A Survey of Some Connections Between Classical, Intuitionistic and Minimal Logic

Abstract

Listen

Translate article

This chapter discusses the relationships among classical, intuitionistic and minimal logic. Classical logic is interpretable (also interpretable with respect to derivability) in intuitionistic and minimal logic by the translation absurdity, conjunctions, implications, and universal formulas. If the intuitionistic natural number theory is consistent, then the classical natural number theory is also consistent. Intuitionistic predicate logic is interpretable (also interpretable with respect to derivability) in classical predicate logic. A classical argument can be understood intuitionistically, if the formulas are interpreted throughout in the weak sense, that is, classical logic can be interpreted in intuitionistic logic by a translation that successively replaces classical formulas by their double negation. The chapter suggests that the axioms for classical and intuitionistic natural number theory are the same.