Semantic Paradoxes

This is the second of a series of three lectures given at Peking University.

Haack explores correspondence theories, the semantic theory, and the Laconicist theory (also known as the "redundancy" theory).

After showing the [problems with the first two kinds of theory, Haack argues for the virtue of the last, but also looks at the several issues still in need of resolution.

This article introduces, studies, and applies a new system of logic which is called 'HYPE'. In HYPE, formulas are evaluated at states that may exhibit truth value gaps (partiality) and truth value gluts (overdeterminedness). Simple and natural semantic rules for negation and the conditional operator are formulated based on an incompatibility relation and a partial fusion operation on states. The semantics is worked out in formal and philosophical detail, and a sound and complete axiomatization is provided both for the propositional and the predicate logic of the system. The propositional logic of HYPE is shown to contain first-degree entailment, to have the Finite Model Property, to be decidable, to have the Disjunction Property, and to extend intuitionistic propositional logic conservatively when intuitionistic negation is defined appropriately by HYPE's logical connectives. Furthermore, HYPE's first-order logic is a conservative extension of intuitionistic logic with the Constant Domain Axiom, when intuitionistic negation is again defined appropriately. The system allows for simple model constructions and intuitive Euler-Venn-like diagrams, and its logical structure matches structures well-known from ordinary mathematics, such as from optimization theory, combinatorics, and graph theory. HYPE may also be used as a general logical framework in which different systems of logic can be studied, compared, and combined. In particular, HYPE is found to relate in interesting ways to classical logic and various systems of relevance and paraconsistent logic, many-valued logic, and truthmaker semantics. On the philosophical side, if used as a logic for theories of type-free truth, HYPE is shown to address semantic paradoxes such as the Liar Paradox by extending non-classical fixed-point interpretations of truth by a conditional as well-behaved as that of intuitionistic logic. Finally, HYPE may be used as a background system for modal operators that create hyperintensional contexts, though the details of this application need to be left to follow-up work.

A dialetheia is a sentence, A, such that both it and its negation, ¬A, are true (we shall talk of sentences throughout this entry; but one could run the definition in terms of propositions, statements, or whatever one takes as her favourite truth-bearer: this would make little difference in the context). Assuming the fairly uncontroversial view that falsity just is the truth of negation, it can equally be claimed that a dialetheia is a sentence which is both true and false...

This article introduces, studies, and applies a new system of logic which is called 'HYPE'. In HYPE, formulas are evaluated at states that may exhibit truth value gaps (partiality) and truth value gluts (overdeterminedness). Simple and natural semantic rules for negation and the conditional operator are formulated based on an incompatibility relation and a fusion operation on states. The semantics is worked out in formal and philosophical detail, and a sound and complete axiomatization is provided both for the propositional and the predicate logic of the system. The propositional logic of HYPE is shown to contain first-degree entailment, to have the Finite Model Property, to be decidable, to have the Disjunction Property, and to extend intuitionis-tic propositional logic conservatively when intuitionistic negation is defined appropriately by HYPE's logical connectives. Furthermore, HYPE's first-order logic is a conservative extension of intuitionistic logic with the Constant Domain Axiom, when intuitionistic negation is again defined appropriately. The system allows for simple model constructions and intuitive Euler-Venn-like diagrams, and its logical structure matches structures well-known from ordinary mathematics, such as from optimization theory, combinatorics, and graph theory. HYPE may also be used as a general logical framework in which different systems of logic can be studied, compared, and combined. In particular, HYPE is found to relate in interesting ways to classical logic and various systems of relevance and paraconsistent logic, many-valued logic, and truthmaker semantics. On the philosophical side, if used as a logic for theories of type-free truth, HYPE is shown to address semantic paradoxes such as the Liar Paradox by extending non-classical fixed-point interpretations of truth by a conditional as well-behaved as that of intuitionistic logic. Finally, HYPE may be used as a background system for modal operators that create hyperintensional contexts, though the details of this application need to be left to follow-up work.

