Crisp Bi-Gödel modal logic and its paraconsistent expansion

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985807%3A_____%2F25%3A00604125" target="_blank" >RIV/67985807:_____/25:00604125 - isvavai.cz</a>

Výsledek na webu

<a href="https://dx.doi.org/10.1093/jigpal/jzad017" target="_blank" >https://dx.doi.org/10.1093/jigpal/jzad017</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1093/jigpal/jzad017" target="_blank" >10.1093/jigpal/jzad017</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Crisp Bi-Gödel modal logic and its paraconsistent expansion

Popis výsledku v původním jazyce

In this paper, we provide a Hilbert-style axiomatization for the crisp bi-Gödel modal logic KbiG. We prove its completeness w.r.t. crisp Kripke models where formulas at each state are evaluated over the standard bi-Gödel algebra on [0, 1]. We also consider a paraconsistent expansion of KbiG with a De Morgan negation ¬, which we dub KG2. We devise a Hilbert-style calculus for this logic and, as a consequence of a conservative translation from KbiG to KG2, prove its completeness w.r.t. crisp Kripke models with two valuations over [0, 1] connected via ¬. For these two logics, we establish that their decidability and validity are PSPACE-complete. We also study the semantical properties of KbiG and KG2. In particular, we show that Glivenko’s theorem holds only in finitely branching frames. We also explore the classes of formulas that define the same classes of frames both in K (the classical modal logic) and the crisp Gödel modal logic BKc. We show that, among others, all Sahlqvist formulas and all formulas ϕ → χ where ϕ and χ are monotone define the same classes of frames in K and BKc.

Název v anglickém jazyce

Crisp Bi-Gödel modal logic and its paraconsistent expansion

Popis výsledku anglicky

In this paper, we provide a Hilbert-style axiomatization for the crisp bi-Gödel modal logic KbiG. We prove its completeness w.r.t. crisp Kripke models where formulas at each state are evaluated over the standard bi-Gödel algebra on [0, 1]. We also consider a paraconsistent expansion of KbiG with a De Morgan negation ¬, which we dub KG2. We devise a Hilbert-style calculus for this logic and, as a consequence of a conservative translation from KbiG to KG2, prove its completeness w.r.t. crisp Kripke models with two valuations over [0, 1] connected via ¬. For these two logics, we establish that their decidability and validity are PSPACE-complete. We also study the semantical properties of KbiG and KG2. In particular, we show that Glivenko’s theorem holds only in finitely branching frames. We also explore the classes of formulas that define the same classes of frames both in K (the classical modal logic) and the crisp Gödel modal logic BKc. We show that, among others, all Sahlqvist formulas and all formulas ϕ → χ where ϕ and χ are monotone define the same classes of frames in K and BKc.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Projekt

<a href="/cs/project/GA22-01137S" target="_blank" >GA22-01137S: Metamatematika substrukturálních modálních logik</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2025

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Logic Journal of the IGPL

ISSN

1367-0751

e-ISSN

1368-9894

Svazek periodika

33

Číslo periodika v rámci svazku

5

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

30

Strana od-do

jzad017

Kód UT WoS článku

001074713800001

EID výsledku v databázi Scopus

2-s2.0-105017431379