Contradictions Logic History Actuality, 2014

Elena FicaraContradictions

Berlin Studiesin Knowledge Research

|Edited byGnter Abel and James Conant

Volume 6

Elena Ficara

Contradictions

|Logic, History, Actuality

Series EditorsProf. Dr. Gnter AbelTechnische Universitt BerlinInstitut fr PhilosophieStrae des 17. Juni 13510623 BerlinGermanye-mail: [email protected]

Prof. Dr. James ConantThe University of ChicagoDept. of Philosophy1115 E. 58th StreetChicago IL 60637USAe-mail: [email protected]

ISBN 978-3-11-033574-3e-ISBN 978-3-11-034082-2

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Bibliographic information published by the Deutsche NationalbibliothekThe Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;detailed bibliographic data are available in the Internet at http://dnb.dnb.de.

2014 Walter de Gruyter GmbH, Berlin/BostonTypesetting: le-tex publishing services GmbH, LeipzigPrinting and binding: Hubert & Co. GmbH & Co. KG, Gttingen Printed on acid-free paperPrinted in Germany

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AcknowledgementsThis book goes back to a conference with the same title that took place at Tech-nical University Berlin in the summer of 2011 and was generously supported bythe Deutsche Forschungsgemeinschaft and the IZW. For help before, during andafter the conference, I amparticularly thankful to Gnter Abel, FrancaDAgostini,Dagfinn Fllesdal, Hans Poser, Claudio Roller, Doris Schps and Elisabeth Simon.Formaking the publication of this volume possible, I am especially grateful to theIZW, in particular to Gnter Abel, for his friendliness and the unconditioned sup-port of the project in every phase of its development. I am also grateful to HadiFaizi, who helped me revising the English texts, and to Peter Remmers for hiseditorial advice. Finally, my thanks go to the publishing house De Gruyter, par-ticularly to Gertrud Grnkorn and Konrad Vorderobermeier for help during theproduction of this volume.

Contents

Elena FicaraIntroduction | 1

Part I: Logic

Graham PriestContradictory Concepts | 13

JC BeallRapunzel Shaves Pinocchios Beard | 27

Franca DAgostiniParadoxes and the Reality of Contradictions | 31

Achille C. VarziLogic, Ontological Neutrality, and the Law of Non-Contradiction | 53

Francesco BertoRepresenting the Contradictory | 81

Part II: History

Enrico BertiObjections to Aristotles Defence of the Principle of Non-Contradiction | 97

Angelica NuzzoThe Justice of Contradiction. Logical Advancement andHistorical Transformations | 109

Luca IlletteratiLimit and Contradiction in Hegel | 127

Klaus ViewegZur Logik moralischer Urteile | 153

Elena FicaraIntroductionThe notion of contradiction is of the greatest importance in several fields. It is acentral topic in the history of ancient philosophy: the very beginning of philos-ophy in history seems to be closely connected to the discovery of contradictionsin Greek language. It is of crucial importance in metaphysics: Aristotles inquiryinto the nature of being is also, if notmainly, inspired by the need of avoiding (anddiagnosing) the occurrence of contradictions. It is evidently, in many senses, oneof the basic concerns of logic. The problem of contradiction is also the problemof disagreement: is it possible that contradictory theses are both true? Under nor-mal circumstances, one of the two theses is false, and this means that one of thedisagreeing parties is wrong. And yet, at least sometimes people disagree with-out any fault (or so it seems), and incompatible positions seem both right. In thissense, the theme of contradiction is also the core of any political reflection aboutdemocratic confrontation, relativism and the role of the concept of truth in polit-ical practice. Not only that, a complete theory about the problematic and heuris-tic relevance of contradictions in any field was typically given by the authors ofGerman Idealism, and specifically by the tradition of Hegelianism, so the issue isalso crucial for the history of philosophy after Kant, and for any inquiry into clas-sical German philosophy. Finally, the problem of the existence, uses, and natureof contradictions is at the core of many contemporary discussions in philosophy:discussions about paradoxes, and the plausibility of paraconsistent logics, butalso about the status of human subjects, as located in social and political con-texts, and about the destiny of Marxism.

All this stated, a typical problem affects contemporary theories on this topic.The main concern is that the authors working on it come from radically differentphilosophical traditions and contexts, and they can only very rarely communicatewith each other, and share their results.

The first aim of the book is thus to stimulate a genuine dialogue between dif-ferent approaches, so that the understanding of the problem of contradictions be-comes as complete as possible. Not only that, the same topic of contradictions, assuggested above, seems to be located at the intersection of different fields, tradi-

1 See the contemporary discussions about Graham Priests dialetheism, which are documentedin many leading philosophical journals, as well as in Priest/Beall/Armour-Garb (2004).2 See Judith Butlers and Slavoy ieks reconsideration ofHegelianism inButler (1987) and iek(2012).

2 | Elena Ficara

tions and schools, so it is particularly apt to overcome the divides between philo-sophical approaches, constituting a common ground of philosophical research.

The papers collected in this volume present some of the most recent resultsof the work about contradictions in philosophical logic, examine the history ofcontradiction in crucial phases of philosophical thought (in ancient philosophyand in German philosophy after Kant), consider the relevance of contradictionsfor political and philosophical current times. Despite the differences between ap-proaches and stiles, a basic question emerges, and it is more or less openly ad-dressed in all the papers. It is the question of the irreducibility, reality, and pro-ductive force of (some) contradictions.

The book has three parts. In the first part, Logic and Metaphysics, leadingexperts of philosophical logic and metaphysics focus on the problem now atthe centre of living debates about non-classical logic, and in particular about di-aletheism of the reality of contradictions, and on the link between logic andmetaphysics. The second part, History, entails papers by specialists of ancientphilosophy and post-Kantian philosophy, the two periods in history where thereflection on contradictions reached probably its greatest development. The pa-pers analyse bothAristotles defence of the lawof non-contradiction, dealingwithsome problems connected with it, and Hegels arguments for the reality and ef-fectiveness of contradictions, and their relevance for practical and political phi-losophy. The last part, Actuality, is devoted to the role and uses of contradictionsfor cultural and political occurrences. It collects papers by eminent contemporaryphilosophersworking in themainly European tradition, on the political, aestheticand biological implications of contradictions.

Graham Priest (Contradictory Concepts) examines a question at the very core ofcontemporary discussionswithin dialetheism. If dialetheism is the view that thereare (some) true contradictions, then the problem is to assess if the contradictions

3 Evidently, the meaning of the term contradiction, as well as that of the expression true con-tradiction is here at stake. For a first overview see Grim (2004), 4972, as well as the paperscollected in this volume (Part One: Logic).4 See Priest/Beall/Armour-Garb (2004).5 The terms dialetheia/dialetheism were coined by Graham Priest and Richard Routley in 1981(see Priest/Routley/Norman (1989), xx) and result from the union of the two Greek words di-(two/double) and aletheia (truth). A dialetheia is a true contradiction (a double truth), i.e. a trueproposition whose negation is also true, and dialetheism is the view according to which there aresome dialetheias (true-and-false propositions) and this does not imply any trivialisation of logic.For a clear overview, see Priest/Berto (2013).

Introduction | 3

are only in our concepts or also in reality. Strictly speaking, dialetheism not onlyimplies that our concepts might entail contradictions, it also implies the meta-physical insight that there are things that satisfy these concepts. But if we holdthat contradictions are in reality, the view that there are facts of the form A andA, and thus that there are not only positive, but also negative facts, seems in-evitable. This view might prove to be so Priest too rich to many stomach.In the paper he focuses on the opposite perspective, the so-called semantic di-aletheism, according to which contradictions are only in our languages, and donot concern reality. Assuming that contradictions are a merely conceptual andlinguistic phaenomenon, Priest examines different strategies in order to get rid ofthem simply by changing our concepts, and shows that they all fail. As amatter offact, semantic dialetheism, so understood, presents many difficulties, first of allexpressive loss, i.e. the impossibility to think and express concepts such as theconcept of totality which classically involve irreducible contradictions.

Jc Bealls Rapunzel Shaves Pinocchios Beard goes back to a discussion, which ap-peared in Analysis, between himself and Peter Eldridge-Smith concerning the socalled Pinocchio paradox, a version of the Liar paradox that involves the empir-ical world. In this paper, Beall presents a paradoxical tale, very similar to Pinoc-chio paradox. In the tale, grass is growing if and only if what Rapunzel says isfalse, and it happens that Rapunzel says that grass is growing. Evidently, this isa liar-like situation, since if what Rapunzel says is true, then grass is not growingand what she says must be false; and if it is false, then grass is growing and whatshe saysmust be true. Pinocchios paradox, similarly, says that Pinocchios nose isgrowing if and only if Pinocchio lies, and Pinocchio says that his nose is growing.At first sight, these versions of the Liar paradox seem to involve empirical-worldgluts, i.e. sentences about the empiricalworld (noses and grass) that are both trueand false. This would ultimately imply a metaphysical dialetheism of some sort,

6 Ed Mares has coined the expressions metaphysical dialetheism, distinguishing it from se-mantic dialetheism. In Mares (2004), 269, he defines the two notions as follows: Both seman-tical and metaphysical dialetheism hold that there are true contradictions, or at least that it ispossible for there to be true contradictions. That is what dialetheism means. The difference be-tween the twoviews concerns the status of these contradictions (. . . ) Themetaphysical dialetheistholds that there are aspects of the world (or of some possible world) for which any accurate de-scription will contain a true contradiction. Semantic dialetheism, on the other hand, maintainsthat it is always possible to redescribe this aspect of the world, using a different vocabulary.7 The Pinocchio paradox was invented by Peter Eldridge-Smiths daughter, Veronique Eldridge-Smith, and occasioned a discussion on empirical-world gluts documented in Analysis. SeeEldridge-Smith/Eldridge-Smith (2010); Eldridge-Smith (2011); Beall (2011); Eldridge-Smith (2012).

