A. Logic A2. Philosophical Logic Proof Theory, Meaning and Paradoxes Luca Tranchini Wilhelm-Schickard-Institut University of Tübingen, Germany

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General description of the topic

The idea of characterizing the meaning of linguistic items and, in particular, of logical constants in terms of the role they play in inferential practices has lead to semantic pictures alternative to the truth-theoretic one developed by the Frege-Tarski tradition. In spite of the efforts initially devoted by authors such as Michael Dummett (1963, 1978, 1991) and Dag Prawitz (1977, 1987) to develop knock-out arguments against the rival view, the claims of a supposed superiority of inferentialist approaches over the truth-theoretic account of meaning are, as yet, far from being settled. The aim of the proposed symposium is that of suggesting a different way of framing the contention. Namely, given a theoretical problem with which both approaches are expected to deal, does one of the two have more to say than the other? Among many possibilities, we choose to engage the theme of semantic paradoxes. The choice is motivated by the belief that a reflection on the nature of paradoxes may prompt the inferentialist perspective to look for new research directions and that it may offer a fruitful ground on which inferentialism may be compared with the more traditional approach to semantics. Paradoxes have accompanied the development of modern logic—and of the semantic theories built thereon—ever since its birth. In the inferentialist case, one could say that a paradox tied to such semantic picture arose even before the formulation of the semantics itself. So was the case when Arthur Prior (1960) showed that to equate the meaning of a linguistic item with its inferential role could easily yield to the problems prototypically represented by the nowadays famous connective ‘tonk’. The response was that several authors devoted efforts in order to give the inferentialist idea more solid basis. Although half a century of work produced a family of related constraints (such as normalization, conservativeness, uniqueness, see among others, Belnap (1962), Dummett (1973) and Read (2000)) that the set of inference rules of a logical system should satisfy in order to avoid bad-behaving features, there 1

is still no general agreement on which constraints, if any, can be imposed on a given set of rules. The situation has indeed become worse due to the incredible development of proof-theoretic means of characterizing logical systems. However, although several types of formalism have been fruitfully developed, it is not always easy to formulate exactly corresponding criteria of meaningfulness in the different settings, in spite of equivalence results concerning the deducibility relations across the several systems. The result is that the attempt at characterizing what is meaningful risks of becoming a formalism-dependent issue. Furthermore, the class of logics which have proof-theoretic significance has sensibly enlarged. Either more implicitly or in a more explicit manner, in every type of formalism it is possible to trace some parameters governing structural properties of the deducibility relation. The degree of freedom in tuning such parameters yielded the characterization of a wide variety of sub-structural logics. In the light of this, it is natural to ask whether the philosophical paradigm at the core of the proof-theoretic version of inferentialism, born before the ‘substructural revolution’, should be revised in order to account for such developments. The contribution of Ole Hjortland to the symposium aims at clarifying the relationship between the sequent and natural deduction settings, by providing a general proof-theoretic semantic framework from which an original characterization of tonk-like operators can be given, thus offering a general analysis of what it is for a linguistic item to be, proof-theoretically speaking, paradoxical. Several strategies have been proposed in order to cope with ‘traditional’ semantic paradoxes such as the Liar Paradox and its relatives in the truththeoretic tradition. Most of them focused on restrictions that a semantic theory should satisfy in order to avoid paradoxical consequences, typical examples of which are the restrictions on the possibility of fully displaying self-reference. More recently, an alternative approach to paradoxes has been developed, aiming at an analysis of the principles needed to actually infer the paradoxical consequences from the semantic theory (Reinhart 1986, Brady 2006, Priest 2006, Field 2008, Beall 2009). The pursue of this strategy has awaken the inferentialist interest in these more traditional semantic paradoxes. For, it turned out that in order to get paradoxes at all, crucial is the appeal to structural principles governing deducibility. In his contribution, Luca Tranchini, considers in particular Neil Tennant’s proposal (1982, 1995) of viewing the proof-theoretic counterparts of paradoxes such as the Liar or Curry’s paradox as being characterized by the fact that, in a natural deduction formulation, the arguments leading to paradoxes fail to normalize. Interestingly, Tennant’s proposal looks like a bridge between the traditional conception of semantic paradoxes and the more typical proof-theoretic one inquired by Hjortland’s contribution, namely the one raised by bad-behaving connectives. For, to request that valid arguments must normalize is indeed the most basic strategy to rule out connectives such as ‘tonk’ as unacceptable. Hence, from the proof-theoretic standoint advocated by Tennant, it could be argued that the paradoxical status of a connective such as ‘tonk’ is structurally analogous to the one of, say, a Liar-like sentence. In his talk, Julien Murzi, will discuss recent views that focus on the role 2

