We have left out negation. We regard ¬φ as a shorthand for φ→ ⊥. A sequent is a pair that packages together some assumptions and a conclusion. We write Γ ` φ for the sequent whose assumptions are the finite set of formulas Γ and whose conclusion is the formula φ. We call Γ the antecedent and φ the consequent of the sequent Γ ` φ. A sequent is satisfied in a model M iff either M |= φ or for some...

Abstract The preferential conditional logic $ \mathbb{PCL} $, introduced by Burgess, and its extensions are studied. First, a natural semantics based on neighbourhood models, which generalizes Lewis’ sphere models for counterfactual logics, is proposed. Soundness completeness of with respect to this class proved directly. Labelled sequent calculi all logics the family then introduced. modular h...

In this work, we explore proof theoretical connections between sequent, nested and labelled calculi. In particular, we show a general algorithm for transforming a class of nested systems into sequent calculus systems, passing through linear nested systems. Moreover, we show a semantical characterisation of intuitionistic, multi-modal and non-normal modal logics for all these systems, via a case...

We consider an extension of bi-intuitionistic logic with the traditional modalities ♦, , and from tense logic Kt. Proof theoretically, this extension is obtained simply by extending an existing sequent calculus for bi-intuitionistic logic with typical inference rules for the modalities used in display logics. As it turns out, the resulting calculus, LBiKt, seems to be more basic than most intui...

Bi-intuitionistic propositional logic (also known as Heyting-Brouwer logic, subtractive logic) extends intuitionistic propositional logic with a connective that is dual to implication, called coimplication. It is the logic of bi-[Cartesian closed] categories and it also has a Kripke semantics. It first got the attention of C. Rauszer who studied it in a number of papers. Later, it has been inve...

Simultaneous quantiier elimination in sequent calculus is an improvement over the well-known skolemization. It allows a lazy handling of instantiations as well as of the order of certain reductions. We prove the soundness of a sequent calculus which incorporates a rule for simultaneous quantiier elimination. The proof is performed by semantical arguments and provides some insights into the depe...

We propose a notion of focusing for nested sequent calculi for modal logics which brings down the complexity of proof search to that of the corresponding sequent calculi. The resulting systems are amenable to specifications in linear logic. Examples include modal logic K, a simply dependent bimodal logic and the standard non-normal modal logics. As byproduct we obtain the first nested sequent c...