June 27-29, 2018 – University of Salzburg

### Speakers

- Andrew Bacon (University of Southern California)
- Roy Cook (University of Minnesota)
- Hartry Field (New York University)
- Hannes Leitgeb (Ludwig-Maximilians-Universität München)
- Lavinia Picollo (Ludwig-Maximilians-Universität München)
- Mark Pinder (University of Bristol)
- Dave Ripley (Monash University)
- Lorenzo Rossi (University of Salzburg)
- Chris Scambler (New York University)
- Gila Sher (University of California, San Diego)
- Neil Tennant (Ohio State University)
- Nicholas Tourville (Rutgers University)
- Elia Zardini (Universidade de Lisboa)

The topic of the workshop is semantic paradox and revenge: the thought that any purported treatment of the semantic paradoxes breeds new paradoxes to which the treatment itself does not apply. We are interested in questions such as: Can there be a revenge-immune solution to the semantic paradoxes? Are inconsistent approaches to semantic paradox better equipped to deal with revenge paradoxes? Is there a general revenge problem for classical approaches to semantic paradox?

### Venue

- University Library (Hauptbibliothek)
- Hofstallgasse 2-4, 5020 Salzburg

### Programme

##### June 27

**Gila Sher**,

*How to elude a paradox by re-examining truth*

**Elia Zardini**,

*Unstable knowledge*

**Chris Scambler**,

*Ineffability and revenge*

**Roy Cook**and

**Nicholas Tourville**,

*Embracing revenge and logical consequence*

##### June 28

**Hartry Field**,

*Kripke and Łukasiewicz: A Synthesis*.

**Lorenzo Rossi**,

*Revenge in context*

**Dave Ripley**,

*What explains the paradoxes?*

**Hannes Leitgeb**,

*HYPE: A System of Hyperintensional Logic (with an application to semantic paradoxes)*

##### June 29

**Lavinia Picollo**,

*Truth in a logic of formal inconsistency: how classical can it get?*

**Andrew Bacon**,

*Propositional granularity and the logical paradoxes*

**Mark Pinder**,

*What is Scharp’s solution to the Liar Paradox? Does it face revenge?*

**Neil Tennant**,

*Proof, paradox, and revenge*

19:00 Dinner at La Campana

### Abstracts

##### Andrew Bacon, Propositional granularity and the logical paradoxes

A number of philosophers have recently argued that certain logical paradoxes, like the Russell-Myhill paradox, are best regarded as limitative results on propositional granularity. In this talk I explore a particular consistent theory of propositional granularity that avoids Russell-Myhill-like paradoxes. I posit the existence of a modality that roughly stands to propositions as logical truth stands to sentences, and explore a theory in which propositions are individuated by “logical equivalence” in that sense. The resulting theory vindicates a variant of the structural idea that propositions can be decomposed into their fundamental constituents and logical operations, and can be used to explicate the Humean idea that fundamental properties and relations are freely recombinable.

##### Roy Cook and Nicholas Tourville, Embracing revenge and logical consequence

After a brief review of the Embracing Revenge account of truth, paradox, and revenge, we will present a number of deductive systems that can be “built” on the semantics – one corresponding to a Kleene-style “gap” approach, one corresponding to a Priest-style “glut” approach, and one corresponding to a Ripley-style “strict-tolerant” approach.

##### Hartry Field, Kripke and Łukasiewicz: A Synthesis

In classical logic the naive theory of truth and satisfaction is inconsistent. Kripke provided a well-known partial solution to the paradoxes in a non-classical logic. But it has a big limitation: it doesn’t work for logics with serious conditionals, or restricted universal quantification. Another partial non-classical solution is given by Łukasiewicz continuum-valued logic. It allows naive truth for sentences containing a rather natural conditional. But it has a different limitation: it doesn’t work for sentences containing even unrestricted quantifiers. (Kripke’s partial solution handled those.) So neither result handles restricted universal quantifiers. It would be nice to synthesize the two: to have an account which handled both unrestricted quantifiers and aŁukasiewicz-like conditional. (And to do so in *essentially* the way that Łukasiewicz and Kripke did. This is not out of the question since theŁukasiewicz and Kripke solutions agree on their common domain.) Such a synthesis would thereby also handle restricted universal quantification, which is interdefinable with the conditional given unrestricted quantification. I’ll show how to do so in this talk. The synthesized approach improves on my previous work on conditonals and restricted quantifiers, in essentially preserving the attractive features of theŁukasiewicz resolution of the quantifier-free semantic paradoxes, including the easy calculation of solutions.

