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This paper addresses the integration of \emph{epistemic entrenchment} into paraconsistent belief revision systems based on Logics of Formal Inconsistency (LFIs). While systems like AGMp and AGM$\circ$ adapt AGM principles to paraconsistency, they lack mechanisms to rank beliefs, primarily due to the absence of properties such as the \emph{replacement property} in the underlying logics. We introduce two novel logics, \textbf{Cbr} and \textbf{RCBr}, with the latter extending the former to fully address these limitations given that it is self-extensional. Using \textbf{RCBr}, we define contraction operations via \emph{epistemic entrenchment}, adhering to key rationality principles. Our framework leverages non-deterministic matrix semantics (Nmatrices), providing a robust foundation for paraconsistent reasoning. These contributions advance the theory of paraconsistent belief revision and pave the way for applications in domains such as multi-agent systems and inconsistent knowledge bases.

In this paper, we investigate proof-theoretic aspects of the logics of evidence and truth LETJ and LETF. These logics extend, respectively, Nelson's logic N and the logic of first-degree entailment FDE, also known as Belnap-Dunn four-valued logic, with a classicality operator that recovers classical logic for formulas in its scope. We will present natural deduction and sequent systems for LETJ and LETF, together with proofs of normalization and cut-elimination theorems, respectively. As a corollary, we obtain decidability for both logics.

The aim of this paper is to combine several Ivlev-like modal systems characterized by 4-valued non-deterministic matrices (Nmatrices) with IDM4, a 4-valued expansion of Belnap-Dunn's logic FDE with an implication introduced by Pynko in 1999. In order to do this, we introduce a new methodology for combining logics which are characterized by means of swap structures, based on what we call superposition of snapshots. In particular, the combination of IDM4 with Tm, the 4-valued Ivlev's version of KT, will be analyzed with more details. From the semantical perspective, the idea is to combine the 4-valued swap structures (Nmatrices) for Tm (and several of its extensions) with the 4-valued twist structure (logical matrix) for IDM4. This superposition produces a universe of 6 snapshots, with 3 of them being designated. The multioperators over the new universe are defined by combining the specifications of the given swap and twist structures. This gives rise to 6 different paradefinite Ivlev-like modal logics, each one of them characterized by a 6-valued Nmatrix, and conservatively extending the original modal logic and IDM4. This important feature allows to consider the proposed construction as a genuine technique for combining logics. In addition, it is possible to define in the combined logics a classicality operator in the sense of logics of evidence and truth (LETs). A sound and complete Hilbert-style axiomatization is also presented for the 6 combined systems, as well as a Prolog program which implements the swap structures semantics for the 6 systems, producing a decision procedure for satisfiability, refutability and validity of formulas in these logics., Comment: Definition of hyperintensionality on p.2 was corrected

We present in this paper an adaptation of the process of combination of logics known as fibring introduced by D. Gabbay. We are focused on the combination of two logics defined by matrix semantics, and based on pairs of functions that relate the logics to be combined. A number of technical results are proved. Among them, we demonstrate that the fibring of two matrix logics is also a matrix one. In addition, we prove that fibring is a (weak) conservative extension of the original logics, and we give conditions for such extension to be strong. We also study the case of fibring identifying two connectives as being the same. Several examples referred to fibring of some well-known matrix logics are shown along this paper., Comment: 28 pages

Belnap-Dunn's relevance logic, BD, was designed seeking a suitable logical device for dealing with multiple information sources which sometimes may provide inconsistent and/or incomplete pieces of information. BD is a four-valued logic which is both paraconsistent and paracomplete. On the other hand, De and Omori while investigating what classical negation amounts to in a paracomplete and paraconsistent four-valued setting, proposed the expansion BD2 of the four valued Belnap-Dunn logic by a classical negation. In this paper, we reintroduce the logic BD2 by means of a primitive weak consistency operator $\copyright$. This approach allows us to state in a direct way that this is not only a Logic of Formal Inconsistency (LFI) but also a Logic of Formal Underterminedness (LFU). After presenting a natural Hilbert-style characterization of BD2 obtained by means of twist-structures semantics, we propose a first-order version of BD2 called QBD2, with semantics based on an appropriate notion of partial structures. We show that in QBD2, $\exists$ and $\forall$ are interdefinable in terms of the paracomplete and paraconsistent negation, and not by means of the the classical negation. Finally, a Hilbert-style calculus for QBD2 is presented, proving the corresponding and soundness and completeness theorems., Comment: 29 pages

