On Strictly Positive Fragments of Modal Logics with Confluence

Abstract

We axiomatize strictly positive fragments of modal logics with the confluence axiom. We consider unimodal logics such as K.2, D.2, D4.2 and S4.2 with unimodal confluence ⋄□p→□⋄p as well as the products of modal logics in the set K,D,T,D4,S4, which contain bimodal confluence ⋄1□2p→□2⋄1p. We show that the impact of the unimodal confluence axiom on the axiomatisation of strictly positive fragments is rather weak. In the presence of ⊤→⋄⊤, it simply disappears and does not contribute to the axiomatisation. Without ⊤→⋄⊤ it gives rise to a weaker formula ⋄⊤→⋄⋄⊤. On the other hand, bimodal confluence gives rise to more complicated formulas such as ⋄1p∧⋄2n⊤→⋄1(p∧⋄2n⊤) (which are superfluous in a product if the corresponding factor contains ⊤→⋄⊤). We also show that bimodal confluence cannot be captured by any finite set of strictly positive implications.