Explicit Abstract Objects in Predicative Settings

IF 0.7 1区哲学 0 PHILOSOPHY

JOURNAL OF PHILOSOPHICAL LOGIC Pub Date : 2024-07-09 DOI:10.1007/s10992-024-09768-1

Sean Ebels-Duggan, Francesca Boccuni

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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 Society 87, 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 \(^2\).

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谓语环境中的显式抽象对象

数学哲学中的抽象主义方案侧重于抽象原则，将其视为运算符范围内对象的隐含定义。在具有谓词理解的二阶逻辑（SOL）中，这些原则是一致的，但在（个别）数学上也是薄弱的。本文受 Boolos（《亚里士多德学会会议录》第 87 卷，137-151，1986 年）和 Zalta（《抽象对象》，Synthese Library 第 160 卷，1983 年）著作的启发，研究了抽象对象的显式定义。这些公理指出，有一个唯一的抽象编码所有满足给定公式 \(\phi(F)\)的概念，F 是一个概念变量。这样一个系统在完全 SOL 中是不一致的。正如扎尔塔（Zalta）所展示的，它可以通过一些复杂的调整变得一致。我们在本文中采用的方法更简单：我们用一种新方法在限制性版本的谓词 SOL 中建立一致性。由此产生的系统 RPEAO 可以解释扩展语境中的一阶 PA，并有一个自然的扩展来传递 PA \(^2\)的特殊解释。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

JOURNAL OF PHILOSOPHICAL LOGIC PHILOSOPHY-

CiteScore

2.50

自引率

20.00%

发文量

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.