Cut-free proof systems for logics of weak excluded middle

Abstract

 In this work we perform a proof-theoretical investigation of some logical systems in the neighborhood of substructural, intermediate and many-valued logics. The common feature of the logics we consider is that they satisfy some weak forms of the excluded-middle principle. We first propose a cut-free hypersequent calculus for the intermediate logic LQ, obtained by adding the axiom *A∨**A to intuitionistic logic. We then propose cut-free calculi for systems W _n , obtained by adding the axioms *A∨(A ⊕ ⋯ ⊕ A) (n−1 times) to affine linear logic (without exponential connectives). For n=3, the system W _n coincides with 3-valued Łukasiewicz logic. For n>3, W _n is a proper subsystem of n-valued Łukasiewicz logic. Our calculi can be seen as a first step towards the development of uniform cut-free Gentzen calculi for finite-valued Łukasiewicz logics.