Pub Date : 2024-10-23DOI: 10.1016/j.apal.2024.103526
Amirhossein Akbar Tabatabai , Raheleh Jalali
We introduce a general and syntactically defined family of sequent-style calculi over the propositional language with the modalities and its fragments as a formalization for constructively acceptable systems. Calling these calculi constructive, we show that any strong enough constructive sequent calculus, satisfying a mild technical condition, feasibly admits all Visser's rules. This means that there exists a polynomial-time algorithm that, given a proof of the premise of a Visser's rule, provides a proof for its conclusion. As a positive application, we establish the feasible admissibility of Visser's rules in sequent calculi for several intuitionistic modal logics, including , , their extensions by the modal axioms T, B, 4, 5, and the axioms for bounded width and depth and their fragments , propositional lax logic and . On the negative side, we show that if a strong enough intuitionistic modal logic (satisfying a mild technical condition) does not admit at least one of Visser's rules, it cannot have a constructive sequent calculus. Consequently, no intermediate logic other than has a constructive sequent calculus.
{"title":"Universal proof theory: Feasible admissibility in intuitionistic modal logics","authors":"Amirhossein Akbar Tabatabai , Raheleh Jalali","doi":"10.1016/j.apal.2024.103526","DOIUrl":"10.1016/j.apal.2024.103526","url":null,"abstract":"<div><div>We introduce a general and syntactically defined family of sequent-style calculi over the propositional language with the modalities <span><math><mo>{</mo><mo>□</mo><mo>,</mo><mo>◇</mo><mo>}</mo></math></span> and its fragments as a formalization for constructively acceptable systems. Calling these calculi <em>constructive</em>, we show that any strong enough constructive sequent calculus, satisfying a mild technical condition, feasibly admits all Visser's rules. This means that there exists a polynomial-time algorithm that, given a proof of the premise of a Visser's rule, provides a proof for its conclusion. As a positive application, we establish the feasible admissibility of Visser's rules in sequent calculi for several intuitionistic modal logics, including <span><math><mi>CK</mi></math></span>, <span><math><mi>IK</mi></math></span>, their extensions by the modal axioms <em>T</em>, <em>B</em>, 4, 5, and the axioms for bounded width and depth and their fragments <span><math><msub><mrow><mi>CK</mi></mrow><mrow><mo>□</mo></mrow></msub></math></span>, propositional lax logic and <span><math><mi>IPC</mi></math></span>. On the negative side, we show that if a strong enough intuitionistic modal logic (satisfying a mild technical condition) does not admit at least one of Visser's rules, it cannot have a constructive sequent calculus. Consequently, no intermediate logic other than <span><math><mi>IPC</mi></math></span> has a constructive sequent calculus.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 2","pages":"Article 103526"},"PeriodicalIF":0.6,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142561348","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 : 2024-10-18DOI: 10.1016/j.apal.2024.103525
S. Jalili , M. Pourmahdian , M. Khani
This paper concerns the study of expansions of models of a geometric theory T by a color predicate p, within the framework of the Fraïssé-Hrushovski construction method. For each , we define a pre-dimension function on the class of Bi-colored models of and consider the subclass consisting of models with hereditary positive . We impose certain natural conditions on T that enable us to introduce a complete -theory for the rich models in . We show how the transfer of certain model-theoretic properties, such as NIP and strong-dependence, from T to , depends on whether α is rational or irrational.