##### Hannes Leitgeb, A System of Hyperintensional Logic (with an application to the semantic paradoxes)

This talk introduces, studies, and applies a new system of logic which is called ’HYPE’. In HYPE, formu- las are evaluated at states that may exhibit truth value gaps (partiality) and truth value gluts (overdeter- minedness). Simple and natural semantic rules for negation and the conditional operator are formulated based on an incompatibility relation and a partial fusion operation on states. The semantics is worked out in formal and philosophical detail, and a sound and complete axiomatization is provided both for the propositional and the predicate logic of the system. The propositional logic of HYPE is shown to con- tain first-degree entailment, to have the Finite Model Property, to be decidable, to have the Disjunction Property, and to extend intuitionistic propositional logic conservatively when intuitionistic negation is defined appropriately by HYPE’s logical connectives. Furthermore, HYPE’s first-order logic is a conser- vative extension of intuitionistic logic with the Constant Domain Axiom, when intuitionistic negation is again defined appropriately. The system allows for simple model constructions and intuitive Euler- Venn-like diagrams, and its logical structure matches structures well-known from ordinary mathematics, such as from optimization theory, combinatorics, and graph theory. HYPE may also be used as a gen- eral logical framework in which different systems of logic can be studied, compared, and combined. In particular, HYPE is found to relate in interesting ways to classical logic and various systems of rel- evance and paraconsistent logic, many-valued logic, and truthmaker semantics. On the philosophical side, if used as a logic for theories of type-free truth, HYPE is shown to address semantic paradoxes such as the Liar Paradox by extending non-classical fixed-point interpretations of truth by a conditional as well-behaved as that of intuitionistic logic. With the help of HYPE’s conditional operator, the T- scheme can be expressed in the object language, and one can prove there to be a HYPE-model in which all instances of the T-scheme for Kripke’s (1975) original object language are satisfied at all states. The HYPE-model in question may be viewed as consisting of all Kripkean truth fixed points taken together as a HYPE-structure. While the T-scheme does not hold unrestrictedly at all states of the model, there is a philosophical rationale for that to be so which concerns the hyperintensionality of the conditional operator.

##### Lavinia Picollo, Truth in a logic of formal inconsistency: how classical can it get?

Weakening classical logic is one of the most popular ways of dealing with semantic paradoxes. Their advocates often claim that such weakening does not affect nonsemantic reasoning. Recently, however, Halbach and Horsten (2006) have shown that this is actually not the case for Kripke’s fixed-point theory based on the Strong Kleene evaluation scheme. Feferman’s axiomatization KF in classical logic is much stronger than its paracomplete counterpart PKF, not only in terms of semantic but also arithmetical con- tent. This paper compares the proof-theoretic strength of an axiomatization of Kripke’s construction based on the paraconsistent evaluation scheme of LP formulated in classical logic with that of an axiom- atization directly formulated in LP extended with a consistency operator. The ultimate goal is to find out whether paraconsistent solutions to the paradoxes that employ consistency operators fare better in this respect than paracomplete ones.

##### Mark Pinder, What is Scharp’s solution to the liar paradox? Does it face revenge?

Kevin Scharp has developed an exciting new version of the inconsistency approach to the liar paradox. Scharp claims that his approach provides a consistent, revenge-free solution. In this talk, I aim to show that: (a) it is unclear whether the approach in fact provides a consistent solution to the paradox; and (b) either way, it does face a revenge paradox.

##### Dave Ripley, What explains the paradoxes?

If two paradoxes have the same explanation, then they should receive the same solution. This is Graham Priest’s ”Principle of Uniform Solution”, and it’s common ground in many discussions of paradox. But just which paradoxes have the same explanation? Following Russell, Priest has proposed the Inclosure Schema as an explanation for a range of para- doxes. Based on this work, a number of theorists have claimed that paradoxes of vagueness have the same explanation as the liar paradox, and so should receive the same solution. On this basis, they have argued against solutions to one of these paradoxes that do not also solve the other. I will argue that this is mistaken: the Inclosure Schema does not succeed in explaining the liar para- dox, and so is not an appropriate tool for deciding whether the liar paradox should receive the same solution as paradoxes of vagueness. I will go on to propose an alternate explanation of these paradoxes, based on Katalin Bimbo’s work in proof theory. If this explanation is right, liar paradoxes and paradoxes of vagueness do not have the same explanation, and so the Principle of Uniform Solution does not apply.

