On Strictly Positive Fragments of Modal Logics with Confluence - Stanislav Kikot; Andrey Kudinov (Mathematics, 2022)
We axiomatize strictly positive fragments of modal logics with the confluence axiom. We consider unimodal logics such as <semantics>K.2</semantics>, <semantics>D.2</semantics>, <semantics>D4.2</semantics> and <semantics>S4.2</semantics> with unimodal confluence <semantics>⋄□p→□⋄p</semantics> as well as the products of modal logics in the set <semantics><mfenced separators="" open="{" close="}">K,D,T,D4,S4</mfenced></semantics>, which contain bimodal confluence <semantics>⋄1□2p→□2⋄1p</semantics>. We show that the impact of the unimodal confluence axiom on the axiomatisation of strictly positive fragments is rather weak. In the presence of <semantics>⊤→⋄⊤</semantics>, it simply disappears and does not contribute to the axiomatisation. Without <semantics>⊤→⋄⊤</semantics> it gives rise to a weaker formula <semantics>⋄⊤→⋄⋄⊤</semantics>. On the other hand, bimodal confluence gives rise to more complicated formulas such as <semantics>⋄1p∧<msubsup>⋄2n</msubsup>⊤→⋄1(p∧<msubsup>⋄2n</msubsup>⊤)</semantics> (which are superfluous in a product if the corresponding factor contains <semantics>⊤→⋄⊤</semantics>). We also show that bimodal confluence cannot be captured by any finite set of strictly positive implications.