played by contraction in formalizing paradoxes. In particular, he will focus on one of the corollaries of Hartry Field’s (2008) truth-theoretic approach to paradoxes, according to which the validity of arguments cannot always be interpreted as the transmission of truth from the assumptions of an argument to its conclusion. Although this claim has been originally presented as induced by Curry’s paradox, Field has recently argued in its favor in a more general fashion, by presenting the irreducibility of validity to truth-transmission as a direct consequence of GËEdel’s ˛ second incompleteness theorem. According to Schroeder-Heister (Schroeder-Heister 2009, Schroeder-Heister forthcoming), one of the dogmas of standard semantics is the idea that the notion of logical consequence, which underlies the one of the validity of an argument, should be analyzed in terms of transmission, or preservation, of a more basic notion (truth, provability). Although from a different perspective, he also suggests, like Field, that the transmission view of consequence should be rejected. In his contribution, Tranchini will try to apply this idea to Dummett and Prawitz’s natural deduction based proof-theoretic approach. In particular, he argues that, by relaxing one of its fundamental assumptions—namely, that valid arguments must always normalize—an account of at least certain types of semantic paradoxes could be given. On the other hand, Murzi’s line of argument aims at casting doubts on the viability of the revisionary approach to the semantic paradoxes. Murzi will argue that validity paradoxes suggest that, in order to preserve consistency, or non-triviality, it is not sufficient to weaken the logic of connectives such as negation and implication. Rather, one needs to weaken the structural rules of the logic, i.e. the rules that characterize the structure of the consequence relation. In particular, Murzi will suggests that a natural way of revisionary logicians to cope with the validity paradoxes, and with the semantic paradoxes more generally, is to weaken the structural rule of contraction. In his contribution, Murzi will raise some worries about contraction-free logics. His criticism will focus on the the difficulties of interpreting as rational the inferential practices subject to contraction-free logics. Without aiming at exhausting the theme, we hope that the symposium will at least suggest a few possible lines or research, along which a seemingly highly-promising field could be further developed. The talks will approach the theme from distinct perspectives: as a result, we hope that the complexity of the problems at stake could be best appreciated and that unexpected convergences may possibly be traced.

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Abstracts Proof-theoretic semantics in the substructural era Ole Hjortland Munich Center for Mathematical Philosophy, Ludwig-Maximilians Universität

Proponents of proof-theoretic semantics have in common that they offer an account of the meaning of logical constants in terms of inference rules. Specifically, they follow some remarks by Gentzen (1935) to the effect that formal rules in a proof-system are implicit definitions of logical constants. However, the proof-theoretic details vary from proposal to proposal. Whereas for example Dummett and Prawitz uses natural deduction as their vantage point, Hacking (1979) preferred sequent calculus with its its distinction between operational and structural rules. Hacking’s idea, which has later been pursued in for example Paoli (2007), is that whereas the operational rules characterise a logical constant, structural rules are properties of the deducibility relation. Borrowing a phrase by Nuel Belnap, the structural rules constitute “the antecedent context of deducibility”. Thus, Hacking requires that operational rules—the (implicit) definitions— are well-behaved in the sense that structural rules remain admissible after introducing them. For Hacking’s purposes, this means admissibility of cut, identity and weakening. These requirements on successful definitions are in a sense natural developments of those introduced in Belnap (1962): conservativeness and uniqueness. In fact, we know that there is a correspondence between normalization (and conservativeness) in natural deduction and admissibility of cut in sequent calculus (see for instance Pottinger (1977) and von Plato (2001)). Similarly, there is a connection between uniqueness (in Belnap’s sense) and admissibility of identity. Unsurprisingly, in both formal frameworks the philosophical motivation is at least partly to rule out rogue connectives such Prior’s tonk. But the framework for proof-theoretic semantics suggested by Hacking has its limitations. The framework is not only exclusively classical, it only allows for logics with so-called standard structural properties. In particular, he admits readily that the way in which the so-called deducibility facts are construed excludes a number of alternative logics. In particular, since admissibility of weakening is held to be a definitional feature of constanthood, there is no scope for logics where structural weakening is rejected, e.g. many relevant and substructural systems. Similarly, even though it never features as a topic in Hacking’s paper, since the relata of sequents are taken to be sets (rather than multisets), another structural rule—contraction—is built into the definitional framework. In other words, characteristically contraction-free logics, e.g. the family of linear logics, receive no attention. With the substructural era in proof-theory, philosophers have become increasingly interested in logics without a Tarskian consequence relation (i.e., a relation which is reflexive, transitive, and monotonic). Substructural logics have been applied in debates about vagueness and semantic paradoxes, but also for example in linguistics and belief revision. The present paper attempts

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to do justice to the importance of substructural logics by developing a prooftheoretic semantics that is sensitive to absence of structural rules. In particular, we want to give an account of so-called proof-theoretic harmony for natural deduction rules that is fine-grained enough to distinguish between additive (context sharing) and multiplicative (non-context sharing) connectives. This distinction, which is critical to both relevant and linear logics, is collapsed in classical logic precisely because of the presence of the structural rules. We start with a well-known approach to proof-theoretic harmony by formulating natural deduction elimination rules in so-called generalised form (see Schroeder-Heister (1984), Read (2000)). Such elimination rules have several advantages: They give all elimination rules a unified form; they are more closely related to sequent calculus than standard natural deduction rules; and since they always involve at least one discharged assumption, they are consistently sensitive to global discharge policies. The correspondences between natural deduction and sequent calculus mentioned above display a tight connection between discharge policies and structural rules. In order to develop an appropriate constraint for substructural connectives, we compare generalised-elimination harmony with a corresponding criterion in sequent calculus: definitional reflection (see Avron (1991) and Sambin et al. (2000)). We show that the substructural distinctions developed for solutions of reflectional equations for sequent calculus can be transferred to natural deduction. It will turn out that the refined notion of proof-theoretic harmony not only disallows well-known problematic connectives such as tonk and its dual, it is also sensitive to connectives that lead to triviality in the presence of certain structural rules, but which are otherwise consistent.