本文在弗拉伊塞-赫鲁晓夫斯基（Fraïssé-Hrushovski）构造方法的框架内，研究用颜色谓词 p 展开几何理论 T 的模型。对于每个 α∈(0,1]，我们在 T∀ 的双色模型类上定义一个前维度函数 δα，并考虑由具有遗传性正 δα 的模型组成的子类 Kα+。我们对 T 施加了某些自然条件，使我们能够为 Kα+ 中的丰富模型引入一个完整的 Π2 理论 Tα。我们展示了某些模型理论性质，如NIP和强依赖性，如何从T转移到Tα，取决于α是有理的还是无理的。
{"title":"Bi-colored expansions of geometric theories","authors":"S. Jalili , M. Pourmahdian , M. Khani","doi":"10.1016/j.apal.2024.103525","DOIUrl":"10.1016/j.apal.2024.103525","url":null,"abstract":"<div><div>This paper concerns the study of expansions of models of a geometric theory <em>T</em> by a color predicate <em>p</em>, within the framework of the Fraïssé-Hrushovski construction method. For each <span><math><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, we define a pre-dimension function <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> on the class of Bi-colored models of <span><math><msup><mrow><mi>T</mi></mrow><mrow><mo>∀</mo></mrow></msup></math></span> and consider the subclass <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>α</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> consisting of models with hereditary positive <span><math><msub><mrow><mi>δ</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>. We impose certain natural conditions on <em>T</em> that enable us to introduce a complete <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-theory <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> for the rich models in <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>α</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math></span>. We show how the transfer of certain model-theoretic properties, such as NIP and strong-dependence, from <em>T</em> to <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>, depends on whether <em>α</em> is rational or irrational.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 2","pages":"Article 103525"},"PeriodicalIF":0.6,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142553194","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 : 2024-10-16DOI: 10.1016/j.apal.2024.103524
Maria Emilia Maietti, Pietro Sabelli
The Minimalist Foundation, for short MF, was conceived by the first author with G. Sambin in 2005, and fully formalized in 2009, as a common core among the most relevant constructive and classical foundations for mathematics. To better accomplish its minimality, MF was designed as a two-level type theory, with an intensional level mTT, an extensional one emTT, and an interpretation of the latter into the first.
Here, we first show that the two levels of MF are indeed equiconsistent by interpreting mTT into emTT. Then, we show that the classical extension is equiconsistent with emTT by suitably extending the Gödel-Gentzen double-negation translation of classical logic in the intuitionistic one. As a consequence, MF turns out to be compatible with classical predicative mathematics à la Weyl, contrary to the most relevant foundations for constructive mathematics.
Finally, we show that the chain of equiconsistency results for MF can be straightforwardly extended to its impredicative version to deduce that Coquand-Huet's Calculus of Constructions equipped with basic inductive types is equiconsistent with its extensional and classical versions too.
{"title":"Equiconsistency of the Minimalist Foundation with its classical version","authors":"Maria Emilia Maietti, Pietro Sabelli","doi":"10.1016/j.apal.2024.103524","DOIUrl":"10.1016/j.apal.2024.103524","url":null,"abstract":"<div><div>The Minimalist Foundation, for short <strong>MF</strong>, was conceived by the first author with G. Sambin in 2005, and fully formalized in 2009, as a common core among the most relevant constructive and classical foundations for mathematics. To better accomplish its minimality, <strong>MF</strong> was designed as a two-level type theory, with an intensional level <strong>mTT</strong>, an extensional one <strong>emTT</strong>, and an interpretation of the latter into the first.</div><div>Here, we first show that the two levels of <strong>MF</strong> are indeed equiconsistent by interpreting <strong>mTT</strong> into <strong>emTT</strong>. Then, we show that the classical extension <span><math><msup><mrow><mi>emTT</mi></mrow><mrow><mi>c</mi></mrow></msup></math></span> is equiconsistent with <strong>emTT</strong> by suitably extending the Gödel-Gentzen double-negation translation of classical logic in the intuitionistic one. As a consequence, <strong>MF</strong> turns out to be compatible with classical predicative mathematics à la Weyl, contrary to the most relevant foundations for constructive mathematics.</div><div>Finally, we show that the chain of equiconsistency results for <strong>MF</strong> can be straightforwardly extended to its impredicative version to deduce that Coquand-Huet's Calculus of Constructions equipped with basic inductive types is equiconsistent with its extensional and classical versions too.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 2","pages":"Article 103524"},"PeriodicalIF":0.6,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142553193","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 : 2024-10-09DOI: 10.1016/j.apal.2024.103523
Marat Faizrahmanov
In this paper, we prove a joint generalization of Arslanov's completeness criterion and Visser's ADN theorem for precomplete numberings which, for the Gödel numbering , has been proved by Terwijn (2018). The question of whether this joint generalization takes place in each precomplete numbering has been raised in his joint paper with Barendregt in 2019. Then we consider the properties of completeness and precompleteness of numberings in the context of the positivity property. We show that no completion of a positive numbering is a minimal cover of that numbering, and that the Turing completeness of any set A is equivalent to the existence of a positive precomplete A-computable numbering of any infinite family with positive A-computable numbering. In addition, we prove that each -computable numbering () of a -computable non-principal family has a -computable minimal cover ν such that for every computable function f there exists an integer n with .