4 | Elena Ficara

for instance the view that there must be something in the empirical world thatboth is and is not, both has and has not a particular physical property. Jc Beall isone of themost eminent contemporary proponents of the view called semantic di-aletheism, according to which contradictions arise because of our language, andbecause of the semantic behaviour of the concept of truth. Accordingly, Beall sug-gests here that Pinocchios and Rapunzels stories aremere tales, and not genuineparadoxes (that is: apparent valid arguments with true premises and an appar-ently false conclusion). In fact, the premises of both arguments are not true, theyare only true according to the story. Truth in a story so Beall is insufficientfor truth at some world.

Evidently, in both Priests and Bealls papers what is at stake is the question:what are themetaphysical and ontological implications of the thesis that there aretrue contradictions? Does the view that there are true contradictions have onto-logical implications? And what is the link between the semantic claim that thereare true contradictions and the ontological one, according to which there effec-tively are contradictory states of affairs, or objects that make contradictions true?These questions are at the core of DAgostinis and Varzis papers.

Franca DAgostini (Paradoxes and the Reality of Contradictions) explicitly ad-dresses the first, examining the problem of the reality of contradictions, and ofthe kind of realism implied by the dialetheists thesis according to which somecontradictions are true. DAgostini considers the expression true contradiction,inquiring into its possible meaning as made true by real (non-constructed, non-fictional) facts. She first considers the kind of evidence involved in epistemic andsemantic paradoxes, then argues for an interpretation of the realism involved bydialetheism as alethic realism, the view according to which a sentence or propo-sition is true if and only if things stand like says. In the perspective of alethicrealism, a fact is simply what can make a proposition true, so there might beindeterminate kinds of facts: universal, as well as conditional, mathematical orphysical, infra-subjective or intra-subjective, physical or intentional facts etc. Allthis stated, DAgostini suggests that alethic realism is perfectly adaptable to threetheses that classically constitute metaphysical realism, namely: the thesis thatthere are facts, that there is a unique true description of these facts, and that wecan formulate true descriptions of facts. From this point of view, alethic realismturns out to be a particular version of metaphysical realism, which only requiresthat the field of what we count as really existing facts must be left open. All this

8 See Beall (2009).

Introduction | 5

stated, DAgostini claims that contradictions are alethically true, which meansthere are many different kinds of contradictions, in accordance with differentkinds of facts. The interpretation of dialetheism in terms of alethic realism, soDAgostini, may settle some controversies affecting the debate on dialetheism andstate the metaphysical (not only semantic) reality of contradictions, without anycommitment to metaphysical trivialism.

Achille Varzi (Logic, Ontological Neutrality, and the Law of Non-Contradiction) ex-amines the connection between the universality of logic and its supposed on-tological neutrality. If logic and its forms have to be universally valid, logic hasto abstract away from content, and to be ontologically neutral. In Varzis recon-struction, Aristotelian logic, precisely because of its ontological presuppositions,was deemed insufficient as a canon of pure logic and overcome bymodern Frege-Russell logic. Alsomodern quantification theory, with its existentially loaded the-orems and patterns, has been claimed to suffer from a defect of logical purity.In particular, Varzi focuses on the law of non-contradictions ontological impli-cations, and on its critique. According to him, the critiques of the law of non-contradiction (beginning with ukasievicz through paraconsistent logics and di-aletheism) are moved by the attempt of overcoming its tacit ontological implica-tions, and directed towards the achievement of a greater universality for logic. Inthis perspective, the main problem is the link between the metaphysical and thesemantic point of view, and more specifically the relation between the (ontologi-cal) law of non-contradiction (it is not possible that both p and not p) and whatVarzi calls the semantic principle of contravalence (it is not possible that a state-ment p both is and is not true). In particular, the failure of contravalence neednot entail genuine ontological overdeterminacy or genuine counterexamples tonon-contradiction. The problem is thus to discriminate between amerely de dictodialetheia and a de re dialetheia. Varzi discusses two options, which can consti-tute an indication in order to argue that the law of non-contradiction is not aninstance of the prejudices from which logic has tried to free itself throughout itshistory in the spirit of even greater ontological neutrality. In fact, every evidencewe might have against the law requires that we deploy other principles, and it isthose principles that might be deemed inadequate as canon of pure logical rea-soning.

Francesco Berto (Representing the Contradictory) asks about the possibility ofrepresenting contradictory states. He therefore develops a semantic and syntac-tic extension of Priests basic relevant logic 4. He introduces, on the syntactic

6 | Elena Ficara

side, the representation operator , which allows both to capture our capacityof seeing or conceiving contradictions and logical impossibilities and to admitparticularly anarchic non-normal worlds, including self-contradictory worlds.As a matter of fact, according to Berto contradictions are human phaenomena,linked to our finite and fallible condition. Thus if we can represent and conceivecontradictions and other absolute impossibilities, non-normal worlds are natu-ral candidates to model this human condition. In contrast, a logic that does notadmit the possibility of representing contradictions is only able to grasp highlyidealized epistemic notions, and cannot mirror the actual situation of humanbeings as finite, fallible, and occasionally self-contradicting cognitive agents.

Enrico Berti (Objections to Aristotles Defence of the Principle of Non-Contra-diction) considers both Aristotles defence of the law of non-contradiction andsome main objections to it (in particular ukasiewiczs and Dancys critiques,and Priests dialetheic arguments against Metaphysics Gamma). In his view,according to Aristotle the law of non-contradiction does not stand in the way ofsaying and thinking contradictions; however, according to the Greek philosopherthe emerging of a contradiction is always so Berti a sign of falsity. Not onlythat, the elimination of a contradiction is a necessary component of the logicaland discussive process of refutation. Thus every dismissal of the law implies achallenge to the very procedure of refutation, and the risk of trivialism (the viewaccording to which everything is contradictory, and therefore everything is trueand can be proved). Dialetheism, arguing for the suspension of the law only inspecial cases, could represent a genuine alternative to trivialism. However soBerti dialetheism does not give a clear criterion to distinguish true contradic-tions from false ones.

Angelica Nuzzo (The Justice of Contradiction: Logical Advancement and Histori-cal Transformations) focuses on the practical and juridical implications of Hegelslogical idea of contradiction, giving a new account of one of the most contentiousHegelian views, the idea of a Weltgeschichte (history of the world) placed as con-clusion of the sphere of objective spirit, and the thesis thatWeltgeschichte ist Welt-gericht (the history of the world is the tribunal of the world). According to thisview, history does not only have the descriptive task of registering events, but

9 See ukasiewicz (1910) and Dancy (1975).10 Priest (2006).

Introduction | 7

it also has a normative import. Nuzzo argues that Hegels view, which has oftenbeen disputed and variously challenged, is the direct and coherent consequenceof the claim that historical processes are structured according to the dialecticallogic of contradiction. According to Nuzzo, conflict and contradiction, rather thanbeing a sign of imbalanceanddisharmony, are themotor of historical change, andthus the very conditions of a just political and practical order. In this perspective,the normative function of contradiction also emerges: conflict is justice andWelt-geschichte isWeltgericht because historical change is produced by strife. This alsomeans that contradictions have a discriminative and ordering power; they do leadneither to chaos nor to nothingness but to epochal transformation.

Luca Illetterati (Limit and Contradiction in Hegel) considers the Hegelian thesis:things are inherently contradictory, arguing for the necessity of interpreting itin a literal, and not metaphorical or conciliatory way. In his reading, there is astrong connection betweenHegels talk of contradiction and the problem of deter-mination. In turn, the problem of determination is, according to Illetterati, linkedto the question of limit: everything is contradictory insofar as everything is de-terminate, i.e. limited. As Kant himself implicitly already acknowledged in somesubparagraphs of the Prolegomena to Any Future Metaphysics, thinking the limitmeans to think a structure that constitutively entails a contradiction. In this sense,if the limit is both an inherently contradictory structure and the locus where ev-erything is what it is, then the view all things are inherently contradictory isunavoidable.

Klaus Vieweg (Zur Logik moralischer Urteile) examines the central role of con-tradictions and antinomies within Hegels practical philosophy, in particular formorality. The concept of morality implies the opposition between particularityand universality (the particularity of the acting and wanting human being andthe universality of the law), an opposition that Hegel interprets as contradiction.According to Hegel, and differently from Kant, the knowledge of the idea of thegood is what allows an adequate understanding of the contradiction involved inmorality. As to Kant, a non-cognitive faith in God is whatmakes possible to recon-cile the opposition between the universality of themoral law and the particularityof the human will. From the Kantian point of view, the contradiction is not con-ceived but only presupposed aswhat has to be overcome. Hegel, differently, shows

11 Kant (1900ff.), AA IV.

8 | Elena Ficara

the contradictory structure of the idea of the good as the identity of the opposites(particularity and universality).

