Essence and causation are fundamental in metaphysics, but little is said about their relations. Some essential properties are of course causal, as it is essential to footprints to have been caused by feet. But I am interested less in causation's role in essence than the reverse: the bearing a thing's essence has on its causal powers. That essencemight make a causal contribution is hinted already by the counterfactual element in causation; and the hint is confirmed by the explanation essence offers (...) of something otherwise mysterious, namely, how events exactly alike in every ordinary respect, like the bolt'ssuddenly snapping and its snapping per se, manage to disagree in what they cause. Some prior difference must exist between these events to make their causal powers unlike. Paradoxically, though, it can only be in point of a property, suddenness, which both events possess in common. Only by postulating a difference in themanner — essential or accidental — of the property's possession is the paradox resolved. Next we need an account of causation in which essence plays an explicit determinative role. That account, based on the idea that causes should becommensurate with their effects, is thatx causesy only if nothing essentially poorer would have done, and nothing essentially richer was needed. (shrink)

This is a defense and extension of Stephen Yablo's claim that self-reference is completely inessential to the liar paradox. An infinite sequence of sentences of the form 'None of these subsequent sentences are true' generates the same instability in assigning truth values. I argue Yablo's technique of substituting infinity for self-reference applies to all so-called 'self-referential' paradoxes. A representative sample is provided which includes counterparts of the preface paradox, Pseudo-Scotus's validity paradox, the Knower, and other (...) enigmas of the genre. I rebut objections that Yablo's paradox is not a genuine liar by constructing a sequence of liars that blend into Yablo's paradox. I rebut objections that Yablo's liar has hidden self-reference with a distinction between attributive and referential self-reference and appeals to Gregory Chaitin's algorithmic information theory. The paper concludes with comments on the mystique of self-reference. (shrink)

It is argued that Yablo’s Paradox is not strictly paradoxical, but rather ‘ω-paradoxical’. Under a natural formalization, the list of Yablo sentences may be constructed using a diagonalization argument and can be shown to be ω-inconsistent, but nonetheless consistent. The derivation of an inconsistency requires a uniform fixed-point construction. Moreover, the truth-theoretic disquotational principle required is also uniform, rather than the local disquotational T-scheme. The theory with the local disquotation T-scheme applied to individual sentences from the Yablo list is (...) also consistent. (shrink)

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

Yablo’s paradox is generated by the following (infinite) list of sentences (called the Yablo list): (s1) For all k > 1, sk is not true. (s2) For all k > 2, sk is not true. (s3) For all k > 3, sk is not true. . . . . . . . .

The blame for the semantic and set-theoretic paradoxes is often placed on self-reference and circularity. Some years ago, Yablo [1985; 1993] challenged this diagnosis, by producing a paradox that's liar-like but does not seem to involve circularity. But is Yablo's paradox really non-circular? In a recent paper, Beall [2001] has suggested that there are no means available to refer to Yablo's paradox without invoking descriptions, and since Priest [1997] has shown that any such description is (...) circular, Beall concludes that Yablo's paradox itself is circular. In this paper, we argue that Beall's conclusion is unwarranted, given that (1) descriptions are not the only way to refer to Yablo's paradox, and (ii) we have no reason to believe that because the description involves self-reference, the denotation of the description is also circular. As a result, for all that's been said so far, we have no reason to believe that Yablo's paradox is circular. (shrink)

The structure of Yablo’s paradox is analysed and generalised in order to show that beginningless step-by-step determination processes can be used to provoke antinomies, more concretely, to make our logical and our on-tological intuitions clash. The flow of time and the flow of causality are usually conceived of as intimately intertwined, so that temporal causation is the very paradigm of a step-by-step determination process. As a conse-quence, the paradoxical nature of beginningless step-by-step determina-tion processes concerns time and causality as (...) usually conceived. (shrink)

