Diego Castaño , José Patricio Díaz Varela , Gabriel Savoy
{"title":"连续 t 规范谓词逻辑的强完备性","authors":"Diego Castaño , José Patricio Díaz Varela , Gabriel Savoy","doi":"10.1016/j.fss.2024.109193","DOIUrl":null,"url":null,"abstract":"<div><div>The axiomatic system introduced by Hájek axiomatizes first-order logic based on BL-chains. In this study, we extend this system with the axiom <span><math><msup><mrow><mo>(</mo><mo>∀</mo><mi>x</mi><mi>ϕ</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>↔</mo><mo>∀</mo><mi>x</mi><msup><mrow><mi>ϕ</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> and the infinitary rule<span><span><span><math><mfrac><mrow><mi>ϕ</mi><mo>∨</mo><mo>(</mo><mi>α</mi><mo>→</mo><msup><mrow><mi>β</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo><mo>:</mo><mi>n</mi><mo>∈</mo><mi>N</mi></mrow><mrow><mi>ϕ</mi><mo>∨</mo><mo>(</mo><mi>α</mi><mo>→</mo><mi>α</mi><mi>&</mi><mi>β</mi><mo>)</mo></mrow></mfrac></math></span></span></span> to achieve strong completeness with respect to continuous t-norms.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"500 ","pages":"Article 109193"},"PeriodicalIF":3.2000,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Strong completeness for the predicate logic of the continuous t-norms\",\"authors\":\"Diego Castaño , José Patricio Díaz Varela , Gabriel Savoy\",\"doi\":\"10.1016/j.fss.2024.109193\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>The axiomatic system introduced by Hájek axiomatizes first-order logic based on BL-chains. In this study, we extend this system with the axiom <span><math><msup><mrow><mo>(</mo><mo>∀</mo><mi>x</mi><mi>ϕ</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>↔</mo><mo>∀</mo><mi>x</mi><msup><mrow><mi>ϕ</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> and the infinitary rule<span><span><span><math><mfrac><mrow><mi>ϕ</mi><mo>∨</mo><mo>(</mo><mi>α</mi><mo>→</mo><msup><mrow><mi>β</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo><mo>:</mo><mi>n</mi><mo>∈</mo><mi>N</mi></mrow><mrow><mi>ϕ</mi><mo>∨</mo><mo>(</mo><mi>α</mi><mo>→</mo><mi>α</mi><mi>&</mi><mi>β</mi><mo>)</mo></mrow></mfrac></math></span></span></span> to achieve strong completeness with respect to continuous t-norms.</div></div>\",\"PeriodicalId\":55130,\"journal\":{\"name\":\"Fuzzy Sets and Systems\",\"volume\":\"500 \",\"pages\":\"Article 109193\"},\"PeriodicalIF\":3.2000,\"publicationDate\":\"2024-11-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Fuzzy Sets and Systems\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0165011424003397\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Fuzzy Sets and Systems","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0165011424003397","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
Strong completeness for the predicate logic of the continuous t-norms
The axiomatic system introduced by Hájek axiomatizes first-order logic based on BL-chains. In this study, we extend this system with the axiom and the infinitary rule to achieve strong completeness with respect to continuous t-norms.
期刊介绍：
Since its launching in 1978, the journal Fuzzy Sets and Systems has been devoted to the international advancement of the theory and application of fuzzy sets and systems. The theory of fuzzy sets now encompasses a well organized corpus of basic notions including (and not restricted to) aggregation operations, a generalized theory of relations, specific measures of information content, a calculus of fuzzy numbers. Fuzzy sets are also the cornerstone of a non-additive uncertainty theory, namely possibility theory, and of a versatile tool for both linguistic and numerical modeling: fuzzy rule-based systems. Numerous works now combine fuzzy concepts with other scientific disciplines as well as modern technologies.
In mathematics fuzzy sets have triggered new research topics in connection with category theory, topology, algebra, analysis. Fuzzy sets are also part of a recent trend in the study of generalized measures and integrals, and are combined with statistical methods. Furthermore, fuzzy sets have strong logical underpinnings in the tradition of many-valued logics.
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