Talk (Work in Progress): Beatrice Buonaguidi (King's College)

Title:

Abstraction, extensionality and hyperintensionality

Abstract:

The logic HYPE has been put forward by Hannes Leitgeb (Leitgeb 2019) both as a framework to study phenomena requiring a weakening of classical logic, among which the semantic paradoxes, and as a framework to model hyperintensional phenomena.

In the investigation of solutions to the semantic paradoxes, it has been shown that HYPE fares better than other non-classical logics as a basis for axiomatisations of Kripkean truth (Fischer et al 2023). However, this result depends on the assumption of classical Peano Arithmetic in the background, indicating that, while HYPE is a strong non-classical logic to formulate a solution to the paradoxes, it does so by exploiting classical recapture results.

Then, to measure the foundational significance of HYPE, it is natural to ask whether abstraction principles obtained via fixed-point models for HYPE are sufficient to develop a nontrivial amount of mathematical objects and concepts. In this talk, I will present a theory based on one such abstraction principle, showing its consistency. Further, I will show that HYPE, being suited to model hyperintensional phenomena, is an illuminating framework to study the role of extensionality axioms over non-classical abstraction. I show that, while standard principles of extensionality are inconsistent with HYPE, weaker axioms of extensionality are consistent with it. This clarifies the significance of what amounts to extensional equivalence in a non-classical setting.

Fischer, Martin; Nicolai, Carlo & Dopico, Pablo (2023). Nonclassical Truth with Classical Strength. A Proof-Theoretic Analysis of Compositional Truth Over Hype. Review of Symbolic Logic 16 (2): 425-448.
Leitgeb, Hannes (2019). HYPE: A System of Hyperintensional Logic. Journal of Philosophical Logic 48 (2): 305-405.