The main aim of this paper is to introduce the logics of evidence and truth LETK+ and LETF+ together with a sound, complete, and decidable six-valued deterministic semantics for them. These logics extend the logics LETK and LETF- with rules of propagation of classicality, which are inferences that express how the classicality operator o is transmitted from less complex to more complex sentences, and vice-versa. The six-valued semantics here proposed extends the 4 values of Belnap-Dunn logic with 2 more values that intend to represent (positive and negative) reliable information. A six-valued non-deterministic semantics for LETK is obtained by means of Nmatrices based on swap structures, and the six-valued semantics for LETK+ is then obtained by imposing restrictions on the semantics of LETK. These restrictions correspond exactly to the rules of propagation of classicality that extend LETK. The logic LETF+ is obtained as the implication-free fragment of LETK+. We also show that the 6 values of LETK+ and LETF+ define a lattice structure that extends the lattice L4 defined by the Belnap-Dunn four-valued logic with the 2 additional values mentioned above, intuitively interpreted as positive and negative reliable information. Finally, we also show that LETK+ is Blok-Pigozzi algebraizable and that its implication-free fragment LETF+ coincides with the degree-preserving logic of the involutive Stone algebras., Comment: 38 pages

It is well known that there is a correspondence between sets and complete, atomic Boolean algebras (CABA's) taking a set to its power-set and, reciprocally, a complete, atomic Boolean algebra to its set of atomic elements. Of course, such a correspondence induces an equivalence between the opposite category of $\textbf{Set}$ and the category of CABA's. We extend this result by taking multialgebras over a signature $\Sigma$, specifically those whose non-deterministic operations cannot return the empty-set, to CABA's with their zero element removed and a structure of $\Sigma$-algebra compatible with its order; reciprocally, one of these "almost Boolean" $\Sigma$-algebras is taken to its set of atomic elements equipped with a structure of multialgebra over $\Sigma$. This leads to an equivalence between the category of $\Sigma$-multialgebras and a category of ordered $\Sigma$-algebras. The intuition, here, is that if one wishes to do so, non-determinism may be replaced by a sufficiently rich ordering of the underlying structures.

Ivlev's pioneering work started in the 1970's showed a new and promissory way in the study of modal logic from the perspective of many-valued logics. Continuing our previous work on Ivlev-like non-normal modal logics with non-deterministic semantics, we present in this paper tableau systems for Tm, S4m and S5m, the non-normal versions of T, S4 and S5, respectively, as well as for their corresponding first-order extensions Tm*, S4m* and S5m*., Comment: 26 pages

In a previous article we introduced the concept of restricted Nmatrices (in short, RNmatrices), which generalize Nmatrices in the following sense: a RNmatrix is a Nmatrix together with a {\em subset} of valuations over it, from which the consequence relation is defined. Within this semantical framework we have characterized each paraconsistent logic $C_{n}$ in the hierarchy of da Costa by means of a $(n+2)$-valued RNmatrix, which also provides a relatively simple decision procedure for each calculus (recalling that $C_1$ cannot be characterized by a single finite Nmatrix). In this paper we extend such RNmatrices for $C_{n}$ by means of what we call {\em restricted swap-structures} over arbitrary Boolean algebras, obtaining so a class of non-deterministic semantical structures which characterizes da Costa's systems. We give a brief algebraic and combinatorial description of the elements of the underlying RNmatrices. Finally, by presenting a notion of category of RNmatrices, we show that the category of RNmatrices for $C_{n}$ is in fact isomorphic to the category of non-trivial Boolean algebras., Comment: 24 pages. arXiv admin note: text overlap with arXiv:2011.10151