Stephen Yablo [23,24] introduces a new informal paradox, constituted by an infinite list of semi-formalized sentences. It has been shown that, formalized in a first-order language, Yablo’s piece of reasoning is invalid, for it is impossible to derive falsum from the sequence, due mainly to the Compactness Theorem. This result casts doubts on the paradoxical character of the list of sentences. After identifying two usual senses in which an expression or set of expressions is said to be paradoxical, since (...) second-order languages are not compact, I study the paradoxicality of Yablo’s list within these languages. While non-paradoxical in the first sense, the second-order version of the list is a paradox in our second sense. I conclude that this suffices for regarding Yablo’s original list as paradoxical and his informal argument as valid. (shrink)

Thought: A Journal of Philosophy, EarlyView.

In 1993, the American logic S. Yablo was proposed an original infinitive formulation of the classical ≪Liar≫ paradox. It questioned the traditional notion of self-reference as the basic structure for semantic paradoxes. The article considers the arguments underlying two different approaches to analysis of proposals of the ≪Infinite Liar≫ and understanding of the genuine sources for semantic paradoxes. The first approach (V. Valpola, G.-H. von Wright, T. Bolander, etc.) imposes responsibility for the emergence of semantic paradoxes on the negation (...) of the truth predicate. It deprives the ≪Infinite Liar≫ sentences of consistent truth values. The second approach is based on a modified version of anaphoric prosententialism (D. Grover, R. Brandom, etc.). The concepts of truth and falsehood are treated as special anaphoric operators. Logical constructs similar to the ≪Infinite Liar≫ do not attribute any definite truth values to sentences from which they are composed, but only state certain types of relations between the semantic content of such sentences. (shrink)

