Results 1 
8 of
8
A systematic proof theory for several modal logics
 Advances in Modal Logic, volume 5 of King’s College Publications
, 2005
"... abstract. The family of normal propositional modal logic systems is given a very systematic organisation by their model theory. This model theory is generally given using frame semantics, and it is systematic in the sense that for the most important systems we have a clean, exact correspondence betw ..."
Abstract

Cited by 26 (1 self)
 Add to MetaCart
abstract. The family of normal propositional modal logic systems is given a very systematic organisation by their model theory. This model theory is generally given using frame semantics, and it is systematic in the sense that for the most important systems we have a clean, exact correspondence between their constitutive axioms as they are usually given in a HilbertLewis style and conditions on the accessibility relation on frames. By contrast, the usual structural proof theory of modal logic, as given in Gentzen systems, is adhoc. While we can formulate several modal logics in the sequent calculus that enjoy cutelimination, their formalisation arises through systembysystem fine tuning to ensure that the cutelimination holds, and the correspondence to the axioms of the HilbertLewis systems becomes opaque. This paper introduces a systematic presentation for the systems K, D, M, S4, and S5 in the calculus of structures, a structural proof theory that employs deep inference. Because of this, we are able to axiomatise the modal logics in a manner directly analogous to the HilbertLewis axiomatisation. We show that the calculus possesses a cutelimination property directly analogous to cutelimination for the sequent calculus for these systems, and we discuss the extension to several other modal logics. 1
Cutelimination and proofsearch for biintuitionistic logic using nested sequents
, 2008
"... We propose a new sequent calculus for biintuitionistic logic which sits somewhere between display calculi and traditional sequent calculi by using nested sequents. Our calculus enjoys a simple (purely syntactic) cutelimination proof as do display calculi. But it has an easily derivable variant cal ..."
Abstract

Cited by 15 (4 self)
 Add to MetaCart
We propose a new sequent calculus for biintuitionistic logic which sits somewhere between display calculi and traditional sequent calculi by using nested sequents. Our calculus enjoys a simple (purely syntactic) cutelimination proof as do display calculi. But it has an easily derivable variant calculus which is amenable to automated proof search as are (some) traditional sequent calculi. We first present the initial calculus and its cutelimination proof. We then present the derived calculus, and then present a proofsearch strategy which allows it to be used for automated proof search. We prove that this search strategy is terminating and complete by showing how it can be used to mimic derivations obtained from an existing calculus GBiInt for biintuitionistic logic. As far as we know, our new calculus is the first sequent calculus for biintuitionistic logic which uses no semantic additions like labels, which has a purely syntactic cutelimination proof, and which can be used naturally for backwards proofsearch.
Modal Sequent Calculi Labelled with Truth Values: Completeness, Duality and Analyticity
 LOGIC JOURNAL OF THE IGPL
, 2003
"... Labelled sequent calculi are provided for a wide class of normal modal systems using truth values as labels. The rules for formula constructors are common to all modal systems. For each modal system, specific rules for truth values are provided that reflect the envisaged properties of the accessi ..."
Abstract

Cited by 8 (5 self)
 Add to MetaCart
(Show Context)
Labelled sequent calculi are provided for a wide class of normal modal systems using truth values as labels. The rules for formula constructors are common to all modal systems. For each modal system, specific rules for truth values are provided that reflect the envisaged properties of the accessibility relation. Both local and global reasoning are supported. Strong completeness is proved for a natural twosorted algebraic semantics. As a corollary, strong completeness is also obtained over general Kripke semantics. A duality result
Nested Sequents for Intuitionistic Logics
, 2011
"... Nested sequent systems for modal logics were introduced by Kai Brünnler, and have come to be seen as an attractive deep reasoning extension of familiar sequent calculi. In an earlier paper I showed there was a connection between modal nested sequents and modal prefixed tableaus. In this paper I exte ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Nested sequent systems for modal logics were introduced by Kai Brünnler, and have come to be seen as an attractive deep reasoning extension of familiar sequent calculi. In an earlier paper I showed there was a connection between modal nested sequents and modal prefixed tableaus. In this paper I extend the nested sequent machinery to intuitionistic logic, both standard and constant domain, and relate the resulting sequent calculi to intuitionistic prefixed tableaus. Modal nested sequent machinery generalizes one sided sequent calculi—the present work similarly generalizes two sided sequents. It is noteworthy that the resulting system for constant domain intuitionistic logic is particularly simple. It amounts to a combination of intuitionistic propositional rules and classical quantifier rules, a combination that is known to be inadequate when conventional intuitionistic sequent systems are used.
Labelled modal sequents
 In Areces and de Rijke [AdR99]. Use Your Logic 7
"... Abstract. In this paper we present a new labelled sequent calculus for modal logic. The proof method works with a more “liberal ” modal language which allows inferential steps where different formulas refer to different labels without moving to a particular world and there computing if the consequen ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper we present a new labelled sequent calculus for modal logic. The proof method works with a more “liberal ” modal language which allows inferential steps where different formulas refer to different labels without moving to a particular world and there computing if the consequence holds. Worldpaths can be composed, decomposed and manipulated through unification algorithms and formulas in different worlds can be compared even if they are subformulas which do not depend directly on the main connective. Accordingly, such a sequent system can provide a general definition of modal consequence relation. Finally, we briefly sketch a proof of the soundness and completeness results. 1
Comparing Modal Sequent Systems
"... abstract. 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 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
abstract. 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 firstorder logic. 1
Nested Sequents for Intuitionistic Logics Melvin Fitting
, 2012
"... Relatively recently nested sequent systems for modal logics have come to be seen as an attractive deep reasoning extension of familiar sequent calculi. In an earlier paper I showed there was a strong connection between modal nested sequents and modal prefixed tableaux. In this paper I show the conne ..."
Abstract
 Add to MetaCart
(Show Context)
Relatively recently nested sequent systems for modal logics have come to be seen as an attractive deep reasoning extension of familiar sequent calculi. In an earlier paper I showed there was a strong connection between modal nested sequents and modal prefixed tableaux. In this paper I show the connection continues to intuitionistic logic, both standard and constant domain, relating nested intuitionistic sequent calculi to intuitionistic prefixed tableaux. Modal nested sequent machinery generalizes one sided sequent calculi—intuitionistic nested sequents similarly generalize two sided sequents. It is noteworthy that the resulting system for constant domain intuitionistic logic is particularly simple. It amounts to a combination of intuitionistic propositional rules and classical quantifier rules, a combination that is known to be inadequate when conventional intuitionistic sequent systems are used.