Validity and the Unprovability of Consistency Julien Murzi Munich Center for Mathematical Philosophy Ludwig-Maximilians Universität Tarski’s Undefinability of Truth Theorem confronts us with a familiar dilemma: if all sentences of the language can be named, one should either give up the naïve theory of truth, according to which (at least) all instances of the T-Schema (T-Schema) T r(pαq) ↔ α hold, or weaken the logic. Recent paracomplete and paraconsistent approaches to semantic paradox opt for the latter option; this is the price, we are told, for preserving the naïve theory of truth, on the one hand, and for being able to theorize about truth-in-L in L, on the other (Reinhart 1986, Brady 2006, Priest 2006, Field 2008, Beall 2009). In spite of their different treatment of the Liar, both paracomplete and paraconsistent logicians agree that Curry’s Paradox (a proof of α from a sentence γ which indirectly says of itself that, if it is true, then α) teaches us that any viable truth-theory should be robustly contraction free (‘rcf’, for short)—free, that is, of any connective satisfying the following three conditions (Restall 1993): 5

C1. α → β ` α β; C2. α, α β ` β; C3. α (α β) ` α β, where → is a detachable conditional occurring in the T-Schema. In this paper, I assume for the sake of argument the viability of the revisionary approach to semantic paradox, and focus on a recently much-discussed corollary of the approach, viz. that, on pain of triviality, one cannot say that valid arguments preserve truth, where truth-preservation is a conditional claim whose consequent is itself a conditional (Field 2006, Beall 2007, Field 2008, Field 2009, Beall 2009): (VTP) If an argument is valid, then, if all of its premises are true, so is its conclusion. The argument is, in essence, Curry’s Paradox: VTP, together with the claim that modus ponens is valid, yields a ‘proof’ of α. Curry’s Paradox is blocked, but its solution comes at a price: VTP must go. Hartry Field, a leading rcf theorist, has recently argued that the argument against VTP can be bolstered, and made independent of Curry’s Paradox. In Field’s view, the failure of VTP is a consequence of Gödel’s Second Incompleteness Theorem (Field 2006, Field 2008, Field 2009). For assuming that all the axioms of our best theory are true, one could run, on the further assumption that our inference rules are valid in the sense of VTP, an induction argument to the conclusion that our theory is sound, and hence consistent. But this is something Gödel taught us to be impossible, if our theory is consistent. Field concludes from this argument—the Consistency Argument, as he calls it—that validity does not satisfy VTP: there are valid arguments with true premises and untrue conclusions. My first main claim in the paper is that the logic of the revisionary approach to the paradoxes, i.e. the logic of paradox, as it is sometimes called, is neither paracomplete nor paraconsistent. As I show, it follows from a validityinvolving version of Curry’s Paradox, which I call Validity Curry, that rcf theories cannot contain a validity predicate, on pain of triviality. That is, rcf theories cannot express validity, at least if ‘valid’ is taken to satisfy what we may call the V-rules: V-I

α1 , . . . , αn ` β V al(pαq, . . . pαn q ∴ pβq) . Val-E V al(pαq, . . . pαn q ∴ pβq) α1 , . . . , αn ` β

Validity can be expressed, on the other hand, that is, Val-I and Val-E can be consistently retained, if we are prepared to restrict the structural rules of the logic (Priest and Routley 1982, Shapiro 2010). Specifically, the most natural option for revisionary theorists is to give up the main structural principle at work in the proof of Curry’s Paradox (and, indeed, Validity Curry, and the semantic paradoxes in general), viz. Structural Contraction: (SC)

Γ, α, α ` β . Γ, α, ` β

If we’re interested in the consequences of the revisionary approach to the semantic paradoxes for our conception of validity, the question to ask is what logicians who reject SC can say about validity and truth-preservation. 6

My second main claim is that there is little to recommend for a substructural revision of our logic—at least, if it is to imply a restriction of SC. To begin with, it is easy to show that, if we reject SC, we can derive T r(pαq) → T r(pβq) from V al(pαq ∴ pβq), though we may not thereby infer, in general, that the argument from V al(pαq ∴ pβq) to T r(pαq) → T r(pβq) is valid: in order to do so, we would need to assume that β is derivable from α. This is puzzling, however: if, for some α’s and β’s, α ` β holds but V al(pαq ∴ pβq) doesn’t, how can the SC-free logician claim to be able to express validity in her language? What is more, dropping SC involves a revision not just of classical logic, but of key logical notions, such as assumption. Dropping SC is tantamount to saying that assuming α, α and assuming α are two different things. Yet, if, as it is often thought, to assume α in the course of a subderivation is to assume that α holds in some situation s, it shouldn’t matter how many times we use α in the course of our subderivation, if we’re reasoning about s, and if, as we’re assuming, α is true in s. Surely α doesn’t cease to be true in s, once we start using it! If, on the other hand, to assume a sentence is to temporarily expand our system by means of a temporary axiom (as argued in Schroeder-Heister 1984), then it is difficult to see why one should impose restrictions on how many times our temporary axiom is to be applied in the course of a subderivation. Surely, if an axiom is part of our system, one should be able to use the axiom, for the time its validity has been assumed. On either conception of what it is to assume α, I suggest, it shouldn’t make a difference how many times the same assumption is used in the course of a subderivation. I conclude that the revisionary approach to the semantic paradoxes is more radical than one might have thought: it does not ultimately allow us to express validity, and it leaves key logical notions in a mystery. I take this to be evidence that our main initial assumption, that the concept of truth is not stratified, should be called into question. I very briefly sketch a account of validity, on which classical logic can be retained, English isn’t fragmented in a hierarchy of languages, and one can truly say, in any context, that valid arguments preserve truth (relative to that context). On my suggested account, VTP, so interpreted, is consistent with Gödel’s Second Incompleteness Theorem, and doesn’t trigger Curry-like paradoxes.

