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Connectives in non-classical logics


Our proposal is to continue the research on connectives in non-classical logics, in particular, intuitionistic logic, as we have been doing since some time ago, i.e. paying attention specially to the concepts of univocity (or strict definability), inter-definability and conservative extension. We will also consider the property that from an algebraic point of view is called compatibility (for extensions with non-compatible extensions, in the case of Hilbert style presentations, it is needed to add another rule to modus ponens), but we are interested both in compatible and non-compatible connectives. In the case of the conservative extension property, we are interested in researching the issue both in propositional and first order logic, taking into consideration that there are extensions with connectives that are conservative over propositional logic, but not in the first order case. We are simultaneously interested in identifying paraconsistent negations, and, in that case, in investigating the strict paraconsistent issue in the sense of Urbas. In this respect, we will consider extensions of intuitionistic logic similar to Heyting-Brouwer Logic.Relating the resulting extensions we will be interested in the usual relevant properties concerning logics, e.g. the finite model property. The research will be done both from a historical-conceptual and systematic point of view. In the latter case, we will consider both the syntactic and the semantic approach.Finally, we will look for applications in the basic teaching of logic. (AU)

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