On Strictly Positive Fragments of Modal Logics with Confluence

Mathematics. 2022. Vol. 10. No. 19. Article 3701.

Kikot S., Kudinov A.

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 $\Diamond\Box p \to \Box\Diamond p$ 
as well as the products of modal logics
in the set {K, D,  T, D4, S4}, which contain bimodal confluence
$\Diamond_1\Box_2 p \to \Box_2\Diamond_1 p$. 
We show that the impact of the unimodal confluence axiom on the 
axiomatisation of strictly positive fragments is rather weak.
In the presence of $\top \to \Diamond \top$, it simply disappears and does not contribute to the axiomatisation. Without $\top \to \Diamond \top$
it gives rise to a weaker formula $\Diamond \top \to \Diamond \Diamond \top$. On the other hand, bimodal confluence gives rise to more complicated formulas such 
as $\Diamond_1 p \land \Diamond_2^n \top \to \Diamond_1 (p \land \Diamond_2^n \top)$ (which are superfluous in a product if the corresponding factor contains $\top \to \Diamond\top$).%kudinov: formula is replaced by an equivalent one 
We also show that bimodal confluence cannot be captured by any 
finite set of strictly positive implications.

Keywords: modal logic, модальная логика, Confluence, направленность, strictly positive logics, строго позитивная логика