Pub Date : 2025-02-18DOI: 10.1016/j.apal.2025.103564
Sapir Ben-Shahar , Heer Tern Koh
A recent area of interest in computable topology compares different notions of effective presentability for topological spaces. In this paper, we show that up to isometry, there is a compact connected Polish space that has both left-c.e. and right-c.e. Polish presentations, but has no computable Polish presentation. We also construct a Polish group that has both left-c.e. and right-c.e. Polish group presentations, but lacks a computable Polish presentation, up to topological isomorphism.
{"title":"Comparing notions of presentability in Polish spaces and Polish groups","authors":"Sapir Ben-Shahar , Heer Tern Koh","doi":"10.1016/j.apal.2025.103564","DOIUrl":"10.1016/j.apal.2025.103564","url":null,"abstract":"<div><div>A recent area of interest in computable topology compares different notions of effective presentability for topological spaces. In this paper, we show that up to isometry, there is a compact connected Polish space that has both left-c.e. and right-c.e. Polish presentations, but has no computable Polish presentation. We also construct a Polish group that has both left-c.e. and right-c.e. Polish group presentations, but lacks a computable Polish presentation, up to topological isomorphism.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 5","pages":"Article 103564"},"PeriodicalIF":0.6,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143463576","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-17DOI: 10.1016/j.apal.2025.103563
Miguel Martins, Tommaso Moraschini
A bi-Heyting algebra validates the Gödel-Dummett axiom iff the poset of its prime filters is a disjoint union of co-trees (i.e., order duals of trees). Bi-Heyting algebras of this kind are called bi-Gödel algebras and form a variety that algebraizes the extension of bi-intuitionistic logic axiomatized by the Gödel-Dummett axiom. In this paper we establish the decidability of the problem of determining if a finitely axiomatizable extension of is locally tabular.
Notably, if L is an axiomatic extension of , then L is locally tabular iff L is not contained in , the logic of a particular family of finite co-trees, called the finite combs. We prove that is finitely axiomatizable. Since this logic also has the finite model property, it is therefore decidable. Thus, the above characterization of local tabularity ensures the decidability of the aforementioned problem.
{"title":"Local tabularity is decidable for bi-intermediate logics of trees and of co-trees","authors":"Miguel Martins, Tommaso Moraschini","doi":"10.1016/j.apal.2025.103563","DOIUrl":"10.1016/j.apal.2025.103563","url":null,"abstract":"<div><div>A bi-Heyting algebra validates the Gödel-Dummett axiom <span><math><mo>(</mo><mi>p</mi><mo>→</mo><mi>q</mi><mo>)</mo><mo>∨</mo><mo>(</mo><mi>q</mi><mo>→</mo><mi>p</mi><mo>)</mo></math></span> iff the poset of its prime filters is a disjoint union of co-trees (i.e., order duals of trees). Bi-Heyting algebras of this kind are called <em>bi-Gödel algebras</em> and form a variety that algebraizes the extension <span><math><mi>bi-GD</mi></math></span> of bi-intuitionistic logic axiomatized by the Gödel-Dummett axiom. In this paper we establish the decidability of the problem of determining if a finitely axiomatizable extension of <span><math><mi>bi-GD</mi></math></span> is locally tabular.</div><div>Notably, if <em>L</em> is an axiomatic extension of <span><math><mi>bi-GD</mi></math></span>, then <em>L</em> is locally tabular iff <em>L</em> is not contained in <span><math><mi>L</mi><mi>o</mi><mi>g</mi><mo>(</mo><mi>F</mi><mi>C</mi><mo>)</mo></math></span>, the logic of a particular family of finite co-trees, called the <em>finite combs</em>. We prove that <span><math><mi>L</mi><mi>o</mi><mi>g</mi><mo>(</mo><mi>F</mi><mi>C</mi><mo>)</mo></math></span> is finitely axiomatizable. Since this logic also has the finite model property, it is therefore decidable. Thus, the above characterization of local tabularity ensures the decidability of the aforementioned problem.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 5","pages":"Article 103563"},"PeriodicalIF":0.6,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143487726","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-02-05DOI: 10.1016/j.apal.2025.103556
Sergio Celani , Rafał Gruszczyński , Paula Menchón
Drawing on the classic paper by Chellas [8], we propose a general algebraic framework for studying a binary operation of conditional that models universal features of the “if …, then …” connective as strictly related to the unary modal necessity operator. To this end, we introduce a variety of conditional algebras, and we develop its duality and canonical extensions theory.