The aim of the paper is to analyze the expressive power of the square operator of Lukasiewicz logic: $\ast x=x\odot x$, where $\odot$ is the strong Lukasiewicz 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 Lukasiewicz square 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 Lukasiewicz's square operator. Finally, we propose an alternative way to account for Lukasiewicz square operator on involutive G\"odel chains. In this setting, we show that such an operator can be captured by a rather intuitive set of equations.

It is well known from universal algebra that, for every signature $\Sigma$, there exist algebras over $\Sigma$ which are absolutely free, meaning that they do not satisfy any identities or, alternatively, satisfy the universal mapping property for the class of $\Sigma$-algebras. Furthermore, once we fix a cardinality of the generating set, they are, up to isomorphisms, unique, and equal to algebras of terms (or propositional formulas, in the context of logic). Equivalently, the forgetful functor, from the category of $\Sigma$-algebras to $\textbf{Set}$, has a left adjoint. This result does not extend to multialgebras. Not only multialgebras satisfying the universal mapping property do not exist, but the forgetful functor $\mathcal{U}$, from the category of $\Sigma$-multialgebras to $\textbf{Set}$, does not have a left adjoint. In this paper we generalize, in a natural way, algebras of terms to multialgebras of terms, whose family of submultialgebras enjoys many properties of the former. One example is that, to every pair consisting of a function, from a multialgebra of terms to another multialgebra, and a collection of choices (which selects how a homomorphism approaches indeterminacies), it corresponds a unique homomorphism, which ressembles the universal mapping property. Another example is that the multialgebras of terms are generated by a set that may be viewed as a strong basis, which we call the ground of the multialgebra. Submultialgebras of multialgebras of terms are what we call weakly free multialgebras. Finally, with these definitions at hand, we offer a simple proof that multialgebras with the universal mapping property for the class of all multialgebras do not exist and that $\mathcal{U}$ does not have a left adjoint., Comment: 17 pages. In this new version, minor typographical and grammatical corrections were performed, as well as some simplifications regarding the use of nomenclature borrowed from universal algebra

In the first part of this paper we analyzed finite non-deterministic matrix semantics for propositional non-normal modal logics as an alternative to the standard Kripke's possible world semantics. This kind of modal systems characterized by finite non-deterministic matrices was originally proposed by Ju. Ivlev in the 70's. The aim of this second paper is to introduce a formal non-deterministic semantical framework for the quantified versions of some Ivlev-like non-normal modal logics. It will be shown that several well-known controversial issues of quantified modal logics, relative to the identity predicate, Barcan's formulas, and de dicto and de re modalities, can be tackled from a new angle within the present framework.

Despite being fairly powerful, finite non-deterministic matrices are unable to characterize some logics of formal inconsistency, such as those found between $\textbf{mbCcl}$ and $\textbf{Cila}$. In order to overcome this limitation, we propose here restricted non-deterministic matrices (in short, RNmatrices), which are non-deterministic algebras together with a subset of the set of valuations. This allows us to characterize not only $\textbf{mbCcl}$ and $\textbf{Cila}$ (which is equivalent, up to language, to da Costa's logic $C_1$) but the whole hierarchy of da Costa's calculi $C_n$. This produces a novel decision procedure for these logics. Moreover, we show that the RNmatrix semantics proposed here induces naturally a labelled tableau system for each $C_n$, which constitutes another decision procedure for these logics. This new semantics allows us to conceive da Costa's hierarchy of $C$-systems as a family of (non deterministically) $(n+2)$-valued logics, where $n$ is the number of "inconsistently true" truth-values and 2 is the number of "classical" or "consistent" truth-values, for every $C_n$., Comment: 34 pages. This new version of the paper removes 2 sections from the old one, one providing RNmatrices for $C_2$ and the other constructing RNmatrices over an arbitrary Boolean algebra as swap structures, and includes 2 new sections, detailing the use of row-branching truth-tables for RNmatrices and tableau systems based on RNmatrices for $C_n$