Gianni Vattimo (Insuperable Contradictions) argues that philosophy has alwaystried to reconcile contradictions, and that the same title Contradictions. Logic,History, Actuality could be intended as the attempt of controlling, and in the endovercoming, contradictions. In explicit polemicwith suchan attempt, Vattimode-fends the view that there are irreducible contradictions, and it is necessary to letcontradictions speak, without controlling them. What he calls ontology of revo-lution constitutes, in his view, the only possible frameable to address irreduciblecontradictions. In this frame, contradictions emerge as the very place where, Hei-deggerianly, being happens (sich ereignet). In Vattimos view, to propose a re-flection on contradictions that does not assume their reduction as its main task,is essential in order to meet the needs of contemporary culture, a culture that ischaracterised by a lack of emergency, rather then by a lack of conciliation.

Federico Vercellone (A Disenchanted Reenchantment. Hermeneutics and Mor-phology) sees in morphology (as the theory or logic of figures) the perspectivethat can best inherit the role of the koin Gadamer wanted for hermeneutics. Notonly that, the logic of figures is also able to deepen Adornos and Horkheimersidea of a dialectic of enlightenment, thus dealing with the peculiar contradic-tions of contemporary culture and science. In particular, Vercellone shows howthe focus on the concept of image (or figure) allows overcoming Heideggers,Gadamers, but also Adornos one-sided critique of technology. According to Ver-cellone, technology, rather than being as Heidegger wanted an instrument ofhiding what truly is, is fundamental in order to stimulate a re-enchantment ofthe world, that is a creativeway of dealing with images, promoting a new humanself-awareness. In this perspective, figures or images are, above all, contradictoryforms, that is: forms of conceptualisation that at the same time show the limits ofevery conceptualisation.

Wolfgang Welsch (Wie wir auf Konsistenz aus sind und warum) analyses therole of contradictions in different dimensions of human culture and life (every-day thought, philosophical and scientific reasoning, biological life), underliningthat they all show an effort of overcoming contradictions. Welsch thus points outthe centrality of consistency (intended as lack of contradictions) in different con-texts, showing that the demand for logical consistency is rooted in more original

Introduction | 9

kinds of demands. He considers some typical cases where we call for consistency,first of all our everyday need for both other peoples consistency and consistencywithin ourselves, and secondly the philosophical claim (Heraclitus, Dgen, Cu-sanus, Hegel) that oppositions on a linguistic or argumentative level should beovercome on a higher argumentative level. In these cases, and despite the emer-gence of apparently irreducible contradictions, the need for consistency seems toprevail, a need that according to Welsch goes beyond the argumentative andlogical dimension. In particular, the pressure for consistency is so Welsch abiological and ontological command: the physical, chemical, biotic forms of self-organisation are all ways of producing consistency.

ReferencesJC Beall, Spandrels of Truth, Oxford, 2009.JC Beall, Dialetheists against Pinocchio, in: Analysis 71, 689691, 2011.J. Butler, Subjects of Desire. Hegelian Reflections in 20th-Century France, New York, 1987.R. M. Dancy, Sense and Contradiction. A Study in Aristotle, Dordrecht and Boston, 1975.P. Eldridge-Smith, Pinocchio against the dialetheists, in: Analysis 71, 306308, 2011.P. Eldridge-Smith, Pinocchio beards the Barber, in: Analysis 72, 749752, 2012.V. Eldridge-Smith, P. Eldridge-Smith, The Pinocchio paradox, in: Analysis 70, 212215, 2010.P. Grim, What is a Contradiction?, in: Graham Priest, JC Beall, and Bradley Armour-Garb

(eds.), The Law of Non-Contradiction, Oxford, 4972, 2004.I. Kant, Kants gesammelte Schriften, edited by the Royal Prussian, later German, Academy of

Sciences, Berlin (quoted as AA, followed by the indication of the volume), 1900ff.J. ukasiewicz, ber den Satz des Widerspruchs bei Aristoteles, in: Bulletin international

de lAcadmie des sciences de Cracovie, Classe dhistoire et de philosophie, 1/2, 1538,1910.

E. D. Mares, Semantic Dialetheism, in: Graham Priest, JC Beall, and Bradley Armour-Garb(eds.), The Law of Non-Contradiction, Oxford, 264275, 2004.

G. Priest, Doubt Truth to Be a Liar, Oxford, 2006.G. Priest, JC Beall, B. Armour-Garb (eds.), The Law of Non-Contradiction, Oxford, 2004.G. Priest, F. Berto, Dialetheism, in: The Stanford Encyclopedia of Philosophy (Summer

2013 Edition), E. N. Zalta (ed.). http://plato.stanford.edu/archives/sum2013/entries/dialetheism/, 2013.

G. Priest, R. Routley, J. Norman (eds.), Paraconsistent Logic. Essays on the Inconsistent, Mu-nich, 1989.

S. iek, Less Than Nothing. Hegel and the Shadow of Dialectical Materialism, London and NewYork, 2012.

|Part I: Logic

Graham PriestContradictory Concepts

1 IntroductionThat we have concepts which are contradictory is not news. That there may bethings which satisfy them, dialetheism, is, by contrast, a contentious view. Myaim here is not to defend it, however; and in what follows, I shall simply assumeits possibility. Those who disagree are invited to assume the same for the sake ofargument. The point of this essay is to think through some issues that the viewraises. In particular, we will be concerned with two inter-related questions:

1. Are the dialetheias simply in our concepts/language, or are they in real-ity? And what exactly does this distinction amount to anyway?

2. Assuming that they are only in our concepts/language, can we get rid ofdialetheias simply by changing these?

I will take up these issues, in the two parts of the paper.

2 Dialetheism, Concepts, and the World2.1 Contradiction by Fiat

A dialetheia is a pair of statements of the form andwhich are both true.Wemay think of statements as (interpreted) sentences expressed in some language a public language, a language of thought, or whatever. In this way they contrast,crucially, with whatever it is that the statements are about. Let us call this, forwant of a better name, the world.

One thing that partly determines the truth value of a statement is its con-stituents: the meanings of the words in the sentence, or the concepts the wordsexpress. (Conceivably, one might draw a distinction here, but not one that seems

1 This is done in Priest (1995), (2006a), (2006b). The topic is discussed by numerous people inthe essays in Priest/Beall/Armour-Garb (2004) and the references cited therein.2 A longer version of this paper will appear as Dialetheism, Concepts and the World, in JokeMeheus, Erik Weber, and DietlindeWouters (eds.), Logic, Reasoning and Rationality, Springer.3 Priest (2006a), 4.

14 | Graham Priest

relevant for present purposes.) Let us call these things, again for want of a betterword, semantic. In certain limit cases, such as Red is a colour, semantic factorsmay completely determine the truth value of a statement. In general, however,the world is also involved in determining the truth value. Thus, the statement thatMelbourne is in Australia is made true, in part, by a certain city, a certain coun-try literally part of this world.

Given that dialetheias are linguistic, one natural way for them to arise issimply in virtue of linguistic/conceptual fiat. Thus, suppose we coin a newword/concept, Adult, and stipulate that it is to be used thus:

if a person is 16 years or over, they are an Adult if a person is 18 years or under, they are not an Adult

Now suppose there is a person, Pat, who is 17. Then we have:

(*) Pat is both an Adult and not an Adult.

Of course, one can contest the claim that the stipulation succeeds in giving thenew predicate a sense. Deep issues lurk here, but I will not go into them, since myconcern is with other matters. I comment only that the stipulation would seem tobe just as successful as the dual kind, endorsed by a number of people, whichunder-determine truth values such as the following, for Child:

if a person is 16 years or under, they are a Child if a person is 18 years or over, they are not a Child

Assuming the stipulation of the kind involved in Adult to work, we have a cer-tain sort of dialetheia here. We might call it, following Mares (2004), a semanticdialetheia. Note that, in terms of the distinction just drawn between semantic andworldly factors, the epithet is not entirely appropriate. The truth of (*) is deter-mined only in part by semantics; some worldly factors are also required, such asPat and Pats age. Still, let us adopt this nomenclature.

4 Quineans would, of course, reject the distinction being made here between semantic andworldly factors. This is not the place to defend the notion of analyticity. I do so in Priest (1979)and Priest (forthcoming).5 See Priest (2001).6 E.g., Soames (1999).

Contradictory Concepts | 15

2.2 Semantic Dialetheism

The dialetheism engendered by the definition of Adult is transparent. There areother examples which are, plausibly, of the same kind, though they are less trans-parent. One of these concerns dialetheias apparently generated by bodies of laws,rules, or constitutions, which can also bemade to hold by fiat. Thus, suppose thatan appropriately legitimated constitution or statute rules that:

every property-holder shall have the right to vote no woman shall have the right to vote

As long as nowoman holds property, all is consistent. But suppose that, for what-ever reason, a woman, Pat, comes to own property, then:

Pat both has and has not got the right to vote.

Examples that are arguably of the same kind are given by multi-criterial terms.Thus, suppose that a criterion for being a male is having male genitalia; and thatanother criterion is the possession of a certain chromosomic structure. These cri-teriamay comeapart, perhaps as the result of surgery of somekind. Thus, supposethat Pat has female genitalia, but a male chromosomic structure. Then:

Pat is a male and not a male.

