COMBINATORY LOGIC AND $ \lambda $-CALCULUS FOR CLASSICAL LOGIC

Since Griffin's work in 1990, classical logic has been an attractive target for extracting computational contents. However, the classical principle used in Griffin's type system is the double-negation-elimination rule, which prevents one to analyze the intuitionistic part and the purely classical part separately. By formulating a calculus with $ \mathrm{J} $ (for the elimination rule of falsehood) and $ \mathrm{P} $ (for Peirce formula which is concerned with purely classical reasoning) combinators, we can separate these two parts. This paper studies the $ \lambda \mathrm{PJ} $ calculus with $ \mathrm{P} $ and $ \mathrm{J} $ combinators and the $ \lambda \mathrm{C} $ calculus with $ \mathrm{C} $ combinator(for the double-negation-elimination rule). We also propose two $ \lambda $-calculi which correspond to $ \lambda \mathrm{PJ} $ and $ \lambda \mathrm{C} $. We give four classes of reduction rules for each calculus, and systematically study their relationship by simulating reduction rules in one calculus by the corresponding one in the other. It is shown that, by restricting the type of $ P $, simulation succeeds for several choices of reduction rules, but that simulating the full calculus $ \lambda \mathrm{PJ} $ in $ \lambda \mathrm{C} $ succeeds only for one class. Some programming examples of our calculi such as encoding of conjunction and disjunction are also given.