# Non-Classical Solutions to the Paradoxes

## Idea & Motivation

The paradoxes pose a challenge to our most basic intuitions about many concepts we use in every day language, such as truth, collections, baldness, and being a heap or Mount Everest. By simple laws of reasonings, we are led from intuitive principles governing these concepts to contradictions or other unwanted consequences. Traditionally, the problem was thought to be found in the aforementioned principles, which were to be restricted in some way. Nowadays, the most popular view is that some of the so-called *simple laws of reasoning* are the ones to blame instead.

Accordingly, there has been a proliferation of weaker systems than classical logic, put forward as alternative ways of reasoning from the intuitive but problematic principles. These non-classical logics block the implication of the unwanted consequences, each of them in its own way. Thus, several questions arise, that we would like to address during the workshop: What's the status of these non-classical logics? Are they, or at least is one of them, to replace classical logic? Are they true logics, in the sense of giving the right verdict on the validity of vernacular arguments? Or perhaps just one of them is? Or are they mere instruments to deal with otherwise trivial principles?

## Speakers

- Guillermo Badia (Johannes Kepler University)
- Eduardo Barrio (University of Buenos Aires/Conicet)
- Jonathan Dittrich (MCMP/LMU Munich)
- Martin Fischer (MCMP/LMU Munich)
- Ole Hjortland (University of Bergen)
- Joao Marcos (Federal University of Rio Grade do Norte/Ruhr University Bochum)
- Carlo Nicolai (MCMP/LMU Munich)
- Federico Pailos (University of Buenos Aires/Conicet)
- Dave Ripley (University of Connecticut)
- Lucas Rosenblatt (University of Buenos Aires/Conicet)
- Paula Teijeiro (University of Buenos Aires/Conicet)
- Elia Zardini (University of Lisbon)

## Program

### Day 1 (26 February 2017)

Time | Event |
---|---|

15:00 - 15:40 | Guillermo Badia: Relevant Languages as Fragments of Boolean Languages |

15:40 - 15:45 | Break |

15:45 - 16:25 | Joao Marcos: Consequence Beyond Truth and Proof |

16:25 - 16:45 | Coffee Break |

16:45 - 18:30 | Discussion |

### Day 2 (27 February 2017)

Time | Event |
---|---|

10:30 - 11:45 | Eduardo Barrio: Models & Proofs: Paraconsistent logics without a Canonical Interpretation |

11:45 - 12:00 | Coffee Break |

12:00 - 13:30 | Martin Fischer and Carlo Nicolai: Truth Has to Live Up to Its Task |

13:30 - 15:00 | Lunch Break |

15:00 - 16:15 | Jonathan Dittrich: Fixed-Point Constructions for Substructural Approaches to Paradox |

16:15 - 16:30 | Coffee Break |

16:30 - 17:45 | Elia Zardini: One, and Only One |

19:30 | Conference Dinner |

### Day 3 (28 February 2017)

Time | Event |
---|---|

10:15 - 11:30 | Ole Hjortland:Theories of Truth and the Maxim of Minimal Mutilation |

11:30 - 11:45 | Coffee Break |

11:45 - 13:00 | Lucas Rosenblatt: Noncontractive Classical Logic |

13:00 - 14:30 | Lunch Break |

14:30 - 15:45 | Paula Teijeiro: What is Tonk |

15:45 - 16:00 | Coffee Break |

16:00 - 17:15 | Federico Pailos: A Logic Without Valid Sentences, Inferences and Metainferences |

17:15 - 17:30 | Coffee Break |

17:30 - 18:45 | Dave Ripley: There Is Such a Thing as a Substructural Approach to Paradox |

## Abstracts

### Guillermo Badia: Relevant Languages as Fragments of Boolean Languages

Under a proper translation, the languages of relevant logic can be model-theoretically characterized as fragments of boolean languages such as first order logic or the logic where we admit arbitrarily large conjunctions and disjunctions while bounding the length of the quantifications. For such characterizations we use the "algebraic" (in the sense that there is no mention of the formal language directly involved) concept of a relevant directed bisimulation. In this talk, I will discuss the place of this relation in helping isolate relevant languages among other semantic creatures. Hopefully, it will become clear the central role that relevant directed bisimulations have for the so called Routley-Meyer semantics for relevant logic.

top

### Carolina Blasio (UNICAMP, BR) & Joao Marcos (UFRN, BRy): Consequence Beyond Truth and Proof

In this contribution we shall experiment with a provocative analysis of logical consequence done neither in terms of preservation of truth nor in terms of preservation of warrant to assert. The analysis will be done instead from an abstract viewpoint, taking the perspective of judgmental agents that entertain certain cognitive attitudes towards the informational content of given sentences. In this approach, neither truth values nor inference rules need to be taken as primitive, for they can be fruitfully explicated in terms of a conceptually prior notion of compatibility between possibly overlapping cognitive attitudes of a given agent. We will discuss some effects of such an approach on the provision of a satisfactory theory of meaning that goes beyond truth and proof, but will also show how the announced analysis maps naturally into a four-valued non-deterministic semantics, and into an analytic bi-dimensional proof system. If time permits, we will also discuss how logical consequence so understood may pave the way towards a novel approach concerning the understanding of gappy and glutty reasoning.

