This article discusses the logic of the Epimenides Paradox using a graphical representation of the logical operators involved. The graphical representation of the Logical Operators is the one used in digital electronics. This method of investigation is very effective in revealing the hidden error in the reasoning that leads to the contradiction. The article explains the vicious circle that the mind goes through when trying to analyze the Paradox. Finally, the article proposes a solution of the Paradox with final (...) conclusions that are not paradoxical but simply counterintuitive. The endless deduction is generated by subsequent evaluations that take place on different levels. In its evaluation the Mind alternates a level of logical evaluation with a level of semantic evaluation. (shrink)

One of Marx‘s and Engels‘ main claims (hereafter ―original Marxism) in their account of the historical ―inevitability of the collapse of capitalism is that one‘s material (economic) conditions, not one‘s ideas, arguments or philosophy, determines one‘s ―consciousness and actions. However, the self-reference in this characterization of philosophical views generates a paradox analogous to the 7th century B.C. Epimenides ―Liar paradox. The Epimenides-paradox arises when Epimenides, a Cretan, states that all Cretans are liars. Epimenides-statement is paradoxical in (...) the sense that if it is true then it is false. Similarly, the Marxist claim that all philosophers who purport to state the cause of all social changes must be wrong, where Marx and Engels are themselves philosophers stating just such a claim, generates a paradox analogous to the ― Epimenides Liar paradox. This leads to what Engels himself calls a―false consciousness, i.e., a failure to understand the true causes of social change. This paradox can only be escaped by abandoning at least one of original Marxism‘s core doctrines. (shrink)

In line with the Principle of Uniform Solution, Graham Priest has challenged advocates like myself of the “multiple-meanings” solution to the paradoxes of truth and knowledge, due to the medieval logician Thomas Bradwardine, to extend this account to a similar solution to the paradoxes of denotation, such as Berry’s, König’s and Richard’s. I here rise to this challenge by showing how to adapt Bradwardine’s principles of truth and signification for propositions to corresponding principles of denotation and signification for descriptive phrases, (...) applying them to give a “multiple-meanings” solution to the denotational paradoxes. (shrink)

Solutions to Russell’s paradox of propositions and to Kaplan’s paradox are proposed based on an extension of von Neumann’s method of avoiding paradox. It is shown that Russell’s ‘anti-Cantorian’ mappings can be preserved using this method, but Kaplan’s mapping cannot. In addition, several versions of the Epimenides paradox are discussed in light of von Neumann’s method.

One way of resolving a paradox is to defuse it to a hypodox. This way is relatively unknown though. The goal of this paper is to explain this way with varied examples. The hypodoxes are themselves a broad class: both the Truth-teller and the 21st birthday of someone born on 29th February can be construed as hypodoxes. The most familiar kind of relation between paradoxes and hypodoxes is exemplified by the relation between the Liar and the Truth-teller. This article concerns (...) a second kind where a paradox is defused to a hypodox by restricting or rejecting some granted principles. The Liar paradox has this second kind of relation to a Liar hypodox, which will be introduced. In some cases, defusing a paradox to a hypodox is only a partial resolution, as the hypodox itself may then need resolving. Even so, such a partial resolution decomposes a complex problem into more easily understood problems. Moreover, I compare the result of defusing a paradox to a hypodox with the results of resolving paradoxes in other ways. I give four examples. The first is mainly pedagogic, concerning a birthday. The second is a lightweight legal case, presenting a parking voucher paradox. The third is a formal system in which a Liar and Liar-like sentences are hypodoxical. The fourth is a philosophical critique of ways of solving Bertrand’s chord paradox. (shrink)

It is proposed that subconscious retro-predictions in conjunction with brain state update cycles are instrumental in the physiological generation of conscious sensations and perceptions, and in all abstract thought. In this paper the hypothesis is supported by conducting a detailed a re-evaluation of the self-referential statements in Set Theory and Formal Logic known as antinomies. This study concludes that the recursive behavior exhibited by abstract enigmas such as "Russell’s Paradox" is analogous to the oscillations typical of bistable perceptual phenomena.