##### Lorenzo Rossi, Revenge in context

In this talk, I will first give a (partial) overview of the FWF project on semantic paradoxes Julien Murzi and I are currently carrying out. Second, I will `zoom in’ on a fragment of this project. In particular, I am going to present the theory of paradoxicality offered in Rossi (2018) as well as our approach to revenge paradoxes.

Our project has two main objectives. First, to provide a uniform analysis of the semantic paradoxes, including the revenge paradoxes. Second, to develop a theory of truth that satisfies some intuitive desiderata and treats the paradoxes in a uniform way. We examine several existing theories of truth, and find them all wanting. In particular, we show that the main non-classical approaches are revenge-prone: they breed new paradoxes they are unable to block. We then develop a novel contextualist theory of truth and paradox, that treats truth-theoretical and revenge paradoxes in a uniform way. Unlike existing contextualist approaches, our theory is compatible with all the absolute generality one could reasonably want.

Our project requires a satisfactory theory of paradoxicality, which can be used to identify all the paradoxes that can be formulated in a given language, including the revenge ones. To this end, we adapt the theory given in Rossi (2018), which shows how several kinds of paradoxical sentences can be distinguished and interpreted in one single model, including revenge ones.

##### Chris Scambler, Ineffability and Revenge

Hartry Field’s logic of truth is offered as a revenge immune solution to the semantic paradoxes. Part of the basis for Field’s claim of revenge immunity for his logic is the presence of a transfinite hierarchy of object-language definable ‘determinacy operators’ Dα ̇ , which collectively allow for the object language classification of many intuitively defective sentences as indeterminateα for some α. In recent work Philip Welch has proved the existence of ineffable liars in Field’s logic. An ineffable liar (relative to a ground model M) is a sentence W that gets ultimate value 1/2 (over M), and so seems in some sense ‘intuitively defective’, but such that for every object language definable determinacy operator (over M) we also have DαW getting ultimate value 1/2, so that the apparent intuitive defectiveness is not measured by the object language determinacy hierarchy. In my talk, I’ll present a simplified version of Welch’s argument that such sentences exist, before discussing the relevance of such sentences to Field’s claims of revenge immunity.

##### Gila Sher, How to elude a paradox by re-examining truth

In this talk I show that the Liar Paradox, in a number of its forms, can be blocked by a ‘material’ principle that focuses on the philosophical content of the theory of truth. I then discuss the relation between this solution and formal solutions to the Liar and show that the material principle can provide a philosophical basis for some of the formal solutions, including one – Tarski’s – that is often criticized for being ad hoc. The talk concludes with an examination of the relation between a philosophical (‘material’) theory and its formal systematization, discussing both the positive contributions of the latter and its distorting effects.

##### Neil Tennant, Proof, Paradox, and Revenge

Strictly classical laws such as LEM are not to blame for the paradoxes. Paradoxicality can be unearthed using Core Logic. There is something special about the proofs or disproofs associated with genuine semantics paradoxes, making them importantly different from straightforward proofs of theoremhood or of inconsistency. That special something is the non-termination of reduction sequences of the proofs or disproofs associated with genuine paradoxes. (Detailed illustrations will be given for the Liar Paradox, Curry’s Paradox, and Grelling’s Paradox.) This insight is robust with regard to the Revenge Liar Paradox. Time permitting, we shall close with some meta-reflections on the Revenge Liar.

##### Elia Zardini, Unstable knowledge

I start by arguing that the Knower paradox offers underappreciated resistance to standard non-classical approaches to the paradoxes of self-reference, because of the divergence they require between rejecting a sentence and accepting its negation. I move on to address a problem which the Knower paradox raises for my own favoured non-contractive, instability-based approach, and which consists in the apparent lack of a suitable principle of ‘epistemic ascent’. Shifting from logic and metaphysics to pragmatics, I de- fend the coherence and reasonableness of accepting sentences (like the Knower sentence or its negation) that express unstable states-of-affairs. Consequently, appealing to unusual conceptual resources, I close by revealing whether paradoxical sentences are true or not . . .

**Organizing committee:**

- Julien Murzi (Salzburg)
- Lorenzo Rossi (Salzburg)

For questions or any other queries, please contact Julien Murzi (julienDOTmurziATsbgDOTacDOTat) or Lorenzo Rossi (lorenzoDOTrossiATsbgDOTacDOTat).

The workshop is hosted by the Philosophy Department (KGW) at the University of Salzburg and is part of the project The Liar and its Revenge in Context, sponsored by the Austrian Science Fund (FWF).