Aim

The aim of this workshop is to discuss and exchange new ideas and recent developments related to nonclassical logics, especially paraconsistent and substructural logics, and its philosophical issues.

Date & Location

• Date: November 19--20, 2015.
• Location: Kyoto University, Yoshida Izumidono. [map]

Speakers

• Colin Caret (Yonsei University)
• Petr Cintula (The Czech Academy of Sciences)
• Yasuo Deguchi (Kyoto University)
• Ryosuke Igarashi (Kyoto University)
• Hidenori Kurokawa (Kobe University)
• Katsuhiko Sano (JAIST)
• Kazushige Terui (Kyoto University)
• Zach Weber (University of Otago)
• Shusuke Yatabe (Kyoto University)

Photos

Few photos from the event, taken by Petr Cintula, can be found here.

Program

Nov. 19:

13:00--14:10: Zach Weber: On closure and truth in substructural theories of truth [slides]

14:10--15:20: Hidenori Kurokawa: Labelled sequent calculi for substructural logics I: relevant logics

15:20--15:40: Coffee break

15:40--16:50: Ryosuke Igarashi: Inferentialism on Negation [slides]

16:50--18:00: Shunsuke Yatabe: A constructive naive set theory and infinity [slides]

Nov. 20:

10:30--11:40: Kazushige Terui: Substructural logics and fixed points [slides]

11:40--13:00: Lunch break

13:00--14:10: Colin Caret: Pluralism and Contextualism [handout]

14:10--15:20: Yasuo Deguchi: Compartmentalizing Trivialism; Nishida's Contradictory Self-identity Viewed from a Non-classical Logic [slides]

15:20--15:40: Coffee break

15:40--16:50: Katsuhiko Sano: Cut-Elimination Theorem for Expansions of Belnap-Dunn's Four Valued Logic via Functional Weak-Completeness [slides]

16:50--18:00: Petr Cintula: Substructural Logics: Deduction theorems and generalized disjunctions [slides]

Abstracts

Zach Weber

Title: On closure and truth in substructural theories of truth

Abstract: Closure is the idea that what is true about a theory of truth should be expressible in it. Commitment to closure under truth motivates non-classical logic; commitment to closure under validity leads to substructural logic (nontransitive or noncontractive). These moves can be thought of as responses to revenge problems. With a focus on truth in mathematics, I will show how a noncontractive approach faces a similar revenge problem with respect to closure under *provability*, and argue that if a noncontractive theory is to be genuinely closed, then it must be `meta'-contraction-free.

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Hidenori Kurokawa

Title: Labelled sequent calculi for substructural logics I: relevant logics

Abstract: Substructural logics have been identified as logics obtained by dropping the structural rules of weakening and/or contraction from the sequent calculus for classical logic formulated in a traditional style. Although traditional relevant logics have slightly different formulations from these cases due to the presence of distributive laws, these logics have also been broadly considered to be in the same family of substructural logics as other typical substructural logics (such as linear logic and BCK logic). This is because relevant logics have the common feature that the monotonicity principle A→(B→A), an axiomatic counterpart of weakening rule, fails to hold. The family of relevant logics has also been semantically characterized by Routley-Meyer semantics, a relational semantics based on ternary accessibility relations. In this talk, we formulate these traditional relevant logics by using labelled sequent calculi with a ternary relational symbol, analogously to the binary labelled sequent calculi for modal logics in (Negri, 2005). In particular, we develop those calculi for relevant logics by adopting G3-style sequent calculi, which are formulated in such a way that all the rules are invertible and all the structural rules (including cut) are admissible. We highlight the fact that, although relevant logics are usually formulated by omitting the structural rules of weakening and contraction, in the labelled sequent calculi presented in this talk, we can show that all the rules are invertible and the structural rules of weakening and contraction are admissible in a height-preserving manner. This is a joint work with Sara Negri.

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Ryosuke Igarashi

Title: Inferentialism on Negation

Abstract: see here.