With the help of renowned logician Taylor Swift, Theresa Helke introduces four fundamental paradoxes: the Liar, Epimenides’, the Truth-Teller, and the No-No.

The title of this book is misleading; the subtitle indicates the content more faithfully. Only the last chapter is concerned with paradoxes, namely, with the semantic paradoxes. But the argument there is based on the general theory of assertion and predication defended in the preceding six chapters, which constitute the heart of the book. Cargile rejects the familiar answers to the semantic paradoxes mainly on the grounds that they require restricting the universality of the laws of logic, which involve self-reference (...) just as much as do the paradoxical sentences. What is Cargile’s answer? His treatment of the Epimenides is representative. A Cretan who asserts that all Cretan assertions are false is asserting a different proposition from the proposition a non-Cretan would assert with the same sentence. What assertion is being made with such a sentence depends on the predications, performed by the assertion, of truth or falsity of the propositions "non-assessably" asserted in the original, "assessable" assertion. The Cretan’s assertion predicates falsity of itself, as well as of every other Cretan assertion, but the non-Cretan’s assertion does not. The latter is presumably false and in any case nonparadoxical. The former is self-contradictory, therefore false, therefore also nonparadoxical, since, among other things, it asserts that it is false that all Cretan assertions are false. Other semantic paradoxes require different treatments, also provided by Cargile, but on the basis of the same general theory of assertion and predication. (shrink)

The main objective of this paper is to examine how theories of truth and reference that are in a broad sense Fregean in character are threatened by antinomies; in particular by the Epimenides paradox and versions of the so-called Russell-Myhill antinomy, an intensional analogue of Russell’s more well-known paradox for extensions. Frege’s ontology of propositions and senses has recently received renewed interest in connection with minimalist theories that take propositions (thoughts) and senses (concepts) as the primary bearers of truth (...) and reference. In this paper, I will present a rigorous version of Frege’s theory of sense and denotation and show that it leads to antinomies. I am also going to discuss ways of modifying Frege’s semantical and ontological framework in order to avoid the paradoxes. In this connection, I explore the possibility of giving up the Fregean assumption of a universal domain of absolutely all objects, containing in addition to extensional objects also abstract intensional ones like propositions and singular concepts. I outline a cumulative hierarchy of Fregean propositions and senses, in analogy with Gödel’s hierarchy of constructible sets. In this hierarchy, there is no domain of all objects. Instead, every domain of objects is extendible with new objects that, on pain of contradiction, cannot belong to the given domain. According to this approach, there is no domain containing absolutely all propositions or absolutely all senses. (shrink)

It seems that the Truth-teller is either true or false, but there is no accepted principle determining which it is. From this point of view, the Truth-teller is a hypodox. A hypodox is a conundrum like a paradox, but consistent. Sometimes, accepting an additional principle will convert a hypodox into a paradox. Conversely, in some cases, retracting or restricting a principle will convert a paradox to a hypodox. This last point suggests a new method of avoiding inconsistency. This article provides (...) a significant example. The Liar paradox can be defused to a hypodox by relatively minimally restricting three principles: the T-schema, substitution of identicals and universal instantiation. These restrictions are not arbitrary. For each, I identify the source of a contradiction given some presumptions. Then I propose each restriction as a reasonable way to deal with that source of contradiction. (shrink)

The story goes that Epimenides, a Cretan, used to claim that all Cretans are always liars. Whether he knew it or not, this claim is odd. It is easy to see it is odd by asking if it is true or false. If it is true, then all Cretans, including Epimenides, are always liars, in which case what he said must be false. Thus, if what he says is true, it is false. Conversely, suppose what Epimenides said (...) is false. Then some Cretan at some time speaks truly. This might not tell us anything about Epimenides. But if, to make the story simple, he were the only Cretan ever to speak, and this was the only thing he ever said, then indeed, he would have to speak truly. And we would then have shown that if what he said was false, it must be true. (shrink)

Most philosophers believe that the Liar Paradox is semantical in character, and arises from difficulties in the predicate “true.” The author argues that the paradox is pragmatic, not semantic, and arises from violations of essential conditions that define statement-making speech acts. The author shows that his solution to the paradox will not only handle the classical Liar sentences that are “necessarily” or “intrinsically” paradoxical, but also sets of Kripke-sentences that are “contingently” paradoxical.

