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.

Thesis (Ph. D.)--University of Texas at Austin, 1999. / Vita. Includes bibliographical references (leaves 195-202). Available also in a digital version from Dissertation Abstracts.

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

Text in Afrikaans, abstract in Afrikaans and English / Die gebruik van simbole maak wiskunde eenvoudiger en kragtiger, maar ook moeiliker verstaanbaar. Laasgenoemde kan voorkom word as slegs eenvoudige en noodsaaklike simbole gebruik word, met die verduidelikings en motiverings in woorde. Die krag van simbole le veral in die feit dat simbole as substitute vir konsepte kan dien. Omdat die krag van simbole hierin le, skuil daar 'n groot gevaar in die gebruik van simbole. Wanneer simbole los is van sinvolle verstandsvoorstellings, is daar geen krag in simbole nie. Dit is die geval met die huidige benadering in skoolalgebra. Voordat voldoende verstandsvoorstellings opgebou is, word daar op die manipulasie van simbole gekonsentreer. Die algebraiese historiese-kenteoretiese perspektief maak algebra meer betekenisvol vir leerders. Hiervolgens moet die leerlinge die geleentheid gegun word om oplossings in prosavorm te skryf en self hul eie wiskundige simbole vir idees spontaan in te voer. Hulle moet self die voordeel van algebraiese simbole beleef. / The use of symbols in algebra both simplifies and strengthens the subject, but it also increases its level of complexity.This problem can be prevented if only simple and essential symbols are used and if the explanations are fully verbalised. The power of symbols stems from their potential to be used as substitutes for concepts. As this constitutes the crux of mathematical symbolic representation, it also presents a danger in that the symbols may not be comprehended. If symbols are not related to mental representations, the symbols are meaningless. This is the case in the present approach to algebra. Before sufficient mental representations are built, there is a concentration on the manipulation of symbols. The algebraic historical epistemological perspective makes algebra more meaningful for learners. Learners should be granted the opportunities to write their solutions in prose and to develop their own symbols for concepts. / Mathematics Education / M. Sc. (Wiskunde-Onderwys)

This thesis presents a unified framework for the analysis of instantaneous kinematics and statics of spatial manipulators. The proposed formulation covers the entire range of kinematic behavior, with kinematic singularity and isotropy appearing as special cases. An analogous treatment of statics is also presented. It is established that the formulations presented are capable of generating exact solutions in closed form for several interesting problems in manipulator analysis. Several such results have been obtained via extensive usage of symbolic computation tools developed for this purpose. The proposed approach is applicable to manipulators of different architectures. However, the focus is on the parallel and hybrid manipulators, as their analysis presents more challenges than their serial counterparts. The theory of screws has been adopted as the basis of the formulation. Instantaneous kinematics and statics are studied in terms of the principal bases of the space of twists, se(3), and the space of wrenches, se* (3), respectively. A dual number parameterisation of the motion space is adopted to make the formulation compact and dimensionally consistent. The properties of the dual combination obviate the need for an explicit scaling between the linear and angular velocities or the forces and moments. Hence the results obtained from the formulation are purely geometrical. The analysis of the twists is performed via the dual velocity Jacobian matrix. The principal basis of se(3) is obtained from the eigenproblem of a symmetrical dual matrix associated with the Jacobian. The notion of a dual eigenproblem is introduced in this context. Solutions are provided for the general case, as well as a few special cases. The computations involve the solution of at most a cubic equation for any arbitrary degree-of-freedom of rigid-body motion, and closed form results are therefore ensured. The results of the eigen-analysis lead to a decoupling of the rotational and the pure translational components of a rigid-body motion. This is termed as the partitioning of degrees-of-freedom. They also motivate an interesting classification of the manipulators based on the instantaneous partition of its degrees-of-freedom. This notion is further extended to analyse the effects of a singularity on the motion characteristics of a manipulator. Due to the duality of se(3) and se*(3), the formulation of statics is completely analogous, and involves, in essence, only the substitution of the dual wrench-transformation matrix for the dual Jacobian. A similar partitioning of the wrench system is introduced based on the eigen-decomposition in the context of statics. It is shown that the principal screws associated with either a system of twists or wrenches can be obtained from a generalised eigenproblem of two symmetric real matrices arising out of the symmetric dual matrix mentioned above. The general 2-and 3-screw systems are analysed in closed form via the generalised characteristic polynomial. Several special screw systems are described in terms of algebraic equations in terms of the coefficients of this polynomial. Principal bases for 4-and 5-systems are obtained in a novel fashion without deriving their reciprocal systems explicitly. Using the same approach based on the analysis of the characteristic polynomial, compact algebraic conditions for singularity and isotropy are derived as the special cases of a single formulation. The above formulations establish the existence of exact closed-form results. However, to implement them symbolically for a real application problem, capabilities in existing computer algebra systems do not suffice in general. Therefore simplification and computational algorithms are developed for dealing with large expressions with algebraic and trigonometric terms typically appearing in kinematics and statics. Three canonical forms of such expressions and the corresponding simplification schemes are presented. The theoretical developments are illustrated with examples of serial, parallel and hybrid manipulators throughout the thesis. However, the most important applications of these are in the kinematic and static analysis of a semi-regular Stewart platform manipulator (in which the top and bottom platforms are semi-regular hexagons). Using the degeneracy of the wrench transformation matrix as the singularity criterion, the singularity manifold of the manipulator is obtained via extensive application of the symbolic simplification algorithms. The constant-orientation singularity manifold is derived in a compact closed form, and a complete geometric characterisation and explicit parameterisation of the same are presented. The constant-position singularity manifold is also obtained in closed form. On the other hand, families of configurations of the manipulator for combined kinematic or static isotropy for a given architecture are derived in closed form. Also, architectural designs are obtained for the manipulator such that it exhibits combined kinematic or static isotropy at a given configuration.

