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...
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MDPI AG
2024-10-01
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| author | Janusz Ciuciura |
| author_facet | Janusz Ciuciura |
| author_sort | Janusz Ciuciura |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-5e7fa1f00dcd457397f0e6ff88412d0d |
| institution | OA Journals |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-5e7fa1f00dcd457397f0e6ff88412d0d2025-08-20T02:26:45ZengMDPI AGAxioms2075-16802024-10-01131173810.3390/axioms13110738Intuitionistic Implication and Logics of Formal InconsistencyJanusz Ciuciura0Department of Logic and Methodology of Science, Institute of Philosophy, University of Łódź, Lindleya 3/5, 90-131 Łódź, PolandLogics 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.https://www.mdpi.com/2075-1680/13/11/738paraconsistent logicda Costa’s logiclogics of formal inconsistencyconsistency operatorintuitionistic implication |
| spellingShingle | Janusz Ciuciura Intuitionistic Implication and Logics of Formal Inconsistency Axioms paraconsistent logic da Costa’s logic logics of formal inconsistency consistency operator intuitionistic implication |
| title | Intuitionistic Implication and Logics of Formal Inconsistency |
| title_full | Intuitionistic Implication and Logics of Formal Inconsistency |
| title_fullStr | Intuitionistic Implication and Logics of Formal Inconsistency |
| title_full_unstemmed | Intuitionistic Implication and Logics of Formal Inconsistency |
| title_short | Intuitionistic Implication and Logics of Formal Inconsistency |
| title_sort | intuitionistic implication and logics of formal inconsistency |
| topic | paraconsistent logic da Costa’s logic logics of formal inconsistency consistency operator intuitionistic implication |
| url | https://www.mdpi.com/2075-1680/13/11/738 |
| work_keys_str_mv | AT januszciuciura intuitionisticimplicationandlogicsofformalinconsistency |