Indexed systems of sequents and cut-elimination

Cut reductions are defined for a Kripke-style formulation of modal logic in terms of indexed systems of sequents. A detailed proof of the normalization (cut-elimination) theorem is given. The proof is uniform for the propositional modal systems with all combinations of reflexivity, symmetry and transitivity for the accessibility relation. Some new transformations of derivations (compared to standard sequent formulations) are needed, and some additional properties are to be checked. The display formulations of the systems considered can be presented as encodings of Kripke-style formulations.     

Cut Elimination for S4C: A Case Study

S4C is a logic of continuous transformations of a topological space. Cut elimination for it requires new kind of rules and new kinds of reductions     

Effective Cut-elimination for a Fragment of Modal mu-calculus

A non-effective cut-elimination proof for modal mu-calculus has been given by G. J?ger, M. Kretz and T. Studer. Later an effective proof has been given for a subsystem M ₁ with non-iterated fixpoints and positive endsequents. Using a new device we give an effective cut-elimination proof for M ₁ without restriction to positive sequents.     

Notes on Constructive Negation

We put together several observations on constructive negation. First, Russell anticipated intuitionistic logic by clearly distinguishing propositional principles implying the law of the excluded middle from remaining valid principles. He stated what was later called Peirce’s law. This is important in connection with the method used later by Heyting for developing his axiomatization of intuitionistic logic. Second, a work by Dragalin and his students provides easy embeddings of classical arithmetic and analysis into intuitionistic negationless systems. In the last section, we present in some detail a stepwise construction of negation which essentially concluded the formation of the logical base of the Russian constructivist school. Markov’s own proof of Markov’s principle (different from later proofs by Friedman and Dragalin) is described.