{"title":"Non-Reflexive Nonsense: Proof Theory of Paracomplete Weak Kleene Logic","authors":"","doi":"10.1007/s11225-023-10086-x","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>Our aim is to provide a sequent calculus whose external consequence relation coincides with the three-valued paracomplete logic ‘of nonsense’ introduced by Dmitry Bochvar and, independently, presented as the weak Kleene logic <span> <span>\\(\\textbf{K}_{\\textbf{3}}^{\\textbf{w}}\\)</span> </span> by Stephen C. Kleene. The main features of this calculus are (i) that it is <em>non-reflexive</em>, i.e., Identity is not included as an explicit rule (although a restricted form of it with premises is derivable); (ii) that it includes rules where <em>no variable-inclusion conditions</em> are attached; and (iii) that it is <em>hybrid</em>, insofar as it includes both left and right operational introduction as well as elimination rules.</p>","PeriodicalId":48979,"journal":{"name":"Studia Logica","volume":"2 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Studia Logica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11225-023-10086-x","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"LOGIC","Score":null,"Total":0}
引用次数: 0
Abstract
Our aim is to provide a sequent calculus whose external consequence relation coincides with the three-valued paracomplete logic ‘of nonsense’ introduced by Dmitry Bochvar and, independently, presented as the weak Kleene logic \(\textbf{K}_{\textbf{3}}^{\textbf{w}}\) by Stephen C. Kleene. The main features of this calculus are (i) that it is non-reflexive, i.e., Identity is not included as an explicit rule (although a restricted form of it with premises is derivable); (ii) that it includes rules where no variable-inclusion conditions are attached; and (iii) that it is hybrid, insofar as it includes both left and right operational introduction as well as elimination rules.
期刊介绍：
The leading idea of Lvov-Warsaw School of Logic, Philosophy and Mathematics was to investigate philosophical problems by means of rigorous methods of mathematics. Evidence of the great success the School experienced is the fact that it has become generally recognized as Polish Style Logic. Today Polish Style Logic is no longer exclusively a Polish speciality. It is represented by numerous logicians, mathematicians and philosophers from research centers all over the world.
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