic, LPm, in his analysis of reasoning in the face of true contradictions, such as those arising from the paradoxes of self-reference. In the course of defending this choice of logic in the face of the criticism that this logic is not truth preserving, Priest argued (2012) that requirement is too much to ask: since LPm is a nonmonotonic logic, it necessarily fails to preserve truth. In this article, I show that this assumption is incorrect, and I explain why nonmonotonic logics can nonetheless be truth preserving. Finally, I diagnose Priest’s error, to explain when nonmonotonic logics do indeed fail to preserve truth.

Les paradoxes i la filosofia: tres visions contemporànies

A paradox is usually described as an apparently false statement supported by an apparently good argument which departs from premises that most people would find trivially true. This survey article presents a brief overview of three different contemporary perspectives on paradoxes. According to the epistemic view, paradoxes play a crucial role in the progress of science and cannot be regarded as sound proofs of a false statement. According to the dialetheist view, the conclusion of some paradoxical reasonings is both, true and false, what means that there must be some true (and false) contradictions. Finally, the mystic view understands recalcitrant paradoxes (antinomies) as unsolvable problems which show the limits of human thought and cognition. The last section of this paper focuses on the paradoxes of self-reference, a family of paradoxes (sharing some structural features) which have had a deep impact in the development of science during the twentieth century. Keywords: paradox, veridic paradox, falsidic paradox, antinomy, dialetheism, selfreference, circularity, inclosure schema.

Social Media: The Unbearable Lightness of Meaning

Milan Kundera, in his classic The Unbearable Lightness of Being undertook a consideration of the ephemeral. So I use it as a springboard for a brief consideration of social media and what is fixed and what is passing about its meaning to us as a self-aware species. Paradoxically that application of the concept slips into a conceptual tesseract as we attempt to hold social media to mean yet another contrivance for fixing our life experiences and then realize that who we were yesterday is not necessarily who we are today nor who we will be tomorrow, nor how we will be remembered once we’re gone. To hold its meaning still is to enter a self-referential paradox where the meaning of the thing contradicts the nature of that which it represents and vice versa. All that to say that while it is impossible to confirm whether social media are simulation or simulacra, it is possible to orient their meaning, through how we use them and design them, toward our better natures. If we succeed, it will fix not individuals but our common humanity, which may be the one thing about us that has never been ephemeral.