Proof-theoretic semantics, paradoxes and Dummett’s fundamental assumption Luca Tranchini Wilhelm-Schickard-Institut University of Tübingen, Germany Dummett’s (1991) proof-theoretic justification of logical laws relies on the assumption—the Fundamental Assumption, as Dummett refers to it—that any valid argument for a proposition must reduce to a so-called ‘canonical’ argument for it. According to Dag Prawitz (1973, 1985), the notion of reduction is best analyzed in terms of functions that globally transform valid arguments into valid canonical ones. The Foundamental Assumption amounts to requiring such 7

functions must be total, i.e. they must yield a valid canonical argument, for all valid arguments to which they apply. We claim that, if the notion of validity is to be the core of a semantic theory, then the Fundamental Assumption should be rejected or, at least, relaxed. We argue in favor of our thesis by considering an analysis of paradoxes proposed by Neil Tennant (1982, 1995). According to Tennant, what identifies paradoxes is their yielding arguments that are impossible to bring into canonical form, due to loops arising in the process of reduction. If this conjecture is correct, the adoption of the Fundamental Assumption is tantamount to invalidate the set-theoretic paradoxes, since the arguments to which they give rise violate the Assumption and hence are automatically not valid. The price to be paid is, however, that paradoxes are deprived of semantic content, at least if, as it is sometimes suggested (see Pratiz 2006), the semantic content of an expression is identified with its contribution to the validity conditions of the arguments in which it may occur. For obviously, by adopting the fundamental assumption, the validity conditions of the arguments in which paradoxical statements play a role are simply trivialized. I suggest that, by rejecting, or at least relaxing, the fundamental assumption, one can indeed ascribe semantic content to paradoxes. By reinterpreting the fundamental assumption as a descriptive (rather than normative) condition, paradoxical arguments could be treated as valid although being still recognizable from the others valid arguments by their distinguishing feature, their irreducibility to canonical form. Although to refer to paradoxical arguments as valid may sound prima facie odd, such an attitude is justified if the notion of validity is viewed as the core of a semantic theory in which meaning conditions are given in terms of validity conditions. For, the resulting semantic framework would then still be capable of identifying paradoxes on the basis of certain proof-theoretic features (e.g. irreducibility), but without ipso facto depriving them of semantic content, since one could still ascribe non-trivial validity conditions to such arguments. I finally remark that the rejection of the Fundamental Assumption actually amounts to conceiving the set of functions characterizing the reduction procedures as partial: such functions do no longer give a valid canonical argument as value whenever applied to a valid argument. I argue that the role of reduction in the proof-theoretic semantics is analogous to the role of the interpretation function in a model-theoretic semantics. In the proof-theoretic framework, canonical arguments may be viewed as the semantic values selected by the semantics. Reductions, on the other hand, may be seen as analogous to the model-theoretic interpretation function, mapping linguistic items onto semantic values. In the light of this analysis, I suggest that a proof-theoretic framework in which the Fundamental Assumption is relaxed in the way indicated yields a semantic picture fully analogous to the one arising from free logics, according to which, in order to deal with certain types of linguistic phenomena (nondenoting singular terms in the free logic case, paradoxes in the proof-theoretic case), the functions codifying the notion of semantic interpretation must be conceived as partial.

