We begin with the hypothetical assumption that Tarski’s 1933 formula ∀ True(x) φ(x) has been defined such that ∀x Tarski:True(x) ↔ Boolean-True. On the basis of this logical premise we formalize the Truth Teller Paradox: "This sentence is true." showing syntactically how self-reference paradox is semantically ungrounded.

Heinz Von Forester characterizes the objects “known” by an autopoietic system as eigen-solutions, that is, as discrete, separable, stable and composable states of the interaction of the system with its environment. Previous articles have presented the FBST, Full Bayesian Significance Test, as a mathematical formalism specifically designed to access the support for sharp statistical hypotheses, and have shown that these hypotheses correspond, from a constructivist perspective, to systemic eigen-solutions in the practice of science. In this article several issues related to (...) the role played by language in the emergence of eigen-solutions are analyzed. The last sections also explore possible connections with the semiotic theory of Charles Sanders Peirce. (shrink)

A paradox, according to Wittgenstein, is something surprising that is taken out of its context. Thus, one way of dealing with paradoxical sentences is to imagine the missing context of use. Wittgenstein formulates what I call the paradigm paradox: ‘one sentence can never describe the paradigm in another, unless it ceases to be a paradigm.’ (PG, p.346) There are several instances of this paradox scattered throughout Wittgenstein’s writings. I argue that this paradox is structurally equivalent to (...) Russell’s paradox. The above quotation is Wittgenstein’s version of the vicious circle principle which counteracts the paradox. The prohibition Wittgenstein describes is, however, limited to a certain language-game. Finally, I argue that there is a structural analogy between a noun being employed as a self-membered set and a paradigmatic sample being included in or excluded from the set it generates. Paradoxical sentences are not prohibited forever; they can indicate a change in our praxis with a given paradigm. (shrink)

This paper decomposes the Liar Paradox into its semantic atoms using Meaning Postulates (1952) provided by Rudolf Carnap. Formalizing truth values of propositions as Boolean properties of these propositions is a key new insight. This new insight divides the translation of a declarative sentence into its equivalent mathematical proposition into three separate steps. When each of these steps are separately examined the logical error of the Liar Paradox is unequivocally shown.

The sentences ‘p but I don’t believe p’ (omissive form) and ‘p but I believe that not-p’ (comissive form) are typical examples of Moore’s paradox. When an agent (sincerely) asserts such sentences under normal circumstances, we consider his statements absurd. The Simple Solution (Moore, Heal, Wolgast, Kriegel, et al.) finds the source of absurdity for such statements in a certain formal contradiction (some kind of like ‘p & not-p’), the presence of which is lexically disguised. This solution is facing (...) criticism since it is obvious that the denial of the contradiction allegedly hidden in Moore’s paradox will not become an expression of tautology for us. Hintikka offers another solution to Moore’s paradox. In his view, the source of absurdity is the practical inconceivability of the situation, in which the agent would be able to hold mutually incompatible (inconsistent) beliefs. The sentences ‘p but I don’t believe that p’ and ‘p but I believe that not-p’ sound absurd because they express high-ordered beliefs that no rational agent can hold. A belief expressed in the sentence ‘he believes that p’, in order to be available to the agent, must be compatible with the truth of everything the agent believes. The problem with Hintikka’s solution is that it relies on an epistemic principle that does not correspond to our general intuitions about the notion of belief. According to this principle, there must be a close connection between the sentence ‘p but I don’t believe that p’ and the sentence ‘I believe that is the case is that p but I don’t believe that p’. However, it is not difficult to imagine the context for situations when the agent asserts the sentence ‘p but I don’t believe that p’ when s/he believes that p, but in parallel does not hold any high-ordered belief about his/her belief in p. For example, the agent does not recognize oneself in a photograph after a hard plastic surgery, and when one asks him ‘who is this?’ it would not be absurd for him/her to accept the following description of his/her beliefs ‘I believe that {I am me} but I don’t believe that I believe that {I am me}’. Wittgenstein provides an alternative solution based on a descriptive analysis of the grammar of verbs we use to express, attribute, and understand each other’s psychological states. In his view, the everyday practice of using the verb ‘believe’ demonstrates a characteristic asymmetry in the semantic and logical properties of the usage of this verb in the first person. The expression ‘I believe falsely’ has no real meaning since no one can meaningfully say the sentence ‘Now I believe falsely that p’ on their own behalf. Therefore, the genuine source of absurdity is associated with situations of self-attribution of false belief by the agent. Despite its originality, Wittgenstein’s solution faces objections. For example, in a certain context, it would not be absurd to assert the sentence ‘There are no beliefs, but I don’t believe in it’, unlike the sentence ‘There are no beliefs and I believe that it is so’. The solution proposed by Wittgenstein does not allow us to correct the situation since one could assert quite reasonably and without any absurdity a sentence like ‘In fact, there are no beliefs and now I falsely believe that it is so’. Based on Evans’ principle of epistemic transparency, some researchers (Atlas, Chan, Kwon, et al.) offer another strategy for solving Moore’s paradox, in which peculiar attention is paid to the dependence of understanding and evaluating the meaning of the sentences ‘p but I do not believe that p’ and ‘p but I believe that not-p’ from the immediate context of their usage. The source of absurdity should be looked for in the established practice of the self-referential use of the pronoun ‘I’. In the researchers’ view, Moore’s paradox arises only in the context of situations where the agent asserting such sentences is conscious speaking about oneself. Therefore, there is an obvious structural analogy between the sentences of the Liar family (‘This very sentence is false’, ‘Now I am lying’, etc.) and Moore’s paradox (‘p but I don’t believe that p’, ‘p but I believe that not-p’) – self-reference is constitutive for both families of paradoxical statements. (shrink)

A dialogue among statements that try to explain to each other the mechanisms and peculiarities of self-referential assertions and, particularly, of their context-dependence.

