Workshop Logical Consequence and Paradox
Bochum, December 23, organized by Graham Priest and Heinrich Wansing, supported by Alexander von Humboldt Foundation.
Venue: Beckmanns Hof, Ruhr University Bochum.
The paradoxes have always been of focal interest in philosophy and logic, and the relation between the paradoxes and logical consequence are especially relevant in solutions which make use of paraconsistent, substructural or other nonclassical logics. The workshop is meant to provide a forum for discussing recent work on semantical, settheoretical and other closely related paradoxes and aspects of logical consequence that play a role for solutions to the paradoxes. Among the topics of the workshop are, the Russell paradox, the Curry paradox, paraconsistent logic, dialetheism, and theories of truth.
The workshop is organized in connection with Graham Priest's stay as a Humboldt Prize winner at Ruhr University Bochum.
Speakers:
 Jc Beall (University of Connecticut, University of Aberdeen)
 Roberto Ciuni (Ruhr University Bochum)
 Ole Hjortland (Munich Centre for Mathematical Philosophy, LMU)
 Julien Murzi (University of Kent and Munich Centre for Mathematical Philosophy, LMU)
 Hitoshi Omori (CUNY, Ruhr University Bochum)
 Francesco Paoli (University of Cagliari)
 Graham Priest (CUNY, Ruhr University Bochum)
 Stephen Read (University of St Andrews)
 Daniel Skurt (Ruhr University Bochum)
 Heinrich Wansing (Ruhr University Bochum)
 Kai Wehmeier (UC Irvine)
 Elia Zardini (University of Barcelona, University of Aberdeen)
Preliminary schedule

Monday, December 2 
Tuesday, December 3 
9:3010:30 
Graham Priest What If ? 
Hitoshi Omori
Towards a fully dialetheic solution to Russell's paradox 

break 
break 
11:0012.00 
Ole Hjortland
An nSided Sequent Calculus for Paraconsistent and Paracomplete Theories of Truth 
Kai Wehmeier
Zalta's Paradox and ModalLogical 
12:0013:00 
Stephen Read
PseudoScottish Curry 
Daniel Skurt
Inconsistencies in NonMonotonic Contexts 

lunch 
lunch 
14:0015:00 
Jc Beall
A new motivation for WK3 
Francesco Paoli
Towards Multiset Consequence Relations 
15:0016:00 
Roberto Ciuni
Paraconsistent Weak Kleene 
Elia Zardini
One, and Only One 

break 
break 
16:3017:30 
Julien Murzi
Dialetheism and Disagreement 
Heinrich Wansing
Remarks on the Curry Paradox 



