Diego Castaño, José Patricio Díaz Varela, Gabriel Savoy
{"title":"Strong completeness for the predicate logic of the continuous t-norms","authors":"Diego Castaño, José Patricio Díaz Varela, Gabriel Savoy","doi":"arxiv-2408.04792","DOIUrl":null,"url":null,"abstract":"The axiomatic system introduced by H\\'ajek axiomatizes first-order logic\nbased on BL-chains. In this study, we extend this system with the axiom\n$(\\forall x \\phi)^2 \\leftrightarrow \\forall x \\phi^2$ and the infinitary rule\n\\[ \\frac{\\phi \\vee (\\alpha \\to \\beta^n):n \\in \\mathbb{N}}{\\phi \\vee (\\alpha \\to\n\\alpha \\& \\beta)} \\] to achieve strong completeness with respect to continuous\nt-norms.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"31 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-08","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-2408.04792","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
The axiomatic system introduced by H\'ajek axiomatizes first-order logic
based on BL-chains. In this study, we extend this system with the axiom
$(\forall x \phi)^2 \leftrightarrow \forall x \phi^2$ and the infinitary rule
\[ \frac{\phi \vee (\alpha \to \beta^n):n \in \mathbb{N}}{\phi \vee (\alpha \to
\alpha \& \beta)} \] to achieve strong completeness with respect to continuous
t-norms.