Neste trabalho estabelecemos uma base teórica para a construção de uma teoria de rep- resentação de lógicas proposicionais. Iniciamos identificando uma relação precisa entre a categoria das lógicas (Blok-Pigozzi) algebrizáveis e a categoria de suas classes de álgebras associadas. Assim obtemos codificações funtoriais para as equipolências e morfismos den- sos entre lógicas. Na tentativa de generalizar os resultados obtidos sobre a codificação dos morfismos entre lógicas algebrizáveis, introduzimos a noção de funtor filtro e sua lógica asso- ciada. Classificamos alguns tipos especiais de lógicas e um estudo da propriedade metalógica de interpolação de Craig via amalgamação em matrizes para lógicas não-protoalgebrizáveis, e estabelecemos a relação entre a categoria dos funtores filtros e a categoria de lógicas. Em seguida, empregamos noções da teoria das instituições para definir instituições para as lógicas proposicionais abstratas, para uma lógica algebrizável e para uma lógica Lindenbaum alge- brizável. Sobre a instituição das lógicas algebrizáveis (lógicas Lindenbaum algebrizáveis), estabelecemos uma versão abstrata do Teorema de Glivenko e que é exatamente o tradi- cional teorema de Glivenko quando aplicado entre a lógica clássica e intuicionista. Por fim, influenciado pela teoria de representação para anéis, apresentamos os primeiros passos da teoria de representação de lógicas. Introduzimos as definições de diagramas modelos à esquerda para uma lógica, Morita equivalência e Morita equivalência estável para lógicas. Mostramos que quaisquer representações para lógica clássica são estavelmente Morita equiv- alentes, entretanto a lógica clássica e intuicionista não são estavelmente Morita equivalentes. / In this work we provide a framework in order to build a representation theory of proposi- tional logics. We begin identifying a precise relation between the category of (Blok-Pigozzi) algebraizable logic and the category of their classes of associated algebras. Then, we have a functorial codification for the equipollence and dense morphisms between logics. Attempt- ing generalize the results found before about codification of morphisms among algebraizable logics, we introduce the notion of filter functor and its associated logic. We classify some special kinds of logics and a study of a meta-logical Craig interpolation property via matri- ces amalgamation for non-protoalgebraizable logics, and we establish a relation between the category of filter functors and the category of logics. In the sequel, we employ notions of institution theory to define the institutions for the abstract propositional logics, for an al- gebraizable logic and Lindenbaum algebraizable logic. On the institutions for algebraizable logics (Lindenbaum algebraizable logics), we introduce the abstract Glivenkos theorem and this notion is exactly the traditional Glivenkos theorem when applied between the classical logic and intuitionistic logic. At last, influenced by the representation theory of rings, we present the first steps on the representation theory of logics. We introduce the definition of left diagram model for a logic, Morita equivalence of logics and stably-Morita equivalence for logics. We have showed that any presentation for classical logic are stably-Morita equivalent, but the classical logic and intuitionistic logic are not stably-Morita equivalent.

De Morgan monoids are algebraic structures that model certain non-classical logics. The variety DMM of all De Morgan monoids models the relevance logic Rt (so-named because it blocks the derivation of true conclusions from irrelevant premises). The so-called subvarieties and subquasivarieties of DMM model the strengthenings of Rt by new logical axioms, or new inference rules, respectively. Meta-logical problems concerning these stronger systems amount to structural problems about (classes of) De Morgan monoids, and the methods of universal algebra can be exploited to solve them. Until now, this strategy was under-developed in the case of Rt and DMM. The thesis contributes in several ways to the filling of this gap. First, a new structure theorem for irreducible De Morgan monoids is proved; it leads to representation theorems for the algebras in several interesting subvarieties of DMM. These in turn help us to analyse the lower part of the lattice of all subvarieties of DMM. This lattice has four atoms, i.e., DMM has just four minimal subvarieties. We describe in detail the second layer of this lattice, i.e., the covers of the four atoms. Within certain subvarieties of DMM, our description amounts to an explicit list of all the covers. We also prove that there are just 68 minimal quasivarieties of De Morgan monoids. Thereafter, we use these insights to identify strengthenings of Rt with certain desirable meta-logical features. In each case, we work with the algebraic counterpart of a meta-logical property. For example, we identify precisely the varieties of De Morgan monoids having the joint embedding property (any two nontrivial members both embed into some third member), and we establish convenient sufficient conditions for epimorphisms to be surjective in a subvariety of DMM. The joint embedding property means that the corresponding logic is determined by a single set of truth tables. Epimorphisms are related to 'implicit definitions'. (For instance, in a ring, the multiplicative inverse of an element is implicitly defined, because it is either uniquely determined or non-existent.) The logical meaning of epimorphism-surjectivity is, roughly speaking, that suitable implicit definitions can be made explicit in the corresponding logical syntax. / Thesis (PhD)--University of Pretoria, 2020. / DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) / Mathematics and Applied Mathematics / PhD / Unrestricted

In this thesis we study completeness properties of infinitary propositional logics from the perspective of abstract algebraic logic. The goal is to under- stand how the basic tool in proofs of completeness, the so called Linden- baum lemma, generalizes beyond finitary logics. To this end, we study few properties closely related to the Lindenbaum lemma (and hence to com- pleteness properties). We will see that these properties give rise to a new hierarchy of infinitary propositional logic. We also study these properties in scenarios when a given logic has some (possibly very generally defined) connectives of implication, disjunction, and negation. Among others, we will see that presence of these connectives can ensure provability of the Lin- denbaum lemma. Keywords: abstract algebraic logic, infinitary logics, Lindenbaum lemma, disjunction, implication, negation

This thesis studies the extensions of the four-valued Belnap-Dunn logic, called super-Belnap logics, from the point of view of abstract algebraic logic. We describe the global structure of the lattice of super-Belnap logics and show that this lattice can be fully described in terms of classes of finite graphs satisfying some closure conditions. We also introduce a theory of so- called explosive extensions and use it to prove new completeness theorems for super-Belnap logics. A Gentzen-style proof theory for these logics is then developed and used to establish interpolation for many of them. Finally, we also study the expansion of the Belnap-Dunn logic by the truth operator ∆. Keywords: abstract algebraic logic, Belnap-Dunn logic, paraconsistent logic, super-Belnap logics

In this master thesis we investigate completeness theorems in the framework of abstract algebraic logic. Our main interest lies in the completeness with respect to the so called relatively (finitely) subdirectly irreducible models. Notable part of the presented theory concerns the difference between finitary and infinitary logical systems. We focus on the well-known fact that the completeness theorem with respect to relatively (finitely) subdirectly irreducible models can be proven in general for all finitary logics and we discuss the possible of generalizing this theorem even to infinitary logics. We show that there are two interesting inter- mediate properties between this completeness and finitarity, namely (completely) intersection-prime extension properties. Based on these notions we define five classes of logics and propose a new hierarchy of finitary and infinitary logics. As a main contribution of this dissertation we present an example of a logic separat- ing some of these classes. Keywords: Abstract algebraic logic, completeness, relatively (finitely) sub- directly irreducible models, RSI-completeness, RFSI-completeness, (completely) intersection-prime extension property, IPEP, CIPEP.