It is customary to expect from a logical system that it can be algebraizable, in the sense that an algebraic companion of the deductive machinery can always be found. Since the inception of da Costa's paraconsistent calculi $C_n$, algebraic equivalents for such systems have been sought. It is known, however, that these systems are not self-extensional (i.e., they do not satisfy the replacement property). More than this, they are not algebraizable in the sense of Blok-Pigozzi. The same negative results hold for several systems of the hierarchy of paraconsistent logics known as Logics of Formal Inconsistency (LFIs). Because of this, several systems belonging to this class of logics are only characterizable by semantics of a non-deterministic nature. This paper offers a solution for two open problems in the domain of paraconsistency, in particular connected to algebraization of LFIs, by extending with rules several LFIs weaker than $C_1$ , thus obtaining the replacement property (that is, such LFIs turn out to be self-extensional). Moreover, these logics become algebraizable in the standard Lindenbaum-Tarski's sense by a suitable variety of Boolean algebras extended with additional operations. The weakest LFI satisfying replacement presented here is called RmbC, which is obtained from the basic LFI called mbC. Some axiomatic extensions of RmbC are also studied. In addition, a neighborhood semantics is defined for such systems. It is shown that RmbC can be defined within the minimal bimodal non-normal logic E+E defined by the fusion of the non-normal modal logic E with itself. Finally, the framework is extended to first-order languages. RQmbC, the quantified extension of RmbC, is shown to be sound and complete w.r.t. the proposed algebraic semantics., Comment: Revised and definitive version

The logics of formal inconsistency (LFIs, for short) are paraconsistent logics (that is, 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 aproaches 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 QLFI1o 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 QLFI1o with a standard equality predicate is also considered.

Boolean-valued models of set theory were independently introduced by Scott, Solovay and Vop\v{e}nka in 1965, offering a natural and rich alternative for describing forcing. The original method was adapted by Takeuti, Titani, Kozawa and Ozawa to lattice-valued models of set theory. After this, L\"{o}we and Tarafder proposed a class of algebras based on a certain kind of implication which satisfy several axioms of ZF. From this class, they found a specific 3-valued model called PS3 which satisfies all the axioms of ZF, and can be expanded with a paraconsistent negation *, thus obtaining a paraconsistent model of ZF. The logic (PS3 ,*) coincides (up to language) with da Costa and D'Ottaviano logic J3, a 3-valued paraconsistent logic that have been proposed independently in the literature by several authors and with different motivations such as CluNs, LFI1 and MPT. We propose in this paper a family of algebraic models of ZFC based on LPT0, another linguistic variant of J3 introduced by us in 2016. The semantics of LPT0, as well as of its first-order version QLPT0, is given by twist structures defined over Boolean agebras. From this, it is possible to adapt the standard Boolean-valued models of (classical) ZFC to twist-valued models of an expansion of ZFC by adding a paraconsistent negation. We argue that the implication operator of LPT0 is more suitable for a paraconsistent set theory than the implication of PS3, since it allows for genuinely inconsistent sets w such that [(w = w)] = 1/2 . This implication is not a 'reasonable implication' as defined by L\"{o}we and Tarafder. This suggests that 'reasonable implication algebras' are just one way to define a paraconsistent set theory. Our twist-valued models are adapted to provide a class of twist-valued models for (PS3,*), thus generalizing L\"{o}we and Tarafder result. It is shown that they are in fact models of ZFC (not only of ZF)., Comment: Abstract improved

In 1988, Ivlev proposed four-valued non-deterministic semantics for modal logics in which the alethic T axiom holds good. Unfortunately, no completeness was proved. In previous work, we proved completeness for some Ivlev systems and extended his hierarchy, proposing weaker six-valued systems in which the T axiom was replaced by the deontic D axiom. Here, we eliminate both axioms, proposing even weaker systems with eight values. Besides, we prove completeness for those new systems. It is natural to ask if a characterization by finite ordinary (deterministic) logical matrices would be possible for all those systems. We will show that finite deterministic matrices do not characterize any of them.

