Maria Beatrice Buonaguidi

A logic is said to be hyperintensional when it admits connectives or operators creating hyperintensional contexts, i.e. sentential contexts not respecting the intersubstitutivity salva veritate of co-intensionals. Odintsov and Wansing (2021) suggest a formal criterion for classifying a logic as hyperintensional based on whether its consequence relation is self-extensional. I argue that satisfaction of Odintsov and Wansing's criterion is neither necessary nor sufficient for characterising a logic as hyperintensional. Indeed, the criterion classifies as hyperintensional logics which are not meant to capture hyperintensional behaviour, and fails to classify as hyperintensional logics which admit some forms of hyperintensional behaviour. I suggest several possible criteria for a logic to count as hyperintensional, arguing that each of them corresponds to a different understanding of hyperintensionality for logics. Then, I weigh these criteria against a family of logics, which display different "nuances'' of hyperintensional behaviour. Finally, I gesture towards a diagnosis of hyperintensionality for logics.

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.

In several papers [6–8], Beall argues that, since we can add to non-classical (including paraconsistent) arithmetics rules that restore classicality, we can effectively recover classical arithmetical reasoning in non-classical systems. According to Halbach and Nicolai [18], however, the move to non-classical arithmetic comes at the expense of proof-theoretic strength, undermining Beall’s claims. Then how can paraconsistent arithmetics be said to recover classical strength? It is not sufficient to prove classical arithmetical consequences in a fragment of the language, as done e.g. by Friedman and Meyer [14], since this would yield strictly weaker theorems. In this paper, I provide a so-called classical recapture result for Zach Weber’s paraconsistent arithmetic $\textsf{subDLQ-A}$, based on the logic $\texttt{subDLQ}$ [43]. I reconstruct a notion of coding and recursion for this paraconsisent arithmetic, and show that the theory, if supplemented with additional forms of induction and classical axioms for identity, supports Gentzen’s classical lower bound proof for transfinite induction up to any ordinal less than $\varepsilon _{0}$.