XIII Workshop on Philosophical Logic

Beatrice Buonaguidi: “What’s the HYPE about hyperintensional logics? Fine-grained criteria for hyperintensionality”

Traditionally, hyperintensional contexts and operators are defined in a negative way, namely, as contexts and operators which do not license unrestricted intersubstitutivity for necessary equivalents, or for logical equivalents (Cresswell 1975, Nolan 2014). I argue that this notion of hyperintensionality is too vague, especially if it needs to be used to judge whether a logical consequence relation counts as hyperintensional, i.e. displays hyperintensional contexts and operators.

Starting by challenging a criterion for hyperintensionality for a logical consequence relation suggested by Odintsov and Wansing (2021), I will present several different criteria for a logic to count as hyperintensional. This will help me draw a diagnosis of hyperintensionality as, simply, some degree of asymmetry between truth preservation and falsity preservation. Furthermore, I will argue that hyperintensionality can show up at different levels for different consequence relations, licensing a spectrum of hyperintensional behaviour: as a case study of this, I will consider N-logics and HYPE (Leitgeb 2019).

Pablo Dopico: “Axiomatic theories of supervaluationist truth: completing the picture”

As is well-known, in his seminal ‘Outline of a theory of truth’, Kripke suggested to run his fixed-point construction for theories of truth over supervaluationist schemes. Three such schemes stood out: (1) the scheme vb, which considers extensions of the truth predicate consistent with the original one; (2) the scheme vc, which considers consistent extensions more generally, and; (3) the scheme mc, which only considers maximally consistent extensions. As is also known, Andrea Cantini proposed an axiomatization of the fixed-point theory constructed over the scheme vc, which he called VF, and proved that the theory was sound with respect to the fixed-point models generated by that scheme. Moreover, he showed that VF was a remarkably strong theory, matching the strength of the impredicative theory ID₁.

In this paper, we complete the picture of axiomatic theories of supervaluationist truth by introducing two new theories that correspond—and are sound with respect—to the schemes vb and mc. In the case of the former scheme, we advance a theory that we call VF⁻, and establish its proof-theoretic strength, which equals that of VF. The most substantial part of the paper, however, is dedicated to the theory which axiomatizes the fixed-point semantic theory over mc, which we call VFM. For the lower-bound, we show that it defines Tarskian ramified truth predicates up to ε₀ (RA_< ε0). For the upper-bound, we provide a cut elimination argument formalized within the theory ID₁^*, which is known to be proof-theoretically equivalent to RA_< ε0.

Finally, we also introduce the schematic reflective closure of the theory VFM, as defined by Feferman. We establish its consistency, and carry out the proof-theoretic analysis for this theory, which confirms that this schematic reflective closure is as proof-theoretically strong as the theory RA_< Γ0.

This is joint work with Daichi Hayashi.

Andrea Iacona: “The Stoic Thesis and its Formalization”

In this talk I develop an idea that goes back to the Stoics, namely, that an argument is valid when the conditional formed by the conjunction of its premises as antecedent and its conclusion as consequent is true. As I will argue, once some basic features of our informal understanding of validity are properly spelled out, and a suitable account of conditionals is adopted, the equivalence between valid arguments and true conditionals makes perfect sense. I will show how this equivalence can be formalized in a first-order language that contains a naive truth predicate and a suitable conditional connective. The validity predicate that turns out to be definable in this language significantly increases our expressive resources and provides a coherent formal treatment of paradoxical arguments.

Carlo Nicolai: “CLassical closures”

I present some observations on the theory of classical determinate truth recently introduced by Fujimoto and Halbach. The observations aim to show that there is a sense in which the primitive determinate predicate of CD+ could be dispensed with without compromising the logical strength and motivation of the theory. In particular, there’s a precise sense in which the axioms of CD+ are a notational variant of the classical closure of Kripke-Feferman truth.

This is joint work with Luca Castaldo.

Francesco Paoli: “Logical Metainferentialism”

Logical inferentialism is the view that the meaning of logical constants is implicitly defined by the operational rules that govern their behaviour in proofs – in particular, sequent calculus proofs, according to an increasingly dominant tendency. A tenable articulation of this view presupposes a clarification of certain crucial aspects, concerning e.g. harmony criteria for rules or what counts as a normal proof. Sequent calculus inferentialists generally do so in terms of proofs from axioms, not of derivations from assumptions. Our version of logical metainferentialism (which is different, in some respects, from the Buenos Aires Plan) calls into question this dogma, against the backdrop of the idea that meaning determination is relative to sequent-to-sequent derivability relations of Gentzen systems. We advance a suggestion towards a metainferentially appropriate reformulation of harmony, and explore its potential by focussing on a case study, the calculi for FDE and its extensions.