Much has been said about intuitionistic and classical logical systems since Gentzen’s seminal work. Recently, Prawitz and others have been discussing how to put together Gentzen’s systems for classical and intuitionistic logic in a single unified system. We call Prawitz’ proposal the Ecumenical System, following the terminology introduced by Pereira and Rodriguez. In this work we present an Ecumenical sequent calculus, as opposed to the original natural deduction version, and state some proof theoretical properties of the system. We reason that (...) sequent calculi are more amenable to extensive investigation using the tools of proof theory, such as cut-elimination and rule invertibility, hence allowing a full analysis of the notion of Ecumenical entailment. We then present some extensions of the Ecumenical sequent system and show that interesting systems arise when restricting such calculi to specific fragments. This approach of a unified system enabling both classical and intuitionistic features sheds some light not only on the logics themselves, but also on their semantical interpretations as well as on the proof theoretical properties that can arise from combining logical systems. (shrink)

Recent works about ecumenical systems, where connectives from classical and intuitionistic logics can co-exist in peace, warmed the discussion on proof systems for combining logics. This discussion has been extended to alethic modalities using Simpson’s meta-logical characterization: necessity is independent of the viewer, while possibility can be either intuitionistic or classical. In this work, we propose a pure, label free calculus for ecumenical modalities, nEK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {nEK}$$\end{document}, where exactly one logical operator figures (...) in introduction rules and every basic object of the calculus can be read as a formula in the language of the ecumenical modal logic EK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {EK}$$\end{document}. We prove that nEK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {nEK}$$\end{document} is sound and complete w.r.t. the ecumenical birelational semantics and discuss fragments and extensions. (shrink)

In 2015 Dag Prawitz proposed a natural deduction ecumenical system, where classical logic and intuitionistic logic are codified in the same system. In his ecumenical system, Prawitz recovers the harmony of rules, but the rules for the classical operators do not satisfy separability. In fact, the classical rules are not pure, in the sense that negation is used in the definition of the introduction and elimination rules for the classical operators. In this work we propose an ecumenical system adapting, to (...) the natural deduction framework, Girard’s mechanism of stoup. This allows for the definition of a pure harmonic natural deduction system for the propositional fragment of Prawitz’s ecumenical logic. We conclude by presenting an extension proposal to the first-order setting and a different approach to purity based on some ideas proposed by Julien Murzi. (shrink)

Benchmarking automated theorem proving (ATP) systems using standardized problem sets is a well-established method for measuring their performance. However, the availability of such libraries for non-classical logics is very limited. In this work we propose a library for benchmarking Girard's (propositional) intuitionistic linear logic. For a quick bootstrapping of the collection of problems, and for discussing the selection of relevant problems and understanding their meaning as linear logic theorems, we use translations of the collection of Kleene's intuitionistic theorems in the (...) traditional monograph "Introduction to Metamathematics". We analyze four different translations of intuitionistic logic into linear logic and compare their proofs using a linear logic based prover with focusing. In order to enhance the set of problems in our library, we apply the three provability-preserving translations to the propositional benchmarks in the ILTP Library. Finally, we generate a comprehensive set of reachability problems for Petri nets and encode such problems as linear logic sequents, thus enlarging our collection of problems. (shrink)