The liar analyzes in detail the genealogy of the paradox expressed by Epimenides when he claimed that all Cretans were liars. As he himself was a native of Crete, this expression was paradoxical since he expressed a truth by lying. Epimenides showed that it is possible to construct perfectly correct sentences according to grammatical and semantic rules, but that they in turn express a contradiction insofar as it is true and false indistinctly. Since the beginning of Western thought, (...) the liar's paradox has been resolved from dissimilar points of view. The book will analyze the different proposals developed in order to deactivate their pernicious consequences for any theory of truth. Thus, Aristotle proposed that the liar should be considered a fallacy, so he attempted to dissolve the presumed validity of his argument through insightful distinctions around the secundum quid et simpliciter fallacy. Subsequently, solutions were tested within the framework of discursive, dialectical and hypothetical models. In the book those propositions that change their truth value over time are analyzed, those called in the metapiptonta tradition. In medieval times the liar was assumed to be insolubious. Special attention was given to the discussion around the transcasus. Kant approached the liar from the point of view of the white lies generated by the antinomies of reason. In modern times, special care was taken to translate the liar into new expressive forms in order to pose it from a merely formal point of view. For this reason the study of him focused on showing and describing his circular, self-referential and reflective character. This work emphasizes the study of the liar within the framework of occasional expressions. The self-reference of the liar's paradox is derived from referential expressions that result from the information expressed circumstantially within the framework of a surrounding context. For this reason we propose a systematic study of the occasional in order to dissolve the liar. Our proposal proposes to characterize the surrounding expressions that determine the liar's referential meaning. (shrink)

i. a poetic solution of the Goldbach Conjecture; ii. several responses to the Epimenides Paradox; iii. the volitional solution to Russell's Paradox.

After a brief and informal explanation of the Gödel’s theorem as a version of the Epimenides’ paradox applied to Elementary Number Theory formulated in first-order logic, Lucas shows some of the most relevant consequences of this theorem, such as the impossibility to define truth in terms of provability and so the failure of Verificationist and Intuitionist arguments. He shows moreover how Gödel’s theorem proves that first-order arithmetic admits non-standard models, that Hilbert’s programme is untenable and that second-order logic is (...) not mechanical. There are furthermore some more general consequences: the difference between being reasonable and following a rule and the possibility that one man’s insight differs from another’s without being wrong. Finally some consequences concerning moral and political philosophy can arise from Gödel’s theorem, because it suggests that – instead of some fundamental principle from which all else follows deductively – we can seek for different arguments in different situations. (shrink)

Lucas introduces this paper by an account of how he began to be interested to questions about Materialism and Mechanism. Then he suggests a simple version of the Incompleteness theorem of Gödel, showing how this theorem proposes a version of the Epimenides’ paradox able to avoid the circularity of this paradox by means of the possibility to express meta-mathematics in terms of arithmetical propositions and by substituting questions concerning truth by questions concerning provability.

Since classical antiquity, theorists have struggled with the problem of self-reference originating in the Liar Paradox of Eubulides: “Does someone who says ‘I am lying’ lie?” Does he speak truly or falsely? The consensus resolution has been to dismiss the contention “This statement is false” as meaningless incoherent on grounds of self-contradiction. Leibniz also deemed the claim incoherent, but not on grounds of self-contradiction incoherence however on grounds of embarking on an infinite and incompletable series of meaningfulness-presuppositions. He thus introduced (...) a novel and constructive mode of semantical analysis. (shrink)

The significance of the semantical paradoxes for natural languages is examined. If Tarski’s reflections on the issue are correct, English is inconsistent. Paul Ziff responds to Tarskian reflections by arguing to the conclusion that no natural language is or can be inconsistent. The authors reject Ziff’s argument, but they defend something similar to its conclusion: no language, natural or otherwise, is or can be inconsistent in the way that Tarski holds languages capable of formulating the Epimenides are inconsistent.