在本文中，我们证明了 Arslanov 的完备性准则和 Visser 的 ADN 定理对预完备数列的联合泛化，对于哥德尔数列 x↦Wx，Terwijn（2018）已经证明了这一联合泛化。关于这一联合泛化是否发生在每一个前完备数列中的问题，在他与巴伦德雷格特（Barendregt）2019年的联合论文中已经提出。然后，我们在实在性性质的背景下考虑编号的完备性和预完备性性质。我们证明，正编号的任何完备都不是该编号的最小盖，而任何集合 A 的图灵完备性都等价于任何具有正 A 可计算编号的无穷族存在正预完备 A 可计算编号。此外，我们还证明了Σn0可计算非主族的每个Σn0可计算编号（n⩾2）都有一个Σn0可计算极小盖ν，从而对于每个可计算函数f都存在一个整数n，且ν(f(n))=ν(n)。
{"title":"Some properties of precompletely and positively numbered sets","authors":"Marat Faizrahmanov","doi":"10.1016/j.apal.2024.103523","DOIUrl":"10.1016/j.apal.2024.103523","url":null,"abstract":"<div><div>In this paper, we prove a joint generalization of Arslanov's completeness criterion and Visser's ADN theorem for precomplete numberings which, for the Gödel numbering <span><math><mi>x</mi><mo>↦</mo><msub><mrow><mi>W</mi></mrow><mrow><mi>x</mi></mrow></msub></math></span>, has been proved by Terwijn (2018). The question of whether this joint generalization takes place in each precomplete numbering has been raised in his joint paper with Barendregt in 2019. Then we consider the properties of completeness and precompleteness of numberings in the context of the positivity property. We show that no completion of a positive numbering is a minimal cover of that numbering, and that the Turing completeness of any set <em>A</em> is equivalent to the existence of a positive precomplete <em>A</em>-computable numbering of any infinite family with positive <em>A</em>-computable numbering. In addition, we prove that each <span><math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msubsup></math></span>-computable numbering (<span><math><mi>n</mi><mo>⩾</mo><mn>2</mn></math></span>) of a <span><math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msubsup></math></span>-computable non-principal family has a <span><math><msubsup><mrow><mi>Σ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msubsup></math></span>-computable minimal cover <em>ν</em> such that for every computable function <em>f</em> there exists an integer <em>n</em> with <span><math><mi>ν</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo><mo>=</mo><mi>ν</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 2","pages":"Article 103523"},"PeriodicalIF":0.6,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142420439","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 : 2024-10-01DOI: 10.1016/j.apal.2024.103522
Noah Schweber
We study Medvedev reducibility in the context of set theory — specifically, forcing and large cardinal hypotheses. Answering a question of Hamkins and Li [6], we show that the Medvedev degrees of countable ordinals are far from linearly ordered in multiple ways, our main result here being that there is a club of ordinals which is an antichain with respect to Medvedev reducibility. We then generalize these results to arbitrary “reasonably-definable” reducibilities, under appropriate set-theoretic hypotheses.
We then turn from ordinals to general structures. We show that some of the results above yield characterizations of counterexamples to Vaught's conjecture; another applies to all situations, assigning an ordinal to any reasonable class of structures and “measure” on that class. We end by discussing some directions for future research.