The aim of this text is to offer an explanation of Gödel's Theorem according to the schemes and notations of the original article. There are many good didactic explanations of the theorem that reveal its central points and implications, but these are difficult to recognize when reading the original work, due to the complexity of its formulation and the author's economical style in explaining the steps of his argument. An exposition of the central concepts will be made, as well as (...) a detailed explanation of the main points of the algebraic development of the proof, which will allow the non-specialist reader to find the well-known paradox from them. (shrink)

A Husserlian phenomenological approach to logic treats concepts in terms of their experiential meaning rather than in terms of reference, sets of individuals, and sentences. The present article applies such an approach in turn to the reasoning operative in various paradoxes: the simple Liar, the complex Liar paradoxes, the Grelling-type paradoxes, and Gödel’s Theorem. It finds that in each case a meaningless statement, one generated by circular definition, is treated as if were meaningful, and consequently as either true or (...) false, although in fact it is neither. The situation is further complicated by the fact that the sentence used to express the meaningless statement is ambiguous, and may also be used to express a meaningful statement. The paradoxes result from a failure to distinguish between the two meanings the sentence may have. (shrink)

The article critically examines the heuristic capacity and methods of using separate tools of Wittgenstein’s philosophical grammar to treat various semantic pathologies (paradoxes of Liar, Truth-Teller, etc.). According to Wittgenstein, philosophical confusion associated with the analysis of such semantic pathologies arises on the grounds of our intuitive faith that we are able to express in language any property that interests us. For instance, we believe that the property “to have a length of exactly one metre” can be meaningfully attributed to (...) any extended object. However, such a faith is fundamentally wrong, since an object like a standard metre, which plays an exclusive role as the criterion of length in our measuring practices based on the metric system, is an example of an extended object about which one can say neither that it is one metre long, nor that it is not one metre long. A correct understanding of the language-game concept shows that the criterion by which we form a set of objects that meet the standards established by the criterion cannot be described in the same way as objects of this set. Language-games using predicates of truth and falsehood are similar to our practice of measuring length using the metric system. In cases of Liar and Truth-Teller sentences, we encounter resembling examples of philosophical confusion. Beyond the context of their usage, “This sentence is false” and “This sentence is true” sentences seem to us paradoxes. But, according to Wittgenstein’s philosophical grammar, a correct method of treating such “paradoxical” sentences demands to find the lack- ing context for their usage, i.e. pay attention to the language-game in which they are emplloyed. Like in the case of the standard metre, we can claim that the sentence “This sentence is false” (“This sentence is true”) is neither false (true), nor not false (true). In an analogous way, Liar and Truth-Teller sentences should be interpreted as peculiar standards of falsehood and truth that we use in our language-games to judge about the truth value of other sentences. (shrink)

This paper gives a definition of self-reference on the basis of the dependence relation given by Leitgeb (2005), and the dependence digraph by Beringer & Schindler (2015). Unlike the usual discussion about self-reference of paradoxes centering around Yablo's paradox and its variants, I focus on the paradoxes of finitary characteristic, which are given again by use of Leitgeb's dependence relation. They are called 'locally finite paradoxes', satisfying that any sentence in these paradoxes can depend on (...) finitely many sentences. I prove that all locally finite paradoxes are self-referential in the sense that there is a directed cycle in their dependence digraphs. This paper also studies the 'circularity dependence' of paradoxes, which was introduced by Hsiung (2014). I prove that the locally finite paradoxes have circularity dependence in the sense that they are paradoxical only in the digraph containing a proper cycle. The proofs of the two results are based directly on König's infinity lemma. In contrast, this paper also shows that Yablo's paradox and its nested variant are non-self-referential, and neither McGee's paradox nor the omega-cycle liar paradox has circularity dependence. (shrink)

What José Luis Bermúdez calls the paradox of self-consciousness is essentially the conflict between two claims: (1) The capacity to use first-personal referential devices like “I” must be explained in terms of the capacity to think first-person thoughts. (2) The only way to explain the capacity for having a certain kind of thought is by explaining the capacity for the canonical linguistic expression of thoughts of that kind. (Bermúdez calls this the “Thought-Language Principle”.) The conflict between (1) and (...) (2) is obvious enough. However, if a paradox is an unacceptable conclusion drawn from apparently valid reasoning from apparently true premises, then Bermúdez’s conflict is no paradox. It is rather a conflict between the view that thought must be explained in terms of language, and the view that first person linguistic reference must be explained in terms of first-person thought. Neither view is immediately obvious, and nor is it obvious that the arguments for either are equally compelling. What we have here is a difference of philosophical opinion, not a paradox. (shrink)

This book offers an original philosophical perspective on exemplarity. Inspired by Wittgenstein’s later work and Derrida’s theory of deconstruction, it argues that examples are not static entities but rather oscillate between singular and universal moments. There is a broad consensus that exemplary cases mediate between singular instances and universal concepts or norms. In the first part of the book, Mácha contends that there is a kind of différance between singular examples and general exemplars or paradigms. Every example is, in part, (...) also an exemplar, and vice versa. Furthermore, he develops a paracomplete approach to the logic of exemplarity, which allows us to say of an exemplar of X neither that it is an X nor that it is not an X. This paradox is structurally isomorphic to Russell’s paradox and can be addressed in similar ways. In the second part of the book, Mácha presents four historical studies that exemplify the ideas developed in the first part. This part begins with Plato’s Forms, understood as standards/paradigms, before considering Kant’s theory of reflective judgment as a general epistemological account of exemplarity. This is then followed by analyses of Hegel’s conceptual moment of particularity and Kuhn’s concept of paradigm. The book concludes by discussing the speculative hypothesis that all our knowledge is based on paradigms, which, following the logic of exemplarity, are neither true nor false. (shrink)

Most paradoxes of self-reference have a dual or ‘hypodox’. The Liar paradox (Lr = ‘Lr is false’) has the Truth-Teller (Tt = ‘Tt is true’). Russell’s paradox, which involves the set of sets that are not self-membered, has a dual involving the set of sets which are self-membered, etc. It is widely believed that these duals are not paradoxical or at least not as paradoxical as the paradoxes of which they are duals. In this (...) paper, I argue that some paradox’s duals or hypodoxes are as paradoxical as the paradoxes of which they are duals, and that they raise neglected and interestingly different problems. I first focus on Richard’s paradox (arguably the simplest case of a paradoxical dual), showing both that its dual is as paradoxical as Richard’s paradox itself, and that the classical, Richard-Poincaré solution to the latter does not generalize to the former in any obvious way. I then argue that my argument applies mutatis mutandis to other paradoxes of self-reference as well, the dual of the Liar (the Truth-Teller) proving paradoxical. (shrink)