In this case, there is no fiat about the matter. One cannot, therefore, argue thatthe contradiction can be avoided by supposing that the act of fiat misfires. Whatone has to do, instead, is to argue that the conditions in question are not criterial.Again, I shall not pursue the matter here.

A final example that is, arguably, in the same camp, is generated by the Ab-straction Principle of naive set theory:

Abs Something is a member of the collection { : ()} iff it satisfies thecondition().

7 The example comes from Priest (2006a), Section 13.2.8 See Priest (2006a), Section 4.8, and Priest/Routley (1989), Section 2.2.1.9 Priest (2006a), ch. 0.

16 | Graham Priest

This leads to contradiction in the form of Russells paradox. Again, there is nofiat here. If one wishes to avoid the contradiction, what one must contest is theclaim that satisfying condition() is criterial for being a member of the set { :()} or, what arguably amounts to the same thing in this case, thatAbs is truesolely in the virtue of the meanings of the words involved, such as is a memberof.

Again, let us not go into this here. The point of the preceding discussion is notto establish that the contradictions involved are true, but to show that dialetheiasmay arise for reasons that are, generally speaking, linguistic/conceptual.

2.3 Contradictions in the World

Some have felt that there may be a more profound sort of dialetheia, a contradic-tion in the world itself, independent of any linguistic/conceptual considerations.Let us call such dialetheias, following Mares again,metaphysical dialetheias.

A major problem here is to see exactly what a metaphysical dialetheia mightbe. Even someone who supposes that all dialetheias are semantic will accede tothe thought that there are contradictions in the world, in one sense. None of thecontradictions we considered in the previous sections, with perhaps the excep-tion of Russells paradox, is generated purely by semantic considerations. In eachcase, the world has to cooperate by producing an object of the appropriate kind,such as the much over-worked Pat. Theworld, then, is such that it renders certaincontradictions true. In that sense, the world is contradictory. But this is not thesense of contradiction that is of interest to metaphysical dialetheism. The contra-dictions in question are still semantically dependent in some way. Metaphysicaldialetheias are not dependent on language at all; only the world.

But how to make sense of the idea? If the world comprises objects, events,processes, or similar things, then to say that the world is contradictory is simplya category mistake, as, then, is metaphysical dialetheism. For the notion to geta grip, the world must be constituted by things of which one can say that they aretrue or false or at least something ontologically equivalent.

10 Take () to be , and to be { : }. Then we have iff . Hence, iff , and so .11 An example of a similar kind, which does have an explicit element of fiat, is that of the Secre-taries Liberation League, given by Chihara (1979).12 Mares (2004). A number of people have taken me (mistakenly) to be committed to this kind ofdialetheism. See Priest (2006a), Section 20.6.13 The point is made in Priest (2006a), Section 11.1.


Are there accounts of the nature of the world of this kind? There are. Themostobvious is a Tractarian view of the world, according to which it is composed offacts. One cannot say that these are true or false, but one can say that they obtainor do not, which is the ontological equivalent. Given an ontology of facts to makesense, metaphysical dialetheism may be interpreted as the claim that there arefacts of the form and , say the facts that Socrates is sitting and that Socratesis not sitting. But as this makes clear, there must be facts of the form , andsince we are supposing that this is language-independent, the negation involvedmust be intrinsic to the fact. That is, there must be facts that are in some sensenegational, negative facts. Now, negative facts have had a somewhat rocky roadin metaphysics, but there are at least certain well-known ways of making sense ofthe notion, so I will not discuss the matter here.

If one accepts an ontology of facts or fact-like structures, then metaphysicaldialetheism makes sense. Note, moreover, that if one accepts such an ontology,metaphysical dialetheism is a simple corollary of dialetheism. Since there are truestatements of the form and then there are facts, or fact-like structures, cor-responding to both of these. All the hard work here is being done by the meta-physics; dialetheism itself is playing only an auxiliary role.

3 Conceptual Revision3.1 Desiderata for Revision

Still, a metaphysics of facts (including negative facts) is too rich for many stom-achs. Suppose that we set this view aside. If we do, all dialetheias are essentiallylanguage/concept dependent. In this way, they are, of course, no different fromany other truths. But some have felt that, if this be so, dialetheias are relatively su-perficial. They can be avoided simply by changing our concepts/language. Com-pare the corresponding view concerning vagueness, held, for example, by Rus-

14 This isnt quite right. Facts may not themselves be intrinsically negative: the relation betweenthe facts that and that must be intrinsic. But this does not change matters much.15 In situation semantics, states of affairs come with an internal polarity bit, 1 or 0. Facts witha 0 bit are negative. Alternatively, a positive fact may be a whole comprising objects and a posi-tive property/relation; whilst a negative fact may be a whole comprising objects and a negativeproperty/relation. For a fuller discussion of a dialetheic theory of facts, see Priest (2006b), ch. 2.16 This assumes that all truths correspond to facts. In principle, anyway, one could endorse aview to the effect that some kinds of sentence are true in virtue of the existence of correspondingfacts, whilst others may have different kinds of truth-makers.

18 | Graham Priest

sell (1923). All vagueness is in language. Reality itself is perfectly precise. Vaguelanguage and its problems may, therefore, be avoided by changing to a languagewhich mirrors this precision.

Contradictions may certainly be resolved sometimes. Thus, consider the le-gal example concerning Pat and her rights. If and when a situation of this kindarises, the law would, presumably, be changed to straighten out the conflictingconditions for being able to vote. Note, however, that this is not to deny dialethe-ism. The situation before the change was dialetheic. The point of the change is torender it not so. Note, also, there is no a priori guarantee thatmaking changes thatresolve this particular contradictionwill guarantee freedom from contradiction intoto. There may well be others. Indeed, making changes to resolve this contra-diction may well introduce others. Laws comprise a complex of conceptual inter-connections, and the concepts apply to an unpredictable world. There is certainlyno decision procedure for consistency in this sort of case; nor, therefore, any guar-antee of success in avoiding dialetheism in practice.

But maybewe could always succeed in principle. Consider the following con-jecture:

Whenever we have a language or set of concepts that are dialetheic, wecan change to another set, at least as good, that is consistent.

The suggestion is, of course, vague, since it depends on the phrase at least asgood. Language has many purposes: conveying information, getting people todo things, expressing emotions. Given the motley of language use, I see no reasonto suppose that an inconsistent language/set of concepts can be replaced by aconsistent set which is just as good for all the things that language does. I donteven know how one could go about arguing for this.

Maybe we stand more chance if we are a little more modest. It might be sug-gested that language has a primary function, namely representation; and, at leastfor this function, given an inconsistent language/set of concepts, one can alwaysreplace it with a consistent one that is just as good. The claim that representationis the primary function of language may, of course, be contested; but let us grantit here. We still have to face the question of what just as good means now, buta natural understanding suggests itself: the replacement is just as good if it can

17 Actually, I think that the change here is not so much a change of concepts as a change of theworld. Arguably, the change of the law does not affect the meanings of vote, right, etc. Thestatement Pat has the right to vote may simply change its truth value, in virtue of a change inthe legal facts.


represent every situation that the old language represents. Let us then considerthe following conjecture:

Any language (set of concepts), , that represents things in a dialetheicway, can be replaced by a consistent language (set of concepts), , thatcan represent every situation that represents, but in a consistent way.

The conjecture is still ambiguous, depending on how one understands the possi-bility of replacement here. Are we to suppose this to be a practical possibility, ora merely theoretical one? If the distinction is not clear, just consider the case ofvagueness again. If there is no such thing as vagueness in re, we could, in prin-ciple, replace our language with vague predicates by one whose only predicatesare crisp. But the result would not be humanly usable.We can perceive that some-thing is red. We cannot perceive that it has a wavelength of between exactly and ngstroms, where and are real numbers. A language with precise colourpredicates would not, therefore, be humanly usable. Any language that can beused only by someonewith superhumanpowers of computation, perception, etc.,would be useless.

To return to the case of inconsistency, we have, then, two questions:

Can the language be replaced in theory? Would the replacement be possible in practice?

A few things I say will bear on the practical question, but by and large I shallrestrict my remarks to the theoretical one. This is because to address the practicalquestion properly one has to understandwhat the theoretical replacement is like.In other words, not only must the answer to the theoretical question be yes, theanswer must provide a sufficiently clear picture of the nature of the replacement.Nothing I go on to say will succeed in doing this. I have stressed the distinctionmainly to point out that even if the answer to the theoretical question is yes, thereplaceability conjecture has another hurdle to jump if the victory for those whourge replacement is to be more than Phyrric.

18 Batens (1999), 267, suggests that a denial of this conjecture is the best way to understand aclaim to the effect that the world is inconsistent. [I]f one claims that the world is consistent,one can only intend to claim that, whatever the world looks like, there is a language and a[correspondence] relation such that the true description of the world as determined by and is consistent. He maintains an agnostic view on the matter. See also Batens (2002), 131.19 I note that Batens (2002), 131, fn. 7, suggests that a consistent replacement for an inconsistentlanguage might well be required to have a non-denumerable number of constants, which wouldmake it humanly unusable.

