Recently, Da Ré, Szmuc, Chemla and Égré (2024) showed that all logics based on Boolean Normal monotonic three-valued schemes coincide with classical logic when defined using a strict-tolerant standard (st). Conversely, they proved that under a tolerant-strict standard (ts), the resulting logics are all empty. Building on these results, we show that classical logic can be obtained by closing under transitivity the union of two logics defined over (potentially different) Boolean normal monotonic schemes, using a strict-strict standard (ss) for one and a tolerant-tolerant standard (tt) for the other, with the first of these logics being paracomplete and the other being paraconsistent. We then identify a notion dual to transitivity that allows us to characterize the logic TS as the dual transitive closure of the intersection of any two logics defined over (potentially different) Boolean normal monotonic schemes, using an ss standard for one and a tt standard for the other. Finally, we expand on the abstract relations between the transitive closure and dual transitive closure operations, showing that they give rise to lattice operations that precisely capture how the logics discussed relate to one another.

This article discusses the definition of paraconsistency understood as the property of a consequence relation that does not trivialize inconsistent theories. Some logicians have argued that standard paraconsistency, the requirement of a non-explosive consequence relation, is insufficient for that purpose. In this article, we have a twofold goal. First, we offer an exposition of some attempts to strengthen standard paraconsistency in the literature. After discussing the shortcomings of those attempts, we examine the concepts of triviality in relation to which those concepts of paraconsistency are defined. Then, as our second goal, we propose an alternative definition of paraconsistency that aims to avoid the trivialization of inconsistent theories in a stricter sense

Radzki has recently claimed the incompleteness of the axioms given by Słupecki for the functionally complete Ł3: some of its tautologies are not provable. In this paper, we provide a new axiom system for this logic (choosing a variant with two propositional constants and the Łukasiewicz implication as primitive symbols) and prove a Completeness Theorem.

This article aims to stimulate interdisciplinary exchange between logicians and cognitive scientists. In particular, I claim that conceptual analogues of hyperintensionality and intensionality can be found when we apply statistical tools to analyse sensorimotor processes in embodied cognition. When considering the functional correlation between the internal state X of an agent, and the external state Y of its environment, I propose that the precise functional form of the correlation has a hyperintensional flavour, while the abstract information carried by the correlation has a purely intensional flavour. Recent work by Kolchinsky and Wolpert attempts to bring ‘semantics’ to physical correlations by analysing the effects of those correlations on task performance. I argue that this ‘semantic information’ framework currently provides a model for intensional, but not hyperintensional, aspects of belief in a hypothetical mental arithmetic scenario. In general, I suggest that cognitive scientists should be more familiar with the intensional/hyperintensional distinction (for instance, I argue that the ‘Bayesian brain’ approach cannot account for hyperintensional aspects of cognition), and that logicians should be aware of analogues of hyperintension in embodied cognition (for instance, I claim that hyperintensional-like phenomena occur as much in bacteria as in humans).

Despite the logical possibility and alleged existence of inconsistent empirical theories, the problem of their testability remains largely unexplored. I develop a testability criterion which makes some of these theories testable, including some observationally inconsistent ones. However, they are not rejectable qua inconsistent by this criterion. These results, while opening the domain of scientific theories to inconsistent ones, challenge the prospects of dialetheism in the philosophy of science.

At the Copernicus University of Toruń, which was founded in 1945 under the most difficult conditions, there were two logic centers from the very beginning, each headed by a prominent representative of the Lvov-Warsaw School. Jaśkowski’s work on formal logic proved to be tradition-defining for the emergence of today’s international center of logical research in Toruń. The article examines the first stages of this process.

This paper introduces nested sequent calculi for modal logics that include non-standard modalities as primitive operators in their languages. By non-standard modalities, we mean non-contingency, contingency, essence, accident, impossibility, and unnecessity. We consider basic normal modal logic K and its serial, reflexive, transitive, and symmetric extensions. Our research begins by using Poggiolesi’s nested sequent calculi as a foundation. These calculi are specifically designed for logics that are formulated in a language that includes the necessity operator. Next, we proceed to modify their rules to accommodate non-standard modalities. We then establish the soundness and completeness of the resulting calculi. As a consequence, we get that the nested sequent calculus for K is cut-free. Subsequently, we provide a constructive cut admissibility proof for K. Finally, we discuss the issues pertaining to the cut admissibility for the extensions of K and their relationships with the so-called special structural rules as well as the potential for considering other forms of non-standard modalities.

Particular reasoning enables the deductive proof of existential properties, such as the satisfiability/consistency of a set of formulas. In this work, we consider particular reasoning in the context of a theory of a given logic. The logic is presented by a semantic constraint specification. From this specification, we induce a particular calculus for the logic at hand. In this calculus we define what is a particular derivation in the context of a theory and show how to extract a model of the theory that satisfies the assertions within the derivation. We demonstrate that the induced particular calculus is both sound and complete with regard to the intended semantics. Our results are applicable to logics with a strong finite model property, including classical, intuitionistic, certain modal logics, and Nelson’s N4 logic, among others.

This paper investigates how the use of different rules for making inferences affects our understanding of what certain Extensible Markup Language (XML) documents do not represent. The aim is to show that we can infer different, contrasting things from the same XML documents, thereby weakening the communication that XML is supposed to support. There are three main reasons why the paper focuses on XML. First, XML, as a metalanguage, has no inherent rules for making inferences, but it also has no constraints on the technologies, systems, or theories that support or define the rules that can be used in conjunction with it. Second, XML is still widely used, and there are many other markup languages based on XML. This means that the critical analysis of these pages is, in principle, extendable to contexts where XML is involved and/or the rules for making inferences are not inherently supported. Third, since XML is explicitly intended to support communication between people, between software applications, and between people and software applications, this analysis may also shed new light on some of the theoretical assumptions behind such communication.

This paper introduces a metainferential version of intuitionistic logic. I work on the framework proposed by some logicians of Buenos Aires, who defend that a logic should be defined in terms of inferences and metainferences of growing complexity. Three logical systems are presented and proved to be adequate from an intuitionistic point of view