Relative Contingency and Bimodality

Abstract

In the first part of the paper it is proved that there exists a one–one mapping between a minimal contingential logic extended with a suitable axiom for a propositional constant τ, named KΔτw, and a logic of necessity ${K\square \tau{w}}$ whose language contains ${\square}$ and τ. The form of the proposed translation aims at giving a solution to a problem which was left open in a preceding paper. It is then shown that the presence of τ in the language of KΔτw allows for the definition, in terms of the non-contingency operator Δ, not only of ${\square}$ but of a second necessity operator O. It is observed that this fact opens the road to an investigation of the τ-free multimodal fragments of KΔτw.