We present a streamlined exposition of a construction by R. Chen, A. Poulin, R. Tao, and A. Tserunyan, which proves the treeability of equivalence relations generated by any locally-finite Borel graph such that each component is a quasi-tree. More generally, we show that if each component of a locally-finite Borel graph admits a finitely-separating Borel family of cuts, then we may 'canonically' construct a forest of special ultrafilters; moreover, if the cuts are dense towards ends, then this forest is a Borel treeing.
我们对 R. Chen、A. Poulin、R. Tao 和 A. Tserunyan 的构造进行了精简阐述。Tao, and A. Tserunyan 的构造，该构造证明了由任何局部有限伯尔图生成的等价关系的可树性，且每个分量都是水树。更广义地说，我们证明了如果局部有限伯尔图的每个分量都有一个有限分离的伯尔切分族，那么我们就可以 "规范地 "构造一个特殊超滤波器森林；此外，如果切分向两端密集，那么这个森林就是伯尔树化。
{"title":"Tree-like graphings of countable Borel equivalence relations","authors":"Zhaoshen Zhai","doi":"arxiv-2409.09843","DOIUrl":"https://doi.org/arxiv-2409.09843","url":null,"abstract":"We present a streamlined exposition of a construction by R. Chen, A. Poulin,\u0000R. Tao, and A. Tserunyan, which proves the treeability of equivalence relations\u0000generated by any locally-finite Borel graph such that each component is a\u0000quasi-tree. More generally, we show that if each component of a locally-finite\u0000Borel graph admits a finitely-separating Borel family of cuts, then we may\u0000'canonically' construct a forest of special ultrafilters; moreover, if the cuts\u0000are dense towards ends, then this forest is a Borel treeing.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"54 3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142266732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Adding Abraham clubs and $α$-properness","authors":"Mohammad Golshani, Rouholah Hoseini Naveh","doi":"arxiv-2409.09697","DOIUrl":"https://doi.org/arxiv-2409.09697","url":null,"abstract":"For every indecomposable ordinal $alpha < omega_1$, we introduce a variant\u0000of Abraham forcing for adding a club in $omega_1$, which is $<alpha$-proper\u0000but not $alpha$-proper.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142266956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Confirming a conjecture of Pa{l}asi'nski and Wro'nski, we show that the bottom of the lattice of subvarieties of BCK is Y-shaped.
为了证实 Pa{l}asi'nski 和 Wro'nski 的猜想，我们证明了 BCK 子变量网格的底面是 Y 形的。
{"title":"The bottom of the lattice of BCK-varieties","authors":"Tomasz Kowalski","doi":"arxiv-2409.09854","DOIUrl":"https://doi.org/arxiv-2409.09854","url":null,"abstract":"Confirming a conjecture of Pa{l}asi'nski and Wro'nski, we show that the\u0000bottom of the lattice of subvarieties of BCK is Y-shaped.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269591","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a self-contained proof of Isa Vialard's formula for $o(Pcdot Q)$ where $P$ and $Q$ are wpos. The proof introduces the notion of a cut of partial order, which might be of independent interest.
{"title":"On maximal order type of the lexicographic product","authors":"Mirna Džamonja, Isa Vialard","doi":"arxiv-2409.09699","DOIUrl":"https://doi.org/arxiv-2409.09699","url":null,"abstract":"We give a self-contained proof of Isa Vialard's formula for $o(Pcdot Q)$\u0000where $P$ and $Q$ are wpos. The proof introduces the notion of a cut of partial\u0000order, which might be of independent interest.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269595","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Shawn Standefer, Shay Allen Logan, Thomas Macaulay Ferguson
The family of relevant logics can be faceted by a hierarchy of increasingly fine-grained variable sharing properties -- requiring that in valid entailments $Ato B$, some atom must appear in both $A$ and $B$ with some additional condition (e.g., with the same sign or nested within the same number of conditionals). In this paper, we consider an incredibly strong variable sharing property of lericone relevance that takes into account the path of negations and conditionals in which an atom appears in the parse trees of the antecedent and consequent. We show that this property of lericone relevance holds of the relevant logic $mathbf{BM}$ (and that a related property of faithful lericone relevance holds of $mathbf{B}$) and characterize the largest fragments of classical logic with these properties. Along the way, we consider the consequences for lericone relevance for the theory of subject-matter, for Logan's notion of hyperformalism, and for the very definition of a relevant logic itself.