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References Avron, A.: 1991, Simple consequence relations, Information and Computation 92, 105–139. Beall, J.: 2007, Truth and paradox: a philosophical sketch, in D. Jacquette (ed.), Philosophy of Logic, Elsevier, Oxford, pp. 325–410. Beall, J.: 2009, Spandrels of Truth, Oxford University Press, Oxford. Belnap, N. D.: 1962, Tonk, plonk and plink, Analysis 22(6), 130–34. Brady, R.: 2006, Universal Logic, CSLI Publications, Stanford. Dummett, M.: 1963, Realism, Truth and Other Enigmas, Duckworth, London. Dummett, M.: 1973, The justification of Deduction, Truth and Other Enigmas, Duckworth, London. Dummett, M.: 1978, Truth and Other Enigmas, Duckworth, London. Dummett, M.: 1991, The Logical Basis of Metaphysics, Harvard University Press, Harvard (Mass.). Field, H.: 2006, Truth and the unprovability of consistency, Mind 115(459), 567– 606. Field, H.: 2008, Saving Truth from Paradox, Oxford University Press, Oxford. Field, H.: 2009, What is the normative role of logic?, Proceedings of the Aristotelian Society 83, 251–68. Gentzen, G.: 1935, Untersuchungen über das logische Schließen, Mathematische Zeitschrift 39, 176–210. eng. transl. ‘Investigations into Logical Deduction’, in The collected papers of Gerhard Gentzen, M. E. Szabo (editor), NorthHolland, 1969. Hacking, I.: 1979, What is logic?, The journal of Philosophy 76(6), 285–319. Paoli, F.: 2007, Implicational paradoxes and the meaning of the logical constants, Australasian Journal of Philosophy 85, 533–79. Pottinger, G.: 1977, Normalization as a homomorphic image of cut-elimination, Annals of mathematical logic 12(3), 223–357. Prawitz, D.: 1973, Towards a fundation of a general proof theory, in P. Suppes, L. Henkin, A. Joja and G. C. Moisil (eds), Logic, Methodology and the Philosophy of Science IV: Proceedings of the Fourth International Congress, North Holland, Amsterdam, pp. 225–50. Prawitz, D.: 1977, Meaning and proofs: on the conflict between classical and intuitionistic logic, Theoria 43, 1–40. Prawitz, D.: 1985, Remarks on some approaches to the concept of logical consequence, Synthese 62, 153–171.

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Prawitz, D.: 1987, Dummett on a theory of meaning and its impact on logic, in B. M. Taylor (ed.), Michael Dummett: contributions to philosophy, Martinus Nijhoff Publisher, Dordrecht. Prawitz, D.: 2006, Meaning approached via proofs, Synthese 148, 507–24. Priest, G.: 2006, In Contradiction, Oxford University Press, Oxford. Expanded edition (first published 1987 Kluwer-Dordrecht). Priest, G. and Routley, R.: 1982, Lessons from Pseudo-Scotus, Philosophical Studies 42, 189–99. Prior, A.: 1960, The runabout inference-ticket, Analysis 21(2), 38–9. Read, S.: 2000, Harmony and autonomy in classical logic, Journal of Philosophical Logic 29, 123–154. Reinhart, W. N.: 1986, Some remarks on extending and interpreting theories,with a partial predicate for truth, Journal of Philosophical Logic 15, 219– 51. Restall, G.: 1993, How to be really contraction free, Studia Logica 52, 381–91. Sambin, G., Battilotti, G. and Faggian, C.: 2000, Basic logic: reflection, symmetry, visibility, Journal of Symbolic Logic 65(3), 979–1013. Schroeder-Heister, P.: 1984, A natural extension of natural deduction, The Journal of Symbolic Logic 49(4), 1284–1300. Schroeder-Heister, P.: 2009, Hypothetical reasoning: A critique of DummettPrawitz-style proof-theoretic semantics, in M. Peliš (ed.), The Logica Yearbook 2008. Schroeder-Heister, P.: forthcoming, The categorical and the hypothetical: A critique of some fundamental assumptions of standard semantics, in S. Lindström, E. Palmgren and D. Westerståhl (eds), The Philosophy of Logical Consequence and Inference. Special Issue of Synthese. Shapiro, L.: 2010, Deflating logical consequence, The Philosophical Quarterly . To appear. Tennant, N.: 1982, Proof and paradox, Dialectica 36, 265–96. Tennant, N.: 1995, Paradox without self-reference, Analysis 55(3), 199–207. von Plato, J.: 2001, Natural deduction with general elimination rules, Archives of mathematical logic 40, 541–67.

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Participants’ Curricula Vitarum

Ole Hjortland Munich Center for Mathematical Philosophy Ludwig-Maximilians-Universität 31, Ludwigstraße 80539, Munich – Germany Email: [email protected]

Academic Employment • Postdoctoral Research Fellow, Munich Center for Mathematical Philosophy, Ludwig-Maximilians-Universität München, January 2011/Present. • Postdoctoral Research Fellow, Arché Research Centre, University of St Andrews, June 2009/December 2010. Education • PhD in Philosophy, University of St. Andrews, viva passed in October 2009. Thesis’ title: “The Structure of Logical Consequence: Proof-theoretic conceptions”. Supervisors: Stephen Read, Crispin Wright, Stewart Shapiro Examiners: Peter Milne and Greg Restall. Articles in Journals • “Inferentialism and the Categoricity Problem: Reply to Raatikainen” (with J. Murzi) - 2009, Analysis 69(3), pp. 480-488. Invited Talks 1. November 2010, Dublin, “Speech acts and the meaning of logical connectives”. Philosophy Department Colloquium, TCD, Dublin. 2. May 2010, Dubrovnik, “Generalised Elimination Harmony and Sequent Calculus Solutions”. IUC, Dubrovnik, Croatia. 3. April 2009, Dublin, “Logic Naturalised”. Symposium on naturalism, TCD, Dublin. 4. April 2009, Dundee, “What does it mean to say that logic is revisable”, University of Dundee, Dundee. 5. January 2009, St Andrews, “The Semantic Role of Proof-Conditions”. 1st FLC workshop, ArchôL, University of St Andrews, St Andrews. 6. September 2008, Melbourne, “The Semantic Role of Proof-Conditions”. Melbourne Logic Seminar, University of Melbourne. Refereed Talks 1. September 2010, Tilburg, “Proof-Theoretic Semantics in the Substructural Era”. René Descartes Lectures by Ian Hacking & Workshop. University of Tilburg. 11

