It is common today to see the world as increasingly unpredictable, and to see that unpredictability as a major source of anxiety. Many of the proposed cures for that anxiety, such as systems like Memex and MyLifeBits, have sought solutions in systems that collect and store a thorough record of events, at a scale from the personal to the global. There the solution to anxiety lies in the ability to play back the record, to turn back the clock and be (...) there then. Both this anxiety and its solution are best seen not simply as remedies for an immediate problem-of terrorism, for example-but rather as evidence of a more deep-seated set of cultural changes, which emerged early in the twentieth century. Paradoxically, the technological solutions offered, whatever the scale, embody the very thing, a lack of a connection to a community, that is both the source of the anxiety and a fundamental impediment to its elimination. (shrink)

The release of Kanye West’s Yeezus was indelibly marked by the provocation of his hit song entitled “New Slaves,” which introduced a pessimistic terminology to capture the paradoxical condition whereby Black freedom from enslavement only resulted in the capturing of Black people psychically in the neo-liberal entanglements of poverty, servitude, and corporatism. His analysis, not unlike currently en vogue theories of Afro-pessimism or Critical Race Theory’s realist lens, maintains that despite all the rhetoric and symbols of progress to the contrary, (...) Black people are simply not free in America. West’s performance of “New Slaves” on Saturday Night Live was only amplified by the “Not For Sale” insignia .. (shrink)

There are theoretical limitations to what can be implemented by a computer program. In this paper we are concerned with a limitation on the strength of computer implemented deduction. We use a version of the Curry paradox to arrive at this limitation.

In this paper, we distinguish two versions of Curry's paradox: c-Curry, the standard conditional-Curry paradox, and v-Curry, a validity-involving version of Curry's paradox that isn’t automatically solved by solving c-curry. A unified treatment of curry paradox thus calls for a unified treatment of both c-Curry and v-Curry. If, as is often thought, c-Curry paradox is to be solved via non-classical logic, then v-Curry may require (...) a lesson about the structure—indeed, the substructure—of the validity relation itself. (shrink)

As it is well known, in the forties of the past century, Curry proved that in any logic S closed under Modus Ponens, uniform substitution of propositional variables and the Contraction Law, the naïve Comprehension axiom trivializes S in the sense that all propositions are derivable in S plus CA. Not less known is the fact that, ever since Curry published his proof, theses and rules weaker than W have been shown to cause the same effect as W (...) causes. Among these, the Contraction rule or the Modus Ponens axiom, for example, are to be noted. But, moreover, as Brady has proved, even the Generalized Modus Ponens axiom or the Generalized Contraction rule give rise to “Curry’s Paradox” under the same circumstances as W does. In some previous work by us, “weak relevant model structures” are defined on “weak relevant matrices” by generalizing Brady’s model structure MCL built upon Meyer’s Crystal matrix CL. We have proved that wr-ms only verify logics with the “depth relevance condition”. The aim of this paper is to show how to falsify gMPa and gRW in certain wr-ms. In particular, it will be shown that gMPa is falisfied in any wr-ms and gRW in any wr-ms verifying Routley and Meyer’s basic positive logic B+. (shrink)

In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this paper I show that a number of logics are susceptible to a strengthened version of Curry's paradox. This can be adapted to provide a proof theoretic analysis of the omega-inconsistency in Lukasiewicz's continuum valued logic, allowing us to better evaluate which logics are suitable for a naïve truth theory. On this basis I identify two natural subsystems of Lukasiewicz logic which (...) individually, but not jointly, lack the problematic feature. (shrink)

“Weak relevant model structures” (wr-ms) are defined on “weak relevant matrices” by generalizing Brady’s model structure ${\mathcal{M}_{\rm CL}}$ built upon Meyer’s Crystal matrix CL. It is shown how to falsify in any wr-ms the Generalized Modus Ponens axiom and similar schemes used to derive Curry’s Paradox. In the last section of the paper we discuss how to extend this method of falsification to more general schemes that could also be used in deriving Curry’s Paradox.