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Shunsuke Yatabe

Title: title: A constructive naive set theory and infinity

Abstract: CONS, a constructive naive set theory in FLew∀ (intuitionistic logic minus the contraction rule) is consistent and we can develop its metamathematics by itself. The significance of CONS is to allow circular definitions of very strong form though it is proof theoretically weak. However, the details of such circularly defined sets are not well-known: we do not know whether they contain non-standard elements in particular. In this paper, we investigate the non-standardness of ω, the set of natural numbers which is also defined circularly, as a testbed, and we give negative answers to the problem whether CONS is ω-consistent, using co-inductive objects essentially.

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Kazushige Terui

Title: Substructural logics and fixed points

Abstract: The talk will consist of two parts. The first part is an introduction to substructural logics from an algebraic perspective. I will in particular focus on the relationship between syntactic and algebraic properties, such as

* local deduction theorem and congruence extension property,

* Robinson property and amalgamation property,

* strong deductive interpolation and transferrable injections.

In the second part, I will discuss substructural logics with fixed points, which are intimately connected to Brouwer's fixed point theorem.

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Colin Caret

Title: Pluralism and Contextualism

Abstract: According to the logical pluralist, there are several distinct relations of logical consequence. Some critics argue that logical pluralism suffers from what I call the collapse problem: that despite its intention to articulate a radically pluralistic doctrine about logic, the view unintentionally collapses into logical monism. In this paper, I propose a contextualist resolution of the collapse problem. Contextualism about validity clarifies the mechanism responsible for a plurality of logics and handles the motivating data better than the original view. It is a major improvement that should be embraced by all logical pluralists.

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Yasuo Deguchi

Title: Compartmentalizing Trivialism; Nishida's Contradictory Self-identity Viewed from a Non-classical Logic

Abstract: The later Nishida advocated the idea of 'Absolutely Contradictory Self-identity'. This idea has disgusted even Nishida-sympathizers because it seems to commit overtly true contradictions. But the situation is even worse because, as we interpret him, he endorsed not merely dialetheism but also trivialism, according to which every contradictions, therefore every propositions are true. Arguably dialetheim was made plausible under an ilk of non-classical logic; paraconsisitent logic. But trivialism needs something more than that so as to render itself making-sense. So let me engineer further non-classical logical devices such as trivializer; i.e., truth-maker that makes every proposition as true, aiming to make the later Nishida's trivilism as more palatable than otherwise.

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Katsuhiko Sano

Title: Cut-Elimination Theorem for Expansions of Belnap-Dunn's Four Valued Logic via Functional Weak-Completeness

Abstract: This talk concerns a syntactic cut-elimination theorem for expansions of Belnap-Dunn's four-valued logic, whose four values are: true, false, neither (gap), and both (glut). First, I introduce the outline of Belnap-Dunn's four-valued logic regarding Dunn's relational semantics, which specifies when a given formula is *related to* the classical truth values (1 and 0) and also provides us with the more intuitive understanding of a particular connective than semantics based on four-valued tables. Moreover, I review the notion of functional weak-completeness from (Omori and Sano 2015) as a generalization of functional completeness in Belnap-Dunn's four-valued logic and also demonstrate which set of connectives becomes functionally weak-complete. Second, I explain how the idea of Dunn's relational semantics enables us to generate effectively inference rules for a given connective to construct a (G3-style) sequent calculus. Finally, I show that the admissibility of cut in a sequent calculus for *any* set of connectives is reduced constructively to the admissibility of cut in a sequent calculus for a weak-functionally complete set of connectives. The content of this talk is a partially joint work with Hitoshi Omori (Kyoto University).

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Petr Cintula

Title: Substructural Logics: Deduction theorems and generalized disjunctions

Abstract: We will discuss generalized forms of deduction theorems and disjunction connectives in a wide context of substructural logics (considered as, not necessarily axiomatic, extensions of non-associative Full Lambek calculus). We will see how both notions are tied together and how they relate to the existence of a suitable Hilbert style calculus for the logic in question. Finally we will present interesting and important consequences of both notions.