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Shameless Classicality
Abstract: A powerful objection against non-classical logics is that they typically employ classical logic in the metalanguage. The problem, roughly put, is that someone who claims to be endorsing a non-classical theory cannot in the same breath accept meta-theoretic results that are available only if one is allowed to reason classically. My first goal in this paper is to analyze three responses to the objection that have been offered in the literature: instrumentalism, the view that one ought to take an instrumentalist attitude towards the metalanguage; full-blown non-classicality, the view that one must be in a position to develop one's meta-theory by only availing oneself of the patterns of reasoning that one's logic affords; and classical recapture, the view that one can use instances of classical patterns of reasoning provided the relevant region of discourse is safe. I offer some reasons for the claim that the third response is superior to the first two. My second goal is to discuss how various prominent non-classical theories can best implement the recapture strategy. As a test case I will consider theories of truth and I will tentatively suggest that a paracomplete account fares better than other accounts, such as those based on a paraconsistent logic or on some substructural logic.
Abstract: The paper addresses the question whether there's any hope of successfully deploying the so-called recapture strategy in the case of the paradoxes of vagueness. The prevailing impression is that the pervasiveness of vague vocabulary across language makes the deployment of this strategy impossible. I argue that this impression is misguided. In fact, my view is that the recapture strategy is available in the case of vagueness-related paradoxes in roughly the same way that it is available in the case of the semantic paradoxes. Once properly understood, the idea that vague expressions are ubiquitous has no bearing on recapture.
The Argument from Proof-Theoretic Strength (with Camillo Fiore and Camila Gallovich)
Abstract: Arguably, the most echoed argument against non-classical theories of truth is that they are deductively weaker than the corresponding classical theories; a typical example of this is that the classical theory KF proves more arithmetical sentences than the non-classical theory PKF, even though both can be seen as adequate axiomatizations of the Kripkean conception of truth. In this paper, we provide two different answers to this argument, focusing on the pair of theories KF and PKF as our test-case. Our first answer is that, despite appearances, the difference in deductive power between KF and PKF is not detrimental to the non-classical theory: PKF proves all the sentences that are theorems of PA, so if the conclusion of the argument is that PKF is an arithmetically weak theory, it follows that PA is also an arithmetically weak theory, which seems at least implausible. The second response is based on an argumentative strategy usually known as classical recapture. There is a natural way to improve the proof-theoretical power of PKF by means of additional axioms. In particular, we shall extend PKF with a subset of the grounded instances of the principle of excluded middle and we will show that the resulting theory has the same proof-theoretical power as KF.
Limitative Results for Naive Paradoxicality (with Luca Castaldo)
Abstract: In this paper, we analyze various principles expressing naive paradoxicality in the context of strong Kleene logic and classical logic. We show that the prospects for a theory of naive paradoxicality within classical logic are rather bleak, as most combinations of naive principles lead to a contradiction. However, we also claim that there’s an asymmetry with strong Kleene logic, where several combinations of naive principles remain consistent.
LP+, K3+, FDE+ and their generalised classical collapse (with Camillo Fiore)
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.
Classical Recapture (book in progress)
Abstract: Comming soon.