20 | Graham Priest

So let us address the theoretical question. Is it true? Yes, but for entirely trivialreasons. canbe the languagewith just one sentence, . is trueof any situation.Thus, every situation is describable, and consistently so. (The language does noteven contain negation.) But this is not an interesting answer to the question, andthe reason is obvious. We have purchased consistency at the cost of the loss ofexpressive power. To make the question interesting, we should require to havethe same expressive power as or more. That is, everything that is able toexpress, is able to express. The idea is vague. What, exactly is it for differentlanguages to be able to express the same thing? But it is at least precise enoughfor us to be able to engage with the question in a meaningful way.

3.2 The Possibility of Revision

Return to the case of multiple criteria. A natural thought here is that we may ef-fect an appropriate revision by replacing the predicate/concept male with twoothers, male1, corresponding to the first criterion, and male2, corresponding tothe second. Pat is a male2, but not a male1, so the contradiction is resolved, andwhat used to be expressed by is male, can now be expressed by is male1 is male2. So far so good; but note that there is no guarantee that in this complexand unpredictable world the result will be consistent. The predicates male1 andmale2 may themselves turn out to behave in the same inconsistent way, due tothe fact that we have different criteria for genitalia or chromosome. More impor-tantly, the resolution of this dialetheia depends on the fact that the old predicatefalls neatly apart into two, individuated by different criteria. This will not be thecase in general. (Just consider the case of Adult, for example, which is not multi-criterial in the same way.)

Wemight attempt amore general wayof resolving dialetheias as follows. Sup-pose we have some predicate, , whose extension (the set of things of which it istrue) and co-extension (the set of things of which it is false) overlap. Given that weare taking it that our predicates do not have to answer to anything in the world,we may simply replace with the three new predicates, , , and , such thatthe things in the extension of are the things that are in the extension of butnot its co-extension; the things in the extension of are the things that are in theco-extension of but not its extension; the things in the extension of are thethings that are in both the extension and co-extension of . The co-extension, ineach case, is simply the complement. The situationmaybe depicted by the follow-ing diagram.For future reference, I call this theQuadrantDiagram. The numbersrefer to the quadrants.


The left-hand side is the extension of. The bottom half is the co-extension of. Quadrant 4 comprises those things of which is neither true nor false, and forpresent purposes we may take this to be empty. The three new predicates haveas extensions the other three quadrants. Each of the new predicates behaves con-sistently. Any dialetheia of the form is expressed by the quite consistent, and the predicate is now expressed, again, as a disjunction, .

So far so good. But recall that the new language must be able to express ev-erything that the old language expressed. A necessary condition for this is thatany situation described by the old language can be described by the new. To keepmatters simple for themoment, let us suppose that the old language contains onlythe predicate and the propositional operators of conjunction, disjunction, andnegation. We have seen how any atomic sentence, , of the old language can beexpressed equivalently by one,+, in the new. If this translation can be extendedto all sentences, then any situation describable in the old language is describablein the new. Thenatural translation is a recursive one. For the positive connectives:

( )+ is+ +

( )+ is+ +

But what of ? We certainly cannot take ()+ to be (+). is true in thebottomhalf of theQuadrantDiagram,whilst() is not true in quadrant 2.

20 Note that, if it is not, the same procedure can be used to get rid of truth value gaps.21 Batens (1999), 271 and (2002), 132 notes this idea. He also notes that in such a transition thetheory expressed in the new language may lose its coherence and conceptual clarity, making itworse.

22 | Graham Priest

In this case there is an easy fix. is equivalent to . So we can dealwith the atomic case. What of the others? There is a simple recipe that works:

(( ))+ is (+) (+)(( ))+ is (+) (+)()

+ is+

In other words, we can drive the negations inwards using De Morgan laws anddouble negation until they arrive at the atoms, where they are absorbed into thepredicate. In this way, every sentence of the old language is equivalent to a con-sistent one in the new language.

The end can therefore be achieved for this simple language. But, for the strat-egy to work, it must be implementable withmuchmore complex and realistic lan-guages. In particular, itmustwork for conditionals, quantifiers of all kinds, modaland other intentional operators; and it is not at all clear that it can be made to doso. At the very least, then, the onus is on the proponent of the strategy to showthat it can.

Moreover, there are general reasons for supposing that it cannot. Intentionaloperators would seem to provide insuperable difficulties. Take an operator suchas Johnbelieves that,B. Howarewe tohandleB?The only obvious suggestionis that (B)+ isB (+), and this will clearly not work. Even logical equivalencedoes not guarantee equivalence of belief: one can believe without believing, for example. Hence, even if and + express the same situation in somesense, one could have B without having B+. The trouble is that belief andsimilar mental states are intentional, directed towards propositions/sentences.These seem to be integral to the intentional state in question, and so cannot beeliminated if we are to describe the intensional state. (Indeed, the same is trueof all conceptual revisions. If peoples thoughts are individuated in terms of oldconcepts, one cannot describe those thoughts if the concepts are junked.)

One possible suggestion at this point is simply to take (B)+ to be B it-self. Of course, if we leave it at that, we have not rid ourselves of the dialetheicconcepts, since these are still occurring in the language. But we might just treatB as a new atomic sentence a single conceptual unit. The problem with thisis clear. There would be an infinite number of independent atomic sentences, andthe language would not be humanly learnable. The construction would fail thepracticality test. And even then, given that the language contains other standardmachinery, there would still be expressive loss. For example, we would no longerhave a way of expressing things such as ( B) or (B ).

Nor is this just a problemaboutmental states. It applies to intensional notionsgenerally. Thus, consider the statement That confirms that. This is not invari-


ant under extensional equivalence. Let us make the familiar assumption that allcreatureswith hearts are creatureswith kidneys. Consider the information that1,. . . , are creatures of kind with a heart. This confirms the claim that all crea-tures of kind have a blood circulation system. The information is extensionallyequivalent to the information that 1, . . ., are creatures of kind with kidneys.But this information does not confirm the claim that all creatures of kind have ablood circulation system.

3.3 Expressive Loss

But worse is yet to come for the conjecture that we can, in theory, always replacean inconsistent language with a consistent one. Suppose that the project of show-ing that every situation describable in the old language can be described in thenew can be carried out, in the way just illustrated or some similar way. This is notsufficient to guarantee that there is no expressive loss.

Consider the naive notion of set again. This is characterised by the schema:

Abs { : ()} ()

which gives rise to inconsistency, as we have noted. Let us suppose that it werereplaced with different notions in the way that we have just considered. Thus, wehave three predicates , , and, where is expressed by .Let us write this as . Given the above schema, we have:

Abs { : ()} ()

and in particular:

{ :

}

Substituting { : } for gives us Russells paradox, as usual. We have not,therefore, avoided dialetheism. Why is this not in conflict with the discussionof the last section? The reason is essentially that the procedure of driving nega-tions inwards, and finally absorbing them in the predicate, produces a languageinwhich there is no negation. The instance of Abs that delivers Russells paradoxcannot, therefore, evenbe formed in this language, since it contains negation. The

22 More generally, relations relevant to confirmation are well known not to be invariant underlinguistic transformations. See, e.g., Miller (1974).23 This is observed by Batens (2002), 132. See also his (1999), 272.

24 | Graham Priest

procedure guarantees, at best, only those instances of Abs where() is positive(negation-free).

We face a choice, then. Either dialetheism is still with us, or we lose the gen-eral schema thatwehadbefore. But the Schema effectively characterizes the naiveconcept of set membership. So if we go the latter way, even if every sentence of theold languagehas anequivalent in thenew, there is still an expressive loss.Wehavelost a conceptwhichwehadbefore.Wehave lost the ability to express arbitrary setformation. Not everything that could be expressed before can still be expressed.

This provides us with an argument as to why we may not always be able toreplace an inconsistent language/conceptual scheme with one that is consistent.There are cases where this can be done only with conceptual impoverishment.That one may achieve consistency by throwing away a concept is not surprising.The notion of truth gives rise to contradictions. No problem: just throw it away!But such a conceptual impoverishment will leave us the poorer. If we were throw-ing away useless things, this might be no loss; but we are not. All the dialetheicconcepts in 2.2 had a use, and so were useful.

Indeed, the concepts may be highly useful contradictions notwithstanding.Thus, for example, the ability to think of the totality of all objects of a certainkind closely related to our ability to quantify over all such objects, and to formthem into a set would seem to be inherent in our conceptual repertoires. It playsan essential role in certain kinds of mathematics (such as category theory), andin our ruminations about the way that language and other conceptual processeswork. But abilities of this kind drive us into contradictions of the sort involved indiscussions of the limits of thought. We could throw away the ability to totalisein this way.Maybe this would restore consistency, but the cost would be to cripplethe kind of mathematical and philosophical investigations that depend on it. Todo so simply in the name of consistency would be like doing so in the name of anarbitrary and repressive government diktat.

The situation is not to be confused with that in which the concept of phlogis-ton was replaced by that of oxygen. We did not, in fact, dispense with the con-cept of phlogiston. We can still use it now. What we rejected was the claim thatsomething satisfies this notion. We now think that nothing does; in consequence,the concept is of no scientific use. (Essentially the same must be said about thenaive notion of set, by defenders of consistent set theories such as.)

