On a Generalization of all Strong Kleene Generalizations of Classical Logic (with P. Cobreros, I. Grábalos, J. S. Toranzo Calderón, J. Viñeta), Studia Logica (2025) [Open Access]
Uniform Weak Kleene Logics (with A. Borzi), Australasian Journal of Logic (forthcoming) [preprint]
Non-deterministic Semantics for logics of Analytic Implication (with D. Szmuc), Erkenntnis, (forthcoming) [preprint]
Non-reflexive Logics (with J. S. Toranzo Calderón), in: Teijeiro P. and Barrio E., eds., Metainferences in Substructural Logics, Springer (forthcoming)
Substructural Routes to Variable Inclusion (with A. Borzi)
This paper examines a range of logical systems within the family of variable inclusion logics—also known as containment logics. We focus on those logics that restrict classically valid inferences to ones meeting specific variable inclusion constraints, hence called variable inclusion companions of classical logic. These constraints can be seen as enforcing varying degrees of relevance between premises and conclusions, placing these systems within the broader tradition of relevance logics. We review established companions of Classical Logic, including Weak Kleene logics (Bochvar 1938, Halldén 1949), pure variable inclusion logics (Paoli et al. 2021), uniform Weak Kleene logics (Borzi and Zirattu forth.), and their intersection—pure uniform logics. We then introduce two new systems named analytic and synthetic companions of Classical Logic. We characterize their consequence relations and interpret them, as their names suggest, as validating only analytic and synthetic inferences. Lastly, we argue that these systems more effectively address the irrelevance cases that motivated earlier proposals, by correctly tracing them to the Monotonicity of the consequence relation.
The Algebra of Analytic Containment (with F. Paoli and D. Szmuc)
We explore certain algebraic structures that naturally emerge within the framework of logics of synonymy, analytic containment, and hyperintensionality. In particular, we argue that Angell’s logic AC, one of the earliest and most successful attempts to analyse the properties of logical constants with a topic-transformative character, can be better understood through a direct algebraic study of De Morgan bisemilattices. Inter alia, we show that a certain 9-element algebra introduced by Ferguson generates De Morgan bisemilattices as a quasivariety, making it the most adequate semantics for AC, as opposed to other 7-element and 16-element algebras considered in the literature.
The Dark Side of a Structural Logic (with A. Borzi)
Some authors (e.g. Cobreros et al. 2020, Scambler 2020, Pailos and Barrio 2022) discussed and developed a two-sided conception of what a logic is, distinguishing between its positive and its negative sides, the former being the set of its valid inferences, the latter being the set of its valid anti-inferences (corresponding to unsatisfiable inferences). In this work we consider a more general account of what could be the negative part of a logic, which includes the former as a particular instance. Building on this idea, we dualize the general construction developed by Szmuc (2023), who showed that for any Tarskian logic, one can define a non-transitive counterpart of it sharing exactly the same positive part, though differing from L as it locally invalidates Cut. Here, we provide a dual result: for every Tarskian logic, we can construct a logic that shares its negative part, but differs from it as reflexivity fails. Our construction makes use of the so-called infectious extension of an algebra and of Malinowski’s q-matrices (Malinowski 1990), namely a generalization of the usual notion of logical matrices where premises and conclusions are given (possibly) different sets of designated values, the latter being a subset of the former.
Compatibility and Implication from a Non-Classical Perspective (with A. Iacona)
This paper draws attention to some key logical relations between two distinct and independently intelligible notions, compatibility and implication. We identify a set of inference rules that express properties of compatibility, and show how some crucial principles concerning conditionals can be derived from these rules on the assumption that implication is definable in terms of compatibility. As will emerge, and this is the main point of the paper, the logical relations considered are to a large extent neutral as to the background logic, for they hold in a significantly wide range of non-classical systems.
De Finetti goes Viral (with M. Rubin)
In this work, our objective is to investigate the logic and the probability logic of the De Finetti's conditional when matched with the Weak Kleene operators. Moreover, we argue that this pairing offers a more natural alternative than others extensively explored in the literature, namely the Strong Kleene and Quasi-connectives.
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