{"title":"Conditional algebras","authors":"Sergio Celani , Rafał Gruszczyński , Paula Menchón","doi":"10.1016/j.apal.2025.103556","DOIUrl":"10.1016/j.apal.2025.103556","url":null,"abstract":"<div><div>Drawing on the classic paper by Chellas <span><span>[8]</span></span>, we propose a general algebraic framework for studying a binary operation of <em>conditional</em> that models universal features of the “if …, then …” connective as strictly related to the unary modal necessity operator. To this end, we introduce a variety of <em>conditional algebras</em>, and we develop its duality and canonical extensions theory.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 5","pages":"Article 103556"},"PeriodicalIF":0.6,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143376529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-23DOI: 10.1016/j.apal.2025.103555
Zoltan A. Kocsis
We introduce a proof-theoretic approach to showing nondefinability of second-order intuitionistic connectives by quantifier-free schemata. We apply the method to prove that Taranovsky's “realizability disjunction” connective does not admit a quantifier-free definition, and use it to obtain new results and more nuanced information about the nondefinability of Kreisel's and Połacik's unary connectives. The finitary and combinatorial nature of our method makes it resilient to changes in metatheory, and suitable for settings with axioms that are explicitly incompatible with classical logic. Furthermore, the problem-specific subproofs arising from this approach can be readily transcribed into univalent type theory and verified using the Agda proof assistant.
{"title":"Proof-theoretic methods in quantifier-free definability","authors":"Zoltan A. Kocsis","doi":"10.1016/j.apal.2025.103555","DOIUrl":"10.1016/j.apal.2025.103555","url":null,"abstract":"<div><div>We introduce a proof-theoretic approach to showing nondefinability of second-order intuitionistic connectives by quantifier-free schemata. We apply the method to prove that Taranovsky's “realizability disjunction” connective does not admit a quantifier-free definition, and use it to obtain new results and more nuanced information about the nondefinability of Kreisel's and Połacik's unary connectives. The finitary and combinatorial nature of our method makes it resilient to changes in metatheory, and suitable for settings with axioms that are explicitly incompatible with classical logic. Furthermore, the problem-specific subproofs arising from this approach can be readily transcribed into univalent type theory and verified using the Agda proof assistant.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103555"},"PeriodicalIF":0.6,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-16DOI: 10.1016/j.apal.2025.103554
Christian d'Elbée
We introduce the model-companion of the theory of fields expanded by a unary function for a multiplicative endomorphism, which we call ACFH. Among others, we prove that this theory is NSOP1 and not simple, that the kernel of the map is a generic pseudo-finite abelian group. We also prove that if forking satisfies existence, then ACFH has elimination of imaginaries.
{"title":"Generic multiplicative endomorphism of a field","authors":"Christian d'Elbée","doi":"10.1016/j.apal.2025.103554","DOIUrl":"10.1016/j.apal.2025.103554","url":null,"abstract":"<div><div>We introduce the model-companion of the theory of fields expanded by a unary function for a multiplicative endomorphism, which we call ACFH. Among others, we prove that this theory is NSOP<sub>1</sub> and not simple, that the kernel of the map is a generic pseudo-finite abelian group. We also prove that if forking satisfies existence, then ACFH has elimination of imaginaries.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103554"},"PeriodicalIF":0.6,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162054","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-14DOI: 10.1016/j.apal.2025.103552
Rodrigo Nicolau Almeida
In this paper we present a general theory of -rules for systems of intuitionistic and modal logic. We introduce the notions of -rule system and of an inductive class, and provide model-theoretic and algebraic completeness theorems, which serve as our basic tools. As an illustration of the general theory, we analyse the structure of inductive classes of Gödel algebras, from a structure theoretic and logical point of view. We show that unlike other well-studied settings (such as logics, or single-conclusion rule systems), there are continuum many -rule systems extending , and show how our methods allow easy proofs of the admissibility of the well-known Takeuti-Titani rule. Our final results concern general questions admissibility in : (1) we present a full classification of those inductive classes which are inductively complete, i.e., where all -rules which are admissible are derivable, and (2) show that the problem of admissibility of -rules over is decidable.
