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Non-Reflexive Nonsense: Proof Theory of Paracomplete Weak Kleene Logic 非反身胡言乱语：准完全弱克莱因逻辑的证明理论
IF 0.7 3区 数学
Studia Logica Pub Date : 2024-03-18 DOI: 10.1007/s11225-023-10086-x
{"title":"Non-Reflexive Nonsense: Proof Theory of Paracomplete Weak Kleene Logic","authors":"","doi":"10.1007/s11225-023-10086-x","DOIUrl":"https://doi.org/10.1007/s11225-023-10086-x","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":0.7,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140156137","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Finite Hilbert Systems for Weak Kleene Logics 弱克莱因逻辑的有限希尔伯特系统
IF 0.7 3区 数学
Studia Logica Pub Date : 2024-03-16 DOI: 10.1007/s11225-023-10079-w
{"title":"Finite Hilbert Systems for Weak Kleene Logics","authors":"","doi":"10.1007/s11225-023-10079-w","DOIUrl":"https://doi.org/10.1007/s11225-023-10079-w","url":null,"abstract":"<h3>Abstract</h3> <p>Multiple-conclusion Hilbert-style systems allow us to finitely axiomatize every logic defined by a finite matrix. Having obtained such axiomatizations for Paraconsistent Weak Kleene and Bochvar–Kleene logics, we modify them by replacing the multiple-conclusion rules with carefully selected single-conclusion ones. In this way we manage to introduce the first <em>finite</em> Hilbert-style single-conclusion axiomatizations for these logics.</p>","PeriodicalId":48979,"journal":{"name":"Studia Logica","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140156160","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Jaśkowski and the Jains 雅斯科夫斯基和耆那教徒
IF 0.7 3区 数学
Studia Logica Pub Date : 2024-03-13 DOI: 10.1007/s11225-024-10096-3
Graham Priest
{"title":"Jaśkowski and the Jains","authors":"Graham Priest","doi":"10.1007/s11225-024-10096-3","DOIUrl":"https://doi.org/10.1007/s11225-024-10096-3","url":null,"abstract":"<p>In 1948 Jaśkowski introduced the first discussive logic. The main technical idea was to take what holds to be what is true at some possible world. Some 2,000 years earlier, Jain philosophers had advocated a similar idea, in their doctrine of <i>syādvāda</i>. Of course, these philosophers had no knowledge of contemporary logical notions; but the techniques pioneered by Jaśkowski can be deployed to make the Jain ideas mathematically precise. Moreover, Jain ideas suggest a new family of many-valued discussive logics. In this paper, I will explain all these matters.</p>","PeriodicalId":48979,"journal":{"name":"Studia Logica","volume":"43 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140115858","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Kripke-Completeness and Sequent Calculus for Quasi-Boolean Modal Logic 准布尔模态逻辑的克里普克完备性和序列微积分
IF 0.7 3区 数学
Studia Logica Pub Date : 2024-03-06 DOI: 10.1007/s11225-024-10095-4
{"title":"Kripke-Completeness and Sequent Calculus for Quasi-Boolean Modal Logic","authors":"","doi":"10.1007/s11225-024-10095-4","DOIUrl":"https://doi.org/10.1007/s11225-024-10095-4","url":null,"abstract":"<h3>Abstract</h3> <p>Quasi-Boolean modal algebras are quasi-Boolean algebras with a modal operator satisfying the interaction axiom. Sequential quasi-Boolean modal logics and the relational semantics are introduced. Kripke-completeness for some quasi-Boolean modal logics is shown by the canonical model method. We show that every descriptive persistent quasi-Boolean modal logic is canonical. The finite model property of some quasi-Boolean modal logics is proved. A cut-free Gentzen sequent calculus for the minimal quasi-Boolean logic is developed and we show that it has the Craig interpolation property.</p>","PeriodicalId":48979,"journal":{"name":"Studia Logica","volume":"22 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140046758","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Variations on the Kripke Trick 克里普克伎俩的变体
IF 0.7 3区 数学
Studia Logica Pub Date : 2024-03-06 DOI: 10.1007/s11225-023-10093-y
{"title":"Variations on the Kripke Trick","authors":"","doi":"10.1007/s11225-023-10093-y","DOIUrl":"https://doi.org/10.1007/s11225-023-10093-y","url":null,"abstract":"<h3>Abstract</h3> <p>In the early 1960s, to prove undecidability of monadic fragments of sublogics of the predicate modal logic <span> <span>(textbf{QS5})</span> </span> that include the classical predicate logic <span> <span>(textbf{QCl})</span> </span>, Saul Kripke showed how a classical atomic formula with a binary predicate letter can be simulated by a monadic modal formula. We consider adaptations of Kripke’s simulation, which we call the Kripke trick, to various modal and superintuitionistic predicate logics not considered by Kripke. We also discuss settings where the Kripke trick does not work and where, as a result, decidability of monadic modal predicate logics can be obtained.</p>","PeriodicalId":48979,"journal":{"name":"Studia Logica","volume":"272 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140046573","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Geometric Implications 几何意义
IF 0.7 3区 数学
Studia Logica Pub Date : 2024-03-06 DOI: 10.1007/s11225-023-10094-x