Actually, it is not even the case that one can give up a concept in the wayrequired. If we have the conceptual ability to totalise, in what sense can this begiven up? One can refuse to exercise the ability, but this would seem to get us

24 A detailed discussion of all this can be found in Priest (1995).


nowhere. (It would be like solving the Liar paradox as follows. : Suppose I saythat I am lying. : Dont.) If you have the ability to think certain thoughts, youcannot, it would seem, lose this without some kind of trauma to the brain, causedby accident or senility. And if this is the case, the recommendation to change ourlanguage/concepts fails the practicality test in this most fundamental way.

4 ConclusionThis has been an essay about contradictory concepts, concepts which generatedialetheias. Assuming there to be such things, two further claims are tempting.1: Dialetheias are merely in our concepts; there are no such things as contradic-tions in re. 2: Dialetheias may always be removed by revising our concepts. Wehave seen that there are grounds for resisting both of these suggestions. I thinkthat Hegel would have been delighted; but that is another matter.

ReferencesD. Batens, Paraconsistency and its Relation to Worldviews, in: Foundations of Science 3,

25983, 1999.D. Batens, In Defence of a Programme for Handling Inconsistencies, in: J. Meheus (ed.), In-

consistency in Science, Dordrecht, 129150, 2002.C. Chihara, The Semantic Paradoxes: A Diagnostic Investigation, in: Philosophical Review 13,

117124, 1979.E. D. Mares, Semantic Dialetheism, in: Graham Priest, JC Beall, B. Armour-Garb (eds.), The

Law of Non-Contradiction, Oxford, 264275, 2004.Dialetheism, Concepts and the World, in: J. Meheus, E. Weber, D. Wouters (eds.), Logic, Rea-

soning and Rationality, Springer, forthcoming.D. Miller, Poppers Qualitative Theory of Verisimilitude, in: British Journal for the Philosophy

of Science 25, 160177, 1974.G. Priest, Two Dogmas of Quineanism, in: Philosophical Quarterly 29, 28930l, 1979.G. Priest, Beyond the Limits of Thought, Cambridge (2nd edition: Oxford 2002), 1995.

25 Versions of this paper, or parts of it, have been given under various titles at a number of philos-ophy departments and conferences over a few years: the University of Melbourne, the Universityof Queensland, the Australasian Association of Philosophy (Australian National University), theUniversity of Chapel Hill (North Carolina), the University of Connecticut, theMassachusetts Insti-tute for Technology, Logic and Reality (Universities of Namur and Louvain la Neuve), the Univer-sity of Gent, the City University of New York (Graduate Center), the Fourth Cambridge GraduateConference on the Philosophy of Logic and Mathematics, and Contradictions: Logic, History, Ac-tuality, Technische Universitt, Berlin. I thank the participants for many lively discussions andhelpful comments.

26 | Graham Priest

G. Priest, Review of Soames (1999), in: British Journal for the Philosophy of Science 52, 211215, 2001.

G. Priest, In Contradiction: A Study of the Transconsistent, 2nd edition, Oxford (1st edition:Dordrecht 1987), 2006a.

G. Priest, Doubt Truth to be a Liar, Oxford, 2006b.G. Priest, Logic Disputes and the a Priori, forthcoming.G. Priest, JC Beall, B. Armour-Garb (eds.), The Law of Non-Contradiction, Oxford, 2004.G. Priest, R. Routley, The Philosophical Significance of Paraconsistent Logic, in: G. Priest,

R. Routley, J. Norman (eds.), Paraconsistent Logic: Essays on the Inconsistent, Munich, ch.18, 1989.

B. Russell, Vagueness, in: Australasian Journal Philosophy 1: 8492; reprinted in: R. Keefeand P. Smith (eds.), Vagueness: A Reader, Cambridge 1999, 61-8, 1923.

S. Soames, Understanding Truth, Oxford, 1999.

JC BeallRapunzel Shaves Pinocchios Beard

1 A Rapunzel StoryStories of Rapunzel are many and varied. A newly discovered one, not publiclyavailable before now, has emerged. The new story involves more gluttiness thanmost known stories indeed, a far-reaching spread of empirical-world gluts (i.e.,sentences that, according to the story, are both true and false of the empiricalworld of the story). The chapters of the new story are many but much the same;each revolves around Rapunzels remarkable nature as a (what else to call it) fal-sity device. Example: according to the chapter on grass growing, we have a minorvariation of the main theme, namely, the given chapters Rapunzel fact:

Grass is growing iff what Rapunzel says is false.

And the chapter ends in grassy gluttiness:

Rapunzel says that grass is growing.

Since, by the falsity principle, what Rapunzel says is false iff its negation is true,the world of the story is one in which, lo and behold, grass both is and is notgrowing a glut involving grass growing!

What the story holds in glutty intrigue, it lacks in variation. The chapter onbirds flying, like all the chapters, continues the theme:

Birds are flying iff what Rapunzel says is false.

And the chapter ends with Rapunzels saying that birds are flying and so moregluttiness results, involving birds flying and not. And so on: a glut in the storysworld for any claim at all including, traditionally, both climbing and not climb-ing long braids of beautiful but bewitched hair, etc.

1 For literary theorists quick to renounce the lawof excluded, letme note that the givenRapunzelstory is post-post-modern: it enjoys the truth of for all in the story.

28 | JC Beall

2 Paradox?The question at hand is beyond literarymerits of the newRapunzel tale. The ques-tion, in light of recent debate (Beall (2011), Eldridge-Smith (2012)), concerns philo-sophical weight. Specifically, the question is: are these new paradoxes?

The new Rapunzel tale is not paradoxical at all. To use Eklunds phrase(2002), theres no pull towards thinking that we have apparently true premisesand apparently valid reasoning backing an apparently false conclusion. Instead,we have a story in which craziness happens. The only apparently true premisesare only apparently true according to the story; and that doesnt make for paradox.

But doesnt the story make for at least possible paradox? Doesnt it carve outsome space of possibility of which such broad, empirical-world gluts are true? Inshort: if we can make even some sense of the story (even if we cant imagine itin the sense of imaging), ought we not conclude that theres a possible world let me just say world of which the story is true? No. Consider a story in whichGoldbachs conjecture is true, and another one inwhich it is false. Both are storieswe can make some sense of (broadly speaking), even though one of them is trueof no world whatsoever.

The point, while perhaps largely uncontroversial, bears repeating: truth in astory is insufficient for truth at some world insufficient, that is, for possibility.Stories are free, subject only to our whims; possibilities are beyond us.

3 Against Pinocchios BeardPlatos beard, according to Quine (1948), purports to take us from (actual) nonex-istence to (actual) existence. Eldridge-Smith (2012) gives us a variant, what wemight call Pinocchios beard, which purports to take us from (impossible) fictionsto (possible) worlds. In particular, the story of Pinocchios paradox is one that issupposed to carry metaphysical import (Eldridge-Smith (2011)). But I dont see it;it is no different, in the end, from the canvassed story of Rapunzel, which showsthe absurdity of Pinocchios beard.

To be fair, Eldridge-Smith isnt relying on a general fiction-to-possibility prin-ciple; his sights are focused on a particular group of theorists, whom he thinksare subject to such a principle. In particular, Eldridge-Smiths concern are conser-vative glut theorists (Beall (2009)) who think that there are no empirical-worldgluts only semantic ones, only spandrels of truth. And his charge, at bottom,is straightforward: since target glut theorists take the Liar paradox to motivategluts, they ought to take every version of the Liar paradox to motivate gluts.

Rapunzel Shaves Pinocchios Beard | 29

Eldridge-Smiths argument (2012) is from a uniform-solution principle: sameparadox, same solution. But what to make of the argument? While Im happy togrant, at least for present argument, the uniform-solution principle, I reject theclaim essential to Eldridge-Smiths argument that the Pinocchio paradox isthe same basic paradox as the Liar. The reason is as above: I reject that Pinoc-chios paradox is even paradoxical. What is required is apparently acceptablereasoning taking us from apparently acceptable premises to an apparently un-acceptable conclusion. But even the conclusion of Pinocchios paradox is ap-parently acceptable: its true that, according to the story, a nose both grows anddoesnt. (Of course, I cant imagine such a story, in the sense of imaging it; but thesurface structure and its implications are clear enough.) Not only do we have anapparently acceptable conclusion (about what, according to the story, goes on inthe world); the premises themselves are apparently acceptable only if embeddedin an according to the story operator (which, if removed, undermines the appar-ently acceptable reasoning).

In sum, to have something count asa version of the Liar paradox, it needs to beparadoxical; it needs to at least appear to imply something about truth or possibletruth. If there were a clear bridge between fiction and possible truth, then Pinoc-chios beard, employed by Eldridge-Smith (2011), would carry whatever weight issupported by the bridge. The trouble, dialectically, is that Eldridge-Smith (2012)relies on the alleged paradoxicality of his Pinocchio story to build such a bridge.But the alleged paradoxicality is only that: namely, alleged paradoxicality.

One might persist in the thought that, at the very least, the Pinocchio para-dox is liar-like enough to put pressure on glut theorists to acknowledge empiricalgluts, gluts beyond the merely semantic liar gluts (the spandrels of truth). Onemight persist, in short, in trying to growPinocchios beard.What Rapunzel shows,I submit, is that theres something deeply flawed in the reasoning behind Pinoc-chios beard: it leads to all-out absurdity.

