A simple Henkin-style completeness proof for Gödel 3-valued logic G3

A simple Henkin-style completeness proof for Gödel 3-valued propositional logic G3 is provided. The idea is to endow G3 with an under-determined semantics (u-semantics) of the type defined by Dunn. The key concept in u-semantics is that of “under-determined interpretation” (u-interpretation). It is shown that consistent prime theories built upon G3 can be understood as (canonical) u-interpretations. In order to prove this fact we follow Brady by defining G3 as an extension of Anderson and Belnap’s positive fragment of First Degree Entailment Logic

Reduced Routley-Meyer semantics for the logics characterized by natural implicative expansions of Kleene's strong 3-valued matrix

15 p.The aim of this paper is to provide a reduced Routley-Meyer semantics for the logics characterized by all natural implicative expansions of Kleene’s strong 3-valued matrix (with two designated values, as well as with only one) susceptible to be interpreted in Routley-Meyer semantics.S

The theory of inconsistency: inconsistant mathematics and paraconsistent logic

Each volume includes author's previously published papers.Bibliography: leaves 147-151 (v. 1).3 v. :Thesis (D.Sc.)--University of Adelaide, School of Mathematical Sciences, 200

Topics in the Proof Theory of Non-classical Logics. Philosophy and Applications

Chapter 1 constitutes an introduction to Gentzen calculi from two perspectives, logical and philosophical. It introduces the notion of generalisations of Gentzen sequent calculus and the discussion on properties that characterize good inferential systems. Among the variety of Gentzen-style sequent calculi, I divide them in two groups: syntactic and semantic generalisations. In the context of such a discussion, the inferentialist philosophy of the meaning of logical constants is introduced, and some potential objections – mainly concerning the choice of working with semantic generalizations – are addressed. Finally, I’ll introduce the case studies that I’ll be dealing with in part II. Chapter 2 is concerned with the origins and development of Jaśkowski’s discussive logic. The main idea of this chapter is to systematize the various stages of the development of discussive logic related researches from two different angles, i.e., its connections to modal logics and its proof theory, by highlighting virtues and vices. Chapter 3 focuses on the Gentzen-style proof theory of discussive logic, by providing a characterization of it in terms of labelled sequent calculi. Chapter 4 deals with the Gentzen-style proof theory of relevant logics, again by employing the methodology of labelled sequent calculi. This time, instead of working with a single logic, I’ll deal with a whole family of them. More precisely, I’ll study in terms of proof systems those relevant logics that can be characterised, at the semantic level, by reduced Routley-Meyer models, i.e., relational structures with a ternary relation between states and a unique base element. Chapter 5 investigates the proof theory of a modal expansion of intuitionistic propositional logic obtained by adding an ‘actuality’ operator to the connectives. This logic was introduced also using Gentzen sequents. Unfortunately, the original proof system is not cut-free. This chapter shows how to solve this problem by moving to hypersequents. Chapter 6 concludes the investigations and discusses the future of the research presented throughout the dissertation

Rethinking inconsistent mathematics

This dissertation has two main goals. The first is to provide a practice-based analysis of the field of inconsistent mathematics: what motivates it? what role does logic have in it? what distinguishes it from classical mathematics? is it alternative or revolutionary? The second goal is to introduce and defend a new conception of inconsistent mathematics - queer incomaths - as a particularly effective answer to feminist critiques of classical logic and mathematics. This sets the stage for a genuine revolution in mathematics, insofar as it suggests the need for a shift in mainstream attitudes about the rolee of logic and ethics in the practice of mathematics

Una expansión implicativa de la matriz tetravaluada de Belnap: una lógica modal tetravaluada carente de las paradojas modales fuertes tipo Łukasiewicz

