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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 firstorder 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
Need to relax? Try brain teasers. I would recommend those marked 'cool'.




