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Abstract: We combine intuitionistic logic and classical logic into a new, first-order logic called polarized intuitionistic logic. This logic is based on a distinction between two dual polarities which we call red and green to distinguish them from other forms of polarization. The meaning of these polarities is defined model-theoretically by a Kripke-style semantics for the logic. Two proof systems are also formulated. The first system extends Gentzenʼs intuitionistic sequent calculus LJ. In addition, this system also bears essential similarities to Girardʼs LC proof system for classical logic. The second proof system is based on a semantic tableau and extends Dragalinʼs multiple-conclusion version of intuitionistic sequent calculus. We show that soundness and completeness hold for these notions of semantics and proofs, from which it follows that cut is admissible in the proof systems and that the propositional fragment of the logic is decidable. [Copyright &y& Elsevier]

Abstract: In this paper we study a version of constructive linear-time temporal logic (LTL) with the “next” temporal operator. The logic is originally due to Davies, who has shown that the proof system of the logic corresponds to a type system for binding-time analysis via the Curry–Howard isomorphism. However, he did not investigate the logic itself in detail; he has proved only that the logic augmented with negation and classical reasoning is equivalent to (the “next” fragment of) the standard formulation of classical linear-time temporal logic. We give natural deduction, sequent calculus and Hilbert-style proof systems for constructive LTL with conjunction, disjunction and falsehood, and show that the sequent calculus enjoys cut elimination. Moreover, we also consider Kripke semantics and prove soundness and completeness. One distinguishing feature of this logic is that distributivity of the “next” operator over disjunction “” is rejected in view of a type-theoretic interpretation. [Copyright &y& Elsevier]