Agustina 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 aim of this work is to introduce a series of requisites for building sound and complete tableaux systems for inferential logics based on non-deterministic valuation schemes and extend them with rules for metainferences, thus yielding metainferential tableaux system capable of proving metainferences of any level n. We show the construction of our tableaux for the logics of the Strong Kleene family and a wide range of different logical systems, most of which have not been studied previously from a metainferential standpoint. Furthermore, we implement this system into a user-friendly, publicly accessible software, designed both for educational purposes as well as assistance in research endeavors.

In a multiple-premise and multiple-conclusion setting, logicians typically use the metalinguistic expression ‘,’ to gather premises and conclusions. In most logical systems, this comma can be freely interchanged with the object-language conjunction on the premise side and with the objectlanguage disjunction on the conclusion side. However, this is not the case for Weak Kleene Logics (WK logics), which include Paraconsistent Weak Kleene (PWK) and its paracomplete counterpart (Kw3). The aim of this work is to define two new substructural systems called uniform Weak Kleene logics (uWK logics), uPWK and uKw3, which reestablish what we call ‘Uniformity’: the correspondence between the structural comma and the object-language WK operators. We begin by providing some philosophical arguments advocating for it as a desirable logical feature. We then characterize their consequence relations and introduce a novel two-sided sequent calculus free of linguistic restrictions for PWK and for Kw3 and show that, with minor adjustments, we can achieve sound and complete sequent calculi for uPWK and uKw3. Finally, we generalize our results for WK logics and their uniform counterparts to other infectious Tarskian logics that meet specific criteria.

In this paper, we present two ways of modelling every epistemic formal conditional commitment that involves (at most) three key epistemic attitudes: acceptance, rejection and neither acceptance nor rejection. The first one consists of adopting the plurality of every mixed Strong Kleene logic (along with an epistemic reading of the truth-values), and the second one involves the use of a unified system of six-sided inferences, named 6SK, that recovers the validities of each mixed Strong Kleene logic. We also introduce a sequent calculus that is sound and complete with respect to both approaches. We compare both accounts, and finally, we suggest that the plurality of Strong Kleene logic as well as the general framework 6SK are linked to formal epistemic norms via bridge principles.

En este capítulo ofrecemos una introducción sistemática e histórica a la lógica, disciplina que contribuyó en gran medida a la producción del conocimiento en general y a la formación del pensamiento científico en particular. La primera sección contiene la introducción sistemática: primero, presentamos las distintas disciplinas que forman parte de la lógica en el sentido amplio del término; luego, identificamos a la lógica en sentido canónico o estricto como el estudio la validez; por último, explicamos en qué sentido la validez es una propiedad formal. La segunda sección contiene la introducción histórica; nos detenemos, en particular, a estudiar dos hitos: el nacimiento de la lógica como disciplina con Aristóteles, y la intervención de Gottlob Frege, a quien se debe, en gran parte, el surgimiento de la lógica tal como la conocemos. Finalmente, damos un breve comentario del estado actual de la disciplina.