19:00 
workshop dinner Restaurant Mutter Wittig 

Titles and abstracts:
JC Beall (University of Connecticut, University of Aberdeen)
A new motivation for WK3
Abstract: I argue that a natural motivation for Weak Kleene (WK3) is as an account of socalled "logic of fiction" or, more accurately, "closure for fiction". I argue that background considerations of what fictions are about motivate WK3.
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Roberto Ciuni (Ruhr University Bochum)
Paraconsistent Weak Kleene
Abstract: In this presentation, I investigate formal and conceptual features of the socalled paraconsistent Weak Kleene (PWK). PWK is a threevalued logic obeying the Principle of Contamination: for every truth function, if any input has the nonclassical truthvalue i, the output also has it; also, every truth fucntion behaves as in classical logic if no input is i. Contrary to the Weak Kleene introduced by Bochvar (1938) and Kleene (1952) PWK is paraconsistent, since i is included in the designated value alongside with `classical truth'. PWK was rst introducted by Soren Hallden (1949) and, independently, Arthur Prior (1957). Remarkably, though the logic is paraconsistent, both Hallden and Prior proposed a gappy reading of the third value, to be understood as meaningless and undefined (due to factual indeterminacy), respectively. The first part of the presenation introduces the logic and focuses on the characterization of logical consequence, some connections with the logic LP by Priest, 1979, and the socalled 'classical collapse' by Beall, 2011. In relation to this, an 'LP collapse' is also provided. Finally, I show that PWK's conjunction can be defined in a variant of LP that includes classical negation beside the usual LP's negation. Interestingly enough, PWK's conjunction is defined in terms of LP's disjunction, conjunction and classical negation, with disjunction being the main connective of the definition. This helps explain a feature of PWK, namely the failure of Conjunction Simplification. The second part of the presentation focuses on the interpretation of the nonclassical truthvalue. First, I question the traditional gappy readings. Second, I investigate the conceptual motiviation for a `glutty' reading that accounts for both the designation of the third value and the failure of Conjunction Simplication.
References
[1] Beall Jc. (2011). Multipleconclusion LP and Default Classicality, Review of Symoblic Logic, 4/2: 326336.
[2] Beall Jc. (2013). LP+, K3+, FDE+ and their Classical Collapse, Review of Symoblic Logic, FirstView Articles.
[3] Bochvar D. (1938). On a Threevalued Calculus and its Application in the analysis of the Paradoxes of the Extended Functional Calculus, Matematicheski Sbornik, 4: 287308.
[4] Hallden S. (1949). The Logic of Nonsense, PhD Thesis. Uppsala, Uppsala University.
[5] Kleene S. (1952). Introduction to Metamathematics. Amsterdam, North Holland.
[6] Priest G. (1979). The Logic of Paradox, Journal of Philosophical Logic, 8: 219241.
[7] Prior A. (1957). Time and Modality. Oxford, OUP.
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Ole Hjortland (Munich Centre for Mathematical Philosophy, LMU)
An nSided Sequent Calculus for Paraconsistent and Paracomplete Theories of Truth
Abstract: I present an nsided sequent calculus for paraconsistent and paracomplete theories of truth, and compare these with previous proof theoretic approaches to such theories, especially that given by Halbach and Horsten (2006). The nsided framework offers a common proof theoretic framework for such theories, and also an interesting inferential diagnosis of paradoxes.
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Julien Murzi (University of Kent and Munich Centre for Mathematical Philosophy, LMU) and Massimiliano Carrara (University of Padua)
Dialetheism and Disagreement
Abstract: In this paper, we cast doubts on the suggestion, recently made by Graham Priest, that dialetheists may express disagreement with the assertion of A by denying A. We argue that, if denial is to serve as a means to express disagreement, it must be exclusive, and that, for this reason, it can’t be expressed in a dialetheist language. Nonexclusive denial is expressible, but won’t in general serve as a means of expressing disagreement. Dialetheists thus face a dilemma: they can express either denial or disagreement, but not both. Along the way, we offer a bilateral logic of exclusive denial for dialetheists—an extension of the logic commonly called LP.
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Hitoshi Omori (CUNY)
Towards a fully dialetheic solution to Russell's paradox
Abstract: Dialethic solutions to paradoxes are based on the metaphysical view which claims that there are true contradictions. And for this purpose, one needs a system of paraconsistent logic, such as Logic of Paradox (LP) developed by Graham Priest, which serves as an underlying logic of the theory. But in some contexts, such as naive set theory, the paraconsistent logic was supposed not to contain classical negation.
Based on these, the aim of the talk is twofold. First, we argue that having classical negation is not as problematic as it seemed to be before. This will be made clear by considering an expansion of LP by the consistency operator that originates in the work of paraconsistent logic by Newton da Costa. The resulting system is known to be equivalent to a threevalued logic called LFI1 in which classical negation can be defined. Second, we argue that LFI1 is still not enough as an underlying logic of dialetheic theories. Indeed, from a certain understanding of logic, it is rather natural to expect to have dialethias already in the level of underlying logic. And to this end, we add a connexive implication considered by Heinrich Wansing. We will present some basic results on the concerned expansion of LP, and make some remarks on the related issues.
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Francesco Paoli (University of Cagliari)
Towards Multiset Consequence Relations
(Joint project with L. Behounek, P. Cintula, J. Gil Férez, E. Mares)
Abstract: In the forthcoming paper [1], Ed Mares and I attempted a solution to the truththeoretical and settheoretical paradoxes based on the following idea: the paradoxes are fallacies of equivocation that arise because each classical logical constant (connective or quantifier) is ambiguous between an extensional and an intensional constant, and moreover, classical logical consequence is ambiguous between an internal and an external notion of consequence. Internal consequence satisfies the deductiondetachment schema but fails structural contraction, while the opposite situation holds for external consequence – however, for the paradoxes to arise both properties need to hold of the same concept of consequence. This position, of course, is only tenable if logical consequence is conceived of as a relation between a multiset of formulas and a formula, contra the standard Tarskian, settheoretical approach. In this talk we consider three possible objections against multiset consequence relations:
1) The natural language examples motivating multisettheoretical arguments and the failure of structural contraction are based on a confusion between deduction and inference (cf. Johnson, C.I. Lewis, Harman, Beall);
2) Internal consequence has no counterpart in mathematical proof procedures;
3) Multiset consequence relations have no theory, in contrast with the wellrehearsed abstract theory of Tarskian consequence relations.
We briefly address the first two objections and report on some preliminary ideas and results concerning the third one.
[1] E. Mares, F. Paoli, "Logical consequence and the paradoxes", Journal of Philosophical Logic, Online First, 2013.
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Graham Priest (CUNY, Ruhr University Bochum)
What If?
Abstract: It is usually assumed that the conditional involved in the TSchema and similar principles is a detachable one. However, Goodship has suggested that it should be understood as a paraconsistent material conditional, and so nondetachable. (Goodship, L. (1996), `On Dialethism', Australiasian Journal of Philosophy 74: 15361.) In this talk I will discuss her proposal.
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Stephen Read (University of St Andrews)
PseudoScottish Curry
Abstract: Recent work on the semantic paradoxes has focussed on VCurry, a validity version of Curry's paradox. Variants on this paradox were known to fourteenthcentury logicians, who proposed their own solutions. Thomas Bradwardine's solution depends on the idea that every sentence signifies many things, and a sentence's truth depends on things' being wholly as it signifies. This idea is underpinned by his claim that a sentence signifies everything that follows from it. But the idea that signification is closed under consequence appears too strong, just as logical omniscience is unacceptable in the logic of knowledge. What is needed is a more restricted closure principle. A clue can be found in indirect speech reports, which are arguably closed under intersubstitutivity of coreferential terms, but not under arbitrary consequences. The upshot is that solving the Curry and other semantic paradoxes does not require revision of logic, thus saving logic from paradox.
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Daniel Skurt (Ruhr University Bochum)
Inconsistencies in NonMonotonic Contexts
Abstract: Paradoxical or inconsistent situations are usually handled within monotonic contexts.Logics like LP or Belnap's FOUR are typical representatives. But these logics per se are not able to handle properly new information, like it is needed e.g. in knowledge representation. Take for instance a medical database, which is feeded with symptoms of a patient and returns based on the symptoms a diagnosis. These symptoms could be contradictory and every new symptom could change the diagnosis. Therefore a nonmonotonic paraconsistent logic is needed. In my talk I will show, how one can construct a nonmonotonic version of a given paraconsistent logic in a canonical way. In order to do this I will introduce a concept which I call iterated preferential models, which is based on Priest's LPm and the nonmonotonic approach circumscription.
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Heinrich Wansing (Ruhr University Bochum)
Remarks on the Curry Paradox
(Joint work with Graham Priest)
Abstract: In the literature one can find the distinction between the Curry Paradox as a connective paradox and the Validity Curry Paradox as a structural paradox. The connective version of the paradox may be solved by abandoning the contraction axiom schema and the validity Curry Paradox associated with internal consequence by giving up the contraction rule. In this talk I will discuss external consequence versions of the Curry paradox. The key to solutions is, again, contraction.
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Kai Wehmeier (UC Irvine)
Zalta's Paradox and ModalLogical Consequence
Abstract: Ed Zalta has argued that modal languages with an actuality operator give rise to contingent logical truths, a result that deserves to be called "Zalta's Paradox," given that logical truths would appear to be paradigm cases of necessary truths. But there is wiggle room with respect to the notion of modallogical truth (more generally, modal logical consequence): Zalta's examples work only if logical truth is realworld, as opposed to general, validity. In the talk, I will survey the state of the debate, argue that extant arguments in favor of general validity as an explication of modallogical truth are unsatisfactory, and propose a novel resolution of Zalta's Paradox.
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Elia Zardini (University of Barcelona, University of Aberdeen)
One, and Only One
Abstract: Standard nonclassical (i.e. nonsubstructural) solutions to the semantic paradoxes of truth deny either the law of excluded middle or the law of noncontradiction; in so doing, they either reject both the truth of a paradoxical sentence and its falsity or accept both the truth of a paradoxical sentence and its falsity. In this sense, both kinds of solutions agree that paradoxical sentences are inconsistent—that such sentences cannot coherently be assigned one and only one truth value. This pattern extends from the semantic paradoxes of truth to the semantic paradoxes of reference: when faced with at least certain particularly recalcitrant paradoxes of naive reference, both kinds of solutions are forced to claim that the paradoxical singular terms in question are inconsistent—that they cannot coherently be assigned one and only one referent. I’ll argue that, contrary to what both kinds of solutions require, under plausible assumptions paradoxical singular terms can be constructed that are forced to refer to a unique object. By considering these and other more traditional paradoxes, I’ll then show how my favoured noncontractive solution to the semantic paradoxes, which generally treats paradoxical entities as consistent rather than as inconsistent, can be so deployed as to offer a unified solution to the semantic paradoxes of truth and to those of reference and definability.