In his article, Vsevolod Ladov poses an important question – might paradoxes admit of a uniform solution? As an answer, I propose looking at an approach in which it is argued that we must recognize the truth predicate as merely analogous to a vague predicate. The proponents of this approach (V. McGee, J. Tap- penden, H. Field, G. Priest and D. Hyde) insist that there is a structural relation between sorites paradoxes and self-reference paradoxes and that they should (...) have a unified solution. (shrink)

A standard problem with the objectivity of social scientific theory in particular is that it is either self-referential, in which case it seems to undermine itself as ideology, or self-excepting, which seem pragmatically self-refuting. Using the example of Marx and his theory of ideology, I show how self-referential theories that include themselves in their scope of explanation can be objective. Ideology may be roughly defined as belief distorted by class interest. I show how Marx thought that (...) natural science was informed by class interest but not therefore necessarily ideology. Capitalists have an interest in understanding the natural world (to a point) so that they can manipulate it for profit. Their survival in the marketplace often depends on their competitive success in doing so. The fact that their interventions into nature, driven by class interest, often succeed, is evidence of the reliability and truth of their natural scientific theories. -/- With social science, capitalists have a partial interest in understanding the world, so that they may maintain their class power, ideological hegemony, and a reasonably stable and self-reproducing economy. Some bourgeois social science, Marx thinks, is therefore vindicated the same way that natural science is, because of, not despite, its success in promoting the class interests that cause it. But the capitalist class, Marx thinks, also has an interest in obfuscating the truth about the social world. It would undermine the stability of capitalism, for example, for the bourgeoisie to adopt what Marx thinks to be the correct social theory, Marxism, and proclaim to the world that capitalism is exploitative and unstable. Capitalism creates a a need for ideological social theories that, driven by class interest, represent capitalism as just, natural, inevitable, and unalterable. If Marxism is correct, while these theories may further capitalist interests, they will not withstand the test of time as capitalist exploitation and instability engenders countersystemic political action. -/- How, then can we know, according to Marx, that Marxism is correct, as applied to itself? In part it is whether it succeeds in further the class interests that cause it. If workers adopt bourgeois theories and Marxism is right, they will find themselves frustrated in pursuit of goals that they have, such as improving their well-being, and if Marxism is wrong, then those theories will help them further their own goals. Similarly with Marxism: if it is right, then workers who adopt it should in the long run have more success in furthering their own goals, and if not, then not. But it is question begging to say that our confidence in the reliability of Marxism is enhanced by its successful promotion of goals that capitalists do not have but (according to Marxism) workers do. To avoid this problem it is necessary to compare the relative success of each sort of theory in promoting the interests and needs that the respective classes have independently of the theory, as well as by the usual criteria of scientific theory choice. -/- Thus the Soviet variant of Marxism was fairly decisively refuted by its failure to generate the support of its own designated constituency necessary to grow and survive. Whether the same is true of other varieties of Marxism or of capitalist social theory, it is, as Zhou En Lai said when asked whether the French Revolution had succeeded, too soon to tell, at least in any definitive way. However, and this is the point of the paper, an objective test of Marxism is possible even though Marxism says that all theory is informed by class interest. The question is whether the interests are in discovering or obfuscating the truth. The point is not limited to Marx, but applies to any self-referential social scientific theory that includes itself in its scope. -/- Keywords: ideology, self-referential theory, self-reference, reliabilism, objectivity, Marx, Marxism, sociology of science, class interest, classes, capitalism, bourgeoisie, workers, working class, social theory of knowledge, confirmation of scientific theory. (shrink)

To counter a general belief that all the paradoxes stem from a kind of circularity (or involve some self--reference, or use a diagonal argument) Stephen Yablo designed a paradox in 1993 that seemingly avoided self--reference. We turn Yablo's paradox, the most challenging paradox in the recent years, into a genuine mathematical theorem in Linear Temporal Logic (LTL). Indeed, Yablo's paradox comes in several varieties; and he showed in 2004 that there are other (...) versions that are equally paradoxical. Formalizing these versions of Yablo's paradox, we prove some theorems in LTL. This is the first time that Yablo's paradox(es) become new(ly discovered) theorems in mathematics and logic. (shrink)

This paper offers an analysis of a hitherto neglected text on insoluble propositions dating from the late XiVth century and puts it into perspective within the context of the contemporary debate concerning semantic paradoxes. The author of the text is the italian logician Peter of Mantua (d. 1399/1400). The treatise is relevant both from a theoretical and from a historical standpoint. By appealing to a distinction between two senses in which propositions are said to be true, it offers an unusual (...) solution to the paradox, but in a traditional spirit that contrasts a number of trends prevailing in the XiVth century. It also counts as a remarkable piece of evidence for the reconstruction of the reception of English logic in italy, as it is inspired by the views of John Wyclif. Three approaches addressing the Liar paradox (Albert of Saxony, William Heytesbury and a version of strong restrictionism) are first criticised by Peter of Mantua, before he presents his own alternative solution. The latter seems to have a prima facie intuitive justification, but is in fact acceptable only on a very restricted understanding, since its generalisation is subject to the so-called revenge problem. (shrink)

