University of Calgary

# Bleg: Systematic Approaches to Generation of Logical Calculi

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Submitted by Richard Zach on Wed, 12/17/2008 - 8:31am

Over lunch the other day, my friend and colleague Agata Ciabattoni told me about her paper at this year's LICS, "From axioms to analytic rules in nonclassical logics". In it, she and her co-authors Nikolaos Galatos and Kazushige Terui present an intriguing and very general result: Suppose you have a logic which can be axiomatized in full Lambek calculus with exchange (that's intuitionistic logic without weakening or contraction, but with exchange) by adding axioms. If the additional axioms aren't too complex, there's a systematic way of generating an analytic hypersequent calculus for your logic, i.e., a systematic way of converting the additional axioms into a structural rule for a hypersequent calculus in such a way that cut is eliminable. This procedure applies to a wide range of substructural logics but also superintuitionistic logics. (UPDATE: more detail in the next post.) So that got us thinking: what other general, systematic approaches to generation of calculi are there? Agata's approach generates calculi of a specific form (hypersequent calculi) from other calculi (Hilbert-style calculi). Then I know of two approaches that systematically generate calculi from a semantics. Arnon Avron, Beata Konikowska, and Anna Zamansky have been doing a lot of work on logics given by what they call non-deterministic matrices. I wrote a while ago about the approach I detailed in my undergraduate thesis, which goes back to work by Rousseau in the 1960s: systematic (i.e., automatic) generation of sequent, tableaux, natural deduction calculi for a logic given by finite truth tables. These are the only systematic results I know of, but that just shows my ignorance! There must be others! I'm sure there are general results in modal correspondence theory, for instance, to obtain axioms and perhaps tableaux systems, etc., for modal logics from conditions on frames. Can anyone help me (and Agata) out here?