"Paradox and perspectivism in the philosophy of language of Bhartrhari : language, thought and reality"

The way the Indian grammarian-philosopher Bhartrhari (5th century CE.) deals with the liar paradox and some other paradoxes is pertinent to Western attempts to cope with similar paradoxes since the early Greeks. The philological study of the relevant verses and a discussion on the identification of the paradoxes and their treatment by Bhartrhari have appeared in earlier publications (Houben 1995a and 1995b).
The present article focuses on the paradox of unsignifiability, that is, the paradox which Herzberger and Herzberger have named the Bhartrhari paradox. Suppose someone maintains: "this is unsignifiable." One may understand that there is a "this" that is unsignifiable, but then one has to admit that "this" has become signifiable as "unsignifiable". Otherwise: taking "this" immediately as "unsignifiable" even its unsignifiability cannot be signified, in other words, its unsignifiability is unsignifiable: again, the unsignifiability of this unsignifiability must be equally unsignifiable, etc., etc...
H. and R. Herzberger (1981) have shown how the points presented by Bhartrhari can be developed into a stable paradox in the modern sense. Bhartrhari, however, identifies a hidden parameter that leads to a paradoxical situation if neglected, and that helps to solve the paradox if it arises. Bhartrhari was never confronted with anything similar to the "strengthened" paradoxes of modern logic, but the hidden parameter identified by him is not only suitable to Barwise's and Etchemendy's solution (1989) to paradoxes such as the liar, but even points to an important improvement of Barwise's and Etchemendy's theory in the area of paradoxical universalistic statements where their current approach flounders.
According to its immediate context, Bhartrhari's treatment of paradoxes applies to "relations" in general and more specifically to "significance relations" (between signifier and signified). The paradox arises when there is class-crossing self-reference -- a word "this" in a sentence referring to that very sentence -- which Bertrand Russell identified as the main cause of the liar and similar paradoxes. In his endeavor to establish a perfect formal language, he avoided self-reference and the appearance of these paradoxes by postulating a hierarchy of languages, an idea that was later refined and developed by Alfred Tarski. Yet the failure of this attempt to prevent self-reference and to establish a consistent formal system was demonstrated by Kurt Goedel. These problems have direct implications for the potential paradoxality of universalistic statements. Here it turns out that Bhartrhari's treatment of paradoxes is of a piece with his perspectivism which implies that, fundamentally, various perspectives on reality are valid. While Barwise and Etchemendy, adopting an Austinian approach to the problem of paradox, demonstrate that the paradoxical statement implies a perspective on reality which is necessarily limited, but - as emphasized in Julian Roberts' reformulation of their approach - which can be reconciled with other perspectives over time, Bhartrhari shows that the change in perspective over time is internal to any process of signification (in statement or even in thought). The shifting parameter (signifier to signified) reveals itself if one pays attention to the pragmatic role played by a statement in the natural situation in which a paradox may arise - and it is easily overseen in graphic representations of circular statements, forming the basis of modern reflections and abstractions which remove the logician more and more from the pragmatic context of a statement precisely while they strengthen his paradox.

Since Saul Kripke's influential work in the 70's, the revisionary approach to semantic paradox, the idea that semantic paradoxes must be solved by weakening classical logic, has been increasingly popular. In this paper, we present a new revenge argument to the effect that the main revisionary approaches breed new paradoxes they are unable to block.

Il presente saggio si propone di caratterizzare per approssimazioni successive alcune delle questioni fondamentali di una teoria della verità per il linguaggio naturale: segnatamente, i problemi posti dai paradossi semantici ad ogni tentativo di definizione del concetto di verità. Nello specifico i temi trattati sono: il problema del significato della verità e il problema del criterio di verità; l’accezione ontologica e l’accezione semantica della verità; la ‘natura’ della verità e la definizione nominale della verità; le cause dell’antinomia del ‘mentitore’; che cosa si deve intendere per soluzione del problema del ‘mentitore’?; le soluzioni di Tarski e di Kripke, e le loro implicazioni sul problema ‘descrittivo’ della verità.

In this paper we take a closer look at Penrose's New Argument for the claim that the human mind cannot be mechanized and investigate whether the argument can be formalized in a sound and coherent way using a theory of truth and absolute provability. Our findings are negative; we can show that there will be no consistent theory that allows for a formalization of Penrose's argument in a straightforward way. In a second step we consider Penrose's overall strategy for arguing for his view and provide a reasonable theory of truth and absolute provability in which this strategy leads to a sound argument for the claim that the human mind cannot be mechanized. However, we argue that the argument is intuitively implausible since it relies on a pathological feature of the proposed theory.