{"title":"Strong reducibilities and set theory","authors":"Noah Schweber","doi":"10.1016/j.apal.2024.103522","DOIUrl":"10.1016/j.apal.2024.103522","url":null,"abstract":"<div><div>We study Medvedev reducibility in the context of set theory — specifically, forcing and large cardinal hypotheses. Answering a question of Hamkins and Li <span><span>[6]</span></span>, we show that the Medvedev degrees of countable ordinals are far from linearly ordered in multiple ways, our main result here being that there is a club of ordinals which is an antichain with respect to Medvedev reducibility. We then generalize these results to arbitrary “reasonably-definable” reducibilities, under appropriate set-theoretic hypotheses.</div><div>We then turn from ordinals to general structures. We show that some of the results above yield characterizations of counterexamples to Vaught's conjecture; another applies to all situations, assigning an ordinal to any reasonable class of structures and “measure” on that class. We end by discussing some directions for future research.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 2","pages":"Article 103522"},"PeriodicalIF":0.6,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142535830","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 : 2024-09-24DOI: 10.1016/j.apal.2024.103521
Hirotaka Kikyo , Akito Tsuboi
We investigate the class of m-hypergraphs in which substructures with l elements have more than s subsets of size m that do not form a hyperedge. The class has a (unique) Fraïssé limit, if . We show that the theory of the Fraïssé limit has SU-rank one if , and dividing and forking will be different concepts in the theory if .
我们研究了 m-hypergraphs 类，在这类图中，有 l 个元素的子结构有多于 s 个大小为 m 的子集不构成一个 hyperedge。如果0≤s<(l-2m-2)，则该类图具有（唯一的）弗雷泽极限。我们证明，如果 0≤s<(l-3m-3), 那么弗拉伊塞极限理论具有 SU-rank one，如果 (l-3m-3)≤s<(l-2m-2), 那么分割和分叉在理论中将是不同的概念。
{"title":"Dividing and forking in random hypergraphs","authors":"Hirotaka Kikyo , Akito Tsuboi","doi":"10.1016/j.apal.2024.103521","DOIUrl":"10.1016/j.apal.2024.103521","url":null,"abstract":"<div><div>We investigate the class of <em>m</em>-hypergraphs in which substructures with <em>l</em> elements have more than <em>s</em> subsets of size <em>m</em> that do not form a hyperedge. The class has a (unique) Fraïssé limit, if <span><math><mn>0</mn><mo>≤</mo><mi>s</mi><mo><</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>l</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>m</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow></math></span>. We show that the theory of the Fraïssé limit has <em>SU</em>-rank one if <span><math><mn>0</mn><mo>≤</mo><mi>s</mi><mo><</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>l</mi><mo>−</mo><mn>3</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>m</mi><mo>−</mo><mn>3</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow></math></span>, and dividing and forking will be different concepts in the theory if <span><math><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>l</mi><mo>−</mo><mn>3</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>m</mi><mo>−</mo><mn>3</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>≤</mo><mi>s</mi><mo><</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>l</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>m</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow></math></span>.</div></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 2","pages":"Article 103521"},"PeriodicalIF":0.6,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142326876","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 : 2024-09-03DOI: 10.1016/j.apal.2024.103512
Bartosz Wcisło
It is an open question whether compositional truth with the principle of propositional soundness: “All arithmetical sentences which are propositional tautologies are true” is conservative over Peano Arithmetic. In this article, we show that the principle of propositional soundness imposes some saturation-like properties on the truth predicate, thus showing significant limitations to the possible conservativity proof.