Higher-order logic, with its type-theoretic apparatus known as the simple theory of types (STT), has increasingly come to be employed in theorizing about properties, relations, and states of affairs—or ‘intensional entities’ for short. This paper argues against this employment of STT and offers an alternative: ordinal type theory (OTT). Very roughly, STT and OTT can be regarded as complementary simplifications of the ‘ramified theory of types’ outlined in the Introduction to Principia Mathematica (on a realist reading). While STT, understood as (...) a theory of intensional entities, retains the Fregean division of properties and relations into a multiplicity of categories according to their adicities and ‘input types’ and discards the division of intensional entities into different ‘orders’, OTT takes the opposite approach: it retains the hierarchy of orders (though with some modifications) and discards the categorisation of properties and relations according to their adicities and input types. In contrast to STT, this latter approach avoids intensional counterparts of the Epimenides and related paradoxes. Fundamental intensional entities lie at the base of the proposed hierarchy and are also given a prominent part to play in the individuation of non-fundamental intensional entities. (shrink)

Russell’s substitutional theory conferred philosophical advantages over the simple type theory it was to emulate. However, it faced propositional paradoxes, and in a 1906 paper “On ‘Insolubilia’ and Their Solution by Symbolic Logic”, he modified the theory to block these paradoxes while preserving Cantor’s results. My aim is to draw out several quandaries for the interpretation of the role of substitution in Russell’s logic. If he was aware of the substitutional (_p_0_a_0) paradox in 1906, why did he advertise “Insolubilia” as (...) a solution to the Epimenides? If he was dissatisfied with the solution, as his correspondence suggests, why did he go on to publish it? Why did substitution reappear with orders in “Mathematical Logic as Based on the Theory of Types” if he had rejected a hierarchy of orders as intolerable? I offer the following as possible explanations: he construed the “logical Epimenides” as a version of the _p_0_a_0 paradox; his dissatisfaction with the “Insolubilia” solution was philosophical, not technical; and substitution re-emerged because he hoped for a new philosophical gloss on orders. Whether or not my explanations are correct, these issues must be addressed in accounting for Russell’s reasons for ramification. (shrink)

Language & Logic -- Glossary -- Aristotle's syllogisms -- Russell's paradox & Frege's logicism -- profile: Aristotle -- Russell's theory of description -- Frege's puzzle -- Gödel's theorem -- Epimenides' liar paradox -- Eubulides' heap -- Science & Epistemology -- Glossary -- I think therefore I am -- Gettier's counter example -- profile: Karl Popper -- The brain in a vat -- Hume's problem of induction -- Goodman's gruesome riddle -- Popper's conjectures & refutations -- Kuhn's scientific revolutions -- (...) Mind & Metaphysics -- Glossary -- Descartes mind-body problem -- Brentano's intentionality -- Fodor's language of thought -- Parfit's persons -- profile: René Descartes -- Chalmer's zombies -- Zeno's paradoxes -- Kant's left hand -- Theseus' ship -- Laplace's demon, determinism, & free will -- Ryle's ghost in the machine -- Ethics & Political Philosophy -- Glossary -- Aristotle's ethics -- States of nature and the social contract -- Kant's categorical imperative -- profile: Immanuel Kant -- Mill's utilitarianism -- Marx's historical materialism -- The trolley problem -- Religion -- Glossary -- Aquinas' five ways -- Anselm's ontological argument -- profile: Thomas Aquinas -- Epicurus' riddle -- Paley's watchmaker -- Pascal's wager -- Hume against miracles -- Grand Moments -- Glossary -- Socrates' method -- Plato's cave -- Aristotle's four causes -- Lucretius' atomism -- profile: Ludwig Wittgenstein -- Berkeley's idealism -- Kant's synthetic a priori -- Hegel's dialectic -- James' pragmatism -- Moore's common sense -- Wittgenstein's picture theory of language -- Continental Philosophy -- Glossary -- Nietzsche's Superman -- profile: Friedrich Nietzsche -- Derrida's deconstruction -- Heidegger's nothing -- Sartre's bad faith. (shrink)