[[[ (Here only the chapters 3 – 8, see *** ) First I argue that the prohibition of linguistic self-reference as a solution to the antinomy problem contains a pragmatic contradiction and is thus not only too restrictive, but just inconsistent (chap.1). Furthermore, the possibilities of non-restrictive strategies for antinomy avoidance are discussed, whereby the explicit inclusion of the – pragmatically presuposed – consistency requirement proves to be the optimal strategy (chap.2). ]]] The central question here is that (...) about the actual reason for antinomic structures. It turns out to be a form of negative self-conditioning (chap.3). This makes it necessary to clarify the status of negative concepts (chap.4). The generalization of these considerations (chap.5) leads to the actual analysis of the antinomic basic structure (chap.6): Decisively for the pragmatics of the concept is that it positively owns a meaning, so that positivity is always constituted by the concept qua concept. Thus a negative concept is characterized by a fundamental ambivalence: From a semantic point of view it has negative character, in its pragmatic status as a concept, however, it has positive character. If the meaning is especially that of a negative self-reference, the ambivalence leads to the antinomic constellation of a negative self-condition – that is the crux of the matter here! The concept thus possesses the property defined by it exactly when it does not possess it and vice versa. A closer analysis shows that the function of reflective structures for the occurrence of antinomies has to be judged much more differentiated than previous opinions suggest. Not only are four forms of reflectivity to be distinguished in this context – ontic, semantic, pragmatic reflectivity, and especially the form of negative self-condition; but it is also apparent that these are intertwined with each other in a way that is difficult to be understood. The astonishing variety of relationships associated with this makes the irritation that has always emanated from the antinomy problem appear more comprehensible. In the developed pragmatic perspective furtheron parallels to the structure of self-consciousness become visible (chap.7). I conclude with considerations about the significance of the antinomic structure for the problem of dialectics, especially for the synthesis formation of mutually exclusive terms (chap.8). (shrink)

In this paper, I start by describing and examining the main results about the option of formalizing the Yablo Paradox in arithmetic. As it is known, although it is natural to assume that there is a right representation of that paradox in first order arithmetic, there are some technical results that give rise to doubts about this possibility. Then, I present some arguments that have challenged that Yablo’s construction is non-circular. Just like that, Priest (1997) has argued that (...) such formalization shows that Yablo’s Paradox involves implicit circularity. In the same direction, Beall (2001) has introduced epistemic factors in this discussion. Even more, Priest has also argued that the introduction of infinitary reasoning would be of little help. Finally, one could reject definitions of circularity in term of fixed-point adopting non-well-founded set theory. Then, one could hold that the Yablo paradox and the Liar paradox share the same non-well-founded structure. So, if the latter is circular, the first is too. In all such cases, I survey Cook’s approach (2006, forthcoming) on those arguments for the charge of circularity. In the end, I present my position and summarize the discussion involved in this volume. En este artículo, describo y examino los principales resultados vinculados a la formalización de la paradoja de Yablo en la aritmética. Aunque es natural suponer que hay una representación correcta de la paradoja en la aritmética de primer orden, hay algunos resultados técnicos que hacen surgir dudas acerca de esta posibilidad. Más aún, presento algunos argumentos que han cuestionado que la construcción de Yablo no sea circular. Así, Priest (1997) ha argumentado que la formalización de la paradoja de Yablo en la aritmética de primer orden muestra que la misma involucra implícitamente circularidad. En la misma dirección, Beall (2001) ha introducido factores epistémicos en esta discusión. Más aún, Priest ha también argumentado que la introducción de razonamiento infinitario como complemento de la formalización en la aritmética sería de poca ayuda. Finalmente, se podría rechazar todo intento de dar definiciones de circularidad en términos de puntos fijos adoptando teoría de conjuntos infundados. Entonces, se podría sostener que la paradoja de Yablo y la del mentiroso comparten la misma estructura infundada. Por eso, si la última es circular, también lo es la primera. En todos los casos, presento el enfoque de Roy Cook (2006, en prensa) sobre estos argumentos que atribuyen circularidad a la construcción de Yablo. En el final, presento mi posición y un breve resumen de la discusión involucrada en este volumen. (shrink)

A novel solution to the knowability paradox is proposed based on Kant’s transcendental epistemology. The ‘paradox’ refers to a simple argument from the moderate claim that all truths are knowable to the extreme claim that all truths are known. It is significant because anti-realists have wanted to maintain knowability but reject omniscience. The core of the proposed solution is to concede realism about epistemic statements while maintaining anti-realism about non-epistemic statements. Transcendental epistemology supports such a view by providing (...) for a sharp distinction between how we come to understand and apply epistemic versus non-epistemic concepts, the former through our capacity for a special kind of reflective self-knowledge Kant calls ‘transcendental apperception’. The proposal is a version of restriction strategy: it solves the paradox by restricting the anti-realist’s knowability principle. Restriction strategies have been a common response to the paradox but previous versions face serious difficulties: either they result in a knowability principle too weak to do the work anti-realists want it to, or they succumb to modified forms of the paradox, or they are ad hoc. It is argued that restricting knowability to non-epistemic statements by conceding realism about epistemic statements avoids all versions of the paradox, leaves enough for the anti-realist attack on classical logic, and, with the help of transcendental epistemology, is principled in a way that remains compatible with a thoroughly anti-realist outlook. (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)

In this paper I consider two paradoxes that arise in connection with the concept of demonstrability, or absolute provability. I assume—for the sake of the argument—that there is an intuitive notion of demonstrability, which should not be conflated with the concept of formal deducibility in a (formal) system or the relativized concept of provability from certain axioms. Demonstrability is an epistemic concept: the rough idea is that a sentence is demonstrable if it is provable from knowable basic (“self-evident”) premises (...) by means of simple logical steps. A statement that is demonstrable is also knowable and a statement that is actually demonstrated is known to be true. By casting doubt upon apparently central principles governing the concept of demonstrability, the paradoxes of demonstrability presented here tend to undermine the concept itself—or at least our understanding of it. As long as we cannot find a diagnosis and a cure for the paradoxes, it seems that the coherence of the concepts of demonstrability and demonstrable knowledge are put in question. There are of course ways of putting the paradoxes in quarantine, for example by imposing a hierarchy of languages a` la Tarski, or a ramified hierarchy of propositions and propositional functions a` la Russell. These measures, however, helpful as they may be in avoiding contradictions, do not seem to solve the underlying conceptual problems. Although structurally similar to the semantic paradoxes, the paradoxes discussed in this paper involve epistemic notions: “demonstrability”, “knowability”, “knowledge”... These notions are “factive” (e.g., if A is demonstrable, then A is true), but similar paradoxes arise in connection with “nonfactive” notions like “believes”, “says”, “asserts”.3 There is no consensus in the literature concerning the analysis of the notions involved—often referred to as “propositional attitudes”—or concerning the treatment of the paradoxes they give rise to. (shrink)

