Labelled Natural Deduction Systems for Propositional Non-Classical Logics
In the previous chapter we investigated labelled natural deduction presentations of propositional modal logics. Here we explore the generalizations needed to build ND systems for large families of propositional non-classical logics, including relevance logics (and, more generally, substructural logics [75, 76, 196]), where we can treat non-classical negation as a modal operator and also consider explicitly positive fragments. (The metatheory of positive logics is different from that for ‘full’ logics; see, e.g., Dunn’s semantic analysis of positive modal logics in .) We generalize our framework to provide a uniform treatment of a wide range of non-classical operators (□, ◊, relevant and intuitionistic implication, non-classical negation, etc.), where we base our presentations on an abstract classification of non-classical operators as ‘universal’ or ‘existential’, and associated general metatheorems. We proceed as follows.
KeywordsModal Logic Intuitionistic Logic Kripke Model Axiom Schema Relevance Logic
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