On Strictly Positive Fragments of Modal Logics with Confluence

We axiomatize strictly positive fragments of modal logics with the confluence axiom. We consider unimodal logics such as K . 2 , D . 2 , D 4 . 2 and S 4 . 2 with unimodal confluence ⋄ □ p → □ ⋄ p as well as the products of modal logics in the set K , D , T , D 4 , S 4 , which contain bimodal confluence ⋄ 1 □ 2 p → □ 2 ⋄ 1 p . 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 ⋄ 1 p ∧ ⋄ 2 n ⊤ → ⋄ 1 ( p ∧ ⋄ 2 n ⊤ ) (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.