{"title":"Saturation properties for compositional truth with propositional correctness","authors":"Bartosz Wcisło","doi":"10.1016/j.apal.2024.103512","DOIUrl":"10.1016/j.apal.2024.103512","url":null,"abstract":"<div><p>It is an open question whether compositional truth with the principle of propositional soundness: “All arithmetical sentences which are propositional tautologies are true” is conservative over Peano Arithmetic. In this article, we show that the principle of propositional soundness imposes some saturation-like properties on the truth predicate, thus showing significant limitations to the possible conservativity proof.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 2","pages":"Article 103512"},"PeriodicalIF":0.6,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168007224001167/pdfft?md5=93f2e704b024dfc73e7a30a7ab95c178&pid=1-s2.0-S0168007224001167-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142168552","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 : 2024-08-30DOI: 10.1016/j.apal.2024.103511
Mauro Di Nasso , Renling Jin
We develop a framework for nonstandard analysis that gives foundations to the interplay between external and internal iterations of the star map, and we present a few examples to show the strength and flexibility of such a nonstandard technique for applications in combinatorial number theory.
{"title":"Foundations of iterated star maps and their use in combinatorics","authors":"Mauro Di Nasso , Renling Jin","doi":"10.1016/j.apal.2024.103511","DOIUrl":"10.1016/j.apal.2024.103511","url":null,"abstract":"<div><p>We develop a framework for nonstandard analysis that gives foundations to the interplay between external and internal iterations of the star map, and we present a few examples to show the strength and flexibility of such a nonstandard technique for applications in combinatorial number theory.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 1","pages":"Article 103511"},"PeriodicalIF":0.6,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142136431","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 : 2024-08-22DOI: 10.1016/j.apal.2024.103510
Daichi Hayashi
Feferman [9] defines an impredicative system of explicit mathematics, which is proof-theoretically equivalent to the subsystem of second-order arithmetic. In this paper, we propose several systems of Frege structure with the same proof-theoretic strength as . To be precise, we first consider the Kripke–Feferman theory, which is one of the most famous truth theories, and we extend it by two kinds of induction principles inspired by [22]. In addition, we give similar results for the system based on Aczel's original Frege structure [1]. Finally, we equip Cantini's supervaluation-style theory with the notion of universes, the strength of which was an open problem in [24].
{"title":"Theories of Frege structure equivalent to Feferman's system T0","authors":"Daichi Hayashi","doi":"10.1016/j.apal.2024.103510","DOIUrl":"10.1016/j.apal.2024.103510","url":null,"abstract":"<div><p>Feferman <span><span>[9]</span></span> defines an impredicative system <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> of explicit mathematics, which is proof-theoretically equivalent to the subsystem <figure><img></figure> of second-order arithmetic. In this paper, we propose several systems of Frege structure with the same proof-theoretic strength as <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>. To be precise, we first consider the Kripke–Feferman theory, which is one of the most famous truth theories, and we extend it by two kinds of induction principles inspired by <span><span>[22]</span></span>. In addition, we give similar results for the system based on Aczel's original Frege structure <span><span>[1]</span></span>. Finally, we equip Cantini's supervaluation-style theory with the notion of universes, the strength of which was an open problem in <span><span>[24]</span></span>.</p></div>","PeriodicalId":50762,"journal":{"name":"Annals of Pure and Applied Logic","volume":"176 1","pages":"Article 103510"},"PeriodicalIF":0.6,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142099250","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 : 2024-08-20DOI: 10.1016/j.apal.2024.103509
Amirhossein Akbar Tabatabai , Raheleh Jalali
We provide a general and syntactically defined family of sequent calculi, called semi-analytic, to formalize the informal notion of a “nice” sequent calculus. We show that any sufficiently strong (multimodal) substructural logic with a semi-analytic sequent calculus enjoys the Craig Interpolation Property, CIP. As a positive application, our theorem provides a uniform and modular method to prove the CIP for several multimodal substructural logics, including many fragments and variants of linear logic. More interestingly, on the negative side, it employs the lack of the CIP in almost all substructural, superintuitionistic and modal logics to provide a formal proof for the well-known intuition that almost all logics do not have a “nice” sequent calculus. More precisely, we show that many substructural logics including , , , Łn (for ), Gn (for ), and almost all extensions of , , , , , , and (except for at most 1, 1, 3, 8, 7, 37, and 6 of them, respectively) do not have a semi-analytic calculus.
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