Algebraic structures formalizing the logic with unsharp implication and negation

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F61989592%3A15310%2F25%3A73633165" target="_blank" >RIV/61989592:15310/25:73633165 - isvavai.cz</a>

Výsledek na webu

<a href="https://academic.oup.com/jigpal/article/33/1/36/7316294" target="_blank" >https://academic.oup.com/jigpal/article/33/1/36/7316294</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1093/jigpal/jzad023" target="_blank" >10.1093/jigpal/jzad023</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Algebraic structures formalizing the logic with unsharp implication and negation

Popis výsledku v původním jazyce

It is well-known that intuitionistic logics can be formalized by means of Heyting algebras, i.e. relatively pseudocomplemented semilattices. Within such algebras the logical connectives implication and conjunction are formalized as the relative pseudocomplement and the semilattice operation meet, respectively. If the Heyting algebra has a bottom element 0, then the relative pseudocomplement with respect to 0 is called the pseudocomplement and it is considered as the connective negation in this logic. Our idea is to consider an arbitrary meet-semilattice with 0 satisfying only the Ascending Chain Condition (these assumptions are trivially satisfied in finite meet-semilattices) and introduce the operators formalizing the connectives negation x0 and implication x → y as the set of all maximal elements z satisfying x ⴷ z = 0 and as the set of all maximal elements z satisfying x ⴷ z ≤ y, respectively. Such a negation and implication is ‘unsharp’ since it assigns to one entry x or to two entries x and y belonging to the semilattice, respectively, a subset instead of an element of the semilattice. Surprisingly, this kind of negation and implication still shares a number of properties of these connectives in intuitionistic logic, in particular the derivation rule Modus Ponens. Moreover, unsharp negation and unsharp implication can be characterized by means of five, respectively seven simple axioms. We present several examples. The concepts of a deductive system and of a filter are introduced as well as the congruence determined by such a filter. We finally describe certain relationships between these concepts.

Název v anglickém jazyce

Algebraic structures formalizing the logic with unsharp implication and negation

Popis výsledku anglicky

It is well-known that intuitionistic logics can be formalized by means of Heyting algebras, i.e. relatively pseudocomplemented semilattices. Within such algebras the logical connectives implication and conjunction are formalized as the relative pseudocomplement and the semilattice operation meet, respectively. If the Heyting algebra has a bottom element 0, then the relative pseudocomplement with respect to 0 is called the pseudocomplement and it is considered as the connective negation in this logic. Our idea is to consider an arbitrary meet-semilattice with 0 satisfying only the Ascending Chain Condition (these assumptions are trivially satisfied in finite meet-semilattices) and introduce the operators formalizing the connectives negation x0 and implication x → y as the set of all maximal elements z satisfying x ⴷ z = 0 and as the set of all maximal elements z satisfying x ⴷ z ≤ y, respectively. Such a negation and implication is ‘unsharp’ since it assigns to one entry x or to two entries x and y belonging to the semilattice, respectively, a subset instead of an element of the semilattice. Surprisingly, this kind of negation and implication still shares a number of properties of these connectives in intuitionistic logic, in particular the derivation rule Modus Ponens. Moreover, unsharp negation and unsharp implication can be characterized by means of five, respectively seven simple axioms. We present several examples. The concepts of a deductive system and of a filter are introduced as well as the congruence determined by such a filter. We finally describe certain relationships between these concepts.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GF20-09869L" target="_blank" >GF20-09869L: Ortomodularita z různých pohledů</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

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

1

Stát vydavatele periodika

GB - Spojené království Velké Británie a Severního Irska

Počet stran výsledku

13

Strana od-do

"36 "- 48

Kód UT WoS článku

001084835400001

EID výsledku v databázi Scopus

2-s2.0-85217628657