{"title":"Topics, Non-Uniform Substitutions, and Variable Sharing","authors":"Shawn Standefer, Shay Allen Logan, Thomas Macaulay Ferguson","doi":"arxiv-2409.08942","DOIUrl":"https://doi.org/arxiv-2409.08942","url":null,"abstract":"The family of relevant logics can be faceted by a hierarchy of increasingly\u0000fine-grained variable sharing properties -- requiring that in valid entailments\u0000$Ato B$, some atom must appear in both $A$ and $B$ with some additional\u0000condition (e.g., with the same sign or nested within the same number of\u0000conditionals). In this paper, we consider an incredibly strong variable sharing\u0000property of lericone relevance that takes into account the path of negations\u0000and conditionals in which an atom appears in the parse trees of the antecedent\u0000and consequent. We show that this property of lericone relevance holds of the\u0000relevant logic $mathbf{BM}$ (and that a related property of faithful lericone\u0000relevance holds of $mathbf{B}$) and characterize the largest fragments of\u0000classical logic with these properties. Along the way, we consider the\u0000consequences for lericone relevance for the theory of subject-matter, for\u0000Logan's notion of hyperformalism, and for the very definition of a relevant\u0000logic itself.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142266955","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that for quasivarieties of p-algebras the properties of (i) having decidable first-order theory and (ii) having decidable first-order theory of the finite members, coincide. The only two quasivarieties with these properties are the trivial variety and the variety of Boolean algebras. This contrasts sharply, even for varieties, with the situation in Heyting algebras where decidable varieties do not coincide with finitely decidable ones.
我们证明，对于 p 矩阵的准变量，(i) 具有可判定的一阶理论和 (ii) 有限成员具有可判定的一阶理论这两个性质是重合的。唯一具有这些性质的两个类群是三元数群和布尔代数数群。这与海廷（Heyting）代数中的情况形成了鲜明的对比，即使对于代数的变种来说，可判定的变种与有限可判定的变种也不重合。
{"title":"Decidable varieties of p-algebras","authors":"Tomasz Kowalski, Katarzyna Słomczyńska","doi":"arxiv-2409.09015","DOIUrl":"https://doi.org/arxiv-2409.09015","url":null,"abstract":"We show that for quasivarieties of p-algebras the properties of (i) having\u0000decidable first-order theory and (ii) having decidable first-order theory of\u0000the finite members, coincide. The only two quasivarieties with these properties\u0000are the trivial variety and the variety of Boolean algebras. This contrasts\u0000sharply, even for varieties, with the situation in Heyting algebras where\u0000decidable varieties do not coincide with finitely decidable ones.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142266953","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate quasivarieties of (distributive) p-algebras. We sharpen some previous results, give a better picture of the subquasivariety lattice, and prove that quasivarieties generated by free p-algebras belong to a rather small quasivariety characterised by excluding all but 3 subdirectly irreducibles.
{"title":"Quasivarieties of p-algebras: some new results","authors":"Tomasz Kowalski, Katarzyna Słomczyńska","doi":"arxiv-2409.08990","DOIUrl":"https://doi.org/arxiv-2409.08990","url":null,"abstract":"We investigate quasivarieties of (distributive) p-algebras. We sharpen some\u0000previous results, give a better picture of the subquasivariety lattice, and\u0000prove that quasivarieties generated by free p-algebras belong to a rather small\u0000quasivariety characterised by excluding all but 3 subdirectly irreducibles.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142266954","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is devoted to the study of analytic equivalence relations which are Borel graphable, i.e. which can be realized as the connectedness relation of a Borel graph. Our main focus is the question of which analytic equivalence relations are Borel graphable. First, we study an equivalence relation arising from the theory of countable admissible ordinals and show that it is Borel graphable if and only if there is a non-constructible real. As a corollary of the proof, we construct an analytic equivalence relation which is (provably in ZFC) not Borel graphable and an effectively analytic equivalence relation which is Borel graphable but not effectively Borel graphable. Next, we study analytic equivalence relations given by the isomorphism relation for some class of countable structures. We show that all such equivalence relations are Borel graphable, which implies that for every Borel action of $S_infty$, the associated orbit equivalence relation is Borel graphable. This leads us to study the class of Polish groups whose Borel actions always give rise to Borel graphable orbit equivalence relations; we refer to such groups as graphic groups. We show that besides $S_infty$, the class of graphic groups includes all connected Polish groups and is closed under countable products. We finish by studying structural properties of the class of Borel graphable analytic equivalence relations and by considering two variations on Borel graphability: a generalization with hypegraphs instead of graphs and an analogue of Borel graphability in the setting of computably enumerable equivalence relations.