Epiphenomenalist dualists hold that certain physical states give rise to non-physical conscious experiences, but that these non-physical experiences are themselves causally inefficacious. Among the most pressing challenges facing epiphenomenalists is the so-called “paradox of phenomenal judgment”, which challenges epiphenomenalism’s ability to account for our knowledge of our own conscious experiences. According to this objection, we lack knowledge of the very thing that epiphenomenalists take physicalists to be unable to explain. By developing an epiphenomenalist theory of subjects and mental states, (...) this paper argues that there is nothing paradoxical or problematic about the epiphenomenalist’s understanding of phenomenal judgments or phenomenal self-knowledge. The appearance of paradox emerges from inconsistently combining (epiphenomenalist) dualism about qualia with a physicalistic conception of subjects of experience. The lesson we should take from this is not that there is anything wrong with epiphenomenalism, but that epiphenomenalist dualists should be “dualists all the way down”—embracing a picture of mind that gives phenomenology a central place, in its understanding of both subjects and their knowledge of their own minds. Epiphenomenalist-friendly accounts of reference and memory are also developed, showing that neither of these issues creates a paradox for the epiphenomenalist. (shrink)

"Thinking about thinking", the topic of this section of our volume, leads us into an Escherian web of self-referential interconnections — which nevertheless make sense, at least if we do not shy away at once in the face of paradox, for fear of inconsistency. How do our critical faculties enter the picture? Adding a few conceptual dimensions to the merely linearly causal one may help us to shed light on the issue, and may help us also to understand (...) some of the basic strategies "Western" thought has developed throughout its historical pathway in its conscious dealings with the 'real'. Of course, in a few pages it is impossible to do justice to this complicated subject. But I'll try to give the reader an as clear as possible description of the, in my view, crucial issues, pointing out the ways they resonate into the many fields of interest represented in the intended readership, and leave it to the reader to dig deeper through the given references. (shrink)

In Zettel, Wittgenstein considered a modified version of Cantor’s diagonal argument. According to Wittgenstein, Cantor’s number, different with other numbers, is defined based on a countable set. If Cantor’s number belongs to the countable set, the definition of Cantor’s number become incomplete. Therefore, Cantor’s number is not a number at all in this context. We can see some examples in the form of recursive functions. The definition "f(a)=f(a)" can not decide anything about the value of f(a). The definiton is incomplete. (...) The definition of "f(a)=1+f(a)" can not decide anything about the value of f(a) too. The definiton is incomplete.<br><br>According to Wittgenstein, the contradiction, in Cantor's proof, originates from the hidden presumption that the definition of Cantor’s number is complete. The contradiction shows that the definition of Cantor’s number is incomplete. <br><br>According to Wittgenstein’s analysis, Cantor’s diagonal argument is invalid. But different with Intuitionistic analysis, Wittgenstein did not reject other parts of classical mathematics. Wittgenstein did not reject definitions using self-reference, but showed that this kind of definitions is incomplete.<br><br>Based on Thomson’s diagonal lemma, there is a close relation between a majority of paradoxes and Cantor’s diagonal argument. Therefore, Wittgenstein’s analysis on Cantor’s diagonal argument can be applied to provide a unified solution to paradoxes. (shrink)

‎A universal schema for diagonalization was popularized by N. S‎. ‎Yanofsky (2003)‎, ‎based on a pioneering work of F.W‎. ‎Lawvere (1969)‎, ‎in which the existence of a (diagonolized-out and contradictory) object implies the existence of a fixed-point for a certain function‎. ‎It was shown that many self-referential paradoxes and diagonally proved theorems can fit in that schema‎. ‎Here‎, ‎we fit more theorems in the universal‎ ‎schema of diagonalization‎, ‎such as Euclid's proof for the infinitude of the primes and new (...) proofs of G. Boolos (1997) for Cantor's theorem on the non-equinumerosity of a set with its powerset‎. ‎Then‎, ‎in Linear Temporal Logic‎, ‎we show the non-existence of a fixed-point in this logic whose proof resembles the argument of Yablo's paradox (1985‎, ‎1993)‎. ‎Thus‎, ‎Yablo's paradox turns for the first time into a genuine mathematico-logical theorem in the framework of Linear Temporal Logic‎. ‎Again the diagonal schema of the paper is used in this proof; and it is also shown that G. Priest's inclosure schema (1997) can fit in our universal diagonal / fixed-point schema‎. ‎We also show the existence of dominating (Ackermann-like) functions (which dominate a given countable set of functions‎, ‎such as primitive recursive functions) in the schema. (shrink)

In this existential reading of Kim Kardashian-West's International Women's Day selfie of 2016, I focus on the rise of selfie culture and public discourse around emerging digital representations of women's bodies. The selfie is a relatively new phenomenon, and is particularly curious because of the subject/object paradox it creates; in taking a selfie, a person asserts control over their own image, but at the same time, becomes object in their own gaze. My argument is that selfies, like other assertions (...) of bodily subjectivity in digital spaces, are a threat to patriarchal structures that paint women as immanent, object, as reflected in public discourse around Kardashian-West's International Women's Day selfie. I draw on both Jean-Paul Sartre and Simone de Beauvoir's work on subjectivity in existentialism and phenomenology, as well as Amy Shields Dobson's work on post-feminism and young women's projections of self, in order to delineate what it is about the selfie that creates this paradox. I also make reference to the work of Elisabeth Grosz and Frantz Fanon in relation to a colonial hierarchy that prioritises body over mind, as well as Laura Mulvey's work on the male gaze. (shrink)

