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The tetravalent modal logic is one of the two logics defined by Font and Rius :481–518, 2000) in connection with Monteiro’s tetravalent modal algebras. These logics are expansions of the well-known Belnap–Dunn’s four-valued logic that combine a many-valued character with a modal character. In fact, $${\mathcal {TML}}$$ TML is the logic that preserves degrees of truth with respect to tetravalent modal algebras. As Font and Rius observed, the connection between the logic $${\mathcal {TML}}$$ TML and the algebras is not so (...) good as in $${{\mathcal {TML}}}^N$$ TML N, but, as a compensation, it has a better proof-theoretic behavior, since it has a strongly adequate Gentzen calculus :481–518, 2000). In this work, we prove that the sequent calculus given by Font and Rius does not enjoy the cut-elimination property. Then, using a general method proposed by Avron et al., we provide a sequent calculus for $${\mathcal {TML}}$$ TML with the cut-elimination property. Finally, inspired by the latter, we present a natural deduction system, sound and complete with respect to the tetravalent modal logic. (shrink)

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We analyze the variety of A. Monteiro’s tetravalent modal algebras under the perspective of two logic systems naturally associated to it. Taking profit of the contrapositive implication introduced by A. Figallo and P. Landini, sound and complete Hilbert-style calculi for these logics are presented.

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Involutive Stone algebras were introduced by R. Cignoli and M. Sagastume in connection to the theory of $n$-valued Łukasiewicz–Moisil algebras. In this work we focus on the logic that preserves degrees of truth associated to S-algebras named Six. This follows a very general pattern that can be considered for any class of truth structure endowed with an ordering relation, and which intends to exploit many-valuedness focusing on the notion of inference that results from preserving lower bounds of truth values, and (...) hence not only preserving the value $1$. Among other things, we prove that Six is a many-valued logic that can be determined by a finite number of matrices. Besides, we show that Six is a paraconsistent logic. Moreover, we prove that it is a genuine Logic of Formal Inconsistency with a consistency operator that can be defined in terms of the original set of connectives. Finally, we study the proof theory of Six providing a Gentzen calculus for it, which is sound and complete with respect to the logic. (shrink)

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The 3-valued paraconsistent logic Ciore was developed by Carnielli, Marcos and de Amo under the name LFI2, in the study of inconsistent databases from the point of view of logics of formal inconsistency (LFIs). They also considered a first-order version of Ciore called LFI2*. The logic Ciore enjoys extreme features concerning propagation and retropropagation of the consistency operator: a formula is consistent if and only if some of its subformulas is consistent. In addition, Ciore is algebraizable in the sense of (...) Blok and Pigozzi. On the other hand, the logic LFI2* satisfies a somewhat counter-intuitive property: the universal and the existential quantifier are inter-definable by means of the paraconsistent negation, as it happens in classical first-order logic with respect to the classical negation. This feature seems to be unnatural, given that both quantifiers have the classical meaning in LFI2*, and that this logic does not satisfy the De Morgan laws with respect to its paraconsistent negation. The first goal of the present article is to introduce a first-order version of Ciore (which we call QCiore) preserving the spirit of Ciore, that is, without introducing unexpected relationships between the quantifiers. The second goal of the article is to adapt to QCiore the partial structures semantics for the first-order paraconsistent logic LPT1 introduced by Coniglio and Silvestrini, which generalizes the semantic notion of quasi-truth considered by Mikeberg, da Costa and Chuaqui. Finally, some important results of classical Model Theory are obtained for this logic, such as Robinson’s joint consistency theorem, amalgamation and interpolation. Although we focus on QCiore, this framework can be adapted to other 3-valued first-order LFIs. (shrink)

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Belnap–Dunn’s relevance logic, \(\textsf{BD}\), was designed seeking a suitable logical device for dealing with multiple information sources which sometimes may provide inconsistent and/or incomplete pieces of information. \(\textsf{BD}\) 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 \(\textsf{BD2}\) of the four valued Belnap–Dunn logic by a classical negation. In this paper, we introduce a four-valued expansion (...) of BD called \({\textsf{BD}^\copyright }\), obtained by adding an unary connective \({\copyright }\,\ \) which is a consistency operator (in the sense of the Logics of Formal Inconsistency, _LFI_s). In addition, this operator is the unique one with the following features: it extends to \(\textsf{BD}\) the consistency operator of LFI1, a well-known three-valued _LFI_, still satisfying axiom _ciw_ (which states that any sentence is either consistent or contradictory), and allowing to define an undeterminedness operator (in the sense of Logic of Formal Undeterminedness, _LFU_s). Moreover, \({\textsf{BD}^\copyright }\) is maximal w.r.t. LFI1, and it is proved to be equivalent to BD2, up to signature. After presenting a natural Hilbert-style characterization of \({\textsf{BD}^\copyright }\) obtained by means of twist-structures semantics, we propose a first-order version of \({\textsf{BD}^\copyright }\) called \({\textsf{QBD}^\copyright }\), with semantics based on an appropriate notion of four-valued Tarskian-like structures called \(\textbf{4}\) -structures. We show that in \({\textsf{QBD}^\copyright }\), the existential and universal quantifiers are interdefinable in terms of the paracomplete and paraconsistent negation, and not by means of the classical negation. Finally, a Hilbert-style calculus for \({\textsf{QBD}^\copyright }\) is presented, proving the corresponding soundness and completeness theorems. (shrink)