The problem with Pinocchios beard is that it demolishes the fictionpossibi-lity divide. Stories are free: make them as you please. Possibility is different; pos-sibility is independent of our creativity. There is no reason for glut theorists (ofany stripe, conservative or not) to think otherwise.

2 Im grateful to Aaron Cotnoir, Michael Hughes, Andrew Parisi, and Ross Vandegrift for feed-back. Im particularly grateful to Elena Ficara for lively conversation at the conference whichoccasioned this (or a variant of this) work, and also grateful for her patience with my delayedsubmission.

30 | JC Beall

ReferencesJC Beall, Spandrels of Truth, Oxford, 2009.JC Beall, Dialetheists against Pinocchio, in: Analysis 71, 689691, 2011.M. Eklund, Deep inconsistency, in: Australasian Journal of Philosophy 80, 321331, 2002.P. Eldridge-Smith, Pinocchio against the dialetheists, in: Analysis 71, 306308, 2011.P. Eldridge-Smith, Pinocchio beards the Barber, in: Analysis 72, 749752, 2012.W. V. O. Quine, On what there is, in: Review of Metaphysics 2, 2138, 1948.

Franca DAgostiniParadoxes and the Reality of Contradictions

1 IntroductionIn a plausible reconstruction, dialetheism, that is the perspective according towhich for some , is true and is also true, is based on some simple theses:

1. the inviolability of the Law of Non-Contradiction (LNC) is not justified bypseudo-Scotos argument, as the argument does not work in paraconsis-tent cases (so the justification is circular)

2. it is not justified either by Aristotles elenctic (discussive) argument, be-cause this does not work for moderate paraconsistency, which admits ofonly some contradictions

3. there are no reasons to accept LNC as inviolable rule, except for the twomentioned in 1 and 2, soLNC is adoptedby classical logic as such,withoutjustification, as a fundamental rule of rationality

4. there is another, primary, rule of reason, and it is the rule of evidence,in virtue of which, if there is some evidence of contradiction, it is rationalto accept it.

They are all acceptable, I suppose. More controversial is the fifth:

5. as there is some evidence of contradiction, in virtue of 14 we should ac-cept that for some , is both true and false: LNC sometimes fails.

In this reconstruction, the crucial point is the evidence of contradictions; or also:the effective occurring of true contradictions, somewhere, in some cases. Usually,dialetheists mention as paradigmatic cases semantic and set-theoretic paradoxes;sorites (more generally: borderline properties); contradictions related to change,motion and infinity; dilemmas and other epistemic inconsistencies.

And yet, in all these cases it is not so clear if the C (contradiction) involvedis real as such. Even in case of micro-physical inconsistencies, one could alwayssay the evidence of C is due to the language of modern physics. It is often con-tended that the modern theory of matter, instantiated by the Standard Model of

1 As I will explain later, I adopt here the empiricist notion of evidence aswhat is grasped as such.So if is evident to A, then A believes that , and is made true by the fact .

32 | Franca DAgostini

Elementary Particles, Quantum Mechanics, and the Special Theory of Relativity,is successful, but not properly true.

All this would mean that semantic dialetheism has a certain primacy, like itwas somehow admitted by Hegel himself. But if it is so, to what extent shouldwe say that semantic contradictions are real, existent, or objectively evident? InHegels use of the term reality (as far as I can say) surely yes, they are, but isthis actually the notion of reality involved in the evidence of Cs, dialetheisticallyspeaking?

Evidently, on this point dialetheism bumps into wide philosophical (namely:epistemological and metaphysical) problems: what really counts for evident?What counts for real? I suggest a good approach to the theme is given by alethicrealism, the realism based on truth. In a word, Cs are alethically real, and thequestion about the reality of Cs turns into the question: is alethic realism onlysemantic, or alsometaphysical realism?

What is interesting in alethic realism (which I interpret as ametaphysical ver-sion of semantic realism), is that it may convey a realistic perspective, withoutrestricting the realm of facts to a specific sort of reality. Facts, intended in alethicperspective, are what makes true, which means that there might be semantic aswell as strictly metaphysical facts, and the latter may be de se facts (such as thefact that I feel tired now) as well as de re (the fact that here and now there is acomputer). To accept this, there is no need to renounce metaphysics (the theoryof what real facts are like); and there is no need to accept that the world is trivial,and that the answer to thequestion what does exist? is everything. What is onlyneeded, is to enlarge the realm of what can make true. In this sense, the perspec-tive is well adaptable to noneism, which not by chance is dialetheists traditionalmetaphysics.

2 This is a controversial point, yet, there are positive suggestions concerning some sort of seman-tic dialetheism in Hegels work. See on this Ficara (2013).3 Hegels idealism meant that the distinction between objects that are experienced and mereobjects of thought has no particular ontological significance. Priest (1987), 3.4 Themost eminent exponent of alethic realism, also called Cornell realism, wasWilliamAlston.See Alston (1993), Alston (1996), and Alston (2002).5 One including the existence of independent, real, facts; the existence of a unique true descrip-tion of these facts; and our possibility of giving sometimes a true description of them, and evalu-ating the truth or falsity of a certain description (see later).6 I tried to show this in DAgostini (2009b) and DAgostini (2012).7 See Priest (2005).

Paradoxes and the Reality of Contradictions | 33

2 Paradoxes and the Evidence of Contradictions2.1 What Is a True Contradiction?A contradiction occurs when is true, and is true as well (where is aproper truthbearer). In realistic perspective, is true means: is made true by somefacts or states of affairs. So ideally, there is a real contradiction when two contra-dictory states of affairs simultaneously occur in the same world, at the same timeand under the same respect. Russell, in The Philosophy of Logical Atomism sug-gested that a same fact makes true the proposition , andmakes false . Sothere would be only one fact, ultimately. The notion of falsemaking is discussed,but the point remains. Logically, it involves the semantics of negation. But meta-physically, and epistemically, we have to reflect on what could be a situation, or acase, whichmakes true: are there two cases, or only one? Is truly evidencethe evidence involved?

In empiricist perspective, Humean as well as Kantian, evidence stands forthe action of reality on an epistemic agents system of beliefs. I have the evidencethat iff a certain real fact causally acts on my cognitive means, making me(truly) believe that . In this account, evident implies true, and true impliesreal: if is evident, then is true, and if is true, then is real. So the notionof empirically evident C implies true and real C: the fact involved is not apparent,and it is not the result of some mistake either.

Empirical evidence may fail, as in the process from (state of affairs) to (belief, or proposition) something may go wrong, so I grasp , but my formed be-lief (for some reasons) becomes, or. Surely, in that case, we cannot saywasstrictly evident to me. My evidence was only believed and not true evidence. Andevidently, this also regards Cs.

Cases of only apparent Cs are frequent, in the usual practice of believing andknowing. Sometimes, what seems to be is in fact something else: becausesay the two are not mutually exclusive, or are not jointly exhaustive, or do notjointly occur (so the same translation in terms of was wrong, actually).In this case, we would say the C is to be dissolved. Some other times, we come

8 See Priest (2006), ch. 2.9 Substantially confirmed by Burge (2010).10 The idea that Kants conception of truth was not realist is a wrong idea: I leave this out, butfor a wider discussion see DAgostini (2012).11 Dissolutions of Cs, as I will explain later, are also to be adopted in those cases in which theC is apparent, as it conceals under-determination, so the truth value glut is in fact a truth valuegap.


to know that in fact only one of the two terms was in fact true. Then we simplysolve the contradiction. Finally, sometimes the apparent C is in fact due to someglitch or mistake, concealed somewhere, or to some false assumption. We get ridof the false assumption, and the C disappears. And this is the reduction to absurd,or more generally: the epistemic reduction of Cs.

A true C is epistemically given as suchwhen all the three procedures fail. Andusually, as ideally the three are ordered (if solution fails, try dissolution, if dissolu-tion fails try reduction), we would say evident Cs are irreducible: the main logicalmethod fails.

2.2 What Is a True Paradox?According to the classical definition (Sainsbury (1995)), a paradox is the appar-ently unacceptable conclusion of an apparently acceptable argument. Acceptablearguments are sound, whichmeans they have true premises, and valid inference.So there are five possibilities: 1. the premises are only apparently (not really) true;2. the inference is only apparently (not really) valid; 3. in fact, premises are un-true and the inference is not valid (both 1 and 2 occur); 4. the conclusion is onlyapparently (not really) unacceptable; 5. premises are truly true, inference is trulyvalid, and yet the conclusion is unacceptable (false).

Only in the last case we have a true paradox. In the other cases, the evidence(in the sense specified above) fails, because there is not really a paradox, rather:a fallacy (cases 1, 2, 3), or what Quine (1962) called veridical paradox: we thoughtthe conclusion was false, but it was true, actually (case 4), and when we come toknow it is true, the paradox disappears.

Now if we ask: what is unacceptable? The reasonable answer is: (if is atruthbearer) is unacceptable iff we know (or believe) it is false, or contradictory.But epistemically, is visibly false when it contradicts something we know (be-lieve) for sure: that . So, ultimately: is unacceptable when it is contradictory;what is unacceptable is the C.

