We investigate the semantics of the logical systems obtained by introducing the modalities 2 and 3 into the family of substructural implication logics (including relevant, linear and intuitionistic implication) . Then, in the spirit of the LDS (Labelled Deductive Systems) methodology, we "import" this semantics into the classical proof system KE. This leads to the formulation of a uniform labelled refutation system for the new logics which is a natural extension of a system for substructural implication developed by the first two authors in a previous paper. Keywords: Kripke semantics, Labelled Deductive Systems, KE system. 1 Introduction The notion of modality is central in pure and applied logic. Many systems presented to formalise some application area require the addition of modality to the language for a variety of reasons: to cater for changes of the system in time, or perhaps for the dependency of the system on the context, or even to bring metalevel notions into the object l...

In this paper a uniform methodology to perform Natural Deduction over the family of linear, relevance and intuitionistic logics is proposed. The methodology follows the Labelled Deductive Systems (LDS) discipline, where the deductive process manipulates declarative units -- formulas label led according to a labelling algebra. In the system described here, labels are either ground terms or variables of a given labelling language and inference rules manipulate formulas and labels simultaneously, generating (whenever necessary) constraints on the labels used in the rules. A set of natural deduction style inference rules is given, and the notion of a derivation is defined which associates a labelled natural deduction style "structural derivation" with a set of generated constraints. Algorithmic procedures, based on a technique called resource abduction, are defined to solve the constraints generated within a derivation, and their termination conditions discussed. A natural deduc...

This paper describes a proof theoretic and semantic approach in which logics belonging to di#erent families can be given common notions of derivability relation and semantic entailment. This approach builds upon Gabbay's methodology of Labelled Deductive Systems (LDS) and it is called the compilation approach for labelled deductive systems (CLDS). Two di#erent logics are here considered, (i) the modal logic of elsewhere (known also as the logic of inequality) and (ii) the moltiplicative fragment of substructural linear logic. A general natural deduction style proof system is given, in which the notion of a theory is defined as a (possibly singleton) structure of points, called a configuration,anda "general" model-theoretic semantic approach is described using a translation technique based on firstorder logic. Then it is shown how both this proof theory and semantics can be directly applied to the logic of elsewhere and to linear logic, illustrating also that the same technique for pro...