Une introduction logique et philosophique au problème de la définition de la vérité. Le chapitre contient une présentation du travail de Tarski sur la vérité. Une présentation du travail de Kripke. Un aperçu sur quelques théories contemporaines plus récentes. Une présentation des débats philosophiques sur la nature de la vérité ouverts par le travail Tarski.

I will use paradox as a guide to metaphysical grounding, a kind of non-causal explanation that has recently shown itself to play a pivotal role in philosophical inquiry. Specifically, I will analyze the grounding structure of the Predestination paradox, the regresses of Carroll and Bradley, Russell's paradox and the Liar, Yablo's paradox, Zeno's paradoxes, and a novel omega plus one variant of Yablo's paradox, and thus find reason for the following: We should continue to characterize grounding as asymmetrical and irreflexive. We should change our understanding of the transitivity of grounding in a certain sense. We should require foundationality in a new, generalized sense, that has well-foundedness as its limit case. Meta-grounding is important. The polarity of grounding can be crucial. Thus we will learn a lot about structural properties of grounding from considering the various paradoxes. On the way, grounding will also turn out to be relevant to the diagnosis (if not the solution) of paradox. All the paradoxes under consideration will turn out to be breaches of some standard requirement on grounding, which makes uniform solutions of large groups of these paradoxes more desirable. In sum, bringing together paradox and grounding will be shown to be of considerable value to philosophy.

Understandably absorbed in technical details, discussion of the semantic paradoxes risks losing sight of broad methodological principles. This essay sketches a general approach to the comparison of rival logics, and applies it to argue that revision of classical propositional logic has much higher costs than its proponents typically recognize.

In recent editions of this journal, Jean-Yves Beziau [8] and Marcin Tkaczyk [42] have criticised a prominent dialetheic logic and common arguments for dialetheism, respectively. While Beziau argues that Priest's logic LP commits the dialetheist to trivialism, the thesis that all propositions are true, Tkaczyk maintains that the arguments traditionally proposed for dialetheism are faulty and ultimately that dialetheism should be rejected as self-refuting. This paper shows that both are mistaken in their contentions. Beziau's argument conflates truth-in-an-interpretation with truth simpliciter and Tkaczyk misconstrues the substance of dialetheic arguments. In the process of identifying these weaknesses of both arguments , the paper clarifies elements of both dialetheic logics and dialethe-ism which these discussions demonstrate are still misunderstood within the literature.

T-biconditionals have often been regarded as insufficient as axioms for truth. This verdict is based on Tarski’s observation that the typed T-sentences suffer from deductive weakness. As indicated by McGee, the situation might change radically if we consider type-free disquotational theories of truth. However, finding a well-motivated set of untyped T-biconditionals that is consistent and recursively enumerable has proven to be very difficult. Moreover, some authors (e.g. Glanzberg) have argued that any solution to the semantic paradoxes necessarily involves ‘inflationary’ means, thus spelling doom to deflationist and minimalist theories of truth in particular. The situation is indeed worrisome as formal theories of minimalist truth are (almost) missing so far. This makes it very hard to properly evaluate the tenets of minimalism. In this article, we will show how to find legitimate instances of the T-schema just by relying on syntactic features of the sentences of our language—in particular, we will explore Quine’s idea of stratification. Based on that, we will introduce some disquotational truth theories that are deductively very strong.

We axiomatize Leitgeb's (2005) theory of truth and show that this theory proves all arithmetical
sentences of the system of ramified analysis up to \epsilon_0. We also give alternative axiomatizations of
Kripke's (1975) theory of truth (Strong Kleene and supervaluational version) and show that they are at
least as strong as the Kripke-Feferman system KF and Cantini's VF, respectively.

The Knower paradox purports to place surprising a priori limitations on what we can know. According to orthodoxy, it shows that we need to abandon one of three plausible and widely-held ideas: that knowledge is factive, that we can know that knowledge is factive, and that we can use logical/mathematical reasoning to extend our knowledge via very weak single-premise closure principles. I argue that classical logic, not any of these epistemic principles, is the culprit. I develop a consistent theory validating all these principles by combining Hartry Field's theory of truth with a modal enrichment developed for a different purpose by Michael Caie. The only casualty is classical logic: the theory avoids paradox by using a weaker-than-classical K3 logic.

