{"title":"Proof-theoretic methods in quantifier-free definability","authors":"Zoltan A. Kocsis","doi":"10.1016/j.apal.2025.103555","DOIUrl":null,"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":0.6000,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Pure and Applied Logic","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0168007225000041","RegionNum":2,"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
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.
期刊介绍：
The journal Annals of Pure and Applied Logic publishes high quality papers in all areas of mathematical logic as well as applications of logic in mathematics, in theoretical computer science and in other related disciplines. All submissions to the journal should be mathematically correct, well written (preferably in English)and contain relevant new results that are of significant interest to a substantial number of logicians. The journal also considers submissions that are somewhat too long to be published by other journals while being too short to form a separate memoir provided that they are of particular outstanding quality and broad interest. In addition, Annals of Pure and Applied Logic occasionally publishes special issues of selected papers from well-chosen conferences in pure and applied logic.
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