2. June 2010, Corte, “Thrice Denied: Speech acts, categoricity, and the meaning of logical connectives”. 2nd Conference on the Square of Opposition, University of Corsica. 3. July 2009, Norwich, “The Semantic Role of Proof-Conditions”. Graduate Session, Joint Session of the Aristotelian Society and the Mind Association, University of East Anglia, Norwich. 4. June 2009, Hejnice, “Categoricity and Non-Classical Logic”. LOGICA ’09, Hejnice, Czech Republic. Keynotes: JC Beall, Nuel Belnap, Stephen Read. 5. January 2009, Cambridge, “Inferentialism and the categoricity problem: reply to Raatikainen” (with Julien Murzi). 2nd Cambridge Graduate Conference in Philosophy of Logic and Mathematics, University of Cambridge. Keynotes: Hannes Leitgeb and Timothy Williamson. 6. June 2008, Hejnice, “Proof-Theoretic Harmony and Structural Assumptions”. LOGICA ’08, Hejnice, Czech Republic. Keynotes: Robert Brandom, Dag Prawitz, Stewart Shapiro. March 2008, Pittsburgh, “Proof-Theoretic Harmony and Structural Assumptions”. 10th Annual Pitt/CMU Graduate Student Philosophy Conference, University of Pittsburgh, Pittsburgh, US. Keynotes: Bas van Fraassen. 7. January 2008, Cambridge, “Proof-Theoretic Harmony and Structural Assumptions”. 1st Cambridge Graduate Conference in Philosophy of Logic and Mathematics, University of Cambridge. Keynotes: Dorothy Edgington and Alan Weir. 8. August 2007, Xi’an, “Inferentialism and Disagreement about Logic”. UNILOG, 2nd World Congress on Universal Logic, Xian, China.

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Julien Murzi Munich Center for Mathematical Philosophy Ludwig-Maximilians-Universität 31, Ludwigstraße 80539, Munich – Germany Phone: +44(0)7830314132 / +49(0)1782085488 Email: [email protected] Academic Employment • Postdoctoral Research Fellow, Munich Center for Mathematical Philosophy, Ludwig-Maximilians-Universität München, November 2010/October 2013. • Teaching Fellow, University of Sheffield, September 2008/February 2009. Education • PhD in Philosophy, University of Sheffield, viva passed in July 2010. Thesis’ title: “Intuitionism and Logical Revision”. Supervisors: Prof. Bob Hale and Dr. Dominic Gregory. Examiners: Prof. Crispin Wright (University of Aberdeen and New York University) and Dr. Rosanna Keefe (University of Sheffield). • PhD in Philosophy, University of Rome “La Sapienza”, viva passed in June 2008. Thesis’ title: “The Paradox of Knowability”. Supervisors: Prof. Cesare Cozzo and Prof. Carlo Cellucci. Examiner: Prof. Gabriele Usberti (University of Siena). The PhD was evaluated as “excellent”, the highest mark in the Italian PhD programme. Articles in English 1. “Manifestability and Unknowability”, in preparation for a special issue of Topoi on epistemic conceptions of truth, edited by G. Usberti and L. Tranchini. 2. “Inferentialism without Verificationism: Reply to Dag Prawitz”, forthcoming in C. Cellucci and E. Ippoliti (eds), Logic and Knowledge. 3. “Knowability and Bivalence: Intuitionistic Solutions to the Paradox of Knowability” - 2010, Philosophical Studies 149(2), pp. 269-81. 4. “Inferentialism and the Categoricity Problem: Reply to Raatikainen” (with O. T. Hjortland) - 2009, Analysis 69(3), pp. 480-488. 5. “The Paradox of Idealization” (with S. Florio) - 2009, Analysis 69(3), pp. 461-469. 6. “How Basic is the Basic Revisionary Argument?” (with L. Incurvati) 2008, Analysis 68(4), pp. 303-309. Articles in Italian 1. “The Realism/Anti-Realism Debate and the Intuitionistic Solution to the Knowability Paradox” (with L. Incurvati) - 2007, in C. Tatasciore, P. Graziani, C. Grimaldi (eds.), Prospettive Filosofiche: Il Realismo, Istituto Italiano per gli Studi Filosofici, Napoli, pp. 99-113. 13

