Abstract
The paper presents a uniform proof-theoretic treatment of several kinds of free logic, including the logics of existence and definedness applied in constructive mathematics and computer science, and called here quasi-free logics. All free and quasi-free logics considered are formalised in the framework of sequent calculus, the latter for the first time. It is shown that in all cases remarkable simplifications of the starting systems are possible due to the special rule dealing with identity and existence predicate. Cut elimination is proved in a constructive way for sequent calculi adequate for all logics under consideration.
Highlights
Free logics come from different sources, appear under many names, and find multiple applications
We provide a systematic treatment of some important free logics in the framework of sequent calculus (SC)
We have presented several sequent calculi for six free logics, both in the classical and the intuitionistic version
Summary
Free logics come from different sources, appear under many names, and find multiple applications. Baaz and Iemhoff [1] provided SC for intuitionistic versions of identity-free PFL and PFL+ with proofs of cut elimination (full for PFL and partial for PFL+) and interpolation. These results were recently improved by Maffezioli and Orlandelli [19]. As for the treatment in terms of SC, Bencivenga [3] contains a formalization of PFL and Gratzl [9] provided SC for NFL= where cut is eliminable partially (so called “inessential cuts”, in Takeuti’s sense, cannot be removed) – both on the classical basis.
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