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

Text in Afrikaans, abstract in Afrikaans and English / Die gebruik van simbole maak wiskunde eenvoudiger en kragtiger, maar ook moeiliker verstaanbaar. Laasgenoemde kan voorkom word as slegs eenvoudige en noodsaaklike simbole gebruik word, met die verduidelikings en motiverings in woorde. Die krag van simbole le veral in die feit dat simbole as substitute vir konsepte kan dien. Omdat die krag van simbole hierin le, skuil daar 'n groot gevaar in die gebruik van simbole. Wanneer simbole los is van sinvolle verstandsvoorstellings, is daar geen krag in simbole nie. Dit is die geval met die huidige benadering in skoolalgebra. Voordat voldoende verstandsvoorstellings opgebou is, word daar op die manipulasie van simbole gekonsentreer. Die algebraiese historiese-kenteoretiese perspektief maak algebra meer betekenisvol vir leerders. Hiervolgens moet die leerlinge die geleentheid gegun word om oplossings in prosavorm te skryf en self hul eie wiskundige simbole vir idees spontaan in te voer. Hulle moet self die voordeel van algebraiese simbole beleef. / The use of symbols in algebra both simplifies and strengthens the subject, but it also increases its level of complexity.This problem can be prevented if only simple and essential symbols are used and if the explanations are fully verbalised. The power of symbols stems from their potential to be used as substitutes for concepts. As this constitutes the crux of mathematical symbolic representation, it also presents a danger in that the symbols may not be comprehended. If symbols are not related to mental representations, the symbols are meaningless. This is the case in the present approach to algebra. Before sufficient mental representations are built, there is a concentration on the manipulation of symbols. The algebraic historical epistemological perspective makes algebra more meaningful for learners. Learners should be granted the opportunities to write their solutions in prose and to develop their own symbols for concepts. / Mathematics Education / M. Sc. (Wiskunde-Onderwys)

Introduced by C.C. Chang in the 1950s, MV algebras are to many-valued (Łukasiewicz) logics what boolean algebras are to two-valued logic. More recently, effect algebras were introduced by physicists to describe quantum logic. In this thesis, we begin by investigating how these two structures, introduced decades apart for wildly different reasons, are intimately related in a mathematically precise way. We survey some connections between MV/effect algebras and more traditional algebraic structures. Then, we look at the categorical structure of effect algebras in depth, and in particular see how the partiality of their operations cause things to be vastly more complicated than their totally defined classical analogues. In the final chapter, we discuss coordinatization of MV algebras and prove some new theorems and construct some new concrete examples, connecting these structures up (requiring a detour through effect algebras!) to boolean inverse semigroups.

O presente trabalho visa entender como o rádio se distingue dos demais sons do mundo. A hipótese aqui formulada é a de que o rádio, em sua definição, é uma linguagem, e não um aparelho. Dessa maneira, há a busca por uma caracterização da linguagem radiofônica seguindo as ideias implicadas em uma Estética da Linguagem (Derrida e antiessencialistas como Ziff, Weitz e Kennick). Com isso, há um estudo detalhado do rádio em seu jogo de linguagem (Wittgenstein) e em seus diferendos (Lyotard), considerados aqui enquanto parergon e ergon, ou seja, enquanto recorte e modelo operacional da linguagem em sua intersecção com o mundo. Para a investigação do jogo de linguagem, foram utilizados conteúdos relacionados à Filosofia Analítica, à Lógica Algébrica e à Teoria dos Jogos para desenvolver um método analítico denonimado Jogo Lógico, voltado para o estudo de jogos de linguagem comunicacionais. Já para a investigação dos diferendos, foram utilizadas as ideias pragmáticas acerca da performatividade e da lógica ilocucionária (Austin e Searle) para analisar os gêneros radiofônicos (a saber: musical, radiojornalismo, esportivo, variedades [talk radio], humorístico, ficção e publicidade). Essas duas investigações formam aquilo que é chamado aqui de Gramática do Rádio - considerando o conceito wittgensteiniano de gramática -, o ponto nodal que nos permite caracterizar o rádio enquanto linguagem. / The present work aims to understand how the radio distinguishes itself from other sounds of the world. The hypothesis formulated here is that the radio, in its definition, is a language, not a machine. Thus, there is the search for a characterization of radio\'s language following the ideas involved in an Aesthetics of Language (Derrida and anti-essentialists like Ziff, Weitz and Kennick). Here, there is a detailed study of the radio in its language-game (Wittgenstein) and their differends (Lyotard), considered in this work as parergon and ergon, i.e. as the cut and the operational model of language in its intersection with the world. For the investigation of the language-game, we used content related to Analytic Philosophy, to Algebraic Logic, and to Game Theory to develop an analytical method called Logic Game, dedicated to the communicative language-games\' study. As for the investigation of differends, we used the pragmatic concepts about the performative and illocutionary logic (Searle and Austin) to analyze the radio genres (ie: music, radio journalism, sports, talk radio, humor, fiction and advertising). These two studies form what is called here the Radio Grammar - considering the Wittgensteinian concept of grammar - the key point that allows us to characterize the radio as a language.

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