{"title":"Π2-rule systems and inductive classes of Gödel algebras","authors":"Rodrigo Nicolau Almeida","doi":"10.1016/j.apal.2025.103552","DOIUrl":"10.1016/j.apal.2025.103552","url":null,"abstract":"<div><div>In this paper we present a general theory of <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-rules for systems of intuitionistic and modal logic. We introduce the notions of <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-rule system and of an inductive class, and provide model-theoretic and algebraic completeness theorems, which serve as our basic tools. As an illustration of the general theory, we analyse the structure of inductive classes of Gödel algebras, from a structure theoretic and logical point of view. We show that unlike other well-studied settings (such as logics, or single-conclusion rule systems), there are continuum many <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-rule systems extending <span><math><mrow><mi>LC</mi></mrow><mo>=</mo><mrow><mi>IPC</mi></mrow><mo>+</mo><mo>(</mo><mi>p</mi><mo>→</mo><mi>q</mi><mo>)</mo><mo>∨</mo><mo>(</mo><mi>q</mi><mo>→</mo><mi>p</mi><mo>)</mo></math></span>, and show how our methods allow easy proofs of the admissibility of the well-known Takeuti-Titani rule. Our final results concern general questions admissibility in <span><math><mi>LC</mi></math></span>: (1) we present a full classification of those inductive classes which are inductively complete, i.e., where all <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-rules which are admissible are derivable, and (2) show that the problem of admissibility of <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-rules over <span><math><mi>LC</mi></math></span> is decidable.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103552"},"PeriodicalIF":0.6,"publicationDate":"2025-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162347","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-13DOI: 10.1016/j.apal.2025.103553
Xiaoyang Wang , Yanjing Wang
Lattice theory has various close connections with modal logic. However, one less explored direction is to view lattices as relational structures based on partial orders, and study the modal logics over them. In this paper, following the earlier steps of Burgess and van Benthem in the 1980s, we use the modal languages of tense logic and polyadic modal logic to talk about lattices via standard Kripke semantics. We first obtain a series of complete axiomatizations of tense logics over lattices, (un)bounded lattices over partial orders or strict orders. In particular, we solve an axiomatization problem left open by Burgess (1984) [8]. The second half of the paper gives a series of complete axiomatizations of polyadic modal logic with nominals over lattices, distributive lattices, and modular lattices, where the binary modalities of infimum and supremum can reveal more structures behind various lattices.
{"title":"Modal logics over lattices","authors":"Xiaoyang Wang , Yanjing Wang","doi":"10.1016/j.apal.2025.103553","DOIUrl":"10.1016/j.apal.2025.103553","url":null,"abstract":"<div><div>Lattice theory has various close connections with modal logic. However, one less explored direction is to view lattices as relational structures based on partial orders, and study the modal logics over them. In this paper, following the earlier steps of Burgess and van Benthem in the 1980s, we use the modal languages of tense logic and polyadic modal logic to talk about lattices via standard Kripke semantics. We first obtain a series of complete axiomatizations of tense logics over lattices, (un)bounded lattices over partial orders or strict orders. In particular, we solve an axiomatization problem left open by Burgess (1984) <span><span>[8]</span></span>. The second half of the paper gives a series of complete axiomatizations of polyadic modal logic with nominals over lattices, distributive lattices, and modular lattices, where the binary modalities of <em>infimum</em> and <em>supremum</em> can reveal more structures behind various lattices.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103553"},"PeriodicalIF":0.6,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-09DOI: 10.1016/j.apal.2024.103551
Eran Alouf
We show that if is a dp-minimal expansion of that defines an infinite subset of , then is interdefinable with . As a corollary, we show the same for dp-minimal expansions of which do not eliminate .