Curry's paradox, so named for its discoverer, namely Haskell B. Curry, is a paradox within the family of so-called paradoxes of self-reference (or paradoxes of circularity). Like the liar paradox (e.g., ‘this sentence is false’) and Russell's paradox , Curry's paradox challenges familiar naive theories, including naive truth theory (unrestricted T-schema) and naive set theory (unrestricted axiom of abstraction), respectively. If one accepts naive truth theory (or naive set theory), then Curry's (...) paradox becomes a direct challenge to one's theory of logical implication or entailment. Unlike the liar and Russell paradoxes Curry's paradox is negation-free; it may be generated irrespective of one's theory of negation. An intuitive version of the paradox runs as follows. (shrink)

“Curry’s paradox”, as the term is used by philosophers today, refers to a wide variety of paradoxes of self-reference or circularity that trace their modern ancestry to Curry (1942b) and Löb (1955). The common characteristic of these so-called Curry paradoxes is the way they exploit a notion of implication, entailment or consequence, either in the form of a connective or in the form of a predicate. Curry’s paradox arises in a number of different domains. (...) Like Russell’s paradox, it can take the form of a paradox of set theory or the theory of properties. But it can also take the form of a semantic paradox, closely akin to the Liar paradox. Curry’s paradox differs from both Russell’s paradox and the Liar paradox in that it doesn’t essentially involve the notion of negation. Common truth-theoretic versions involve a sentence that says of itself that if it is true then an arbitrarily chosen claim is true, or—to use a more sinister instance—says of itself that if it is true then every falsity is true. The paradox is that the existence of such a sentence appears to imply the truth of the arbitrarily chosen claim, or—in the more sinister instance—of every falsity. In this entry, we show how the various Curry paradoxes can be constructed, examine the space of available solutions, and explain some ways Curry’s paradox is significant and poses distinctive challenges. (shrink)

The paradoxes of self-reference are genuinely paradoxical. The liar paradox, Russell’s paradox and their cousins pose enormous difficulties to anyone who seeks to give a comprehensive theory of semantics, or of sets, or of any other domain which allows a modicum of self-reference and a modest number of logical principles. One approach to the paradoxes of self-reference takes these paradoxes as motivating a non-classical theory of logical consequence. Similar logical principles are used in each of the paradoxical inferences. (...) If one or other of these problematic inferences are rejected, we may arrive at a consistent (or at least, a coherent) theory. In this paper I will show that such approaches come at a serious cost. The general approach of using the paradoxes to restrict the class of allowable inferences places severe constraints on the domain of possible propositional logics, and on the kind of metatheory that is appropriate in the study of logic itself. Proof-theoretic and model-theoretic analyses of logical consequence make provide different ways for non-classical responses to the paradoxes to be defeated by revenge problems: the redefinition of logical connectives thought to be ruled out on logical grounds. Non-classical solutions are not the “easy way out” of the paradoxes. (shrink)

We propose contractionless constructive logic which is obtained from Nelson's constructive logic by deleting contractions. We discuss the consistency of a naive set theory based on the proposed logic in relation to Curry's paradox. The philosophical significance of contractionless constructive logic is also argued in comparison with Fitch's and Prawitz's systems.

Curry's paradox, sometimes described as a general version of the better known Russell's paradox, has intrigued logicians for some time. This paper examines the paradox in a natural deduction setting and critically examines some proposed restrictions to the logic by Fitch and Prawitz. We then offer a tentative counterexample to a conjecture by Tennant proposing a criterion for what is to count as a genuine paradox.

