{"title":"Explicit Abstract Objects in Predicative Settings","authors":"Sean Ebels-Duggan, Francesca Boccuni","doi":"10.1007/s10992-024-09768-1","DOIUrl":null,"url":null,"abstract":"<p>Abstractionist programs in the philosophy of mathematics have focused on abstraction principles, taken as <i>implicit</i> definitions of the objects in the range of their operators. In second-order logic (<span>SOL</span>) with predicative comprehension, such principles are consistent but also (individually) mathematically weak. This paper, inspired by the work of Boolos (<i>Proceedings of the Aristotelian Society</i> <i>87</i>, 137–151, 1986) and Zalta (<i>Abstract Objects</i>, vol. 160 of <i>Synthese Library</i>, 1983), examines <i>explicit</i> definitions of abstract objects. These axioms state that there is a unique abstract encoding all concepts satisfying a given formula <span>\\(\\phi (F)\\)</span>, with <i>F</i> a concept variable. Such a system is inconsistent in full <span>SOL</span>. It can be made consistent with several intricate tweaks, as Zalta has shown. Our approach in this article is simpler: we use a novel method to establish consistency in a restrictive version of predicative <span>SOL</span>. The resulting system, <span>RPEAO</span>, interprets first-order <span>PA</span> in extensional contexts, and has a natural extension delivering a peculiar interpretation of <span>PA</span> <span>\\(^2\\)</span>.</p>","PeriodicalId":51526,"journal":{"name":"JOURNAL OF PHILOSOPHICAL LOGIC","volume":"51 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"JOURNAL OF PHILOSOPHICAL LOGIC","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s10992-024-09768-1","RegionNum":1,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"0","JCRName":"PHILOSOPHY","Score":null,"Total":0}
引用次数: 0
Abstract
Abstractionist programs in the philosophy of mathematics have focused on abstraction principles, taken as implicit definitions of the objects in the range of their operators. In second-order logic (SOL) with predicative comprehension, such principles are consistent but also (individually) mathematically weak. This paper, inspired by the work of Boolos (Proceedings of the Aristotelian Society87, 137–151, 1986) and Zalta (Abstract Objects, vol. 160 of Synthese Library, 1983), examines explicit definitions of abstract objects. These axioms state that there is a unique abstract encoding all concepts satisfying a given formula \(\phi (F)\), with F a concept variable. Such a system is inconsistent in full SOL. It can be made consistent with several intricate tweaks, as Zalta has shown. Our approach in this article is simpler: we use a novel method to establish consistency in a restrictive version of predicative SOL. The resulting system, RPEAO, interprets first-order PA in extensional contexts, and has a natural extension delivering a peculiar interpretation of PA\(^2\).
期刊介绍：
The Journal of Philosophical Logic aims to provide a forum for work at the crossroads of philosophy and logic, old and new, with contributions ranging from conceptual to technical. Accordingly, the Journal invites papers in all of the traditional areas of philosophical logic, including but not limited to: various versions of modal, temporal, epistemic, and deontic logic; constructive logics; relevance and other sub-classical logics; many-valued logics; logics of conditionals; quantum logic; decision theory, inductive logic, logics of belief change, and formal epistemology; defeasible and nonmonotonic logics; formal philosophy of language; vagueness; and theories of truth and validity. In addition to publishing papers on philosophical logic in this familiar sense of the term, the Journal also invites papers on extensions of logic to new areas of application, and on the philosophical issues to which these give rise. The Journal places a special emphasis on the applications of philosophical logic in other disciplines, not only in mathematics and the natural sciences but also, for example, in computer science, artificial intelligence, cognitive science, linguistics, jurisprudence, and the social sciences, such as economics, sociology, and political science.
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