[ES]Hacia el final de su vida el gran lógico polaco J. Lukasiewicz desarrollaría el sistema modal tetravaluado conocido como L. Este sistema verificaba tesis como (MA ^MB) ! M (A ^ B) ó L(A v B) ! (LA v LB), enclavadas dentro de las paradojas modales fuertes tipo Lukasiewicz. El sistema fue ampliamente criticado por la verificación de dichas tesis. Por otra parte, en [Brady, 1982], R. T. Brady presenta la lógica relevante BN4, una versión tetravaluada del clásico sistema del condicional relevante R. Con ambos antecedentes, el objetivo de la presente investigación pasa por desarrollar un sistema que actúe como compañero de BN4 tal y como E lo es con respecto a R y que carezca de las paradojas que asolan el sistema de Lukasiewicz. Para ello, en primer lugar, definiremos la matriz M4, que servirá como base para los sistemas que posteriormente desarrollaremos. Una vez definida la matriz daremos dos semánticas distintas: la semántica tetravaluada intrínseca a la matriz y una semántica bivalente tipo Belnap-Dunn, demostrando que las dos semánticas son equivalentes. Posteriormente definiremos un sistema basado en FDE al que llamaremos FDF4. Daremos para este sistema pruebas de corrección, completud y probaremos que surge de la matriz M4. Adicionalmente definiremos un sistema con E como base al que denominaremos EF4, para el que también daremos pruebas de corrección, completud y además probaremos que se trata de una axiomatización de la matriz M4, todo ello apoyándonos en el hecho de tratarse de un sistema equivalente a FDF4. Con respecto a EF4 desarrollaremos dos modalidades distintas: Una a través de las extensiones interdefinicionales de Lukasiewicz que, en este caso, resulta equivalente a la modalidad inherente a E, y otra basada en la propuesta de J. Y. Beziau que entronca con la propuesta de J. M. Font y M. Rius, a su vez ligada a la tradición de los algebristas portugueses encabezados por A. Monteiro. De esta manera definiremos dos sistemas modales diferentes, EF4-M y EF4-L. Para el primero daremos una única axiomatización, como es habitual, mientras que para el segundo daremos cuatro axiomatizaciones distintas. Para cada uno de ellos, EF4-M y EF4-L, desarrollaremos tanto una prueba de corrección como de completud. En último término, desarrollaremos tanto una semántica relacional ternaria de modelos reducidos, como una semántica relacional ternaria basada en 2 set-up para EF4, ofreciendo, de nuevo, pruebas de corrección y completud con respecto a ambas. Para concluir, probaremos que FDF4 es también correcto y completo con respecto a las semánticas relacionales que hemos definido y probaremos que la semántica de 2 set-up es un caso particular de la semántica de modelos reducidos.[EN]Towards the end of his life, the great Polish logician J. Lukasiewicz developed the fourth-valued modal system known as L. This system validated theses as (MA ^MB) ! M (A ^ B) or L(A v B) ! (LA v LB), which are part of what it is known as strong modal Lukasiewicz-type paradoxes. Because of this, this system was strongly criticized. On the other hand, in [Brady, 1982], R. T. Brady presents his relevant logic BN4, a fourth-valued version of the relevant implication system R. Taking this background into account, the main goal of this research is to build a system that works as a companion of BN4 (just like E does with respect to R) and lacks the paradoxes that can be found in Lukasiewicz's system. Firstly, we define the matrix M4, which is the base for all the systems that we develop later. We then introduce two different semantics, i.e., the fourth-valued semantics related to the matrix and a bivalent Belnap- Dunn type semantics, and we show that both semantics are equivalent. Next, the system that we have labeled FDF4, which is based on FDE, is defined. We prove that this system is both sound and complete in the strong sense and that it is indeed an axiomatization of the M4 matrix. Afterwards, we define a system based on E that we name EF4, for which we also prove strong soundness and completeness and how it originates from the M4 matrix, all of this based on the fact that EF4 is a system equivalent to FDF4. With respect to EF4, two different modalities are presented: the first one, which in this case is equivalent to the inherent modality of E, is developed from the interdefinitional extensions used by Lukasiewicz, and the second one, from the proposal of J. Y. Beziau related to the approach of J. M. Font and M. Rius that in its turn is linked to the Portuguese algebraic tradition led by A. Monteiro. This way, we get two different modal systems, EF4-M and EF4-L. For the former, we give just one axiomatization, while for the latter, we supply up to four dfferent ones. For both systems, EF4-M and EF4-L, we prove soundness and completeness. Furthermore, EF4 is provided with a reduced ternary relational semantics, as well as with a 2-set-up ternary relational semantics, and it is proved that it is sound and complete with respect to both semantics. Finally, it is shown that the system FDF4 is also sound and complete with respect to both aforementioned relational semantics and that the 2-set-up semantics is a particular case of the reduced semantics