I then assess the philosophical merits of this approach. I argue that, unlike the traditional semantic paradoxes involving extensional notions like truth, its plausibility depends on the way in which sentences are referred to—whether in natural languages via direct sentential reference, or in mathematical theories via indirect sentential reference by Gödel coding. In particular, I argue that from the perspective of natural language, my non-classical treatment of knowledge as a predicate is plausible, while from the perspective of mathematical theories, its plausibility depends on unresolved questions about the limits of our idealized deductive capacities.

In  this  article,  we  will  show  that  uncomputability  is  a  relative property  not  only  of  oracle  Turing  machines,  but  also  of  subrecursive classes. We will define the concept of a Turing submachine, and a recursive relative  version  for  the  Busy Beaver  function which we  will  call  Busy Beaver Plus function. Therefore, we will prove that the computable Busy Beaver Plus function defined on any Turing submachine is not computable by any program running on this submachine. We will thereby demonstrate the existence of a “paradox” of computability a la Skolem: a function is computable when “seen from the outside” the subsystem, but uncomputable when “seen from within” the same subsystem. Finally, we will raise the possibility of defining universal submachines,  and a hierarchy of negative Turing degrees.

Ttler=‘Ttler is true’ says of itself that it is true. It is a truth-teller. I argue that we have equally telling arguments (i) to the effect that all truth-tellers must have the same truth-value (ii) and the effect that truth-tellers differ in truth-value. This is what I call the Truth-Tellers paradox. This paradox stems from the fact that the truth-value of a truth-teller like Ttler should be determined by the fact that it says of itself that it is true (which entails (i)) but that it cannot be determined by that fact (as witnessed by (ii)). The Truth-Tellers paradox resem- bles the classical semantic paradoxes like the Liar. In both cases, a form of self-reference allows us to derive a contradiction from oth- erwise plausible semantic and logical principles. Furthermore the Truth-Tellers paradox can be formulated without using sentences which are in an intuitive sense ungrounded, it thus severs the link, almost universally taken for granted, between the semantic para- doxes and ungroundedness. Finally, some classical solutions to the Liar do not generalize to the Truth-Tellers paradox.

Metabiology is a mathematical theory mainly based on algorithmic information theory and allows us to study an open-ended evolution of programs, that is, to study how fast the organisms/programs become more complex or more creative without never stagnate at a maximum level of complexity or creativity. We take, as starting point, the first chaitinian metabiological models in which the organisms are computable systems, Nature (or environment) is a hyper-computable system, mutations are randomly generated computable systems (that is, random algorithmic mutations) and the fitnesses are measured by how big is the output of the organisms. In this context, this paper focuses on two questions: Assuming that organisms are uncomputable systems, should metabiology become unsuitable or unsound? Or, on the other hand, assuming that Nature itself is a computable system, should metabiology become unsuitable or unsound? Thus, the main purpose is to demonstrate that metabiology walks towards a general theory of open-ended evolutionary systems, whether these systems are sub-computable, computable or hyper-computable. As a secondary purpose we show that scientific and philosophical consequences of our mathematical results spread across fundamental questions about biology and artificial life, as well as, about physics and artificial intelligence. To satisfy these two objectives, we build two new metabiological models for two new different configurations of organisms and nature. First, we demonstrate it is possible to build a metabiological evolution in which nature is a computable system and, hence, it makes the organisms and mutations to be sub-computable systems. To show this we introduce the recursive relative uncomputability (a computable/recursive version of the relative uncomputability that occurs among different Turing degrees). We do with Turing, Radó and Chaitin the same Skolem did with Cantor by making a relativization of the Busy-Beaver´s uncomputability in relation to the Omega number, so that this new uncomputability
can exist between a computer and any of his sub-computers. As a consequence, we introduce the “paradox” of uncomputability which says that exists at least one function that is not computable if we “look from the inside”, but that is computable if we “look from the outside”. Further, we correlate this pseudo-paradox with problems of human creativity and artificial intelligence and with problems of consciousness, artificial life and the computability of the Universe. Second, we build a metabiological model for the evolution of hyper-programs in which the organisms are hyper-computable systems of any finite order and nature is a hyper-program of ordinal order omega. In this last model we demonstrate it is possible an evolution of organisms/hyper-programs, with just “blind” random algorithmic mutations (it means, that do not take into consideration previous organisms/hyper-programs), which progressively leads the organisms/hyper-programs to be able of solving any problem in the arithmetical hierarchy. Allied to this, we discuss about computability or uncomputability of “living beings” and upon the possibility of mathematically studying metabiological evolutionary systems that can surpass, as much as we want, the “frontier” of the computable.