In this paper the 3-valued paraconsistent first-order logic QCiore is studied from the point of view of Model Theory. The semantics for QCiore is given by partial structures, which are first-order structures in which each n-ary predicate R is interpreted as a triple of paiwise disjoint sets of n-uples representing, respectively, the set of tuples which actually belong to R, the set of tuples which actually do not belong to R, and the set of tuples whose status is dubious or contradictory. Partial structures were proposed in 1986 by I. Mikenberg, N. da Costa and R. Chuaqui for the theory of quasi-truth (or pragmatic truth). In 2014, partial structures were studied by M. Coniglio and L. Silvestrini for a 3-valued paraconsistent first-order logic called LPT1, whose 3-valued propositional fragment is equivalent to da Costa-D'Otaviano's logic J3. This approach is adapted in this paper to QCiore, and some important results of classical Model Theory such as Robinson's joint consistency theorem, amalgamation and interpolation are obtained. Although we focus on QCiore, this framework can be adapted to other 3-valued first-order logics., Comment: The Introduction was improved; some typos were fixed, and some notation was improved

In this paper we consider the logics $L_n^i$ obtained from the (n+1)-valued Lukasiewicz logics $L_{n+1}$ 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 analysed. We present a very general theorem which provides sufficient conditions for maximality between logics. As a consequence of this theorem it is shown that $L_n^i$ is maximal w.r.t. CPL whenever n is prime. Concerning strong maximality between the logics $L_n^i$ (that is, maximality w.r.t. rules instead of axioms), we provide algebraic arguments in order to show that the logics $L_n^i$ are not strongly maximal w.r.t. CPL, even for n prime. Indeed, in such case, we show there is just one extension between $L_n^i$ and CPL obtained by adding to $L_n^i$ a kind of graded explosion rule. Finally, using these results, we show that the logics $L_n^i$ with n prime and i/n < 1/2 are ideal paraconsistent logics.

Multialgebras (or hyperalgebras, or non-deterministic algebras) have been very much studied in Mathematics and in Computer Science. In 2016 Carnielli and Coniglio introduced a class of multialgebras called swap structures, as a semantic framework for dealing with several logics of formal inconsistency (or LFIs) which cannot be semantically characterized by a single finite matrix. In particular, these LFIs are not algebraizable by the standard tools of abstract algebraic logic. In this paper, the first steps towards a theory of non-deterministic algebraization of logics by swap structures are given. Specifically, a formal study of swap structures for LFIs is developed, by adapting concepts of universal algebra to multialgebras in a suitable way. A decomposition theorem similar to Birkhoff's representation theorem is obtained for each class of swap structures. Moreover, when applied to the 3-valued algebraizable logic J3 the usual class of algebraic models is recovered, and the swap structures semantics became twist-structures semantics (as introduced by Fidel-Vakarelov). This fact, together with the existence of a functor from the category of Boolean algebras to the category of swap structures for each LFI, which is closely connected with Kalman's functor, suggests that swap structures can be considered as non-deterministic twist structures, opening so interesting possibilities for dealing with non-algebraizable logics by means of multialgebraic semantics.

In this paper sequent calculi for the classical fragment (that is, the conjunction-disjunction-implication-negation fragment) of the nonsense logics B3, introduced by Bochvar, and H3, introduced by Halld\'en, are presented. These calculi are obtained by restricting in an appropriate way the application of the rules of a sequent calculus for classical propositional logic CPL. The nice symmetry between the provisos in the rules reveal the semantical relationship between these logics. The Soundness and Completeness theorems for both calculi are obtained, as well as the respective Cut elimination theorems., Comment: In Proceedings LSFA 2012, arXiv:1303.7136