New Insights on the Intuitionistic Interpretation of Default Logic

In this work we further investigate the relation, first found by Truszczyński, between modal logic S4F and Default Logic (DL), analyzing some interesting properties and showing its application to other general non-monotonic formalisms. For comparison purposes, we start defining a subset of S4F we called Intuitionistic Default Logic (IDL), which consists in incorporating an additional set of propositional operators. These operators are translated into modal formulas using Gödel’s pattern for encoding Intuitionistic Logic into S4. Then, we prove that, under a particular models minimization policy, IDL generalizes Turner’s Nested Default Logic (NDL), in the sense that the latter does not allow nesting or combining the rule conditional operator. This result is also used to show that strong equivalence of default theories can be reduced to a simple S4F equivalence test. Finally, we prove that IDL also generalizes Pearce’s Equilibrium Logic, which encodes logic programs into the intermediate logic of Hereand-There.