Podemos chamar o famoso “paradoxo” de Skolem de um pseudoparadoxo metalinguístico da denumerabilidade: de dentro do sistema um conjunto não é enumerável enquanto que de fora do sistema ele pode ser. O mesmo fenômeno aparece em relação à computabilidade, ou seja, existe um pseudoparadoxo metalinguístico da computabilidade? No qual “de dentro” algo é incomputável enquanto que “de fora” esse algo pode ser computável? É na metabiologia que esse fenômeno surge para ajudar a responder a pergunta sobre como ficaria a evolução dos organismos/programas sujeitos a mutações algorítmicas aleatórias numa natureza computável, visto que nos primeiros modelos demonstrados por Gregory Chaitin isso se dá numa natureza oracular, um hipercomputador.

Palavras-Chave: Metabiologia, Paradoxo de Skolem, Teoria da Computabilidade, Computabilidade da Vida, Hipótese do Universo Computável, Teoria da Informação Algorítmica.

In this paper I introduce Horwich's deflationary theory of truth, called 'Minimalism', and I present his proposal of how to cope with the Liar Paradox. The proposal proceeds by restricting the T-schema and, as a consequence of that, it needs a constructive specification of which instances of the T-schema are to be excluded from Minimalism. Horwich has presented, in an informal way, one construction that specifies the Minimalist theory. The main aim of the paper is to present and scrutinize some formal versions of Horwich's construction.

We show that any coherent complete partial order (ccpo) is obtainable as the fixed-point poset of the strong Kleene jump of a suitably chosen first-order ground model. This is a strengthening of Visser's result that any finite ccpo is obtainable in this way. The same is true for the van Fraassen supervaluation jump, but not for the weak Kleene jump.

Zahlen standen am Beginn des mathematischen Denkens und bilden bis heute einen syste-matischen Bezugspunkt im Aufbau und Verständnis der Mathematik. Für jede logische und philosophische Untersuchung der Mathematik spielt die Idee der Zahl eine zentrale Rolle. Im vorliegenden Buch wird anhand der natürlichen Zahlen, und später der reellen Zahlen, auf Fragen im Umfeld der Unendlichkeit der Zahlenreihe und der Eindeutigkeit des axiomatisch definierten Zahlbegriffs eingegangen, bis hin zur konkreten Einführung von Nichtstandardzah-len. An zentraler Stelle stehen Auseinandersetzungen um die Bedeutung und Tragweite der klassischen formalen Logik, der Mengenlehre und der modernen symbolischen (mathemati-schen) Logik als Darstellungsmittel mathematischer Zusammenhänge. Dies schliesst eine originelle philosophische Diskussion von Paradoxien und von verschiedenartigen Grenzen logischer und symbolischer Darstellbarkeit mit ein, die auch Paul Finslers Gedanken zur Grundlegung der Mathematik einbeziehen. Im Anhang findet sich ein Nachwort von Gregor Schneider, welches die vorliegenden Untersuchungen mit einigen modernen Tendenzen in der Philosophie der Mathematik konfrontiert, sowie ein bisher nicht auf Deutsch erschiener Aufsatz von Thoralf Skolem aus dem Jahre 1941 über die Grundlegung der Mengenlehre

La presente rubrica si propone di dare un'esposizione contestualizzata di un particolare problema filosofico, attraverso la rilettura di un articolo, saggio, o libro particolarmente influente e ricco di sviluppi per l'argomento in questione. Essa non vuole (né potrebbe) essere in alcun modo esaustiva degli argomenti trattati; ciò che qui intendiamo offrire è semplicemente uno strumento che consenta al lettore di approfondire una tematica filosofica eminente. A tale scopo includiamo una breve bibliografia ragionata al termine della presentazione.

This chapter provides a methodological case for maintaining classical logic even in the face of the semantic paradoxes. It advocates an abductive methodology for choosing, or adjudicating, logics (or, more specifically, logical theories), and notes that the semantic paradoxes constitute promising grounds for an abductive critique of classical logic. As the chapter notes, there is a strong prima facie abductive case for classical logic. This is not due to a principle of conservativism. It does not appeal to the benefits of familiarity with classical logic or the costs of changing the logic. It concerns, rather, intrinsic features of classical logic, such as its simplicity and strength. The chapter concludes that classical logic is doing fine by ordinary criteria and has no need for further justification, so we should keep it.

