Swap structures semantics for Ivlev-like modal logics

In 1988, J. Ivlev proposed some (non-normal) modal systems which are semantically characterized by four-valued non-deterministic matrices in the sense of A. Avron and I. Lev. Swap structures are multialgebras (a.k.a. hyperalgebras) of a special kind, which were introduced in 2016 by W. Carnielli and M. Coniglio in order to give a non-deterministic semantical account for several paraconsistent logics known as logics of formal inconsistency, which are not algebraizable by means of the standard techniques. Each swap structure induces naturally a non-deterministic matrix. The aim of this paper is to obtain a swap structures semantics for some Ivlev-like modal systems proposed in 2015 by M. Coniglio, L. Fariñas del Cerro and N. Peron. Completeness results will be stated by means of the notion of Lindenbaum–Tarski swap structures, which constitute a natural generalization to multialgebras of the concept of Lindenbaum–Tarski algebras

Non-deterministic algebraization of logics by swap structures1

Multialgebras have been 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 that 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 logics J3 and Ciore, their classes of algebraic models are retrieved, and the swap structures semantics become twist structures semantics. This fact, together with the existence of a functor from the category of Boolean algebras to the category of swap structures for each LFI, suggests that swap structures can be seen as non-deterministic twist structures. This opens new avenues for dealing with non-algebraizable logics by the more general methodology of multialgebraic semantics

First-order swap structures semantics for some Logics of Formal Inconsistency

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

A basic epistemic logic and its algebraic model

In this paper we propose an algebraic model for a modal epistemic logic. Although it is known the existence of algebraic models for modal logics, considering that there are so many different modal logics, so it is not usual to give an algebraic model for each such system. The basic epistemic logic used in the paper is bimodal and we can show that the epistemic algebra introduced in the paper is an adequate model for it. In this paper we propose an algebraic model for a modal epistemic logic. Although it is known the existence of algebraic models for modal logics, considering that there are so many different modal logics, so it is not usual to give an algebraic model for each such system. The basic epistemic logic used in the paper is bimodal and we can show that the epistemicalgebra introduced in the paper is an adequate model for it.

Elementos algébricos para a noção de poucos e sua formalização em sistemas lógicos dedutivos

Grácio (1999), em sua tese de doutorado intitulada “Lógicas moduladas e raciocínio sob in-certeza”, estabeleceu uma formalização no ambiente quantificacional para o termo da lingua-gem natural: “muitos”. Buscando a formalização desse conceito no ambiente proposicional, Feitosa, Nascimento e Grácio (2009) no artigo “Algebraic elements for the notions of „many‟”, apresentam uma estrutura matemática denominada conjuntos fechados superior-mente que torna possível o desenvolvimento de uma álgebra para “muitos” e também de uma lógica proposicional para “muitos”. De modo similar ao trabalho apresentado por Feitosa, Nascimento e Grácio (2009) para a noção de “muitos”, este trabalho investiga os elementos algébricos necessários para a formalização da noção de “poucos” e desenvolve uma álgebra para “poucos”, que tem como base uma estrutura matemática denominada conjuntos quase fechados inferiormente. A partir dessa álgebra para “poucos”, este trabalho apresenta uma lógica proposicional para “poucos” (LPP) nos sistemas dedutivos: hilbertiano e tableauxGrácio (1999), in her doctorate thesis entitled “Lógicas moduladas e raciocínio sob incerteza”, provided a formalization of the term “many”, whose can be met in natural language, inside a quantificational context. To formalize this concept in a propositional environment, Feitosa, Nascimento and Grácio (2009) presented another mathematical structure entitled upper closed sets in the paper “Algebraic elements for the notions of „many‟ ”, whose allows the develop-ment of an algebra for “many” and also a propositional logic for many. In a similar way, this paper investigates the necessary algebraic elements for the formalization of the notion of few. We also develop an algebra for “few” which is based on a mathematical structure called lower almost closed sets. From this algebra for “few, we present a propositional logic for few (LPP) in a Hilbert system. After that we present the LPP in tableauxCoordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES

Non-deterministic matrices : theory and applications to algebraic semantics

Orientador: Marcelo Esteban ConiglioTese (doutorado) - Universidade Estadual de Campinas, Instituto de Filosofia e Ciências HumanasResumo: Chamamos de multioperação qualquer operação que retorna, para cada argumento, um conjunto de valores ao invés de um único valor. Através das multioperações podemos definir uma estrutura algébrica munida com pelo menos uma multioperação. Esta estrutura é chamada de multiálgebra. O estudo delas começou em 1934, com a publicação de um artigo de Marty. No âmbito da Lógica, as multiálgebras foram consideradas por Avron e seus colaboradores sob o nome de matrizes não-determinísticas (ou Nmatrizes) e utilizadas como ferramenta semântica para a caracterização de algumas lógicas que não podem ser modeladas por uma única matriz finita. Carnielli e Coniglio introduziram a semântica de estruturas swap para LFIs (Lógicas da Inconsistência Formal), que são Nmatrizes definidas sobre ternas em uma álgebra booleana, que generaliza a semântica de Avron. Nesta Tese iremos apresentar um novo método de algebrização de lógicas, baseado em multiálgebras e em estruturas swap, que é similar ao método clássico de algebrização de Lindenbaum-Tarski, porém mais abrangente, porque podemos aplicá-lo a sistemas em que alguns operadores não são congruenciais. Em particular, este método será aplicado à uma família de lógicas modais não-normais e à algumas LFIs que não são algebrizáveis por nenhum método bem conhecido, incluindo a teoria geral de Blok e Pigozzi. Também obteremos teoremas de representação para alguns LFIs e provaremos que, sob a nossa abordagem, as classes de estruturas de swap para algumas extensões axiomáticas de mbC são subclasses da classe das estruturas swap para a lógica mbCAbstract: We call multioperation any operation that return for even argument a set of values instead of a single value. Through multioperations we can define an algebraic structure equipped with at least one multioperation. This kind of structure is called multialgebra. The study of them began in 1934 with the publication of a paper of Marty. In the realm of Logic, multialgebras were considered by Avron and his collaborators under the name of non-deterministic matrices (or Nmatrices) and used as semantics tool for characterizing some logics which cannot be characterized by a single finite matrix. Carnielli and Coniglio introduced the semantics of swap structures for LFIs (Logics of Formal Inconsistency), which are Nmatrices defined over triples in a Boolean algebra, generalizing Avron¿s semantics. In this thesis we will introduce a new method of algebraization of logics based on multialgebras and swap structures that is similar to classical algebraization method of Lindenbaum-Tarski, but more extensive because it can be applied to systems such that some operators are non-congruential. In particular, this method will be applied to a family of non-normal modal logics and to some LFIs that are not algebraizable by method as Blok and Pigozzi general theory. We also will obtain representation theorems for some LFIs and we will prove that, under out approach, the classes of swap structures for some axiomatic extensions of mbC are a subclass of the class of swap structures for the logic mbCDoutoradoFilosofiaDoutora em Filosofia2013/04568-1FAPES

A LÓGICA TK EM DEDUÇÃO NATURAL, CÁLCULO DE SEQUENTES E TABLEAUX

(Feitosa, Grácio, Nascimento, 2007) introduziram uma nova lógica, a Lógica TK, que foi apresentada inicialmente no estilo hilbertiano. O objetivo deste trabalho é apresentar a Lógica TK em sistemas de dedução natural, cálculo de sequentes e tableaux assim como demonstrar a equivalência entre esses novos sistemas e o original

Non-deterministic algebraization of logics by swap structures

Multialgebras (or hyperalgebras or non-deterministic algebras) have been 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) that 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 logics J3 and Ciore, their classes of algebraic models are retrieved, and the swap structures semantics become twist structures semantics (as independently introduced by M. Fidel and D. 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 seen as non-deterministic twist structures. This opens new avenues for dealing with non-algebraizable logics by the more general methodology of multialgebraic semantics285Volume II: New Advances in Logics of Formal Inconsistency 1 Introduction10211059CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO - CNPQFUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO - FAPESP308524/2014-4; 150064/2018-72013/04568-1; 2016/21928-

On the closure operator, a concept involving several and distinct topics of Mathematics

We present the closure operators and detach several distinct topics of mathematics in which we can observe the action of these closure operators. So, we can see these operators as a trans-topic concept. Considering that these mathematics theories can be applied to so many fields, then we enhance the closure operators as a transdisciplinary notion.We present the closure operators and detach several distinct topics of mathematics in which we can observe the action of these closure operators. So, we can see these operators as a trans-topic concept. Considering that these mathematics theories can be applied to so many fields, then we enhance the closure operators as a transdisciplinary notion