Class Theory in HYPE

Maria Beatrice Buonaguidi; Carlo Nicolai

doi:10.1215/00294527-2025-0009

Class Theory in HYPE

Philosophy

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Abstract

This paper studies class theory over the logic HYPE recently introduced by Hannes Leitgeb. We formulate suitable abstraction principles and show their consistency by displaying a class of fixed-point (term) models. By adapting a classical result by Brady, we show their inconsistency with standard extensionality principles, as well as the incompatibility of our semantics with weak extensionality principles introduced in the literature. We then formulate our version of weak extensionality (appropriate to the behavior of the conditional in HYPE) and show its consistency with one of the abstraction principles previously introduced. We conclude with observations and examples supporting the claim that, although arithmetical axioms over HYPE are as strong as classical arithmetical axioms, the behavior of classes over HYPE is akin to the one displayed by classes in other nonclassical class theories.

Original language	English
Pages (from-to)	433-454
Number of pages	22
Journal	Notre Dame Journal of Formal Logic
Volume	66
Issue number	4
Early online date	10 Jul 2025
DOIs	https://doi.org/10.1215/00294527-2025-0009
Publication status	Published - Nov 2025

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10.1215/00294527-2025-0009

CLASS THEORY IN HYPE_BUONAGUIDI_2025_GREEN AAMAccepted author manuscript, 546 KBLicence: CC BY

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AB - This paper studies class theory over the logic HYPE recently introduced by Hannes Leitgeb. We formulate suitable abstraction principles and show their consistency by displaying a class of fixed-point (term) models. By adapting a classical result by Brady, we show their inconsistency with standard extensionality principles, as well as the incompatibility of our semantics with weak extensionality principles introduced in the literature. We then formulate our version of weak extensionality (appropriate to the behavior of the conditional in HYPE) and show its consistency with one of the abstraction principles previously introduced. We conclude with observations and examples supporting the claim that, although arithmetical axioms over HYPE are as strong as classical arithmetical axioms, the behavior of classes over HYPE is akin to the one displayed by classes in other nonclassical class theories.

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JO - Notre Dame Journal of Formal Logic

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Class Theory in HYPE

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