Wright 的严格有穷主义一阶逻辑
Wright's First-Order Logic of Strict Finitism
AI总结：
本文将 Yamada 对 Wright 严格有穷主义逻辑的经典命题重构扩展为一阶逻辑，给出可靠完备的 Kripke 式语义和自然演绎系统，并说明其与既有结果的关系。
摘要：
本文提出了对 Wright 的严格有穷主义一阶逻辑的一种经典重构。严格有穷主义是一种比直觉主义更具限制性的数学建构主义立场。Wright 在其严格有穷主义元理论中，于 Wright（《实在论、意义与真理》，第4章，第2版，1993年；Blackwell Publishers，Oxford、Cambridge，第107–75页，1982年）中勾勒了该逻辑的语义。Yamada（J Philos Log，https://doi.org/10.1007/s10992-022-09698-w，2023年）在一个使该逻辑与直觉主义命题逻辑相对应的辅助条件下，提出了严格有穷主义的命题逻辑，作为其经典重构。本文将该命题逻辑扩展为一种不假设该条件的一阶逻辑。我们将给出一对可靠且完备的 Kripke 式语义与自然演绎系统，并表明若施加该条件，该逻辑便呈现为 Yamada（2023）结果的自然扩展。
英文摘要：
A classical reconstruction of Wright's first-order logic of strict finitism is presented. Strict finitism is a constructive standpoint of mathematics that is more restrictive than intuitionism. Wright sketched the semantics of said logic in Wright (Realism, Meaning and Truth, chap 4, 2nd edition in 1993. Blackwell Publishers, Oxford, Cambridge, pp.107-75, 1982), in his strict finitistic metatheory. Yamada (J Philos Log. https://doi.org/10.1007/s10992-022-09698-w, 2023) proposed, as its classical reconstruction, a propositional logic of strict finitism under an auxiliary condition that makes the logic correspond with intuitionistic propositional logic. In this paper, we extend the propositional logic to a first-order logic that does not assume the condition. We will provide a sound and complete pair of a Kripke-style semantics and a natural deduction system, and show that if the condition is imposed, then the logic exhibits natural extensions of Yamada (2023)'s results.
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