Intuitionistic Implication and Logics of Formal Inconsistency

Intuitionistic Implication and Logics of Formal Inconsistency

Logics of Formal Inconsistency (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mi>F</mi><mi>I</mi></mrow></semantics></math></inline-for...

Full description

Saved in:
Bibliographic Details
Main Author: Janusz Ciuciura
Format: Article
Language:English
Published: MDPI AG 2024-10-01
Series:Axioms
Subjects:
paraconsistent logic
da Costa’s logic
logics of formal inconsistency
consistency operator
intuitionistic implication
Online Access:https://www.mdpi.com/2075-1680/13/11/738
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:Logics of Formal Inconsistency (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mi>F</mi><mi>I</mi></mrow></semantics></math></inline-formula> for short) are a class of paraconsistent logics that validate the principle of gentle explosion, meaning that any formula can be derived from the set of formulas: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∘</mo><mi>α</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∼</mo><mi>α</mi></mrow></semantics></math></inline-formula>. A unique feature of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mi>F</mi><mi>I</mi></mrow></semantics></math></inline-formula> is the use of the symbol ‘∘’ to represent notions of consistency at the object-language level. These logics are simple in essence, built upon all the axiom schemas of positive classical logic, axioms for negation and the so-called ‘consistency operator’ ∘, with the only inference rule being detachment. In this paper, we propose an alternative foundation for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><mi>F</mi><mi>I</mi></mrow></semantics></math></inline-formula>, which is the positive fragment of intuitionistic propositional logic. We present bi-valuational ‘Loparić-like’ semantics for the resulting logics and discuss their potential extensions.
ISSN:2075-1680

Similar Items