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Abstract: Non-classical approaches to paradox are sometimes criticized on the grounds that they require abandoning core principles of classical logic, thereby threatening the adequacy of ordinary, scientific and mathematical reasoning. A common response, in the case of the paradoxes of truth, appeals to the strategy of classical recapture, according to which classical reasoning can be preserved for non-problematic statements. However, it is often argued that this strategy cannot be extended to the paradoxes of vagueness, since vagueness is ubiquitous across natural language and thus cannot be isolated in the required way. This paper examines and challenges this argument. I claim that the inference from the ubiquity of vagueness to the unavailability of recapture rests on an unacknowledged ambiguity, and that once this ambiguity is exposed, the argument fails. I then develop a positive case for recapture by presenting a framework in which non-classical reasoning is confined to a limited fragment of the language, while substantial portions of classical reasoning are recovered.
Truth, give me strength (with Camillo Fiore and Camila Gallovich)
Abstract: One of the most challenging arguments against non-classical logics is that they give rise to weaker theories than classical logic. This concern is especially pressing in mathematics, where classical reasoning is often seen as indispensable. The thought is that, other things being equal, if a non-classical theory is mathematically weaker than the corresponding classical theory, then the latter should be preferred. In this work we offer two responses to the argument from strength, taking the classical theory Kripke-Feferman (KF) and the non-classical theory Partial Kripke-Feferman (PKF) as our test case. Although both theories aim to axiomatize Kripke’s semantic construction for truth over Peano arithmetic, KF is arithmetically stronger than PKF. Our first response to the argument is that comparing KF to PKF requires adopting not only a conception of truth but also a conception of arithmetic. After reviewing various conceptions, we conclude that PKF fares at least as well as KF under most of them. Our second response is based on an argumentative strategy known as classical recapture. PKF can be strengthened by selectively adding safe instances of excluded middle. Together, these responses suggest that, contrary to what is often assumed, the argument from strength does not undermine non-classical logics.
Limitative Results for Naive Paradoxicality (with Luca Castaldo)
Abstract: The naive notion of paradoxicality is itself inherently prone to paradox. Recently, several authors have argued that if we attempt to supplement theories of truth with an account of what makes a sentence paradoxical, we risk generating so-called revenge paradoxes. These limitative results suggest that the prospects for a naive conception of paradoxicality are rather bleak. However, they also give rise to a natural question: what can we consistently say about naive paradoxicality? This paper aims to take some initial steps toward answering that question. We examine various principles expressing naive paradoxicality in the contexts of classical logic and paracomplete strong Kleene logic. Our results reveal unexpected asymmetries between the two frameworks. In particular, the form that a consistent theory of paradoxicality must take in classical logic differs substantially from its form in a paracomplete setting. Conversely, the paths to inconsistency in classical theories of paradoxicality differ significantly from those in paracomplete theories.
LP+, K3+, FDE+ and their generalised classical collapse (with Camillo Fiore) -- Slides
Abstract: Jc Beall devised so-called classical recapture results for logics LP+, K3+ and FDE+. Each of these results says that an inference is valid in classical logic just in case there is a corresponding inference (with some additional premises and/or conclusions) that is valid in the target non-classical system. However, the philosophical significance of Beall’s results has been put into question. One of the main criticisms is that they presuppose that a non-logical theory is just a set of statements closed under consequence, but many theories based on paracomplete logics (such as K3+ or FDE+) cannot be plausibly formulated in this guise. In this paper, we present generalised versions of Beall’s results. Our results eschew the criticism just mentioned, because they are compatible with understanding a non-logical theory as including not only statements, but also inferences and/or meta-inferences of arbitrary level.
Struggling with Coherence (with Bruno Da Ré and Jonathan Erenfryd)
Abstract: We analyze the debate within bilateralist inferentialism concerning the relative priority of incoherence over coherence. Building on the idea that coherence and incoherence should be treated as equally primitive, we challenge Rea Golan’s recent arguments for privileging incoherence. We argue, first, that Golan’s objections rely on assumptions that are questionable from an inferentialist perspective. Second, we introduce a new hybrid calculus that axiomatizes both coherence and incoherence while avoiding the limitations of previous systems. We conclude, contra Golan, that coherence is no less fundamental than incoherence, and that bilateralist approaches should treat them on a par.
Classical Reflection (with Carlo Nicolai)
Abstract: Reflection principles formalize soundness assertions. In most cases, adding reflection principles to a theory results in a genuine increase in strength. This observation has been put to a variety of philosophical uses other than its original application to Gödelian incompleteness. Recently, Nicolai, Horsten, and Fischer have argued that reflection may be relevant to the comparison of classical and non-classical theories of truth. On their view, reflection provides a way of understanding how a non-classical theorist may legitimately extend the resources of an initially weaker theory, thereby allowing it, in certain contexts, to match the strength of its classical counterpart. Unfortunately, standard reflection principles cannot generally be formulated in conditional form within non-classical theories, since they are unsound in that setting. We argue, however, that unrestricted reflection may not be necessary for these purposes. More modest forms of reflection may suffice. To illustrate this possibility, we focus on (a paracomplete version of) the axiomatic theory of truth $\pkf$ and show that a suitably defined autonomous progression of restricted reflection principles, combined with a legitimate jump operation on ordinals, suffices to recover the strength of comparable classical systems.
Classical Recapture (book in progress)