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### Martin Fischer and Carlo Nicolai: Truth Has to Live Up to Its Task

A crucial desideratum for the truth predicate is to increase one's expressive resources. This requirement can be spelled out in different ways: one option is to consider the amount of non-semantic reasoning that can be gained by having truth in one's toolbox. According to one popular argument, this desideratum can be satisfactorily met by theories of truth formulated in classical logic, whereas theories of truth formulated in nonclassical logics such as FDE or K3 fare much worse in this respect. More generally, the transparency of the truth predicate is often considered to be at odds with the need for expressive power: whereas nonclassical theories of the sort just mentioned are fully transparent, classical theories cannot be. In our talk we will discuss different options to close the expressive gap existing between nonclassical and classical theories: in particular, we will first consider the possibility of strengthening the nonlogical components of a nonclassical theory via reflection principles while keeping the underlying logic fixed; secondly, we strengthen the underlying logic by adding a suitable (nonclassical) conditional to it and leave the nonlogical part unchanged. We conclude by exploring the delicate balance between transparency and expressive power in these two options.top

### Federico Pailos: A Logic Without Valid Sentences, Inferences and Metainferences

I will present a three-valued matrix logic, MMI. MMI, as TS, has no valid sentence or inference. But, unlike TS, has no valid metainference either. This is possible because MMI’s defines only a consequence relation for metainferences. MMI’s consequence relation for metainferences is very similar to TS’ consequence relation for inferences. Basically, a metainference is valid in MMI iff for every valuation v, if v satisfies every premise of the metainference according to ST, v satisfies the conclusion according to TS. This determinines a consequence relation for inferences, which is just TS'. As a result, MMI is “informative” with respect to the meta-metainference: there are valid meta-metainferences in MMI, but there are also invalid ones. Finally, I will present a sound but deeply incomplete proof system for MMI, called PMMI. Though PMMI may be easily expanded with operational rules, it is not obvious how to capture every valid meta-metainference that is not an instance of a structural rule, in a finite set of operational rules.

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### Dave Ripley: There is Such a Thing as a Substructural Approach to Paradox

In recent work, Lionel Shapiro has argued that "there is no useful explication of the idea of a substructural approach to paradox". His argument is based on giving a seemingly-substructural reformulation of Jc Beall's recently-favoured approach to paradox, which seems to be nonsubstructural if anything is. In this talk, I explore the prospects for a particular explication of the idea of a substructural approach to paradox---one I take to be useful. It turns out that on this explication, Beall's approach is just as it seems: nonsubstructural, whether formulated in Shapiro's style or not.top

### Lucas Rosenblatt: Noncontractive Classical Logic

One of the most fruitful applications of substructural logics stems from their capacity to deal with self-referential paradoxes. Both Contraction and Cut play a crucial role in typical paradoxical arguments. In this talk I address a number of difficulties affecting noncontractive approaches to paradox that have been discussed in the recent literature. The situation was roughly this: if you decide to go substructural, the nontransitive approach offers a lot of benefits that are not available in the noncontractive setting. I sketch a new noncontractive account that has these benefits. In particular, it has both a proof- and a model-theoretic presentation, it can be extended to a first-order language and it retains every classically valid argument.top

### Paula Teijeiro: What is Tonk

The argument from Tonk was proposed by Prior as a rebuttal of the idea that offering introduction and elimination rules for a symbol is enough to make it meaningful. Nevertheless, Belnap noticed that in order to prove A⊢B from the Tonk-rules, it is necessary to appeal to the transitivity of ⊢, suggesting then the requirements of conservativity and uniqueness . We know now that Tonk is conservative with respect to a number of non transitive logics, but it has only been shown to be completely determined in a non reflexive setting. My goal is, first, to explore to what extent is Tonk ill-defined in reflexive logics, that is, which requirements on the determinacy of meaning must be relaxed in order to consider Tonk meaningful in those contexts. And second, to establish what degree of determinacy can be obtained if, instead of adapting the context to fit Tonk, one adapts Tonk to better fit the context.top

### Elia Zardini: One, and Only One

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.top

## Acknowledgement

This workshop is generously funded by the Deutsche Forschungsgemeinschaft (DFG) and organized by the Munich Center for Mathematical Philosophy (MCMP, LMU Munich), as part of the collaboration project "Logics of Truth" between the MCMP and the Buenos Aires Logic Group (University of Buenos Aires).

## Contact

For questions about all aspects of the conference, please contact Lavinia Picollo: ncsp2017@lrz.uni-muenchen.de.

## Venue

Main University Building

Geschwister-Scholl-Platz 1

D-80539 München

Room B015

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