The Russell–Prawitz modality

Abstract

In his 1903, Principles of Mathematics, Bertrand Russell mentioned possible definitions of conjunction, disjunction, negation and existential quantification in terms of implication and universal quantification, exploiting impredicative universal quantifiers over all propositions. In his 1965 Ph.D. thesis Dag Prawitz showed that these definitions hold in intuitionistic second order logic. More recently, these definitions have been used to represent logic in various impredicative type theories. This treatment of logic is distinct from the more standard Curry–Howard representation of logic in a dependent type theory.The main aim of this paper is to compare, in a purely logical, non type-theoretic setting, this Russell–Prawitz representation of intuitionistic logic with other possible representations. It turns out that associated with the Russell–Prawitz representation is a lax modal operator, which we call the Russell–Prawitz modality, and that any lax modal operator can be used to give a translation of intuitionistic logic into itself that generalises both the double negation interpretation, double negation being a paradigm example of a lax modality, and the Russell–Prawitz representation.