Logical consequence and paradox


Curry_poster

 

Workshop Logical Consequence and Paradox

Bochum, December 2-3, 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 non-classical logics. The workshop is meant to provide a forum for discussing recent work on semantical, set-theoretical 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

 

9:30-10:30

 

Graham Priest

 

What If ?

 

Hitoshi Omori

 

Towards a fully dialetheic solution to Russell's paradox

 

 

break

 

break

 

11:00-12.00

 

Ole Hjortland

 

An n-Sided Sequent Calculus for Paraconsistent and Paracomplete Theories of Truth

 

Kai Wehmeier

 

Zalta's Paradox and Modal-Logical

 

 

12:00-13:00

 

Stephen Read

 

Pseudo-Scottish Curry

 

Daniel Skurt

 

Inconsistencies in Non-Monotonic Contexts

 

 

lunch

 

lunch

 

14:00-15:00

 

Jc Beall

 

A new motivation for WK3

 

Francesco Paoli

 

Towards Multiset Consequence Relations

 

15:00-16:00

 

Roberto Ciuni

 

Paraconsistent Weak Kleene

 

Elia Zardini

 

One, and Only One

 

 

break

 

break

 

16:30-17:30

 

Julien Murzi

 

Dialetheism and Disagreement

 

Heinrich Wansing

 

Remarks on the Curry Paradox

 

 

 

 

 19:00

workshop dinner

Restaurant Mutter Wittig
Bongardstr. 35, 44787 Bochum,

Tel: 0234/12141

 

 

 

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 so-called "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 so-called paraconsistent Weak Kleene (PWK). PWK is a three-valued logic obeying the Principle of Contamination: for every truth function, if any input has the non-classical truth-value 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 unde fined (due to factual indeterminacy), respectively. The first part of the presenation introduces the logic and focuses on the characterization of logical consequence, some connec-tions with the logic LP by Priest, 1979, and the so-called 'classical collapse' by Beall, 2011. In relation to this, an 'LP collapse' is also provided. Finally, I show that PWK's conjunction can be defi ned in a variant of LP that includes classical negation beside the usual LP's negation. Interestingly enough, PWK's conjunction is defi ned in terms of LP's disjunction, conjunction and classical negation, with disjunction being the main connective of the de finition. This helps explain a feature of PWK, namely the failure of Conjunction Simplifi cation. The second part of the presentation focuses on the interpretation of the non-classical truth-value. 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 Simpli cation.

References
[1] Beall Jc. (2011). Multiple-conclusion LP and Default Classicality, Review of Symoblic Logic, 4/2: 326-336.
[2] Beall Jc. (2013). LP+, K3+, FDE+ and their Classical Collapse, Review of Symoblic Logic, FirstView Articles.
[3] Bochvar D. (1938). On a Three-valued Calculus and its Application in the analysis of the Paradoxes of the Extended Functional Calculus, Matematicheski Sbornik, 4: 287-308.
[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: 219-241.
[7] Prior A. (1957). Time and Modality. Oxford, OUP.

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Ole Hjortland (Munich Centre for Mathematical Philosophy, LMU)

An n-Sided Sequent Calculus for Paraconsistent and Paracomplete Theories of Truth

Abstract: I present an n-sided 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 n-sided 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. Non-exclusive 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 three-valued 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 truth-theoretical and set-theoretical 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 deduction-detachment 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, set-theoretical approach. In this talk we consider three possible objections against multiset consequence relations:

1)  The natural language examples motivating multiset-theoretical 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 well-rehearsed 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 T-Schema and similar principles is a detachable one. However, Goodship has suggested that it should be understood as a paraconsistent material conditional, and so non-detachable. (Goodship, L. (1996), `On Dialethism', Australiasian Journal of Philosophy 74: 153-61.) In this talk I will discuss her proposal.

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Stephen Read (University of St Andrews)

Pseudo-Scottish Curry

Abstract: Recent work on the semantic paradoxes has focussed on V-Curry, a validity version of Curry's paradox. Variants on this paradox were known to fourteenth-century 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 co-referential 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 Non-Monotonic 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 non-monotonic paraconsistent logic is needed. In my talk I will show, how one can construct a non-monotonic 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 non-monotonic 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 Modal-Logical 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 modal-logical truth (more generally, modal logical consequence): Zalta's examples work only if logical truth is real-world, 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 modal-logical 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 non-classical (i.e. non-substructural) solutions to the semantic paradoxes of truth deny either the law of excluded middle or the law of non-contradiction; 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 non-contractive 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.