This article examines the various Liar paradoxes and their near kin, Grelling’s paradox and Gödel’s Incompleteness Theorem with its self-referential Gödel sentence. It finds the family of paradoxes to be generated by circular definition–whether of statements, predicates, or sentences–a manoeuvre that generates pseudo-statements afflicted with the Liar syndrome: semantic vacuity, semantic incoherence, and predicative catalepsy. Such statements, e.g., the self-referential Liar statement, are meaningless, and hence fail to say anything, a point that invalidates the reasoning on which (...) the various paradoxes rest. The seeming plausibility of the paradoxes is due to the fact that often the sentence used to make the pseudo-statement is ambiguous in that it may also be used to make a genuine statement about the pseudo-statement. Hence, if a formal system is to avoid ambiguity and consequent paradox and contradiction, it must distinguish between the two statements the sentence may be used to make. Gödel’s Theorem presents a further complication in that the self-reference involved is sentential rather than statemental. Nevertheless, on the intended interpretation of the system as a formalization of arithmetic, the self-referential Gödel sentence can only be an ambiguous statement, one that is both a pseudo-statement and its genuine double. Consequently, the conclusions commonly drawn from Gödel’s theorem must be deemed unwarranted. Arithmetic might well be formalized in a proper system that either excludes circular definition or introduces disambiguators. (shrink)

The semantic paradoxes are associated with self-reference or referential circularity. However, there are infinitary versions of the paradoxes, such as Yablo's paradox, that do not involve this form of circularity. It remains an open question what relations of reference between collections of sentences afford the structure necessary for paradoxicality -- these are the so-called "dangerous" directed graphs. Building on Rabern, et. al (2013) we reformulate this problem in terms of fixed points of certain functions, thereby boiling (...) it down to get a purely mathematical problem. (shrink)

In some recent articles, Cobreros, Egré, Ripley, & van Rooij have defended the idea that abandoning transitivity may lead to a solution to the trouble caused by semantic paradoxes. For that purpose, they develop the Strict-Tolerant approach, which leads them to entertain a nontransitive theory of truth, where the structural rule of Cut is not generally valid. However, that Cut fails in general in the target theory of truth does not mean that there are not certain safe instances of Cut (...) involving semantic notions. In this article we intend to meet the challenge of answering how to regain all the safe instances of Cut, in the language of the theory, making essential use of a unary recovery operator. To fulfill this goal, we will work within the so-called Goodship Project, which suggests that in order to have nontrivial naïve theories it is sufficient to formulate the corresponding self-referential sentences with suitable biconditionals. Nevertheless, a secondary aim of this article is to propose a novel way to carry this project out, showing that the biconditionals in question can be totally classical. In the context of this article, these biconditionals will be essentially used in expressing the self-referential sentences and, thus, as a collateral result of our work we will prove that none of the recoveries expected of the target theory can be nontrivially achieved if self-reference is expressed through identities. (shrink)

From my ongoing "Metalogical Plato" project. The aim of the diagram is to make reasonably intuitive how the Socratic elenchos (the logic of refutation applied to candidate formulations of virtues or ruling knowledges) looks and works as a whole structure. This is my starting point in the project, in part because of its great familiarity and arguable claim to being the inauguration of western philosophy; getting this point less wrong would have broad and deep consequences, including for philosophy’s self-understanding. (...) -/- (i.) is the first pass at elenchos in which the Socratic interlocutor does not reflect on knowledge being the crux of the problem. (ii.) is the second, rarer, reflective pass in which they are, making the investigation explicitly about knowledge. Its centrality in the Charmides makes that neglected dialogue of superlative importance. This structure is also the gateway through which the discussion/dialectic crosses into the Agathology (discussion of the form of the good) at Republic 505. -/- The problem of elenchos, then, grasped as a whole structure, is that it seems that knowledge can neither satisfactorily be included in nor excluded from its own scope. The development of the ti esti (“what is ---?”) question leads to the introduction of knowledge into its own scope (i. implicitly, as goodness contrasted with blind rule-following ii. explicitly qua knowledge) while the development of the dual peri tinos (“----about what?) question leads to K’s elimination from its own scope. The introductions are motivated to avoid contradiction, but produce regress; the eliminations are motivated to avoid regress, but produce contradiction. In scholarship, and in the history of philosophy, the ti esti question is universally recognized, to the point of being identified with philosophy’s origin and essence; the peri tinos question is neglected textually and never recognized as the equal dual to the ti esti. This, I claim, has blocked the development of a nontrivial logical appreciation of what Plato's Socrates is up to. (One rather disastrous effect of this neglect is taking Aristotle as the beginning of the development of logic, rather than, correctly, for the beginning of logic’s fatal separation from mathematics and dialectic.) -/- Further, because of this monopticism of the ti esti, the function of consistency in the elenchos has not been understood, even with respect to the ti esti. The ti esti is actually in search of completeness, given a norm of consistency; the peri tinos is in search of consistency, given a norm of completeness. Only appreciating the two questions as dual allows space in the structure to clarify these different orientations relative to consistency. (And, dually, to completeness, whose function in the elenchos is generally entirely missed by scholars and relegated to discussions of eros; it's not out of place there, of course, but its significance is secured here.) Recognizing the duality of the ti esti and peri tinos questions is thus the royal road, in the Socratic-Platonic context, to catching sight of what we post-Cantorians can recognize as the metalogical duality of consistency and completeness. (The salutary disruptive effects of this Plato-Cantor proximity have, of course, been traced in complementary ways by Badiou.) -/- Note that the diagram is supposed to provide a relatively accessible orientation, not to stand on its own, and certainly not to be the last word on any subject. An important qualification (telegraphed in the previous paragraph) is that what elenchos shows is not finally circular or paradoxical, though the problem first presents as such (stubbornly, obdurately, as "difficult" Plato's Socrates always says with characteristic understatement). What is depicted here is meant, at a first pass, to be the shape of that first presentation, the form of the problem of elenchos, rather than of its solution. It's not an accident that this problem strongly resembles "Russell's paradox" (not Russell's not a paradox.) Problem is to solution as RP is to the diagonal theorems. (shrink)