Conference Proceedings 1. “Knowledge, Lotteries and Vagueness” - forthcoming in P. Hanna (ed.), Proceedings of the 2nd International Conference of Philosophy, Athens. 2. “Do We Really Need Relativism About Truth?” - 2006, in G. Gasser, C. Kanzian, and E. Runggaldier (eds), Proceedings of the XXIX Wittgenstein Symposium, ALWS, Kirchberg am Wechsel, pp. 210-12. Short Pieces and Reviews 1. J. Salerno and J. Murzi (eds), “Church’s Referee Reports on Fitch’s ‘A Definition of Value’ ” - 2009, in J. Salerno (ed.) New Essays on the Knowability Paradox, Oxford University Press. 2. “Intuitionism” - forthcoming in Key Terms in Logic, J. Williamson and F. Russo (eds), Continuum Books; also published in The Reasoner, vol. 3, Number 9, September 2009, p. 17. 3. Review of Jonathan Kvanvig, The Paradox of Knowability - 2006, 2R, vol 1., pp. 96-110. Invited Talks 1. April 2011, St Andrews: “Validity Curry”, fifth Foundations of Logical Consequence Workshop, Archè, University of St Andrews. 2. November 2010, Padova: “Two Flavours of Curry Paradox”, “Nuove ricerche sulla conseguenza logica”, Workshop on Logical Consequence. 3. September 2010, Siena: “Manifestability and Decidability”, International Workshop on Epistemic Conceptions of Truth, University of Siena. 4. June 2010, Rome: “Inferentialism without Verificationism: Reply to Dag Prawitz”, Logic and Knowledge, International Conference, University of Rome “La Sapienza”. 5. May 2010, Dubrovnik: “Validity and the Unprovability of Consistency”, OSU/Arché/Maribor conference on logical consequence. 6. November 2009, St Andrews: “Notions of Harmony in Proof-Theoretic Semantics”, Gentzen Centenary Symposium, University of St Andrews. 7. January 2007, Geneva: “Church-Fitch, Idealization and Knowability”, Conférence Phileas, University of Geneva. Refereed Talks 1. September 2010, Aberdeen: “Inferentialism and Incompleteness”, Logical and Language Conference, Northern Institute of Philosophy and Institute of Philosophy. 2. June 2010, St Andrews: “Validity and the Unprovability of Consistency”, Foundations of Logical Consequence Conference, Arché.

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3. March 2010, London: “Validity and the Unprovability of Consistency”, Philosophical Consequences of Second-Order Modal Logic, International Graduate Workshop with Oystein Linnebo, Ian Rumfitt, Gabriel Uzquiano, and Timothy Williamson. 4. October 2009, Amsterdam: ‘Carnap’s Categoricity Problem and the Meaning of the Logical Constants”, Amsterdam Graduate Philosophy Conference. Commentator: Prof. Dick de Jongh. 5. September 2009, L’Aquila: “Coming True: A Note on Truth and Actuality”, 2009 Latin Meeting in Analytic Philosophy: Language and Temporality. Commentator: Dr. Neftalí Villanueva. 6. July 2009, London: “Harmony and invertibility”, BPPC, King’s College. 7. June 2009, Hejnice: “Carnap’s Categoricity Problem and the Meaning of the Logical Constants”, Logica 2009. 8. January 2009, Cambridge: “Inferentialism and the Categoricity Problem: Reply to Raatikainen” (with O. T. Hjortland), Graduate Conference on the Philosophies of Logic and Mathematics. Commentator: Dr. Peter Smith. 9. October 2008, St Andrews: “Inferentialism and the Categoricity Problem”, Arché, University of St Andrews. 10. June 2008, Canterbury: “Knowability and Bivalence”, BPPC, University of Kent. (Awarded best paper of the 2008 edition of the BPPA.) 11. January 2008, Cambridge: “The Paradox of Idealization”, Graduate Conference on the Philosophies of Logic and Mathematics. Commentator: Luca Incurvati. 12. December 2007, Baltimore: “The Paradox of Idealization” (with S. Florio), Eastern Division of the APA. Commentator: Prof. Joe Salerno. 13. November 2007, Oxford: “The Paradox of Idealization”, 11th Oxford Graduate Conference. Commentator: Prof. Timothy Williamson. 14. November 2007, Paris: “Actuality in Branching Time”, Context– Dependence, Perspective and Relativity in Language and Thought, Ecole Normale Supérieure and Institut Jean Nicod. 15. November 2007, St Andrews: “How Basic is the Basic Revisionary Argument?” (with L. Incurvati), 4th Arché/CSMM Graduate Conference. Commentator: Dr. Marcus Rossberg. 16. September 2007, Bergamo: “Actuality in Branching Time”, International Conference on the Metaphysics of Time, University of Bergamo. 17. July 2007, Norwich: “The Paradox of Idealization”, BPPC, University of East Anglia. 18. July 2007, Bristol: “The Paradox of Idealization”, Joint Session of the Aristotelian Society and of the Mind Association, University of Bristol. 15

19. July 2007, Leeds: “The Paradox of Idealization”, Second Graduate Conference of the CMM, Center for the Metaphysics and Mind, University of Leeds. 20. June 2007, Athens: “Lotteries, Knowledge and Vagueness”, International Philosophy Conference. 21. April 2007, San Francisco: “The Church-Fitch Paradox and the Problem of Transworld Knowability”, Pacific Division of the APA. Commentator: Prof. Roy Cook. 22. January 2007, Miami: “Lotteries, Knowledge and Vagueness”, International Graduate Conference in Epistemology, University of Miami. Commentator: Ben Burgis. 23. December 2006, Washington DC: “Lotteries, Knowledge and Vagueness”, SRAD, Society for the Realist and Anti-Realist Discussion, Eastern Division of the APA. 24. September 2006, Aix-en-Provence: “The Church-Fitch Paradox and the Problem of Transworld Knowability”, SOPHA, International Meeting of the French Society for Analytical Philosophy. 25. August 2006, Kirchberg am Wechstel: “Do We Really Need Relativism about Truth?” Cultures: Conflict-Analysis-Dialog, XXIX International Wittgenstein Symposium. 26. June 2006, Nancy: “Possible Knowledge of Actual Truths”, Réalisme(s) et Anti-Réalisme(s), International Philosophy Conference, University of Nancy 2. 27. June 2006, Lisbon: “Possible Knowledge of Actual Truths”, ENFA, Portuguese National Conference for Analytic Philosophy.