This short note is about how to fit Language into the constraints of Logic. Such a move should not be always possible, but if we want to say The Liar is a logical problem, then we have to be able to fit the piece of human language it deals with entirely inside of Logic. If we talk about logical systems, we need to find the right logical embedment of the problem and that will then determine if it belongs to Classical or Nonclassical Logic. We search for the right embedment using linguistic tools. The main finding there is that opposing and negating are very different things, despite holding some intersection: the truth may never be told but that does not mean one always lies. The flexibility of human language leads to several places and one of them is the impossibility of applying logical systems to it. Only in a very special universe, where Language is limited in several ways, could we think of applying logical systems to it in a successful manner. If Language is considered per se, no logical system can deal with its entire complexity. We can use (philosophical) Logic to deal with it, though, just not logical systems. The note then presents a proof that no logical system can deal with Language per se, so without artificial restrictions, and a solution to The Liar inside of Philosophical Logic.

The article contains a critical analysis of Wittgenstein’s theory of logical symbolism. According to an influential interpretation, Wittgenstein presented in the Tractatus a new method of solving paradoxes. This method seems a simple and effective alternative to Russell’s type theory. Wittgenstein’s theory of logical symbolism is based on the requirement of clear notation and the context principle: the type of a symbol only “shows” itself in the way we use the signs of our language. The function sign φ(φx) does not express any paradox, because the syntactic rules for its use, written in clear notation, should “show” us that φ(φx) = ψ(φx). Many researchers (Davant, Ishiguro, Mounce, Ruffino, Friedlander, Jolley, Livingston, Ladov, et al.) follow this interpretation. However, the difficulty of such a view on Wittgenstein’s theory of logical symbolism is that there hides the fallacy of petitio principii. Indeed, in examples of a functional sign of the form φ(φx), we are interested not only in the question of whether the functions φ are different symbols, but also in how this functional sign φ(φx) itself excludes the symbolization of the same object by different ways. This interpretation is contrasted with the idea that Wittgenstein’s theory of logical symbolism is in fact a modified analogue of Russell’s simple theory of types. The reciprocality principle becomes the core of Wittgenstein’s theory: the combinatorial potential of the “prototype” of a functional sign is identical to the combinatorial potential of the “prototype” of an argument. According to Wittgenstein, only describing the combinatorial potential of linguistic expressions (symbols) can vanish the illusion of paradoxes. The function cannot be its argument, because the function sign φ(φx) already contains the “prototype” of its argument, “showing” us that φ(φx) = φx. The correctness of this interpretation does not exclude the possibility that the differences between Russell and Wittgenstein are in fact nothing more than façon de parler.

The perhaps most important criticism of the nontransitive approach to semantic paradoxes is that it cannot truthfully express exactly which metarules preserve validity. I argue that this criticism overlooks that the admissibility of metarules cannot be expressed in any logic that allows us to formulate v-Curry sentences and that is formulated in a classical metalanguage. Hence, the criticism applies to all approaches that do their metatheory in classical logic. If we do the metatheory of nontransitive logics in a nontransitive logic, however, there is no reason to think that the argument behind the criticism goes through. In general, asking a logic to express its own admissible metarules may not be a good idea.

There are a variety of reasons why we would want a  paraconsistent account of logic, that is, an account of logic where an inconsistent theory does not have every sentence as a consequence. The one which will occupy our attention is metaphysical or semantic. One might, for various reasons, endorse that there are ‘true contradictions’, or as they are sometimes called, truth-value gluts – true sentences of the form p ^ not-p, claims which are both true and false. In this paper, we explore how various metaphysical commitments can motivate the adoption of different paraconsistent logics.

Tarski’s undefinability theorem of arithmetic truth says that any first-order language so rich that contains the arithmetic can not contain its own truth predicate satisfying the T-scheme. Its proof is based upon the idea that under the hypothesis of the T-scheme, the Liar paradox can be represented in the language and so the contradiction ensues, which implies, by reductio ad absurdum, the non-existence of the truth predicate satisfying the T-scheme. By a scrutiny of three truth theories followed from Tarski’s theorem, this paper advances a new scheme for truth—the relativized T-scheme, i.e., the relativization of T-scheme on a frame (that is, a digraph in graph theory). The thesis of this paper is: the paradoxes can be nontrivially used in the proof of Tarski’s undefinability theorem under the relativized T-scheme, just as they had been employed in the proof of Go ̈del’s first incompleteness theorem.