It is proved that Yablo’s paradox and the Liar paradox are equiparadoxical, in the sense that their paradoxicality is based upon exactly the same circularity condition—for any frame ${\mathcal{K}}$ , the following are equivalent: (1) Yablo’s sequence leads to a paradox in ${\mathcal{K}}$ ; (2) the Liar sentence leads to a paradox in ${\mathcal{K}}$ ; (3) ${\mathcal{K}}$ contains odd cycles. This result does not conflict with Yablo’s claim that his sequence is non-self-referential. Rather, it gives Yablo’s (...) paradox a new significance: his construction contributes a method by which we can eliminate the self-reference of a paradox without changing its circularity condition. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:January 22, 2009 (8:41 pm) G:\WPData\TYPE2802\russell 28,2 048red.wpd russell: the Journal of Bertrand Russell Studies n.s. 28 (winter 2008–09): 127–42 The Bertrand Russell Research Centre, McMaster U. issn 0036-01631; online 1913-8032 YABLO’S PARADOX AND RUSSELLIAN PROPOSITIONS Gregory Landini Philosophy / U. of Iowa Iowa City, ia 52242–1408, usa [email protected] Is self-reference necessary for the production of Liar paradoxes? Yablo has given an argument that self-reference is not necessary. (...) He hopes to show that the indexical apparatus of self-reference of the traditional Liar paradox can be avoided by appealing to a list, a consecutive sequence, of sentences correlated one-one withnaturalnumbers.Yabloopens his “Paradox without Self-Reference” (Analysis, 1993) with the assumption that there is a sequence such that: Snz: “(;k)(k > n.!. True Sk)” Each sentence on Yablo’s list is supposed to be correlated one-one with number n. Each sentence is supposed to say that for every natural number kz greater than n, the k-th sentence on the list is not true. By comparing Yablo’s construction to an analogous construction with early Russellian propositions, we show that Yablo has failed to generate a paradox. 1. vicious circularity in yablo’s paradox S elf-reference has been taken to be the source of paradoxes. Poincaré and Russell are commonly associated with this thesis. In an amusing passage, the mathematician Jourdain recalls something of the dialogue between them: Nearly all mathematicians agreed that the way to solve these paradoxes was simply not to mention them; but there was some divergence of opinion as to how they were to be unmentioned. It was clearly unsatisfactory merely not to mention them. Thus Poincaré was apparently of the opinion that the best way of avoiding such awkward subjects was to mention that they were not to be mentioned. But [as Russell put it] “one might as well, in talking to a man with January 22, 2009 (8:41 pm) G:\WPData\TYPE2802\russell 28,2 048red.wpd 128 gregory landini 1 Philip Jourdain, The Philosophy of Mr. B*rtr*nd R*ss*llz (London: Allen & Unwin, 1918), p. 77. Russell is quoted from “Mathematical Logic as Based on the Theory of Types” (1908), LK, p. 63. 2 “On ‘Insolubilia’ and Their Solution By Symbolic Logic”, in B. Russell, Essays in Analysis, ed. Douglas Lackey (London: Allen & Unwin, 1973), p. 196. 3 I believe that this is Russell’s view in 1906. See Gregory Landini, “Russell’s Separation of the Logical and Semantic Paradoxes”, Revue internationale de philosophie 58 (2004): 257–94 (issue titled “Russell en héritage / Le centenaire des Principes”, ed. Philippe de Rouilhan). a long nose, say: ‘When I speak of noses, I except such as are inordinately long’, which would not be a very successful eTort to avoid a painful topic.”1 Poincaré maintained that one should exclude the oTending cases and therebyavoid“viciouscircles”ofself-referencewhichgenerateparadoxes. Russell objected: We may illustrate this by what M. Poincaré says concerning Richard’s paradox. Having Wrst put E = “all numbers deWnable in a Wnite number of words” we arrive at a paradox, due, says M. Poincaré, to our having included a number only deWnable in a Wnite number of words by means of E. This vicious circle he proposes to avoid by deWning E as “all numbers deWnable in a Wnite number of words without mentioning Ey”. To the uninitiated, this deWnition looks more circular than ever.2 Russell held that some paradoxes such as those of classes require, in order to exclude oTending cases without mentioning them, a “reconstruction of logical Wrst principles”. Others, such as Richard’s, are to be dismissed because they involve confused and viciously circular notions of “deWnability ”.3 Is self-reference necessary for paradoxes? The question is not well crafted. There are quite diTerent notions ofz “self-reference” involved in paradoxes. The self-reference involved in the Liar “This sentence is false” is provided by an apparatus of indexicals. This apparatus is quite distinct from the self-reference involved in Russell’s early ontology of propositions (as mind- and language-independent states of aTairs). This is an ontological self... (shrink)

This is a response to a paper “Paradox without satisfaction”, Analysis 63, 152-6 (2003) by Otavio Bueno and Mark Colyvan on Yablo’s paradox. I argue that this paper makes several substantial mathematical errors which vitiate the paper. (For the technical details, see [12] below.).

Some years ago, Yablo gave a paradox concerning an infinite sequence of sentences: if each sentence of the sequence is 'every subsequent sentence in the sequence is false', a contradiction easily follows. In this paper we suggest a formalization of Yablo's paradox in algebraic and topological terms. Our main theorem states that, under a suitable condition, any continuous function from 2N to 2N has a fixed point. This can be translated in the original framework as follows. Consider (...) an infinite sequence of sentences, where any sentence refers to the truth values of the subsequent sentences: if the corresponding function is continuous, no paradox arises. (shrink)

Yablo’s Paradox, an infinite-sentence version of the Liar Paradox, aims to show that semantic paradox can emerge even without circularity. I will argue that the lack of meaning/content of the sentences involved is the source of the paradoxical outcome.I will introduce and argue for a Moderate Antirealist approach to truth and meaning, built around the twin principles that neither truth nor meaning can outstrip knowability. Accordingly, I will introduce a MAR truth operator that both forges an explicit (...) connection between truth and knowability and distinguishes between truth and factuality. I will also argue that the meaning/content of propositions should be identified not with the set of possible worlds in which the propositions are true/factual, but rather in which they are known.I will show that our MAR framework dissolves Yablo’s Paradox and also confirms our intuition that these sentences are all devoid of content/meaning. (shrink)

