A logic of judgmental existence and its relation to proof irrelevance

Ivo Pezlar

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arXiv · 2405.14481

A logic of judgmental existence and its relation to proof irrelevance

Ivo Pezlar

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Abstract

We introduce a simple natural deduction system for reasoning with judgments of the form "there exists a proof of $\varphi$" to explore the notion of judgmental existence following Martin-L\"{o}f's methodology of distinguishing between judgments and propositions. In this system, the existential judgment can be internalized into a modal notion of propositional existence that is closely related to truncation modality, a key tool for obtaining proof irrelevance, and lax modality. We provide a computational interpretation in the style of the Curry-Howard isomorphism for the existence modality and show that the corresponding system has some desirable properties such as strong normalization or subject reduction.

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Ivo Pezlar. 2024-05-23. A logic of judgmental existence and its relation to proof irrelevance. https://arxiv.org/abs/2405.14481

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