{"title":"On dp-minimal expansions of the integers","authors":"Eran Alouf","doi":"10.1016/j.apal.2024.103551","DOIUrl":"10.1016/j.apal.2024.103551","url":null,"abstract":"<div><div>We show that if <span><math><mi>Z</mi></math></span> is a dp-minimal expansion of <span><math><mo>(</mo><mi>Z</mi><mo>,</mo><mo>+</mo><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> that defines an infinite subset of <span><math><mi>N</mi></math></span>, then <span><math><mi>Z</mi></math></span> is interdefinable with <span><math><mo>(</mo><mi>Z</mi><mo>,</mo><mo>+</mo><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo><</mo><mo>)</mo></math></span>. As a corollary, we show the same for dp-minimal expansions of <span><math><mo>(</mo><mi>Z</mi><mo>,</mo><mo>+</mo><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> which do not eliminate <span><math><msup><mrow><mo>∃</mo></mrow><mrow><mo>∞</mo></mrow></msup></math></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103551"},"PeriodicalIF":0.6,"publicationDate":"2025-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162324","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-12DOI: 10.1016/j.apal.2024.103550
Emanuele Frittaion
We analyze Coquand's game-theoretic interpretation of Peano Arithmetic [6] through the lens of elementary descent recursion [8]. In Coquand's game semantics, winning strategies correspond to infinitary cut-free proofs and cut elimination corresponds to debates between these winning strategies. The proof of cut elimination, i.e., the proof that such debates eventually terminate, is by transfinite induction on certain interaction sequences of ordinals. In this paper, we provide a direct implementation of Coquand's proof, one that allows us to describe winning strategies by descent recursive functions. As a byproduct, we obtain yet another proof of well-known results about provably recursive functions and functionals.
{"title":"Peano arithmetic, games and descent recursion","authors":"Emanuele Frittaion","doi":"10.1016/j.apal.2024.103550","DOIUrl":"10.1016/j.apal.2024.103550","url":null,"abstract":"<div><div>We analyze Coquand's game-theoretic interpretation of Peano Arithmetic <span><span>[6]</span></span> through the lens of elementary descent recursion <span><span>[8]</span></span>. In Coquand's game semantics, winning strategies correspond to infinitary cut-free proofs and cut elimination corresponds to <em>debates</em> between these winning strategies. The proof of cut elimination, i.e., the proof that such debates eventually terminate, is by transfinite induction on certain <em>interaction</em> sequences of ordinals. In this paper, we provide a direct implementation of Coquand's proof, one that allows us to describe winning strategies by descent recursive functions. As a byproduct, we obtain yet another proof of well-known results about provably recursive functions and functionals.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103550"},"PeriodicalIF":0.6,"publicationDate":"2024-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162345","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-10DOI: 10.1016/j.apal.2024.103549
Matthew Harrison-Trainor , Dhruv Kulshreshtha
We work in the setting of Zermelo-Fraenkel set theory without assuming the Axiom of Choice. We consider sets with the Boolean operations together with the additional structure of comparing cardinality (in the Cantorian sense of injections). What principles does one need to add to the laws of Boolean algebra to reason not only about intersection, union, and complementation of sets, but also about the relative size of sets? We give a complete axiomatization.
A particularly interesting case is when one restricts to the Dedekind-finite sets. In this case, one needs exactly the same principles as for reasoning about imprecise probability comparisons, the central principle being Generalized Finite Cancellation (which includes, as a special case, division-by-m). In the general case, the central principle is a restricted version of Generalized Finite Cancellation within Archimedean classes which we call Covered Generalized Finite Cancellation.
{"title":"The logic of cardinality comparison without the axiom of choice","authors":"Matthew Harrison-Trainor , Dhruv Kulshreshtha","doi":"10.1016/j.apal.2024.103549","DOIUrl":"10.1016/j.apal.2024.103549","url":null,"abstract":"<div><div>We work in the setting of Zermelo-Fraenkel set theory without assuming the Axiom of Choice. We consider sets with the Boolean operations together with the additional structure of comparing cardinality (in the Cantorian sense of injections). What principles does one need to add to the laws of Boolean algebra to reason not only about intersection, union, and complementation of sets, but also about the relative size of sets? We give a complete axiomatization.</div><div>A particularly interesting case is when one restricts to the Dedekind-finite sets. In this case, one needs exactly the same principles as for reasoning about imprecise probability comparisons, the central principle being Generalized Finite Cancellation (which includes, as a special case, division-by-<em>m</em>). In the general case, the central principle is a restricted version of Generalized Finite Cancellation within Archimedean classes which we call Covered Generalized Finite Cancellation.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 4","pages":"Article 103549"},"PeriodicalIF":0.6,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162346","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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