El contenido de la presente discusión de Análisis Filosófico surge a partir de diversas actividades organizadas por mí en SADAF y en la UBA. En primer lugar, Roy Cook dictó en SADAF el seminario de investigación intensivo On Yablo's Paradox durante la última semana de julio de 2011. En el seminario, el profesor Cook presentó el manuscrito aún sin finalizar de su libro The Yablo Paradox: An Essay on Circularity, Oxford, Oxford UP, (en prensa). Extensas y apasionantes (...) discusiones ocurrieron durante esos encuentros sobre circularidad y construcciones infinitarias. Fue en ese tiempo, donde me surgió la idea de editar una discusión sobre las ideas que Cook defiende en ese trabajo. El proyecto era una extensión natural del trabajo que veníamos realizando con mi grupo de investigación en temas vinculados al concepto de verdad, autorreferencia y paradojas. Luego, durante el segundo cuatrimestre de 2011, dicté el seminario La paradoja de Yablo, en el instituto de filosofía de la UBA. Algunos de los borradores de los artículos que aparecen en el presente volumen tienen su origen en este curso. Finalmente, invité por segunda vez al profesor Cook al Symposium on Yablo's Paradox realizado en SADAF en julio de 2012. En esta oportunidad, se presentaron las versiones finales de los artículos de Lavinia Picollo, Paula Teijeiro, Federico Pailos, Diego Tajer, Lucas Rosenblatt e Ignacio Ojea que se incluyen a continuación. El encuentro incluyó las inteligentes réplicas del profesor Cook y profundas discusiones sobre los mencionados temas lógicosemánticos. Quiero agradecer a todos los integrantes del Gaf[log] que participaron activamente en las mencionadas actividades, ya sea en la publicación posterior o en los coloquios y seminarios que le dieron origen. Agradezco al Comité editorial de Análisis Filosófico, en especial a Alberto Moretti, quienes apoyaron desde sus comienzos este proyecto. Finalmente, y de manera especial, quiero expresar mi gratitud al profesor Roy Cook, quien no sólo apoyó e inspiró el proyecto desde sus comienzos, sino que además compartió generosamente sus ideas y las discutió con estimulante pasión. (shrink)

In this paper, we strengthen Hardy’s [1995] and Ketland’s [2005] arguments on the issues surrounding the self-referential nature of Yablo’s paradox [1993]. We first begin by observing that Priest’s [1997] construction of the binary satisfaction relation in revealing a fixed point relies on impredicative definitions. We then show that Yablo’s paradox is ‘ω-circular’, based on ω-inconsistent theories, by arguing that the paradox is not self-referential in the classical sense but rather admits circularity at the least transfinite countable (...) ordinal. Hence, we both strengthen arguments for the ω-inconsistency of Yablo’s paradox and present a compromise solution of the problem emerged from Yablo’s and Priest’s conflicting theses. (shrink)

Temporal logic is of importance in theoretical computer science for its application in formal verification, to state requirements of hardware or software systems. Linear temporal logic is an appropriate logical environment to formalize Yablo’s paradox which is seemingly non-self-referential and basically has a sequential structure. We give a brief review of Yablo’s paradox and its various versions. Formalization of these paradoxes yields some theorems in Linear Temporal Logic (LTL) for which we give syntactic proofs using an appropriate axiomatization (...) of LTL. (shrink)

It is widely considered that Gödel’s and Rosser’s proofs of the incompleteness theorems are related to the Liar Paradox. Yablo’s paradox, a Liar-like paradox without self-reference, can also be used to prove Gödel’s first and second incompleteness theorems. We show that the situation with the formalization of Yablo’s paradox using Rosser’s provability predicate is different from that of Rosser’s proof. Namely, by using the technique of Guaspari and Solovay, we prove that the undecidability of each instance (...) of Rosser-type formalizations of Yablo’s paradox for each consistent but not Σ1-sound theory is dependent on the choice of a standard proof predicate. (shrink)