In approaching Ch. 4 of Saving Truth from Paradox, it might be helpful first to revisit Curry’s original paper, and to revisit Lukasiewicz too, to provide more of the scenesetting that Field doesn’t himself fill in. So in §1 I’ll say something about Curry, in §2 we’ll look at what Lukasiewicz was up to in his original three-valued logic, and in §3 we’ll look at the move from a three-valued to a many-valued Lukasiewicz logic. In §4, I (...) move on to announce a theorem by H´. (shrink)

Curry’s paradox is well known. The original version employed a conditional connective, and is not forthcoming if the conditional does not satisfy contraction. A newer version uses a validity predicate, instead of a conditional, and is not forthcoming if validity does not satisfy structural contraction. But there is a variation of the paradox which uses “external validity”. And since external validity contracts, one might expect the appropriate version of the Curry paradox to be inescapable. In (...) this paper we show that this is not the case. We consider two ways of formalising the notion of external validity, and show that in both of these the paradox is not forthcoming without the appropriate forms of contraction. (shrink)

Although formal analysis provides us with interesting tools for treating Curry’s paradox, it certainly does not exhaust every possible reading of it. Thus, we suggest that this paradox should be analysed with non-formal tools coming from pragmatics. In this way, using Grice’s logic of conversation, we will see that Curry’s sentence can be reinterpreted as a peculiar conditional sentence implying its own consequent.

Curry's paradox for "if.. then.." concerns the paradoxical features of sentences of the form "If this very sentence is true, then 2+2=5". Standard inference principles lead us to the conclusion that such conditionals have true consequents: so, for example, 2+2=5 after all. There has been a lot of technical work done on formal options for blocking Curry paradoxes while only compromising a little on the various central principles of logic and meaning that are under threat. -/- Once (...) we have a sense of the technical options, though, a philosophical choice remains. When dealing with puzzles in the logic of conditionals, a natural place to turn is independently motivated semantic theories of the behaviour of "if... then...". This paper argues that the closest-worlds approach outlined in Nolan 1997 offers a philosophically satisfying reason to deny conditional proof and so block the paradoxical Curry reasoning, and can give the verdict that standard Curry conditionals are false, along with related "contraction conditionals". (shrink)

Several authors have argued that a version of Curry's paradox involving validity motivates rejecting the structural rule of contraction. This paper criticizes two recently suggested alternative responses to “validity Curry.” There are three salient stages in a validity Curry derivation. Rejecting contraction blocks the first, while the alternative responses focus on the second and third. I show that a distinguishing feature of validity Curry, as contrasted with more familiar forms of Curry's paradox, is (...) that paradox arises already at the first stage. Accordingly, blocking the second or third stages won't suffice for resolving the paradox. (shrink)

Beall and Cotnoir (2017) argue that theists may accept the claim that God's omnipotence is fully unrestricted if they also adopt a suitable nonclassical logic. Their primary focus is on the infamous Stone problem (i.e., whether God can create a stone too heavy for God to lift). We show how unrestricted omnipotence generates Curry‐like paradoxes. The upshot is that Beall and Cotnoir only provide a solution to one version of the Stone problem, but that unrestricted omnipotence generates other problems (...) which they do not adequately address. (shrink)

A Curry paradox about truth is generated by the following sentence, written on the board in room 101:If the sentence on the board in room 101 is true then 1 ≠ 1.A Curry paradox about validity is generated by the following argument, written on the board in room 102:The argument on the board in room 102 is valid. Therefore, 1 ≠ 1.Though the sentence and the argument generate Curry paradoxes, they also generate more basic paradoxes, (...) in a sense to be made clear. I argue that if we solve these more basic paradoxes, we have solutions to both kinds of Curry paradox. The positive proposal is in part inspired by a brief remark of Gödel’s, that the paradoxes might appear “as something analogous to dividing by zero”—so that the concepts of truth and validity, for example, are everywhere applicable except for certain singular points or singularities. A second central claim is that 'true' and 'valid' are context-sensitive predicates. This contextual-singularity approach to the Curry paradoxes applies also to other paradoxes of truth, validity, denotation, and predicate-extension. So a more general aim of the paper is to provide a unified response to semantic paradox. (shrink)

