Some Implicational Semilinear Gaggle Logics: (Dual) Residuated-Connected Logics
Implicational partial Galois logics and some of their <i>semilinear</i> extensions, such as semilinear extensions satisfying abstract Galois and dual Galois connection properties, have been introduced together with their relational semantics. However, similar extensions satisfying residu...
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| Format: | Article |
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MDPI AG
2022-04-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/11/4/183 |
| _version_ | 1797436906534862848 |
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| author | Eunsuk Yang |
| author_facet | Eunsuk Yang |
| author_sort | Eunsuk Yang |
| collection | DOAJ |
| description | Implicational partial Galois logics and some of their <i>semilinear</i> extensions, such as semilinear extensions satisfying abstract Galois and dual Galois connection properties, have been introduced together with their relational semantics. However, similar extensions satisfying residuated, dual residuated connection properties have not. This paper fills the gaps by introducing those <i>semilinear</i> extensions and their relational semantics. To this end, the class of implicational (dual) residuated-connected prelinear gaggle logics is defined and it is verified that these logics are <i>semilinear</i>. In particular, associated with the contribution of this work, we note the following two: One is that implications can be introduced by <i>residuated connection</i> in semilinear logics. This shows that the residuated, dual residuated connection properties are important and so need to be investigated in semilinear logics. The other is that <i>set-theoretic</i> relational semantics can be provided for semilinear logics. Semilinear logics have been dealt with extensively in algebraic context, whereas they have not yet been performed in the set-theoretic one. |
| first_indexed | 2024-03-09T11:09:18Z |
| format | Article |
| id | doaj.art-a7cf5fc926f144b9b7a8c2063bd83224 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-09T11:09:18Z |
| publishDate | 2022-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-a7cf5fc926f144b9b7a8c2063bd832242023-12-01T00:48:45ZengMDPI AGAxioms2075-16802022-04-0111418310.3390/axioms11040183Some Implicational Semilinear Gaggle Logics: (Dual) Residuated-Connected LogicsEunsuk Yang0Center for Humanities & Social Sciences, Department of Philosophy, Institute of Critical Thinking and Writing, Jeonbuk National University, Rm 417, Jeonju 54896, KoreaImplicational partial Galois logics and some of their <i>semilinear</i> extensions, such as semilinear extensions satisfying abstract Galois and dual Galois connection properties, have been introduced together with their relational semantics. However, similar extensions satisfying residuated, dual residuated connection properties have not. This paper fills the gaps by introducing those <i>semilinear</i> extensions and their relational semantics. To this end, the class of implicational (dual) residuated-connected prelinear gaggle logics is defined and it is verified that these logics are <i>semilinear</i>. In particular, associated with the contribution of this work, we note the following two: One is that implications can be introduced by <i>residuated connection</i> in semilinear logics. This shows that the residuated, dual residuated connection properties are important and so need to be investigated in semilinear logics. The other is that <i>set-theoretic</i> relational semantics can be provided for semilinear logics. Semilinear logics have been dealt with extensively in algebraic context, whereas they have not yet been performed in the set-theoretic one.https://www.mdpi.com/2075-1680/11/4/183fuzzy logics(dual) residuated connectionsemilinear logicgagglesRoutley–Meyer-style semantics |
| spellingShingle | Eunsuk Yang Some Implicational Semilinear Gaggle Logics: (Dual) Residuated-Connected Logics Axioms fuzzy logics (dual) residuated connection semilinear logic gaggles Routley–Meyer-style semantics |
| title | Some Implicational Semilinear Gaggle Logics: (Dual) Residuated-Connected Logics |
| title_full | Some Implicational Semilinear Gaggle Logics: (Dual) Residuated-Connected Logics |
| title_fullStr | Some Implicational Semilinear Gaggle Logics: (Dual) Residuated-Connected Logics |
| title_full_unstemmed | Some Implicational Semilinear Gaggle Logics: (Dual) Residuated-Connected Logics |
| title_short | Some Implicational Semilinear Gaggle Logics: (Dual) Residuated-Connected Logics |
| title_sort | some implicational semilinear gaggle logics dual residuated connected logics |
| topic | fuzzy logics (dual) residuated connection semilinear logic gaggles Routley–Meyer-style semantics |
| url | https://www.mdpi.com/2075-1680/11/4/183 |
| work_keys_str_mv | AT eunsukyang someimplicationalsemilineargagglelogicsdualresiduatedconnectedlogics |