The aim of this paper is to show that it’s not a good idea to have a theory of truth that is consistent but ω-inconsistent. In order to bring out this point, it is useful to consider a particular case: Yablo’s Paradox. In theories of truth without standard models, the introduction of the truth-predicate to a first order theory does not maintain the standard ontology. Firstly, I exhibit some conceptual problems that follow from so introducing it. Secondly, I show (...) that in second order theories with standard semantics the same procedure yields a theory that doesn’t have models. So, while having an ω- inconsistent theory is a bad thing, having an unsatisfiable theory of truth is actually worse. This casts doubts on whether the predicate in question is, after all, a truthpredicate for that language. Finally, I present some alternatives to prove an inconsistency adding plausible principles to certain theories of truth. (shrink)

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

Analytic semantics got its start when Frege pointed out differences in cognitive content between sentences that in some good sense “say the same.” Frege put cognitive content (in the form of sense) at the heart of semantic content. Most prefer nowadays to see cognitive contents as generated by semantic contents in context; a sentence's cognitive significance is an aspect rather of the information imparted by its use. I argue for a particular version of this idea. Semantic contents generate cognitive contents (...) by operating levers in a larger piece of machinery, involving also the presupposition P one is leaning on and the subject matter M under discussion. A sentence's cognitive content is what its literal-content-restricted-to-P-worlds says about M. It's by triggering different presuppositions and/or different subject matters that coreferential names make their distinctive mark on a sentence's felt information value. (shrink)

Aboutness has been studied from any number of angles. Brentano made it the defining feature of the mental. Phenomenologists try to pin down the aboutness-features of particular mental states. Materialists sometimes claim to have grounded aboutness in natural regularities. Attempts have even been made, in library science and information theory, to operationalize the notion. But it has played no real role in philosophical semantics. This is surprising; sentences have aboutness-properties if anything does. Aboutness is the first book to examine through (...) a philosophical lens the role of subject matter in meaning. A long-standing tradition sees meaning as truth-conditions, to be specified by listing the scenarios in which a sentence is true. Nothing is said about the principle of selection--about what in a scenario gets it onto the list. Subject matter is the missing link here. A sentence is true because of how matters stand where its subject matter is concerned. Stephen Yablo maintains that this is not just a feature of subject matter, but its essence. One indicates what a sentence is about by mapping out logical space according to its changing ways of being true or false. The notion of content that results--directed content--is brought to bear on a range of philosophical topics, including ontology, verisimilitude, knowledge, loose talk, assertive content, and philosophical methodology. Written by one of today's leading philosophers, Aboutness represents a major advance in semantics and the philosophy of language. (shrink)

A reply to Fine’s critique of Aboutness. Fine contrasts two notions of truthmaker, and more generally two notions of “state.” One is algebraic; states are sui generis entities grasped primarily through the conditions they satisfy. The other uses set theory; states are sets of worlds, or, perhaps, collections of such sets. I try to defend the second notion and question some seeming advantages of the first.

ontology metaontology wright platonism fregean existence epistemology.

[Stephen Yablo] The usual charge against Carnap's internal/external distinction is one of 'guilt by association with analytic/synthetic'. But it can be freed of this association, to become the distinction between statements made within make-believe games and those made outside them-or, rather, a special case of it with some claim to be called the metaphorical/literal distinction. Not even Quine considers figurative speech committal, so this turns the tables somewhat. To determine our ontological commitments, we have to ferret out all traces of (...) nonliterality in our assertions; if there is no sensible project of doing that, there is no sensible project of Quinean ontology. /// [Andre Gallois] I discuss Steve Yablo's defence of Carnap's distinction between internal and external questions. In the first section I set out what I take that distinction, as Carnap draws it, to be, and spell out a central motivation Carnap has for invoking it. In the second section I endorse, and augment, Yablo's response to Quine's arguments against Carnap. In the third section I say why Carnap's application of the distinction between internal and external questions runs into trouble. In the fourth section I spell out what I take to be Yablo's version of Carnap. In the last I say why that version is especially vulnerable to the objection raised in the second. (shrink)