12 This may happen in two ways, as I specified in DAgostini (2009b), corresponding to the twomain kinds of paradoxes: sorites and antinomies. The first is ( ) and ( ),both from a proposition and from its negation we get a C. The second is , and ,from a proposition we get its negation, and vice versa.13 Lycan (2010) has recently proposed the idea that paradoxes come in degree, and it is fairlysimilar to my point: on this, and for the definition of paradox as resistant contradiction seeDAgostini (2009b), first part.14 A third possibility is that is unacceptable insofar as meaningless. But in this case, onecouldnt say is properly a truthbearer.


Now we have here a more restrictive definition: a true paradox, properlyspeaking, is a contradiction. And the evidence involved, is somehow unquiet ev-idence, of the kind But this isnt how it is we say. Yet this is how it has tobe! But this is how it is , as Wittgenstein wrote in the Philosophical Investi-gations.

When do we actually have this sort of unquiet evidence? Graham Priest, in InContradiction and elsewhere, mentions the following paradigmatic cases of trueCs: 1. logical (semantic, set-theoretic) paradoxes, conveying the schema ;2. transition states, or borderline cases (change, motion, borders and cut offs);3. contradictory results due to the notion of infinity; 4. epistemic undecidable sit-uations, such as the case in which I believe that and I believe that , but I cometo believe that ( ); 5. normative conflicts, to say conflicts occurring in nor-mative systems, when we have apparently consistent laws, which in some appli-cations reveal not being simultaneously observable.

This somehow confirms that effective-real contradictions are (conveyed by)what we usually call paradoxes: antinomies, sorites, infinity paradoxes, epis-temic paradoxes, and those practical paradoxes that are dilemmas. Now thequestion is: are all these Cs effective, which is to say: evident, and evident insofaras real and truly irreducible?

As to borderline situations (borders, cut offs, transition states, etc.) there isno doubt, or so it seems. Evidence is against LNC. The example Priest most fre-quently mentions is: when I am on the threshold of a room, I am in and out at thesame time. Whether we truly have double phenomenic evidence or not, in thesecases, is a controversial issue, because usually these aremicro-phenomena, notcaptured by our perception. But this only holds for what Tyler Burge has called,in his monumental work on the Origins of Objectivity, individual representation-alism: the idea that cognition is a strictly individual phenomenon. Burge notes:there is a structural difference between perception and propositional attitudes,but the representational content of perception is a veridical condition, thatwhenmet by an appropriately sensed subject matter is veridical (Burge (2010), 539).So we would say: border-contradictions seem to be evident insofar as real, and

15 Wittgenstein (1985), 112113.16 Priest (1987).17 Otherwise, the idea that paradoxes ultimately are epistemic or logical structures conveying Cs,is implicit in most of the literature about non classical logic, and it is fairly reasonable to admitthat we call paradoxical situations that are undecidable by over-determination.18 See Priest (2006), 5764. On the possibility or impossibility of perceptive Cs, see Sorensen(2003) and Sorensen (2008).


so because the conception of them is veridical, even if we cannot grasp them byconscious perceptive means.

In other cases, and namely as to epistemic-doxastic Cs (I believe that and Ibelieve that ), or as to liar-like Cs ( iff ), there might be some doubts. Thereare Cs, but one may say they are only apparent, or are the result of some glitch ormistakenassumptions, or occur in a special systemor afield (say: a semantic field,intended as fictional or conventional), which we would not properly call real, assuch.

3 Paradoxes in Epistemic PerspectiveThe objective occurring of epistemic Cs is controversial, basically for two reasons:because natural epistemic systems are fragmented, or intrinsically heteroge-neous, or irreducibly dynamic (so the two contradictory terms do not properlyoccur at the same time, and in the same context or field); or because very of-ten what seems to be (or what passes off as) over-determination is in fact under-determination.

The first account somehow legitimates non-adjunctive approaches, that isthose, which admit that and may occur, but not jointly (the basic rule ofAdjunction fails: you get , but not ). Knowingly, some interpreta-tions of Hegels logic resolutely deny that it is a paraconsistent logic, stressingthe dynamic nature of Hegels dialectics. In Sorensens perspective epistemic in-fra-subjective systems may be inconsistent, as beliefs are so to say produced bydifferent bureaus in a firma: the results obtained by one bureaumay be ignoredby another.

However, I think the main point with epistemic Cs is the second explanation:that often, what epistemically seems to be a glut is in fact a gap, and over-determi-nacy is the superficial effect of underlying under-determinacy. Lets see now someexamples.

19 See Varzi (2004).20 Sorensen (2001)hasproducedanelenctic argument to the effect that epistemic contradictionsare somewhat unavoidable. The argument can be discussed, but I do not treat the problem here.


3.1 There Is No Preface Paradox

Here is a version of the preface paradox:

= In this list (in this book) there is some false sentence.

In Priests account (1987), the author of the preface presumably believes each sen-tence in the book is true (insofar as she has written it), then she believes that. But she says that in the book there are some false sentences, so shedoes not believe that ( ). Contradiction.

Beall and Restall (2006) suggest what we would call a dissolution by under-determination. Suppose theauthor believes each sentenceof thebook is very likelytrue, so, say: is 0.8 true, is 0.9 true, and is 1 true. Now it happens she is alsoentitled to believe that there might be some false sentence in the book, as theconjunction will totalize 0.8 0.9 1 = 0.72. More precisely, letssuppose the authors epistemic standard is:

(ES) I accept iff (|) 0.8

I accept if and only if the probability of given the evidence is 0.8 or more.Given the evaluations above, the author must rationally accept each proposition, , , as each singularly satisfies (ES), but she cannot accept , whichis 0.72, and therefore she alsomust accept as true (0.8 or more). So there is no Cin the authors beliefs.

Surely, the probabilistic account of truth is a positive resource for epistemol-ogy. Rather, it is the bestway to connect logic and epistemology.What Christensen(2004) has called gradualistic model of beliefs, as opposed to binary modelis reasonable, and well captured by probabilistic calculus. In this case, Beallsand Restalls approach also gives good arguments in favour of non-adjunctive ac-counts, because what fails, in the authors beliefs, is namely the adjunction rule.

So we see that epistemic gradualism is able to dissolve epistemic Cs, at leastin case of intra-subjective Cs, such as those conveyed by the preface paradox, oralso: the lottery paradox.


3.2 Reliability Conflicts

In the preface paradox, there is no doxastic C: the author does not exactly knowwhat is the false sentence in the book (or to what extent the proposed sentencesare true), so he or she gives asymmetric evaluations. But are there cases in whichA positively 0.8 believes that and also 0.8 believes that ?

In principle, thismayhappen in twoways: because there is a conflict betweenthe sources of belief, or because there is a real conflict between the pieces of evi-dencewe get.

An example of the first case is the reliabilist paradox. Suppose I am perfectlyconfident in both the Catholic Church and scientists. But it happens the CatholicPope says that , for instance: a 14 day human proto-embryo is a human being;and scientists say that not: a 14 day humanproto-embryo is not a humanbeing.What should I believe? Also in this case the main point is under-determinacy: theconflict arises simply because I do not know what is true, as we simply lack ashared definition of what is a human being, and I have not enough information tojudge which the preferable definition is.

If the epistemic constraint conveyed by the two sources is strong enough, Imay be possibly led to give a value 0.8 or so and a value 0.8 as well. And itshould benoted that beliefs are contrastive: if I 0.8 believe that, then I 0.2 believethat. So as amatter of fact, my evaluations are: both 0.8 and 0.2 for and both0.8 and 0.2 for .

However, this hardly captures the effective dynamic of beliefs of an epistemicagentwho is cultivating some ideas regarding 14-day humanproto-embryos. First,possibly the two sources will act one against the other, producing some global di-minishing of the reliability of both. Second, if the agent has to take some deci-sions, she will produce some rational re-arrangement of the comparative reliabil-ity of one or the other source. In other terms, if I (Catholic, but fond of science)have to vote for a law concerning scientific research about stem cells, stated theopposite judgements of my sources of belief, I will diminish my confidence in theabsolute reliability of the Pope (or science); or else, I will ignore the metaphysicalquestion concerning the effective being human of the embryo, and I wont voteon the basis of metaphysical considerations. So the contrastive belief concerningone term will increase, or both and will result not designed. In any case,there wont be any true doxastic C.


3.3 Conflicting Pieces of Evidence

In case of epistemic Cs conveyed by contradictory evidence, the in ES is double.In virtue of my evidence, P() = 0.8 (or more), and P() = 0.8 (or more) as well.Fermi-Hart paradox is a good example. The proposition E: Extraterrestrial intel-ligent people do exist is typically over-evaluated, in virtue of the two arguments:

(1) If advanced extraterrestrial civilizations existed, they would have con-tacted usThere is no clear evidence that such a contact has taken placeTherefore: it is highly improbable that civilizations of this kind do exist.

(2) The universe is huge, or even infiniteTherefore: it is highly probable that advanced extraterrestrial civiliza-tions do exist.

The two arguments seem to be compelling. We have the evaluation = 0.9 orso, in virtue of (1) and = 0.9 or so, in virtue of (2). Defenders of E will fiercelyoppose the upholders of , and vice versa. But what about the third person, theepistemic observer A, who takes into account both (1) and (2)?

Surely, a rational A wont admit ( ) = 0.81, to say: that it is highlyprobable that ETs do exist and do not exist at the same time. Rather, she will thinkthe issue is fundamentally under-determined. Possibly, the contrastive methodwould work, because the two arguments will act one against the other, yielding aglobal diminishing of the belief involved in the

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