{"title":"逻辑学的关系伴侣","authors":"Sankha S. Basu, Sayantan Roy","doi":"arxiv-2408.17019","DOIUrl":null,"url":null,"abstract":"The variable inclusion companions of logics have lately been thoroughly\nstudied by multiple authors. There are broadly two types of these companions:\nthe left and the right variable inclusion companions. Another type of\ncompanions of logics induced by Hilbert-style presentations (Hilbert-style\nlogics) were introduced in a recent paper. A sufficient condition for the\nrestricted rules companion of a Hilbert-style logic to coincide with its left\nvariable inclusion companion was proved there, while a necessary condition\nremained elusive. The present article has two parts. In the first part, we give\na necessary and sufficient condition for the left variable inclusion and the\nrestricted rules companions of a Hilbert-style logic to coincide. In the rest\nof the paper, we recognize that the variable inclusion restrictions used to\ndefine variable inclusion companions of a logic\n$\\langle\\mathcal{L},\\vdash\\rangle$ are relations from\n$\\mathcal{P}(\\mathcal{L})$ to $\\mathcal{L}$. This leads to a more general idea\nof a relational companion of a logical structure, a framework that we borrow\nfrom the field of universal logic. We end by showing that even Hilbert-style\nlogics and the restricted rules companions of these can be brought under the\numbrella of the general notions of logical structures and their relational\ncompanions that are discussed here.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Relational Companions of Logics\",\"authors\":\"Sankha S. Basu, Sayantan Roy\",\"doi\":\"arxiv-2408.17019\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The variable inclusion companions of logics have lately been thoroughly\\nstudied by multiple authors. There are broadly two types of these companions:\\nthe left and the right variable inclusion companions. Another type of\\ncompanions of logics induced by Hilbert-style presentations (Hilbert-style\\nlogics) were introduced in a recent paper. A sufficient condition for the\\nrestricted rules companion of a Hilbert-style logic to coincide with its left\\nvariable inclusion companion was proved there, while a necessary condition\\nremained elusive. The present article has two parts. In the first part, we give\\na necessary and sufficient condition for the left variable inclusion and the\\nrestricted rules companions of a Hilbert-style logic to coincide. In the rest\\nof the paper, we recognize that the variable inclusion restrictions used to\\ndefine variable inclusion companions of a logic\\n$\\\\langle\\\\mathcal{L},\\\\vdash\\\\rangle$ are relations from\\n$\\\\mathcal{P}(\\\\mathcal{L})$ to $\\\\mathcal{L}$. This leads to a more general idea\\nof a relational companion of a logical structure, a framework that we borrow\\nfrom the field of universal logic. We end by showing that even Hilbert-style\\nlogics and the restricted rules companions of these can be brought under the\\numbrella of the general notions of logical structures and their relational\\ncompanions that are discussed here.\",\"PeriodicalId\":501306,\"journal\":{\"name\":\"arXiv - MATH - Logic\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Logic\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2408.17019\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.17019","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
The variable inclusion companions of logics have lately been thoroughly
studied by multiple authors. There are broadly two types of these companions:
the left and the right variable inclusion companions. Another type of
companions of logics induced by Hilbert-style presentations (Hilbert-style
logics) were introduced in a recent paper. A sufficient condition for the
restricted rules companion of a Hilbert-style logic to coincide with its left
variable inclusion companion was proved there, while a necessary condition
remained elusive. The present article has two parts. In the first part, we give
a necessary and sufficient condition for the left variable inclusion and the
restricted rules companions of a Hilbert-style logic to coincide. In the rest
of the paper, we recognize that the variable inclusion restrictions used to
define variable inclusion companions of a logic
$\langle\mathcal{L},\vdash\rangle$ are relations from
$\mathcal{P}(\mathcal{L})$ to $\mathcal{L}$. This leads to a more general idea
of a relational companion of a logical structure, a framework that we borrow
from the field of universal logic. We end by showing that even Hilbert-style
logics and the restricted rules companions of these can be brought under the
umbrella of the general notions of logical structures and their relational
companions that are discussed here.