[Stephen Yablo] The usual charge against Carnap's internal/external distinction is one of 'guilt by association with analytic/synthetic'. But it can be freed of this association, to become the distinction between statements made within make-believe games and those made outside them-or, rather, a special case of it with some claim to be called the metaphorical/literal distinction. Not even Quine considers figurative speech committal, so this turns the tables somewhat. To determine our ontological commitments, we have to ferret out all traces of (...) nonliterality in our assertions; if there is no sensible project of doing that, there is no sensible project of Quinean ontology. /// [Andre Gallois] I discuss Steve Yablo's defence of Carnap's distinction between internal and external questions. In the first section I set out what I take that distinction, as Carnap draws it, to be, and spell out a central motivation Carnap has for invoking it. In the second section I endorse, and augment, Yablo's response to Quine's arguments against Carnap. In the third section I say why Carnap's application of the distinction between internal and external questions runs into trouble. In the fourth section I spell out what I take to be Yablo's version of Carnap. In the last I say why that version is especially vulnerable to the objection raised in the second. (shrink)

Mark Colyvan (2010) raises two problems for ‘easy road’ nominalism about mathematical objects. The first is that a theory’s mathematical commitments may run too deep to permit the extraction of nominalistic content. Taking the math out is, or could be, like taking the hobbits out of Lord of the Rings. I agree with the ‘could be’, but not (or not yet) the ‘is’. A notion of logical subtraction is developed that supports the possibility, questioned by Colyvan, of bracketing a theory’s (...) mathematical aspects to obtain, as remainder, what it says ‘mathematics aside’. The other problem concerns explanation. Several grades of mathematical involvement in physical explanation are distinguished, by analogy with Quine’s three grades of modal involvement. The first two grades plausibly obtain, but they do not require mathematical objects. The third grade is likelier to require mathematical objects. But it is not clear from Colyvan’s example that the third grade really obtains. (shrink)

Kripke, argued like this: it seems possible that E; the appearance can't be explained away as really pertaining to a "presentation" of E; so, pending a different explanation, it is possible that E. Textbook Kripkeans see in the contrast between E and its presentation intimations of a quite general distinction between two sorts of meaning. E's secondary or a posteriori meaning is the set of all worlds w which E, as employed here, truly describes. Its primary or a priori meaning (...) is the set of all w such that if w is actual, then E is true. "Conceivability error" occurs when a primary possibility is mistaken for a secondary one. Textbook Kripkeanism is rejected on the grounds that it makes meaning too modal and modality too much a matter of meaning. (shrink)

Many topics have not been covered, in most cases because I don't know quite what to say about them. Would it be possible to add a decidability predicate to the language? What about stronger connectives, like exclusion negation or Lukasiewicz implication? Would an expanded language do better at expressing its own semantics? Would it contain new and more terrible paradoxes? Can the account be supplemented with a workable notion of inherent truth (see note 36)? In what sense does stage semantics (...) lie “between” fixed point and stability semantics? In what sense, exactly, are our semantical rules inconsistent? In what sense, if any, does their inconsistency resolve the problem of the paradoxes?The ideals of strength, grounding, and closure together define an intuitively appealing conception of truth. Nothing would be gained by insisting that it was the intuitive conception of truth, and in fact recent developments make me wonder whether such a thing exists. However that may be, until the alternatives are better understood it would be foolish to attempt to decide between them. Truth gives up her secrets slowly and grudgingly, and loves to confound our presumptions. (shrink)