{"title":"On Geometric Implications","authors":"","doi":"10.1007/s11225-023-10094-x","DOIUrl":"https://doi.org/10.1007/s11225-023-10094-x","url":null,"abstract":"<h3>Abstract</h3> <p>It is a well-known fact that although the poset of open sets of a topological space is a Heyting algebra, its Heyting implication is not necessarily stable under the inverse image of continuous functions and hence is not a geometric concept. This leaves us wondering if there is any stable family of implications that can be safely called geometric. In this paper, we will first recall the abstract notion of implication as a binary modality introduced in Akbar Tabatabai (Implication via spacetime. In: Mathematics, logic, and their philosophies: essays in honour of Mohammad Ardeshir, pp 161–216, 2021). Then, we will use a weaker version of categorical fibrations to define the geometricity of a category of pairs of spaces and implications over a given category of spaces. We will identify the greatest geometric category over the subcategories of open-irreducible (closed-irreducible) maps as a generalization of the usual injective open (closed) maps. Using this identification, we will then characterize all geometric categories over a given category <span> <span>({mathcal {S}})</span> </span>, provided that <span> <span>({mathcal {S}})</span> </span> has some basic closure properties. Specially, we will show that there is no non-trivial geometric category over the full category of spaces. Finally, as the implications we identified are also interesting in their own right, we will spend some time to investigate their algebraic properties. We will first use a Yoneda-type argument to provide a representation theorem, making the implications a part of an adjunction-style pair. Then, we will use this result to provide a Kripke-style representation for any arbitrary implication.</p>","PeriodicalId":48979,"journal":{"name":"Studia Logica","volume":"13 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140046803","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Angell and McCall Meet Wansing 安格尔和麦考尔会见万辛
IF 0.7 3区 数学
Studia Logica Pub Date : 2024-02-17 DOI: 10.1007/s11225-023-10083-0
Hitoshi Omori, Andreas Kapsner
{"title":"Angell and McCall Meet Wansing","authors":"Hitoshi Omori, Andreas Kapsner","doi":"10.1007/s11225-023-10083-0","DOIUrl":"https://doi.org/10.1007/s11225-023-10083-0","url":null,"abstract":"<p>In this paper, we introduce a new logic, which we call <b>AM3</b>. It is a connexive logic that has several interesting properties, among them being <i>strongly connexive</i> and validating the <i>Converse Boethius Thesis</i>. These two properties are rather characteristic of the difference between, on the one hand, Angell and McCall’s <b>CC1</b> and, on the other, Wansing’s <b>C</b>. We will show that in other aspects, as well, <b>AM3</b> combines what are, arguably, the strengths of both <b>CC1</b> and <b>C</b>. It also allows us an interesting look at how connexivity and the intuitionistic understanding of negation relate to each other. However, some problems remain, and we end by pointing to a large family of weaker logics that <b>AM3</b> invites us to further explore.</p>","PeriodicalId":48979,"journal":{"name":"Studia Logica","volume":"3 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139768159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Independence Results for Finite Set Theories in Well-Founded Locally Finite Graphs 有据局部有限图中有限集理论的独立性结果
IF 0.7 3区 数学
Studia Logica Pub Date : 2024-02-16 DOI: 10.1007/s11225-023-10087-w
{"title":"Independence Results for Finite Set Theories in Well-Founded Locally Finite Graphs","authors":"","doi":"10.1007/s11225-023-10087-w","DOIUrl":"https://doi.org/10.1007/s11225-023-10087-w","url":null,"abstract":"<h3>Abstract</h3> <p>We consider all combinatorially possible systems corresponding to subsets of finite set theory (i.e., Zermelo-Fraenkel set theory without the axiom of infinity) and for each of them either provide a well-founded locally finite graph that is a model of that theory or show that this is impossible. To that end, we develop the technique of <em>axiom closure of graphs</em>.</p>","PeriodicalId":48979,"journal":{"name":"Studia Logica","volume":"26 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139773637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Ecumenical Propositional Tableau 全基督教命题表
IF 0.7 3区 数学
Studia Logica Pub Date : 2024-02-16 DOI: 10.1007/s11225-023-10091-0
{"title":"Ecumenical Propositional Tableau","authors":"","doi":"10.1007/s11225-023-10091-0","DOIUrl":"https://doi.org/10.1007/s11225-023-10091-0","url":null,"abstract":"<h3>Abstract</h3> <p>Ecumenical logic aims to peacefully join classical and intuitionistic logic systems, allowing for reasoning about both classical and intuitionistic statements. This paper presents a semantic tableau for propositional ecumenical logic and proves its soundness and completeness concerning Ecumenical Kripke models. We introduce the Ecumenical Propositional Tableau (<span> <span>(E_T)</span> </span>) and demonstrate its effectiveness in handling mixed statements.</p>","PeriodicalId":48979,"journal":{"name":"Studia Logica","volume":"34 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139768212","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Woodruff’s Constructive Nonsense Logic 论伍德拉夫的建构式废话逻辑
IF 0.7 3区 数学
Studia Logica Pub Date : 2024-01-22 DOI: 10.1007/s11225-023-10092-z
Jonas R. B. Arenhart, Hitoshi Omori
{"title":"On Woodruff’s Constructive Nonsense Logic","authors":"Jonas R. B. Arenhart, Hitoshi Omori","doi":"10.1007/s11225-023-10092-z","DOIUrl":"https://doi.org/10.1007/s11225-023-10092-z","url":null,"abstract":"<p>Sören Halldén’s logic of nonsense is one of the most well-known many-valued logics available in the literature. In this paper, we discuss Peter Woodruff’s as yet rather unexplored attempt to advance a version of such a logic built on the top of a constructive logical basis. We start by recalling the basics of Woodruff’s system and by bringing to light some of its notable features. We then go on to elaborate on some of the difficulties attached to it; on our way to offer a possible solution to such difficulties, we discuss the relation between Woodruff’s system and two-dimensional semantics for many-valued logics, as developed by Hans Herzberger.</p>","PeriodicalId":48979,"journal":{"name":"Studia Logica","volume":"116 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139515405","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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