Specifically, whenever a truth-theoretic paradox is used in the proof of Tarski’s theorem, whether it deduces a contradiction in a frame is related directly to the property of this frame— usually the circularity of this frame. This shows that each paradox is contradictory in and only in some certain frames. This discovery defines the relative contradictoriness of the paradox. The main enterprise of this paper is to establish a series of generalizations of Tarski’s theorem by the use of paradoxes, and so to characterize the frames fixing the relative contradictoriness of a paradox, compare the strength of the relative contradictoriness between different paradoxes, give an analysis of how the paradox generally depends on the self-reference and the circularity, and finally, in the spirit of the relativization, offer an intuitionistic characterization of the truth for classical logic.
The introduction of this paper presents some typical truth-theoretic paradoxes, puts for- ward a basic problem to be solved about Tarski’s theorem, and introduces some basic theoretic tools for problem setup. In chapter 2, I first outline the truth theories of Tarski, Kripke and Herzberger-Gupta, explain how Tarski’s T-scheme had been revised in the three theories, and then abstract from these revisions into the relativized T-scheme as a new starting point of the truth theory.

In chapter 3, I give a series of generalizations of Tarski’s theorem under the relativized T-scheme􏳊characterizing and comparing the relative contradictoriness of the typical truth-
theoretic paradoxes. It is proved that for any positive integer n = 2^i(2j +1), the n-card paradox must imply the contradiction in a frame when it is used in the proof of Tarski’s theorem, iff. this frame contains a cycle such that its height is not divisible by 2^(i+1). And for any positive integers n and m, the n-card paradox is no more contradictory than the m-card paradox (it means that the former can not be contradictory in a frame unless the latter is so in this frame, when they are used in the proof of Tarski’s theorem), iff. (n)_2 ≤ (m)_2, where (n)_2 denotes the multiplicity of 2 in the (unique) prime factorization of n. In particularly, when the Liar paradox is used in the proof of Tarski’s theorem, it deduces contradiction in and only in the frames containing odd cycles; the Liar paradox is strictly less contradictory than Jourdain’s card paradox. It is also proved that the Yablo’s paradox is precisely the same contradictory as the Liar paradox.

In chapter 4, a propositional language is devised in virtue of the cross-reference shared generally by the truth-theoretic paradoxical sentences. There is no truth predicate in this lan- guage and the relativized T-scheme is indirectly embodied in the interpretation of a special bi- nary operator. I here show that the previous treatment of paradoxes can be reformulated in this language but pay more attention to how the paradoxes depend upon the self-reference and the circularity. The main result of this chapter is that any (natural) paradox is non-self-referential and independent of the proper circularity unless it contains at least infinite sentences. In par- ticular, for any integer n ≥ 1, n-row Yabloesque paradox is non-self-referential; and for any transfinite ordinal α, α-ary Herzberger’s paradox is independent of the proper circularity.


In conclusion, the undertaking of approaching the truth-theoretic paradoxes provides suffi- cient evidence to demonstrate the elegance and power of the relativized T-scheme in the analysis of these paradoxes. The relativized T-scheme, as a new starting point of the truth theory, not only is a form of the intuitive justification and ontological source, but also is a principle of mathematical beauty and logical strength. It can be expected that this scheme inaugurates a new way for the future development of the truth theory.

Key Words: Circularity, Relativized T-scheme, Self-reference, Tarski’s theorem, Truth- theoretic Paradox

The Unexpected Hanging Problem is also known as the Surprise Examination Problem. We here
solve it by isolating what is logical reasoning from the rest of the human psyche. In a not-soorthodox analysis, following our tradition (The Liar, Dichotomy, The Sorites and Russell’s Paradox),
we talk about the problem from a perspective that is more distant than all the known perspectives.
From an observational point that is in much farther than all the observational points used until now,
the reader can finally see why the problem has been perpetuated as a problem and can also see
that the problem was never an actual problem: Once more, we have an allurement. The allurement
this time makes us start paying attention to all the complexity of the human psyche when studying
problems that involve human feelings. The main finding could be told to be that we have to
understand and study more the human psyche, in all its intricacies, also when dealing with
problems that seem to belong with exclusivity to Mathematics or Logic