In abstract algebraic logic, many systems, such as those paraconsistent logics taking inspiration from da Costa's hierarchy, are not algebraizable by even the broadest standard methodologies, as that of Blok and Pigozzi. However, these logics can be semantically characterized by means of non-deterministic algebraic structures such as Nmatrices, RNmatrices and swap structures. These structures are based on multialgebras, which generalize algebras by allowing the result of an operation to assume a non-empty set of values. This leads to an interest in (...) exploring the foundations of multialgebras applied to the study of logic systems. It is well known from universal algebra that, for every signature \, there exist algebras over \ which are absolutely free, meaning that they do not satisfy any identities or, alternatively, satisfy the universal mapping property for the class of \-algebras. Furthermore, once we fix a cardinality of the generating set, they are, up to isomorphisms, unique, and equal to algebras of terms. Equivalently, the forgetful functor, from the category of \-algebras to 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 \, from the category of \-multialgebras to 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 submultialgebra of a multialgebra of terms to another multialgebra, and a collection of choices, there corresponds a unique homomorphism, what resembles 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 \ does not have a left adjoint. (shrink)

Despite being fairly powerful, finite non-deterministic matrices are unable to characterize some logics of formal inconsistency, such as those found between mbCcl and 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 mbCcl and 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. (shrink)

The main aim of this paper is to introduce the logics of evidence and truth $$LET_{K}^+$$ and $$LET_{F}^+$$ together with sound, complete, and decidable six-valued deterministic semantics for them. These logics extend the logics $$LET_{K}$$ and $$LET_{F}^-$$ with rules of propagation of classicality, which are inferences that express how the classicality operator $${\circ }$$ 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 $$LET_{K}$$ is obtained by means of Nmatrices based on swap structures, and the six-valued semantics for $$LET_{K}^+$$ is then obtained by imposing restrictions on the semantics of $$LET_{K}$$. These restrictions correspond exactly to the rules of propagation of classicality that extend $$LET_{K}$$. The logic $$LET_{F}^+$$ is obtained as the implication-free fragment of $$LET_{K}^+$$. We also show that the 6 values of $$LET_{K}^+$$ and $$LET_{F}^+$$ 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 $$LET_{K}^+$$ is Blok-Pigozzi algebraizable and that its implication-free fragment $$LET_{F}^+$$ coincides with the degree-preserving logic of the involutive Stone algebras. (shrink)

One of the most expected properties of a logical system is that it can be algebraizable, in the sense that an algebraic counterpart of the deductive machinery could be found. Since the inception of da Costa's paraconsistent calculi, an algebraic equivalent for such systems have been searched. It is known that these systems are non 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, these logics are uniquely characterized by semantics of non-deterministic kind. This paper offers a solution for two open problems in the domain of paraconsistency, in particular connected to algebraization of LFIs, by obtaining several LFIs weaker than C1, each of one is algebraizable in the standard Lindenbaum-Tarski's sense by a suitable variety of Boolean algebras extended with operators. This means that such LFIs satisfy the replacement property. 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, and 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. BALFI semantics. (shrink)

In a previous article we introduced the concept of restricted Nmatrices (in short, RNmatrices), which generalize non-deterministic (in short, Nmatrices) in the following sense: a RNmatrix is a Nmatrix together with a subset of valuations over it, from which the consequence relation is defined. Within this semantical framework we have characterized each paraconsistent logic Cn 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 C1 (...) cannot be characterized by a single finite Nmatrix). In this paper we extend such RNmatrices for Cn by means of what we call 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 Cn is in fact isomorphic to the category of non-trivial Boolean algebras. (shrink)