Distribution-Free Normal Modal Logics
This article initiates the semantic study of distribution-free normal modal logic systems, laying the semantic foundations and anticipating further research in the area. The article explores roughly the same area, though taking a different approach, as a recent article by Bezhanishvili, de Groot, Dm...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-04-01
|
| Series: | Logics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2813-0405/3/2/3 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849431857519656960 |
|---|---|
| author | Chrysafis Hartonas |
| author_facet | Chrysafis Hartonas |
| author_sort | Chrysafis Hartonas |
| collection | DOAJ |
| description | This article initiates the semantic study of distribution-free normal modal logic systems, laying the semantic foundations and anticipating further research in the area. The article explores roughly the same area, though taking a different approach, as a recent article by Bezhanishvili, de Groot, Dmitrieva and Morachini, who studied a distribution-free version of Dunn’s positive modal logic (PML). Unlike PML, we consider logics that may drop distribution and that are equipped with both an implication connective and modal operators. We adopt a uniform relational semantics approach, relying on recent results on representation and duality for normal lattice expansions. We prove canonicity and completeness in the relational semantics of the minimal distribution-free normal modal logic, assuming just the K-axiom, as well as those of its axiomatic extensions obtained by adding any of the D, T, B, S4 or S5 axioms. Adding distribution can be easily accommodated and, as a side result, we also obtain a new semantic treatment of intuitionistic modal logic. |
| format | Article |
| id | doaj-art-c560c629188648aabb4e7b1689ca5a76 |
| institution | Kabale University |
| issn | 2813-0405 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Logics |
| spelling | doaj-art-c560c629188648aabb4e7b1689ca5a762025-08-20T03:27:29ZengMDPI AGLogics2813-04052025-04-0132310.3390/logics3020003Distribution-Free Normal Modal LogicsChrysafis Hartonas0Department of Digital Systems, University of Thessaly, 382 21 Volos, GreeceThis article initiates the semantic study of distribution-free normal modal logic systems, laying the semantic foundations and anticipating further research in the area. The article explores roughly the same area, though taking a different approach, as a recent article by Bezhanishvili, de Groot, Dmitrieva and Morachini, who studied a distribution-free version of Dunn’s positive modal logic (PML). Unlike PML, we consider logics that may drop distribution and that are equipped with both an implication connective and modal operators. We adopt a uniform relational semantics approach, relying on recent results on representation and duality for normal lattice expansions. We prove canonicity and completeness in the relational semantics of the minimal distribution-free normal modal logic, assuming just the K-axiom, as well as those of its axiomatic extensions obtained by adding any of the D, T, B, S4 or S5 axioms. Adding distribution can be easily accommodated and, as a side result, we also obtain a new semantic treatment of intuitionistic modal logic.https://www.mdpi.com/2813-0405/3/2/3sub-classical modal logicintuitionistic modal logicdistribution-free modal logiccompleteness via canonicity |
| spellingShingle | Chrysafis Hartonas Distribution-Free Normal Modal Logics Logics sub-classical modal logic intuitionistic modal logic distribution-free modal logic completeness via canonicity |
| title | Distribution-Free Normal Modal Logics |
| title_full | Distribution-Free Normal Modal Logics |
| title_fullStr | Distribution-Free Normal Modal Logics |
| title_full_unstemmed | Distribution-Free Normal Modal Logics |
| title_short | Distribution-Free Normal Modal Logics |
| title_sort | distribution free normal modal logics |
| topic | sub-classical modal logic intuitionistic modal logic distribution-free modal logic completeness via canonicity |
| url | https://www.mdpi.com/2813-0405/3/2/3 |
| work_keys_str_mv | AT chrysafishartonas distributionfreenormalmodallogics |