Recursion or self-reference is a key feature of contemporary research and writing in semiotics. The paper begins by focusing on the role of recursion in poststructuralism. It is suggested that much of what passes for recursion in this field is in fact not recursive all the way down. After the paradoxical meaning of radical recursion is adumbrated, topology is employed to provide some examples. The properties of the Moebius strip prove helpful in bringing out the dialectical nature of (...) radical recursion. The Moebius is employed to explore the recursive interplay of terms that are classically regarded as binary opposites: identity and difference, object and subject, continuity and discontinuity, etc. To realize radical recursion in an even more concrete manner, a higher-dimensional counterpart of the Moebius strip is utilized, namely, the Klein bottle. The presentation concludes by enlisting phenomenological philosopher Maurice Merleau-Ponty’s concept of depth to interpret the Klein bottle’s extra dimension. (shrink)

The semantic paradoxes are a family of arguments – including the liar paradox, Curry’s paradox, Grelling’s paradox of heterologicality, Richard’s and Berry’s paradoxes of definability, and others – which have two things in common: first, they make an essential use of such semantic concepts as those of truth, satisfaction, reference, definition, etc.; second, they seem to be very good arguments until we see that their conclusions are contradictory or absurd. These arguments raise serious doubts concerning the (...) coherence of the concepts involved. This article will offer an introduction to some of the main theories that have been proposed to solve the paradoxes and avert those doubts. Included is also a brief history of the semantic paradoxes from Eubulides to Tarski and Curry. (shrink)

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. (shrink)

The purpose of this paper is to open for investigation a range of phenomena familiar from dynamical systems or chaos theory which appear in a simple fuzzy logic with the introduction of self-reference. Within that logic, self-referential sentences exhibit properties of fixed point attractors, fixed point repellers, and full chaos on the [0, 1] interval. Strange attractors and fractals appear in two dimensions in the graphing of pairs of mutually referential sentences and appear in three dimensions in (...) the graphing of mutually referential triples. (shrink)

The paper begins by acknowledging that weakened systematic precision in phenomenology has made its application in philosophy of science obscure and ineffective. The defining aspirations of early transcendental phenomenology are, however, believed to be important ones. A path is therefore explored that attempts to show how certain recent developments in the logic of self-reference fulfill in a clear and more rigorous fashion in the context of philosophy of science certain of the early hopes of phenomenologists. The resulting dual (...) approach is applied to several problems in the philosophy of science: on the one hand, to proposed rejections of scientific objectivity, to the doctrine of radical meaning variance, and to the Quine-Duhem thesis, and or. the other, to an analysis of hidden variable theory in quantum mechanics. (shrink)

As part of our small contribution in dialogue toward better peace development and reconciliation studies, and following Toffler & Toffler’s War and Antiwar (1993), the present article delves into a realm of logic beyond the traditional confines of negation and the excluded middle principle, exploring the nuances of "Otherness" that transcend classical and Nagatomo logics. Departing from the foundational premises of classical Aristotelian logic systems, this exploration ventures into alternative realms of reasoning, specifically examining Neutrosophic Logic and Klein bottle logic (...) (cf. Smarandache, 2005). The study challenges conventional boundaries and explores the implications of embracing paradoxes and self-reference in logic systems, aiming to redefine approaches to understanding truth and reasoning. The paper investigates how these alternative logics open avenues for philosophical inquiry, redefining entropy, and potentially influencing innovative perspectives in free energy systems. Through this exploration, it seeks to expand the discourse on logic, welcoming a broader spectrum of thought beyond established frameworks; and we also discuss shortly a number of possible implementations including in risk management and also Klein bottle entropy redefinition (Tang et al, 2018). (shrink)

If you were to build a world from Nothing, how would you do it? By investigating the nature of self-reference, we will show how this can be achieved, how starting from Nothing, Everything can be obtained. Various implications of the definition of self-reference will be investigated, showing how it can account for various aspects of the phenomenology of consciousness, thus showing how starting from only 1 principle, a world of infinite complexity can be obtained. Parallels with (...) set theory will be made along the way. (shrink)

If you were to build a world from Nothing, how would you do it? By investigating the nature of self-reference, we will show how this can be achieved, how starting from Nothing, Everything can be obtained. Various implications of the definition of self-reference will be investigated, showing how it can account for various aspects of the phenomenology of consciousness, thus showing how starting from only 1 principle, a world of infinite complexity can be obtained. Parallels with (...) set theory will be made along the way. (shrink)

Three distinct turning points (“bottleneck breakings”) in universal evolution are discussed at some length in terms of “self-reference” and (corresponding) “Reality Principles.” The first (origin and evolution of animate Nature) and second (human consciousness) are shown to necessarily precede a third one, that of Marxist philosophy. It is pointed out that while the previous two could occupy a natural (so in a sense neutral) place as parts of human science, the self-reference of Marxism, as a _social_ (...) human phenomenon, through its direct bearings on the _practice_ of society, did have a stormy history. I conclude that the fall of Bolshevism was unavoidable, and still, we might uphold our hope for a truly free society of humankind, just on the very basis of what we have learned of the fate of Marxist philosophy as such, as a _recursively evolving_ social _practice_: the freedom of humankind of its own ideological burdens (constraints). (shrink)

The phenomenon of self-reference combines self-refutation in the case of Plato’s critics of Protagoras in “Theaetetus” and self-predication in the case of a difficulty which Plato himself faces developing the theory of ideas in “Parmenides”. The author of the article asserts that self-predication does not produce a negative impact on Plato’s metaphysics and in no way destroys the integrality of Plato’s philosophy: It is logically correct as are both his criticism of Protagoras’ relativism and the (...) theory of ideas. (shrink)

Postmodernists claim that there is no truth. However, the statement 'there is no truth ' is self -contradictory. This essay shows the following: One cannot state the idea 'there is no truth ' universally without creating a paradox. In contrast, the statement 'there is truth ' does not produce such a paradox. Therefore, it is more logical that truth exists.

This paper argues that a certain type of self-referential sentence falsifies the widespread assumption that a declarative sentence's meaning is identical to its truth condition. It then argues that this problem cannot be assimilated to certain other problems that the assumption in question is independently known to face.