{"title":"Borel graphable equivalence relations","authors":"Tyler Arant, Alexander S. Kechris, Patrick Lutz","doi":"arxiv-2409.08624","DOIUrl":"https://doi.org/arxiv-2409.08624","url":null,"abstract":"This paper is devoted to the study of analytic equivalence relations which\u0000are Borel graphable, i.e. which can be realized as the connectedness relation\u0000of a Borel graph. Our main focus is the question of which analytic equivalence\u0000relations are Borel graphable. First, we study an equivalence relation arising\u0000from the theory of countable admissible ordinals and show that it is Borel\u0000graphable if and only if there is a non-constructible real. As a corollary of\u0000the proof, we construct an analytic equivalence relation which is (provably in\u0000ZFC) not Borel graphable and an effectively analytic equivalence relation which\u0000is Borel graphable but not effectively Borel graphable. Next, we study analytic\u0000equivalence relations given by the isomorphism relation for some class of\u0000countable structures. We show that all such equivalence relations are Borel\u0000graphable, which implies that for every Borel action of $S_infty$, the\u0000associated orbit equivalence relation is Borel graphable. This leads us to\u0000study the class of Polish groups whose Borel actions always give rise to Borel\u0000graphable orbit equivalence relations; we refer to such groups as graphic\u0000groups. We show that besides $S_infty$, the class of graphic groups includes\u0000all connected Polish groups and is closed under countable products. We finish\u0000by studying structural properties of the class of Borel graphable analytic\u0000equivalence relations and by considering two variations on Borel graphability:\u0000a generalization with hypegraphs instead of graphs and an analogue of Borel\u0000graphability in the setting of computably enumerable equivalence relations.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142266958","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The main aim of this paper is to show that the topics of {L}ukasiewicz logic, semirings and tropical structures fruitfully meet. This gives rise to a topos theoretic perspective to {L}ukasiewicz logic. A functorial tropicalization of MV-algebras in the variety V(C) is proposed. We further consider a logic based on perfect MV-algebras and having truth values that are perturbations of boolean values, and we show how this logic can exhibit models functorially connected with points of a non-commutative geometry.
{"title":"Tropicalization through the lens of Łukasiewicz logic, with a topos theoretic perspective","authors":"Antonio Di Nola, Giacomo Lenzi, Brunella Gerla","doi":"arxiv-2409.08682","DOIUrl":"https://doi.org/arxiv-2409.08682","url":null,"abstract":"The main aim of this paper is to show that the topics of {L}ukasiewicz\u0000logic, semirings and tropical structures fruitfully meet. This gives rise to a\u0000topos theoretic perspective to {L}ukasiewicz logic. A functorial\u0000tropicalization of MV-algebras in the variety V(C) is proposed. We further\u0000consider a logic based on perfect MV-algebras and having truth values that are\u0000perturbations of boolean values, and we show how this logic can exhibit models\u0000functorially connected with points of a non-commutative geometry.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142266957","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
How can we reason around logical paradoxes without falling into them? This paper introduces grounded deduction or GD, a Kripke-inspired approach to first-order logic and arithmetic that is neither classical nor intuitionistic, but nevertheless appears both pragmatically usable and intuitively justifiable. GD permits the direct expression of unrestricted recursive definitions - including paradoxical ones such as 'L := not L' - while adding dynamic typing premises to certain inference rules so that such paradoxes do not lead to inconsistency. This paper constitutes a preliminary development and investigation of grounded deduction, to be extended with further elaboration and deeper analysis of its intriguing properties.
我们怎样才能绕过逻辑悖论进行推理而不陷入悖论呢？GD 允许直接表达无限制的递归定义--包括 "L := not L "这样的悖论定义--同时为某些推理规则添加了动态类型预设，从而使这类悖论不会导致不一致。本文是对基础演绎法的初步发展和研究，我们还将进一步阐述和深入分析其引人入胜的特性。
{"title":"Reasoning Around Paradox with Grounded Deduction","authors":"Bryan Ford","doi":"arxiv-2409.08243","DOIUrl":"https://doi.org/arxiv-2409.08243","url":null,"abstract":"How can we reason around logical paradoxes without falling into them? This\u0000paper introduces grounded deduction or GD, a Kripke-inspired approach to\u0000first-order logic and arithmetic that is neither classical nor intuitionistic,\u0000but nevertheless appears both pragmatically usable and intuitively justifiable.\u0000GD permits the direct expression of unrestricted recursive definitions -\u0000including paradoxical ones such as 'L := not L' - while adding dynamic typing\u0000premises to certain inference rules so that such paradoxes do not lead to\u0000inconsistency. This paper constitutes a preliminary development and\u0000investigation of grounded deduction, to be extended with further elaboration\u0000and deeper analysis of its intriguing properties.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142224553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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