On Strictly Positive Fragments of Modal Logics with Confluence

Stanislav Kikot () and Andrey Kudinov
Additional contact information
Stanislav Kikot: Institute for Information Transmission Problems, 127051 Moscow, Russia
Andrey Kudinov: Institute for Information Transmission Problems, 127051 Moscow, Russia

Mathematics, 2022, vol. 10, issue 19, 1-14

Abstract: 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.

Keywords: modal logic; strictly positive logics; confluence (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/10/19/3701/pdf (application/pdf)
https://www.mdpi.com/2227-7390/10/19/3701/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:10:y:2022:i:19:p:3701-:d:937559

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().