I here investigate the sense in which diagonalization allows one to construct sentences that are self-referential. Truly self-referential sentences cannot be constructed in the standard language of arithmetic: There is a simple theory of truth that is intuitively inconsistent but is consistent with Peano arithmetic, as standardly formulated. True self-reference is possible only if we expand the language to include function-symbols for all primitive recursive functions. This language is therefore the natural setting for investigations of (...) class='Hi'>self-reference. (shrink)

In his Paradoxes (1995: Cambridge University Press: 149) Mark Sainsbury presents the following pair of sentences: Line 1: The sentence written on Line 1 is nonsense. Line 2: The sentence written on Line 1 is nonsense. Sainsbury (1995: 149, 154) here makes three assertions: (1) The sentence in Line 1 is so viciously self-referential that it falls into the truth-value gap. The sentence is really nonsense. (2) The sentence in Line 2 is by contrast true. For it states precisely (...) that the sentence in Line 1 is nonsense. (3) The two sentences in Lines 1 and 2 are an example of the principle that two sentence tokens of the same sentence-type can have different truth-values, although they have the same reference and state the same property of the object of reference. In this paper, I argue that Sainsbury’s assumptions are false in all three cases. (shrink)

This paper examines theories of first person authority proposed by Dorit Bar-On (2004), Crispin Wright (1989a) and Sydney Shoemaker (1988). What all three accounts have in common is that they attempt to explain first person authority by reference to the way our language works. Bar-On claims that in our language self-ascriptions of mental states are regarded as expressive of those states; Wright says that in our language such self-ascriptions are treated as true by default; and Shoemaker suggests (...) that they might arise from our capacity to avoid Moore-paradoxical utterances. I argue that Bar-On’s expressivism and Wright’s constitutive theory suffer from a similar problem: They fail to explain how it is possible for us to instantiate the language structures that supposedly bring about first person authority. Shoemaker’s account does not suffer from this problem. But it is unclear whether the capacity to avoid Moore-paradoxical utterances really yields self-knowledge. Also, it might be that self-knowledge explains why we have this capacity rather than vice versa. (shrink)

This is the introduction to Self-reference: Reflections on Reflexivity, edited by Steven James Bartlett and Peter Suber. The introduction identifies and describes a wide range of varieties of self-reference, some which have become important topics of investigation in philosophy, and others which are of significance in other disciplines. /// The anthology is the first published collection of essays to give a sense of depth and breadth of current work on this fascinating and important set of issues. (...) The volume contains 13 essays by well-known authors in this field, written on special invitation for this collection. In addition, the book includes the first general bibliography of works on self-reference, comprising more than 1,200 citations. (shrink)

A CONSTRUCTIVE CRITIQUE OF THE FOUNDATIONS OF PHILOSOPHY: Abstract of the Ph.D. Dissertation By Ramesh N. Patel June, 1970 -/- Philosophy pursues a rational explication of our understanding, experiences, and values in terms of objective truth and reality. Conspicuously, its view of rationality has been rigid and preconceived. Application of the preconceived reason in the explication of the essential features of our world fails and issues in a network of dialectical tangles. These artificially created tangles pose a unique intellectual challenge, (...) but the perennial failure to resolve them limits the intellectual response of philosophers to remaining caught in the tangles, or to taking intuitively favorable sides, or to simply denouncing the tangles as meaningless. Each is understandable. -/- Superficially, traditional philosophical reason looks like a correct tool for the explication of our world of common sense, but its applications to the world result in logical tangles showing its inadequacy. The philosopher’s faith in such reason ends up in one of two positions: abandoning our world in the interests of such reason, or embodying itself in the world through a special form of reason. Plato took the first and Aristotle took the second position. My analysis shows that Plato failed to explicate the relation between his world of Forms and our world of particulars, and, more importantly, between the Forms and the intuitive reason which grasps them. Aristotle, in my analysis, cannot conceal that the law of noncontradiction is a mere exhortation and has no descriptive necessity with respect to thought, things, or meaning. We also find such dialectical difficulties in Augustine, Hume and Strawson. -/- “Microphilosophical feasibilism” is our proposal for the best treatment, if not the definitive solution, of these problems. Instead of separating philosophical method into logic, epistemology, axiology, and metaphysics on the one side, and partitioning the projected results of philosophizing into validity, truth, valuation, and reality on the other, we need to start philosophizing with a general distinction between “method employed” and “intended result.” This distinction is not pre-colored by claims to ultimate verities. It is a direct aid to understanding. I consider the basic inevitable paradox that a philosopher should explain not only our understanding of experience but also his philosophizing itself. The circularity between “method employed” and “intended result” needs to be admitted as an initial fact about reason rather than circumvented by an implausibly claimed capacity of philosophical reason to formally transcend everything including this circularity. -/- Presuppositions are reduced to a minimum inevitable number. Being our primary microphilosophical criteria, they are reason, experience, and value in their minimal sense. They cannot be plausibly substituted or further reduced. They cannot be preferred one or more to others arbitrarily. We show how, in microphilosophy they coincide or synchronize. Jointly and basically, they form the minimum method employed. The initial, basic, intended result is called minimum feasibility. The fundamental circularity between method and result leaves no way out but to ground self-reference in such a manner that method and result “logico-genetically” coincide. Two extreme situations are avoided: “the logical zero-situation” where nothing is true or real, and “the credulous open situation” where everything is true or real. Finding a mean manifests a basic value. If the meaning of truth and reality is to be preserved, the basic value must have a clear impact and express descriptive rather than revisionary reason. -/- Several other stringent demands are raised as conditions which microphilosophy should fulfil to achieve feasibility. The final, joint, outcome of both “method” and “result,” in the thesis of microphilosophical feasibilism, is formulated thus: Self refers freely and symbolically to itself, own person, own body, material bodies, other persons’ bodies, other persons, and other selves. -/- Ramesh N. Patel Doctor of Philosophy in Philosophy conferred in June, 1970 University of New Mexico, Albuquerque, N.M. (shrink)