Harman’s lottery paradox, generalized by Vogel to a number of other cases, involves a curious pattern of intuitive knowledge ascriptions: certain propositions seem easier to know than various higher-probability propositions that are recognized to follow from them. For example, it seems easier to judge that someone knows his car is now on Avenue A, where he parked it an hour ago, than to judge that he knows that it is not the case that his car has been stolen and driven (...) away in the last hour. Contextualists have taken this pattern of intuitions as evidence that ‘knows’ does not always denote the same relationship; subject-sensitive invariantists have taken this pattern of intuitions as evidence that non-traditional factors such as practical interests figure in knowledge; still others have argued that the Harman Vogel pattern gives us a reason to abandon the principle that knowledge is closed under known entailment. This paper argues that there is a psychological explanation of the strange pattern of intuitions, grounded in the manner in which we shift between an automatic or heuristic mode of judgment and a controlled or systematic mode. Understanding the psychology behind the pattern of intuitions enables us to see that the pattern gives us no reason to abandon traditional intellectualist invariantism. The psychological account of the paradox also yields new resources for clarifying and defending the single premise closure principle for knowledge ascriptions. (shrink)

This paper is concerned with formal solutions to the lottery paradox on which high probability defeasibly warrants acceptance. It considers some recently proposed solutions of this type and presents an argument showing that these solutions are trivial in that they boil down to the claim that perfect probability is sufficient for rational acceptability. The argument is then generalized, showing that a broad class of similar solutions faces the same problem. An argument against some formal solutions to the lottery paradox The (...) argument generalized Some variations Adding modalities Anticipated objections. (shrink)

Most discussions frame the Liar Paradox as a formal logical-linguistic puzzle. Attempts to resolve the paradox have focused very little so far on aspects of cognitive psychology and processing, because semantic and cognitive-psychological issues are generally assumed to be disjunct. I provide a motivation and carry out a cognitive-computational treatment of the liar paradox based on a cognitive-computational model of language and conceptual knowledge within the Predictive Processing framework. I suggest that the paradox arises as a failure of synchronization between (...) two ways of generating the liar situation in two different PP sub-models, one corresponding to language processing and the other to the processing of meaning and world-knowledge. In this way, I put forward the claim that the liar sentence is meaningless but has an air of meaningfulness. I address the possible objection that the proposal violates the Principle of Unrestricted Compositionality, which purportedly regulates the conceptual competence of thinkers. (shrink)

Global ignorance about Africa continues to sustain inappropriate global interventions to resolve public health crises, often with disastrous consequences. To explain why this continues to happen, I marshal two theorems that predict basic statistical properties, called ‘the doctrinal paradox’ and ‘the discursive dilemma’, which underlie scientific consensus formation and evidence-based decision making on a global scale. These mathematical results illuminate the epistemic and material injustices committed by the protocols of medical research conducted at the highest level of global knowledge production (...) in the service of international humanitarian aid for Africans. These social choice theorems reveal that a global scientific consensus projecting claims and proposing policies about Africa’s disease burden is likely to yield a low degree of reliability, in that the probability of its accuracy is less than chance. The solution to this anomaly is to remove from the global scientific agenda the statistically unrealisable demand of satisfying too many multinational corporate and foreign governments’ priorities as equally entitled to benefit from the knowledge produced to improve Africa’s public health sector. Instead, foreign funding targeted to support medical science and policy in Africa should be directed by those specialists in situ who are most familiar with their own national health challenges and potential solutions, rather than relying upon foreign decision makers to interpret Africa’s emergency public health care needs. Keywords Doctrinal paradox, discursive dilemma, scientific consensus, epistemic trust, global health ethics, group agency. (shrink)