Any theory of truth must find a way around Curry’s paradox, and there are well-known ways to do so. This paper concerns an apparently analogous paradox, about validity rather than truth, which JC Beall and Julien Murzi call the v-Curry. They argue that there are reasons to want a common solution to it and the standard Curry paradox, and that this rules out the solutions to the latter offered by most “naive truth theorists.” To (...) this end they recommend a radical solution to both paradoxes, involving a substructural logic, in particular, one without structural contraction. In this paper I argue that substructuralism is unnecessary. Diagnosing the “v-Curry” is complicated because of a multiplicity of readings of the principles it relies on. But these principles are not analogous to the principles of naive truth, and taken together, there is no reading of them that should have much appeal to anyone who has absorbed the morals of both the ordinary Curry paradox and the second incompleteness theorem. (shrink)

In ‘Currying omnipotence: A reply to Beall and Cotnoir’, Andrew Tedder and Guillermo Badia argue that Jc Beall and A. J. Cotnoir’s gappy solution to the traditional paradox of unrestricted omnipotence does not extend to a Curry-like version of the paradox. In this paper, we show that it does extend to it.

In “God of the Gaps: A Neglected Reply to God’s Stone Problem”, Jc Beall and A. J. Cotnoir offer a gappy solution to the paradox of (unrestricted) omnipotence that is typified by the classic stone problem. Andrew Tedder and Guillermo Badia, however, have recently argued that this solution could not be extended to a more serious Curry-like version of the paradox. In this paper, we show that such a gappy solution does extend to it.

Gregory Currie is one of the world’s preeminent philosophers of art and a highly-respected philosopher of mind. Imagining and Knowing: the Shape of Fiction is his seventh book, with his conspicuous contributions to the analytic tradition of philosophy including the first systematic philosophical aesthetics in no less than two fields, film (Image and Mind: Film, Philosophy and Cognitive Science, 1995) and narrative (Narratives and Narrators: A Philosophy of Stories, 2010). Currie’s trademark approach is the seamless integration of art criticism and (...) aesthetic theory on the one hand with empirical psychology and evolutionary biology on the other and Imagining and Knowing follows suit. His stated aim is twofold: to make a convincing case for the significance of the imagination to the understanding of and engagement with fiction and to make an equally convincing case against the significance of fiction as a source of knowledge. The monograph consists of an introduction, eleven chapters divided into three parts, and a coda. Part I is concerned with making a case for imagining fiction, defending fiction as an utterance over fiction as a genre (Chapter 1) and the role of the imagination in the engagement with fiction (Chapter 2). This role is fleshed out by means of Currie’s conceptions of ‘desires in imagination, or i-desires’ (Chapter 3) and i-emotions (Chapter 4), which also provide neat solutions to both the paradox of fiction and the imaginative resistance debates (56). Part II sets the scene for the case against fictional knowledge, establishing an epistemic taxonomy (Chapter 5), arguing that claims about the cognitive value of fiction should cohere with the findings of experimental psychology (Chapter 6), and debunking evolutionary claims for the significance of fiction (Chapter 7). Part III presents Currie’s deflationary account, tackling each of what he considers to be the four most convincing arguments for learning from fiction in turn: fictional thought experiments (Chapter 8), truth in literature (Chapter 9), the psychology of authors (Chapter 10), and the relationship between fiction and empathy (Chapter 11). (shrink)

This paper presents a new puzzle for certain positions in the theory of truth. The relevant positions can be stated in a language including a truth predicate T and an operation that takes sentences to names of those sentences; they are positions that take the T-schema A ↔ T to hold without restriction, for every sentence A in the language. As such, they must be based on a nonclassical logic, since paradoxes that cannot be handled classically will arise. The bestknown (...) of these paradoxes is probably the liar paradox – a sentence that says of itself that it is not true – but our concern here is not with the liar. Instead, our focus is a variant of Curry ’s paradox – a sentence that says of itself that if it is true, everything is [3, 5, 7, 11]. In §1, we present the standard version of Curry ’s paradox and the strain of response to it we wish to focus on. This strain of response crucially invokes non-normal worlds: worlds at which the laws of logic differ from the laws that actually hold. In §2, we go on to argue that, in light of temporal curry paradox, this strain of response ought also to accept non-normal times: times at which the laws of logic in the actual world differ from the laws that hold now. We then consider, in §3, what this would mean for the theorists in question. (shrink)

