We formulate a unified display calculus proof theory for the four principal varieties of bunched logic by combining display calculi for their component logics. Our calculi satisfy cut-elimination, and are sound and complete with respect to their standard presentations. We show how to constrain applications of display-equivalence in our calculi in such a way that an exhaustive proof search need be only finitely branching, and establish a full deduction theorem for the bunched logics with classical additives, BBI and CBI. We (...) also show that the standard sequent calculus for BI can be seen as a reformulation of its display calculus, and argue that analogous sequent calculi for the other varieties of bunched logic are very unlikely to exist. (shrink)

In this paper, we examine a number of relevant logics’ variable sharing properties from the perspective of theories of topic or subject-matter. We take cues from Franz Berto’s recent work on topic to show an alignment between families of variable sharing properties and responses to the topic transparency of relevant implication and negation. We then introduce and defend novel variable sharing properties stronger than strong depth relevance—which we call cn-relevance and lossless cn-relevance—showing that the properties are satisfied by the weak (...) relevant logics and , respectively. We argue that such properties address a sort of semantic lossiness of strong depth relevance. (shrink)

This paper presents the notion of multiset-multiset frame (mm-frame for short), a frame equipped with a relation between (finite) multisets over the set of points which satisfies the condition called compositionality. This notion is an extension of Restall and Standefer’s multiset frame, a frame that relates a multiset to a single point. While multiset frames serve as frames for the positive fragments of relevant logics RW and R, mm-frames are for the full RW and R with negation. We show this (...) by presenting a way of constructing an mm-frame from any GS-frame, a frame with two dual ternary relations in which the Routley star is definable. (shrink)

The aim of the present book is to give a comprehensive account of the ‘state of the art’ of substructural logics, focusing both on their proof theory and on their semantics (both algebraic and relational. It is for graduate students in either philosophy, mathematics, theoretical computer science or theoretical linguistics as well as specialists and researchers.

We present substructural negations, a family of negations classified in terms of structural rules of an extended kind of sequent calculus, display calculus. In considering the whole picture, we emphasize the duality of negation. Two types of negative modality, impossibility and unnecessity, are discussed and "self-dual" negations like Classical, De Morgan, or Ockham negation are redefined as the fusions of two negative modalities. We also consider how to identify, using intuitionistic and dual intuitionistic negations, two accessibility relations associated with impossibility (...) and unnecessity. (shrink)

This is a purely conceptual paper. It aims at presenting and putting into perspective the idea of a proof-theoretic semantics of the logical operations. The first section briefly surveys various semantic paradigms, and Section 2 focuses on one particular paradigm, namely the proof-theoretic semantics of the logical operations.

This is an exploratory and expository paper, comparing display logic formulations of normal modal logics with labelled sequent systems. We provide a translation from display sequents into labelled sequents. The comparison between different systems gives us a different way to understand the difference between display systems and other sequent calculi as a difference between local and global views of consequence. The mapping between display and labelled systems also gives us a way to understand labelled systems as properly structural and not (...) just as systems encoding modal logic into first-order logic. (shrink)

In previous work we gave a new proof-theoretical method for establishing upper-bounds on the space complexity of the provability problem of modal and other propositional non-classical logics. Here we extend and refine these results to give an O -space decision procedure for the basic positive relevance logic B+. We compute this upper-bound by first giving a sound and complete, cut-free, labelled sequent system for B+, and then establishing bounds on the application of the rules of this system.

The work is concerned with the so called display property of display logic. The motivation behind it is discussed and challenged. It is shown using one display calculus for intuitionistic logic as an example that the display property can be abandoned without losing subformula, cut elimination and completeness properties in such a way that results in additional expressive power of the system. This is done by disentangling structural connectives so that they are no longer context-sensitive. A recipe for characterizing structural (...) extensions of display calculi is provided. (shrink)

A star-free relational semantics for relevant logic is presented together with a sound and complete sequent proof theory. It is an extension of the dualist approach to negation regarded as modality, according to which de Morgan negation in relevant logic is better understood as the confusion of two negative modalities. The present work shows a way to define them in terms of implication and a new connective, co-implication, which is modeled by respective ternary relations. The defined negations are confused by (...) a special constraint on ternary relation, called the generalized star postulate, which implies definability of the Routley star in the frame. The resultant logic is shown to be equivalent to the well-known relevant logic R. Thus it can be seen as a reconstruction of R in the dualist framework. (shrink)

This paper presents an overview of the methods of hypersequents and display sequents in the proof theory of non-classical logics. In contrast with existing surveys dedicated to hypersequent calculi or to display calculi, our aim is to provide a unified perspective on these two formalisms highlighting their differences and similarities and discussing applications and recent results connecting and comparing them.

It is known that propositional relevant logics can be conservatively extended by the addition of a Heyting (intuitionistic) implication connective. We show that this same conservativity holds for a range of first-order relevant logics with strong identity axioms, using an adaptation of Fine’s stratified model theory. For systems without identity, the question of conservatively adding Heyting implication is thereby reduced to the question of conservatively adding the axioms for identity. Some results in this direction are also obtained. The conservative presence (...) of Heyting implication allows the development of an alternative model theory for quantified relevant logics. (shrink)

It is shown that the constructive four-valued logic N4 can be faithfully embedded into the modal logic S4. This embedding is used to obtain complete, cut-free display sequent calculi for N4 and C4, the modal logic of consistency over N4. C4 is a natural monotonic base system for semantics-based non-monotonic reasoning.

In this paper we introduce and compare four different syntactic methods for generating sequent calculi for the main systems of modal logic: the multiple sequents method, the higher-arity sequents method, the tree-hypersequents method and the display method. More precisely we show how the first three methods can all be translated in the fourth one. This result sheds new light on these generalisations of the sequent calculus and raises issues that will be examined in the last section.

A new combined temporal logic called synchronized linear-time temporal logic (SLTL) is introduced as a Gentzen-type sequent calculus. SLTL can represent the n -Cartesian product of the set of natural numbers. The cut-elimination and completeness theorems for SLTL are proved. Moreover, a display sequent calculus δ SLTL is defined.

We define cut-free display calculi for knowledge logics wherean indiscernibility relation is associated to each set of agents, andwhere agents decide the membership of objects using thisindiscernibility relation. To do so, we first translate the knowledgelogics into polymodal logics axiomatised by primitive axioms and thenuse Kracht's results on properly displayable logics to define thedisplay calculi. Apart from these technical results, we argue thatDisplay Logic is a natural framework to define cut-free calculi for manyother logics with relative accessibility relations.