Algebraic structures formalizing the logic with unsharp implication and negation

Result code in 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>

Result on the web

<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>

Result language

angličtina

Original language name

Algebraic structures formalizing the logic with unsharp implication and negation

Original language description

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.

Czech name

—

Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10101 - Pure mathematics

Project

<a href="/en/project/GF20-09869L" target="_blank" >GF20-09869L: The many facets of orthomodularity</a><br>

Continuities

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

Publication year

2025

Confidentiality

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

Name of the periodical

LOGIC JOURNAL OF THE IGPL

ISSN

1367-0751

e-ISSN

1368-9894

Volume of the periodical

33

Issue of the periodical within the volume

1

Country of publishing house

GB - UNITED KINGDOM

Number of pages

13

Pages from-to

"36 "- 48

UT code for WoS article

001084835400001

EID of the result in the Scopus database

2-s2.0-85217628657