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 validity-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. (shrink)

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

We extend the Embracing Revenge account of the semantic paradoxes by constructing two distinct consequence relations that reflect, in different ways, the transfinitely-many-valued semantics developed in earlier work. In particular, we adapt the underlying ideas of “gappy” approaches based on K3, and “glutty” approaches based on LP, to the Embracing Revenge framework, by treating the infinitely many non-classical truth values as infinitely many ways that a sentence might fail to receive a classical truth value in the former case, and as (...) infinitely many ways that a sentence might receive both classical truth values in the latter. In order to obtain deductive systems that are very nearly classical, we need to utilize different conditionals (and different negations) in each logic. (shrink)

The Liar paradox is the directly self-referential Liar statement: This statement is false.or : " Λ: ∼ T 1" The argument that proceeds from the Liar statement and the relevant instance of the T-schema: " T ↔ Λ" to a contradiction is familiar. In recent years, a number of variations on the Liar paradox have arisen in the literature on semantic paradox. The two that will concern us here are the Curry paradox, 2 and the (...) Yablo paradox. 3The Curry paradox demonstrates that neither negation nor a falsity predicate is required in order to generate semantic paradoxes. Given any statement Φ whatsoever, we need merely consider the statement: If this statement is true, then Φor: " Ξ: T → Φ" Here, via familiar reasoning, one can ‘prove’ Φ merely through consideration of statement Ξ and the Ξ-instance of the T-schema.Interestingly, the Liar paradox can be viewed as nothing more than a special case of the Curry paradox. If we define negation in terms of the conditional and a primitive absurdity constant ‘⊥’: 4" ∼ Ψ = df Ψ → ⊥" then the Liar paradox is simply the instance of the Curry paradox obtained by substituting ‘⊥’ for Φ.The Yablo paradox demonstrates that circularity is also not required in order to generate semantic paradox. 5 The paradox proceeds by considering an infinite ω-sequence of statements of the form: S 1: ) S 2: ) S 3: ) : : : : : S i: ) : : : : : :– that is, the set of …. (shrink)

This article shows that there is a liar-like paradox that arises for rational credence that relies only on very weak logical and credal principles. The paradox depends on a weak rational reflection principle, logical principles governing conjunction, and principles governing the relationship between rational credence and proof. To respond to this paradox, we must either reject even very weak rational reflection principles or reject some highly plausible logical or credal principle.

We group the existing variants of the familiar set-theoretical and truth-theoretical paradoxes into two classes: connective paradoxes, which can in principle be ascribed to the presence of a contracting connective of some sort, and structural paradoxes, where at most the faulty use of a structural inference rule can possibly be blamed. We impute the former to an equivocation over the meaning of logical constants, and the latter to an equivocation over the notion of consequence. Both equivocation sources are tightly related, (...) and can be cleared up by adopting a particular substructural logic in place of classical logic. We then argue that our perspective can be justified via an informational semantics of contraction-free substructural logics. (shrink)

Consider the following argument written on the board in room 227: 1 = 1. So, the argument on the board in room 227 is not valid. This argument generates a paradox. The aim of this paper is to present a resolution of this paradox and related paradoxes of validity, including a version of the Curry paradox. The proposal stresses the close connections between these validity paradoxes and paradoxes of truth and paradoxes of denotation. So a more (...) general aim is to provide a unified response to semantic paradox. The positive proposal is in part inspired by a brief, tantalizing suggestion of Gödel’s, that the paradoxes might appear “as something analogous to dividing by zero”—so that the concept of validity, for example, is everywhere applicable except for certain singular points or singularities. A second central claim is that ‘valid’ is a context-sensitive predicate. The key notions of this contextual-singularity theory are presented and applied to a variety of cases. Any purported solution to paradox must deal with the phenomenon of revenge, and a response to revenge is outlined. The paper closes with remarks about the accommodation of Tarski’s claim that natural languages are universal. (shrink)

In this paper a principle of substitutivity of logical equivalents salve veritate and a version of Leibniz’s law are formulated and each is shown to cause problems when combined with naive truth theories.