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Luca Tranchini Wilhelm-Schickard-Institut für Informatik Eberhard-Karls-Universität Tübingen Sand 13, 72076, Tübingen, Germany E-mail: [email protected] Academic Employment • Postdoctoral Research Fellow at the computer science department of Tübingen Univeristy (Germany), within the “Dialogical Foundations of Semantics” (DiFoS) project, funded by the ESF EUROCORES program “LogICCC - Modelling Intelligent Interaction”, November 2010 - May 2011. Education • Joint Ph.D Degree in Philosophy and Cognitive Sciences, Tübingen University and Siena University, viva passed in September 2010. Thesis title: “Proof and Truth. An anti-realist perspective”. Supervisors: Prof. Dr. Peter Schroeder-Heister (Tübingen) and Prof. Gabriele Usberti (Siena). Examiners: Prof. Enrico Moriconi (Pisa University) and Prof. Dr. Heinrich Wansing (Bochum University). Articles in Journals • “Truth from a proof-theoretic perspective.” in preparation for a special issue of Topoi on epistemic conceptions of truth, edited by G. Usberti and L. Tranchini. Conference Proceedings • “The role of negation in proof-theoretic semantics: a proposal”, in M. Peliš (ed.), The Logica Yearbook 2008, College Publications, London, 2009; • “Truth: an Anti-realist Adequacy Condition”, in Piotr Stalmaszczyk (ed.), Philosophy of Language and Linguistics, Volume I: The Formal Turn, Ontos Verlag, Heusenstamm bei Frankfurt, 2010. Articles in Italian • “Refutation: a proof-theoretic account”, in C. Marletti (ed.), First Pisa Colloquium in Logic, Language and Epistemology, ETS, Pisa, 2010. Pre-prints • “A proof of Cut-elimination for linear logic”, Rapporto interno del dipartimento di Scienze Matematiche e Informatiche “R. Magari” (Siena University Mathematics Department preprint series), n. 483, Siena University, October 2006. Invited Talks 1. September 2010, Siena: “Truth from a proof-theoretic perspective”, Antirealistic notions of truth, workshop organized by the Italian PRIN project “Aspects of truth”, Siena, Italy, 10-12th September, 2010. 17

2. March 2010, Amsterdam: “Truth, Proof and Dialogues”, Modelling Interaction DIalogues SOcial choice and VAgueness (MIDISOVA), a workshop jointly organized by ESF funded LogICCC Projects, Amsterdam, the Netherland, 26-28th March, 2010. 3. February 2010, Lille: “Truth, Proof and Dialogues”, Dialogues and Games, workshop organized by the ESF funded LogiCCC Project DiFos, Lille III University, France, 8-9th February, 2010. 4. October 2009, St. Andrews: “Dualizing Proof-Theoretic Semantics”, The Logic of Denial, workshop organized by the ARCHÉ research center, St. Andrews University, United Kingdom, 24-25th October, 2009. Refereed talks 1. April 2010, Estoril: “Falsificationism. Dualizing Proof-theoretic semantics”, UNILOG III, third world conference on universal logic, Estoril, Portugal, 22-25th April, 2010; 2. September 2009, Florence: “Proof and Truth. An anti-realist perspective”, Seminario Congiunto SUM-Coordinamento Nazionale Dottorati di Ricerca in Filosofia, workshop organized by the Italian Institute of Human Sciences (SUM) and by the coordination of the Italian Ph.D. programs in Philosophy, Florence, Italy, 30th September-1st October, 2009; 3. September 2009, Bremen: “Truth: an Anti-realist Adequacy Condition”, G.A.P. 7, biennial conference of the German Society for Analytic Philosophy, Bremen, Germany, 14-17th September 2009; 4. May 2009, Łód´z: “Truth: an Anti-realist Adequacy Condition”, PhiLang2009, conference organized by the Department of English and General Linguistics at University of Łód´z, Poland, 16-18th May, 2009; 5. September 2008, Bergamo: “The meaning of negation: a proposal” Sifa 2008, 8th conference organized by the Italian Society for Analytic Philosophy, Bergamo, Italy, 25-27th September, 2008; 6. July 2008, Lisbon: “The meaning of negation: a proposal”, Negation and Denial, workshop organized by the Centro de Filosofia da Universidade de Lisboa, Lisbon, Portugal, 8-10th July, 2008; 7. June 2008, Hejnice: “The role of negation in proof-theoretic semantics: a proposal”, Logica 2008, conference held in Hejnice, Czech Republic, 1620th June, 2008.

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