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Median Logic

The question 'Which of the two logics - classical and intuitionistic - is better?' has been around for about a century. Each has certain advantages over the other in certain dimensions. Both intuitionistic and classical logic have issues, too. Classical logic has decent models but existence proofs are not constructive in it. Intuitionistic logic has disjunction and existence properties and thus constructive proofs but the propositional fragment of intuitionistic logic while being finite does not have a finite model. It seems that classical logic has a 'better' propositional part whereas intuitionistic logic is 'better' suited for purely predicate statements.

Is it possible to combine the best of both worlds? Research in this domain was initiated by Godel and Tarski almost as soon as intuitionistic logic emerged. Since derivable intuitionistic formulas constitute a subset of derivable classical formulas, the focus of this research is on investigating properties of the logics lying between the two. Such logics are called intermediate or superintuitionistic. Intermediate logics are usually defined by adding one or more axiom schemas weaker than LEM to intuitionistic logic.

I introduced a bizarre intermediate logic that coincides with classical logic in its propositional part and coincides with intuitionistic logic in its purely predicate part. This logic is closed under modus ponens and closed under propositional substitution. This logic is a minimal intermediate logic that coincides with classical logic in its propositional part and coincides with intuitionistic logic on the set of formulas not containing propositional symbols. The minimality of median logic is critical because it implies that no other extension covering classical propositional logic can be made more ‘intuitionistic’ than median logic. Whereas supersets of median logic are less intuitionistic, its subsets are not fully classical in the propositional part.

This logic will be called median logic because, in a sense, it lies right in the middle between intuitionistic and classical logic. Considering the formulas that are derivable classically but not intuitionistically, median logic comprises all of them among purely propositional formulas, none of them among purely predicate formulas, and somewhat ‘half’ of them in between. Median logic has a balanced mix of classical and intuitionistic characteristics. Derivable propositional and first-order formulas smoothly blend in median logic.

Investigation of the properties of median logic is based on using sequent calculus. Admissibility of cu is retained for median logic. I define constraints to Kripke structes so that these constrained Kripke structures can be used to model median logic. The extent of the disjunction and existence properties within median logic is investigated.

My paper on this topic is available online at the preprint server of St. Petersburg Mathematical Society:
Abstract       Full Text

Alternatively, same results can be obtained within the framework of intuitionistic logic without defining a new logic. This is achieved by assuming that all propositional atoms (symbols) are decidable. A paper exploiting this latter approach is available at the Arxiv preprint server:
Abstract and Full Text
 

Copyright © 2004 Alexander Sakharov

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