After an introduction to set the stage, we consider some variations on the reasoning behind Curry's Paradox arising against the background of classical propositional logic and of BCI logic and one of its extensions, in the latter case treating the "paradoxicality" as a matter of nonconservative extension rather than outright inconsistency. A question about the relation of this extension and a differently described (though possibly identical) logic intermediate between BCI and BCK is raised in a final section, which (...) closes with a handful of questions left unanswered by our discussion. (shrink)

Curry claims that the positive paradox principle, ` A ⊃ in his elementary statement presentation, ‘is valid in any normal interpretation’ . By previous definition, ‘an interpretation of a system S is a normal interpretation just when the proposition A is true when and only when ` A’ . But his argument to normal validity is interestingly, and relevantly, invalid.

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)

The paper offers a solution to the semantic paradoxes, one in which (1) we keep the unrestricted truth schema “True(A)↔A”, and (2) the object language can include its own metalanguage. Because of the first feature, classical logic must be restricted, but full classical reasoning applies in “ordinary” contexts, including standard set theory. The more general logic that replaces classical logic includes a principle of substitutivity of equivalents, which with the truth schema leads to the general intersubstitutivity of True(A) with A (...) within the language.The logic is also shown to have the resources required to represent the way in which sentences (like the Liar sentence and the Curry sentence) that lead to paradox in classical logic are “defective”. We can in fact define a hierarchy of “defectiveness” predicates within the language. Contrary to claims that any solution to the paradoxes just breeds further paradoxes (“revenge problems”) involving defectiveness predicates, there is a general consistency/conservativeness proof that shows that talk of truth and the various “levels of defectiveness” can all be made coherent together within a single object language. (shrink)

In this paper we present two new approaches for dealing with semantic paradoxes and soritical predicates based on fuzzy logic. We show that both of them have conceptual advantages over the more traditional Łukasiewicz approach, and that the second one even avoids standard proofs of ω-nconsistency.

In this note, we review paradoxes like Russell’s, the Liar, and Curry’s in the context of intuitionistic logic. One may observe that one cannot blame the underlying logic for the paradoxes, but has to take into account the particular concept formations. For proof-theoretic semantics, however, this comes with the challenge to block some forms of direct axiomatizations of the Liar. A proper answer to this challenge might be given by Schroeder-Heister’s definitional freedom.

There is significant interest in type-free systems that allow flexible self-application. Such systems are of interest in property theory, natural language semantics, the theory of truth, theoretical computer science, the theory of classes, and category theory. While there are a variety of proposed type-free systems, there is a particularly natural type-free system that we believe is prototypical: the logic of recursive algorithms. Algorithmic logic is the study of basic statements concerning algorithms and the algorithmic rules of inference between such statements. (...) As shown in [1], the threat of paradoxes, such as the Curry paradox, requires care in implementing rules of inference in this context. As in any type-free logic, some traditional rules will fail. The first part of the paper develops a rich collection of inference rules that do not lead to paradox. The second part identifies traditional rules of logic that are paradoxical in algorithmic logic, and so should be viewed with suspicion in type-free logic generally. (shrink)

Rejecting structural contraction has been proposed as a strategy for escaping semantic paradoxes. The challenge for its advocates has been to make intuitive sense of how contraction might fail. I offer a way of doing so, based on a “naive” interpretation of the relation between structure and logical vocabulary in a sequent proof system. The naive interpretation of structure motivates the most common way of blaming Curry-style paradoxes on illicit contraction. By contrast, the naive interpretation will not as easily (...) motivate one recent noncontractive approach to the Liar paradox. (shrink)