I am going to assume, in what follows, that when we engage with a fiction we are participating in a game of make-believe; that is, that we are engaging in an imaginative effort. In this paper I shall attempt to identify the kind of game we are playing. I begin with two words of caution. First, identifying the kind of game will be a matter of finding a game whose structure best reflects the facts about our engagement with fiction. The (...) fit, however, will not be exact. In a game of mud pies, the fact that the cardboard box holds a maximum of six globs of mud may make it true in the game that the oven holds a maximum of six pies, but the fact that the box has "Fyffes bananas" written on the side of it will probably not make it true in the game that the oven has the same. Second the variety of works of fiction makes it unwise to assume that one kind of game will cover all cases. These two provisos might be thought to vitiate my project before it begins. In the face of an objection to my first proviso, namely, that the kind of make-believe I maintain we play with fiction does not mirror the facts of our relations with fiction, I can reply that the structure is not isomorphic in that particular respect. In reply to the claim that in some particular instance we do not play the game of make-believe that I suggest we do, I can simply agree and count it as one of the many exceptions to the rule. (shrink)

We begin with a prepositional languageLpcontaining conjunction (Λ), a class of sentence names {Sα}αϵA, and a falsity predicateF. We (only) allow unrestricted infinite conjunctions, i.e., given any non-empty class of sentence names {Sβ}βϵB,is a well-formed formula (we will useWFFto denote the set of well-formed formulae).The language, as it stands, is unproblematic. Whether various paradoxes are produced depends on which names are assigned to which sentences. What is needed is a denotation function:For example, theLPsentence “F(S1)” (i.e.,Λ{F(S1)}), combined with a denotation functionδsuch (...) thatδ(S1)“F(S1)”, provides the (or, in this context, a)Liar Paradox.To give a more interesting example,Yablo's Paradox[4] can be reconstructed within this framework.Yablo's Paradoxconsists of an ω-sequence of sentences {Sk}kϵωwhere, for eachnϵω:WithinLPan equivalent construction can be obtained using infinite conjunction in place of universal quantification - the sentence names are {Si}iϵωand the denotation function is given by:We can express this in more familiar terms as:etc. (shrink)

A number of authors have argued that Peano Arithmetic supplemented with a logical validity predicate is inconsistent in much the same manner as is PA supplemented with an unrestricted truth predicate. In this paper I show that, on the contrary, there is no genuine paradox of logical validity—a completely general logical validity predicate can be coherently added to PA, and the resulting system is consistent. In addition, this observation lead to a number of novel, and important, insights into the nature (...) of logical validity itself. (shrink)

Paradox Lost covers ten of philosophy’s most fascinating paradoxes, in which seemingly compelling reasoning leads to absurd conclusions. The following paradoxes are included: The Liar Paradox, in which a sentence says of itself that it is false. Is the sentence true or false? The Sorites Paradox, in which we imagine removing grains of sand one at a time from a heap of sand. Is there a particular grain whose removal converts the heap to a non-heap? The Puzzle of the Self-Torturer, (...) in which a series of seemingly rational choices has us accepting a life of excruciating pain, in exchange for millions of dollars. Newcomb’s Problem, in which we seemingly maximize our expected profit by taking an unknown sum of money, rather than taking the same sum plus $1000. The Surprise Quiz Paradox, in which a professor finds that it is impossible to give a surprise quiz on any particular day of the week... but also that if this is so, then a surprise quiz can be given on any day. The Two Envelope Paradox, in which we are asked to choose between two indistinguishable envelopes, and it is seemingly shown that each envelope is preferable to the other. The Ravens Paradox, in which observing a purple shoe provides evidence that all ravens are black. The Shooting Room Paradox, in which a deadly game kills 90% of all who play, yet each individual’s survival turns on the flip of a fair coin. Each paradox is clearly described, common mistakes are explored, and a clear, logical solution offered. Paradox Lost will appeal to professional philosophers, students of philosophy, and all who love intellectual puzzles. (shrink)

Many have the intuition that the right response to the Lottery Paradox is to deny that one can justifiably believe of even a single lottery ticket that it will lose. The paper shows that from any theory of justification that solves the paradox in accordance with this intuition, a theory not of that kind can be derived that also solves the paradox but is more conducive to our epistemic goal than the former. It is argued that currently there is no (...) valid reason not to give preference to the derived accounts over the accounts from which they come. (shrink)