Abstract We present a non-deterministic semantic framework for all modal logics in the modal cube, extending prior works by Kearns and others. Our approach introduces modular and uniform multi-valued non-deterministic matrices (Nmatrices) for each logic, where necessitation is captured by the systematic use of level valuations. The semantics is grounded in an eight-valued system and provides a sound and complete decision procedure for each modal logic, extending and refining earlier semantics as particular cases. Additionally, we propose a novel model-theoretic perspective that links our framework to relational (Kripke-style) semantics, addressing longstanding questions regarding the correspondence between modal axioms and semantic conditions in non-deterministic settings. This yields a philosophically robust and technically modular alternative to traditional possible-world semantics.

Abstract In this paper, we investigate proof-theoretic aspects of the logics of evidence and truth $LET_{J}$ and $LET_{F}$. These logics extend, respectively, Nelson’s logic N4 and the logic of first-degree entailment, also known as Belnap–Dunn four-valued logic, with a classicality operator ${{\circ }}$ that recovers classical logic for formulas in its scope. We will present natural deduction and sequent systems for $LET_{J}$ and $LET_{F}$, together with proofs of normalization and cut-elimination theorems, respectively. As a corollary, we obtain decision procedures, which guarantees bottom-up proof search for both logics.

Abstract The aim of the paper is to analyze the expressive power of the square operator of Łukasiewicz logic: $\ast x=x\odot x$, where $\odot $ is the strong Łukasiewicz conjunction. In particular, we aim at understanding and characterizing those cases in which the square operator is enough to construct a finite MV-chain from a finite totally ordered set endowed with an involutive negation. The first of our main results shows that, indeed, the whole structure of MV-chain can be reconstructed from the involution and the Łukasiewicz square operator if and only if the obtained structure has only trivial subalgebras and, equivalently, if and only if the cardinality of the starting chain is of the form $n+1$ where $n$ belongs to a class of prime numbers that we fully characterize. Secondly, we axiomatize the algebraizable matrix logic whose semantics is given by the variety generated by a finite totally ordered set endowed with an involutive negation and Łukasiewicz square operator. Finally, we propose an alternative way to account for Łukasiewicz square operator on involutive Gödel chains. In this setting, we show that such an operator can be captured by a rather intuitive set of equations.

Abstract The logics of formal inconsistency (LFIs, for short) are paraconsistent logics (i.e. logics containing contradictory but non-trivial theories) having a consistency connective which allows to recover the ex falso quodlibet principle in a controlled way. The aim of this paper is considering a novel semantical approach to first-order LFIs based on Tarskian structures defined over swap structures, a special class of multialgebras. The proposed semantical framework generalizes previous approaches to quantified LFIs presented in the literature. The case of QmbC, the simpler quantified LFI expanding classical logic, will be analyzed in detail. An axiomatic extension of QmbC called $\textbf{QLFI1}_\circ $ is also studied, which is equivalent to the quantified version of da Costa and D’Ottaviano 3-valued logic J3. The semantical structures for this logic turn out to be Tarkian structures based on twist structures. The expansion of QmbC and $\textbf{QLFI1}_\circ $ with a standard equality predicate is also considered.

In this paper we consider the logics |$\mathsf{L}_n^i$| obtained from the |$(n+1)$|-valued Łukasiewicz logics Ł|$_{n+1}$| by taking the order filter generated by |$i/n$| as the set of designated elements. In particular, the conditions of maximality and strong maximality among them are analyzed. We present a very general theorem that provides sufficient conditions for maximality between logics. As a consequence of this theorem, it is shown that |$\mathsf{L}_n^i$| is maximal w.r.t. CPL whenever |$n$| is prime. Concerning strong maximality (i.e. maximality w.r.t. rules instead of only axioms), we provide algebraic arguments in order to show that the logics |$\mathsf{L}_n^i$| are not strongly maximal w.r.t. CPL, even for |$n$| prime. Indeed, in such case, we show that there is just one extension between |$\mathsf{L}_n^i$| and CPL obtained by adding to |$\mathsf{L}_n^i$| a kind of graded explosion rule. Finally, using these results, we show that the logics |$\mathsf{L}_n^i$| with |$n$| prime and |$i/n < 1/2$| are ideal paraconsistent logics.

This article intends to contribute to the debate about the uses of paraconsistent reasoning in the foundations of set theory, by means of using the logics of formal inconsistency and by considering consistent and inconsistent sentences, as well as consistent and inconsistent sets. We establish the basis for new paraconsistent set-theories (such as ZFmbC and ZFCil ) under this perspective and establish their non-triviality, provided that ZF is consistent. By recalling how George Cantor himself, in his efforts towards founding set theory more than a century ago, not only used a form of ‘inconsistent sets’ in his mathematical reasoning, but regarded contradictions as beneficial, we argue that Cantor's handling of inconsistent collections can be related to ours.

Capitalising on the graph-theoretic account of fibring proposed in [31], we show that fibring preserves the finite model property under mild conditions. Illustrations are provided for modal, deontic, paraconsistent and linear logics.

A graph-theoretic account of logics is explored based on the general notion of m-graph (i.e; a graph where each edge can have a finite sequence of nodes as source). Signatures, interpretation structures and deduction systems are seen as multi-graphs (m-graphs). After defining a category freely generated by a m-graph, formulas and expressions in general can be seen as morphisms. Moreover, derivations involving rule instantiation are also morphisms. Soundness and completeness theorems are proved. As a consequence of the generality of the approach our results apply to very different logics encompassing, among others, substructural logics as well as logics with non-deterministic semantics, and subsume all logics endowed with an algebraic semantics.

A graph-theoretic account of fibring of logics is developed, capitalizing on the interleaving characteristics of fibring at the linguistic, semantic and proof levels. Fibring of two signatures is seen as a multi-graph (m-graph) where the nodes and the m-edges include the sorts and the constructors of the signatures at hand. Fibring of two models is a multi-graph (m-graph) where the nodes and the m-edges are the values and the operations in the models, respectively. Fibring of two deductive systems is an m-graph whose nodes are language expressions and the m-edges represent the inference rules of the two original systems. The sobriety of the approach is confirmed by proving that all the fibring notions are universal constructions. This graph-theoretic view is general enough to accommodate very different fibrings of propositional based logics encompassing logics with non-deterministic semantics, logics with an algebraic semantics, logics with partial semantics and substructural logics, among others. Soundness and weak completeness are proved to be preserved under very general conditions. Strong completeness is also shown to be preserved under tighter conditions. In this setting, the collapsing problem appearing in several combinations of logic systems can be avoided.

The concept of fibring is extended to higher-order logics with arbitrary modalities and binding operators. A general completeness theorem is established for such logics including HOL and with the meta-theorem of deduction. As a corollary, completeness is shown to be preserved when fibring such rich logics. This result is extended to weaker logics in the cases where fibring preserves conservativeness of HOL-enrichments. Soundness is shown to be preserved by fibring without any further assumptions.