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Tetravalent modal logic (TML) was introduced by Font and Rius in 2000. It is an expansion of the Belnap-Dunn four-valued logic FOUR, a logical system that is well-known for the many applications found in several fields. Besides, TML is the logic that preserves degrees of truth with respect to Monteiro’s tetravalent modal algebras. Among other things, Font and Rius showed that TML has a strongly adequate sequent system, but unfortunately this system does not enjoy the cut-elimination property. However, in a (...) previous work we presented a sequent system for TML with the cut-elimination property. Besides, in this same work, it was also presented a sound and complete natural deduction system for this logic. In the present article we continue with the study of TML under a proof-theoretic perspective. In the first place, we show that the natural deduction system that we introduced before admits a normalization theorem. In the second place, taking advantage of the contrapositive implication for the tetravalent modal algebras introduced by A. V. Figallo and P. Landini, we define a decidable tableau system adequate to check validity in the logic TML. Finally, we provide a sound and complete tableau system for TML in the original language. These two tableau systems constitute new (proof-theoretic) decision procedures for checking validity in the variety of tetravalent modal algebras. (shrink)

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In order to develop efficient tools for automated reasoning with inconsistency (theorem provers), eventually making Logics of Formal inconsistency (_LFI_) a more appealing formalism for reasoning under uncertainty, it is important to develop the proof theory of the first-order versions of such _LFI_s. Here, we intend to make a first step in this direction. On the other hand, the logic _Ciore_ was developed to provide new logical systems in the study of inconsistent databases from the point of view of _LFI_s. (...) An interesting fact about _Ciore_ is that it has a _strong_ consistency operator, that is, a consistency operator which (forward/backward) propagates inconsistency. Also, it turns out to be an algebraizable logic (in the sense of Blok and Pigozzi) that can be characterized by means of a 3-valued logical matrix. Recently, a first-order version of _Ciore_, namely _QCiore_, was defined preserving the spirit of _Ciore_, that is, without introducing unexpected relationships between the quantifiers. Besides, some important model-theoretic results were obtained for this logic. In this paper we study some proof–theoretic aspects of both _Ciore_ and _QCiore_ respectively. In first place, we introduce a two-sided sequent system for _Ciore_. Later, we prove that this system enjoys the cut-elimination property and apply it to derive some interesting properties. Later, we extend the above-mentioned system to first-order languages and prove completeness and cut-elimination property using the well-known Shütte’s technique. (shrink)

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Belnap–Dunn’s relevance logic, \(\textsf{BD}\), was designed seeking a suitable logical device for dealing with multiple information sources which sometimes may provide inconsistent and/or incomplete pieces of information. \(\textsf{BD}\) 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 \(\textsf{BD2}\) of the four valued Belnap–Dunn logic by a classical negation. In this paper, we introduce a four-valued expansion (...) of BD called \({\textsf{BD}^\copyright }\), obtained by adding an unary connective \({\copyright }\,\ \) which is a consistency operator (in the sense of the Logics of Formal Inconsistency, _LFI_s). In addition, this operator is the unique one with the following features: it extends to \(\textsf{BD}\) the consistency operator of LFI1, a well-known three-valued _LFI_, still satisfying axiom _ciw_ (which states that any sentence is either consistent or contradictory), and allowing to define an undeterminedness operator (in the sense of Logic of Formal Undeterminedness, _LFU_s). Moreover, \({\textsf{BD}^\copyright }\) is maximal w.r.t. LFI1, and it is proved to be equivalent to BD2, up to signature. After presenting a natural Hilbert-style characterization of \({\textsf{BD}^\copyright }\) obtained by means of twist-structures semantics, we propose a first-order version of \({\textsf{BD}^\copyright }\) called \({\textsf{QBD}^\copyright }\), with semantics based on an appropriate notion of four-valued Tarskian-like structures called \(\textbf{4}\) -structures. We show that in \({\textsf{QBD}^\copyright }\), the existential and universal quantifiers are interdefinable in terms of the paracomplete and paraconsistent negation, and not by means of the classical negation. Finally, a Hilbert-style calculus for \({\textsf{QBD}^\copyright }\) is presented, proving the corresponding soundness and completeness theorems. (shrink)

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Hypersequents are a natural generalization of ordinary sequents which turn out to be a very suitable tool for presenting cut-free Gentzent-type formulations for diverse logics. In this paper, an alternative way of formulating hypersequent calculi (by introducing meta-variables for formulas, sequents and hypersequents in the object language) is presented. A suitable category of hypersequent calculi with their morphisms is defined and both types of fibring (constrained and unconstrained) are introduced. The introduced morphisms induce a novel notion of translation between logics (...) which preserves metaproperties in a strong sense. Finally, some preservation features are explored. (shrink)

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In this paper we propose to enrich the four-valued modal logic associated to Monteiro's Tetravalent modal algebras (TMAs) with a deductive implication, that is, such that the Deduction Meta-theorem holds in the resulting logic. All this lead us to establish some new connections between TMAs, symmetric (or involutive) Boolean algebras, and modal algebras for extensions of S5, as well as their logical counterparts.

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