Ricardo Da Silva

La noción de verdad juega un papel fundamental dentro del marco lógico- semántico expuesto por Gottlob Frege en sus ensayos “Función y concepto” y “Sentido y referencia”. En su artículo El pensamiento. Una investigación lógica, el filósofo alemán nos dice que la lógica es la ciencia que estudia lo verdadero y las leyes que en cuanto verdadero le competen, pero, ¿qué debemos entender por “verdad”? Esta interrogante será uno de los puntos centrales que Frege expondrá en su artículo. ¿La verdad es una relación? ¿La verdad es una propiedad? Y, en caso de que sea una propiedad, ¿qué nos ofrece la verdad como predicado?, ¿qué nuevo conocimiento sobre aquello que predica nos aporta?Ahora bien, tras explicar y responder las preguntas con respecto a la verdad, nuestro autor termina afirmando en El pensamiento…, que la verdad es un concepto sui generis e indefinible, y es entonces cuando la noción de verdad expuesta en El pensamiento…, parece generar una tensión con la noción de verdad que sustenta todo el aparato lógico-semántico de la filosofía de Frege. El trabajo que expondré busca responder a las preguntas anteriores con respecto a la naturaleza de la verdad, rigiéndome por las respuestas de Frege y siguiendo la guía de autores como Joseph Salerno y Anthony Kenny y, además, busco examinar las interpretaciones existentes que pueden darse al “conflicto” que se genera debido a las dos diferentes nociones del concepto de verdad que aparecen a lo largo de la obra de Frege.The notion of truth plays a key role within the semantic logical framework outlined by Gottlob Frege in his essays “Function and concept” and “On Sense and Reference”. In his article “The Thought: a logical investigation”, the German philosopher tells us that logic is the science of truth and the laws, as true, which are of its competence, but what we must understand by “truth”? This question will be one of the central points that Frege shall exhibit in his article.”Is truth a relationship? Is truth a property?”. And, if it is a property, what86 episteme ns, vol. 31, nº 1, 2011, pp. 85-90gives us the truth as predicate? what new knowledge about what he predicates brings us?Now, after explaining and answering questions about the truth, our author ends by stating, in “The thought…”, that truth is a concept sui generis and indefinable, and that’s when the notion of truth presented in “Thought...”, seems to generate tension with the notion of truth which underlies all the semantic-logical apparatus of the philosophy of Frege. These essay looks to answer the above questions regarding the nature of truth, guided by the responses of Frege and following the direction of authors like Joseph Salerno and Anthony Kenny. It also seeks to examine existing interpretations that may occur to the “conflict” that is generated due to the two different notions of the concept of truth that appear throughout the work of Frege.

En este artículo realizamos una reconstrucción del Programa original de Hilbert antes del surgimiento de los teoremas limitativos de la tercera década del siglo pasado. Para tal reconstrucción empezaremos por mostrar lo que Torretti llama los primeros titubeos formales de Hilbert, es decir, la defensa por el método axiomático como enfoque fundamentante. Seguidamente, mostraremos como estos titubeos formales se establecen como un verdadero programa de investigación lógico-matemático y como dentro de dicho programa la inquietud por la decidibilidad de los problemas matemáticos y en específico la decidibilidad de la Lógica de primer orden cobra peso. Luego pasamos a analizar como la inquietud por la decibilidad toma lugar dentro del pensamiento filosófico-matemático de Hilbert presentándose como uno de los grandes problemas a los cuales la metamatemática debe encontrar una solución, esto lo hacemos mostrando un contraste con autores, como John von Neumann y Roberto Torretti, quienes de alguna u otra manera no interpretan el problema de la decidibilidad de la Lógica de primer orden como un problema de peso dentro del programa original de Hilbert. Finalmente argumentamos que el resultado meta-teórico de Church puede entenderse como una refutación del optimismo intelectual que permea a todo el programa original de Hilbert.

The present paper has three objectives: Presenting an actualization of a proof of the decidability of monadic predicates logic in the contemporary model theory context; Show examples of decidable and undecidable fragments inside First order logic, offering an original proof of the following theorem: Any formula of First of order logic is decidable if its prenex normal form is in the following form: ∀x1,…,∀xn∃y1,…,∃ymφ; Presenting a theorem that characterizes the validity of First order logic by the tautologicity of Propositional logic, said result is interesting since immediately arises the doubt of how to conciliate said characterization with Alonzo Church’s Undecidability Theorem for First Order Logic.

Church's Undecidability Theorem is one of the meta-theoretical results of the mid-third decade of the last century, which along with other limiting theorems such as those of Gödel and Tarski have generated endless reflections and analyzes, both within the framework of the formal sciences, that is, mathematics, logic and theoretical computation, as well as outside them, especially the philosophy of mathematics, philosophy of logic and philosophy of mind. We propose, as a general purpose of this article, to formulate Church's Undecidability Theorem and present the main ideas of its demonstration. In order to carry out the first objective, we need to introduce and explain the notions of recursive function and numbering used by Gödel, which will allow to formally and rigorously enunciate Church's Theorem. After we enunciate Church's Theorem of Unspeakability in a formal and rigorous manner, we will present the main ideas of the proof of Church's Undecidability Theorem for First Order Logic, which uses Robinson's axiomatic system for arithmetic and four facts about himself: In Robinson's system for arithmetic recursive functions are representable Robinson's system is undecidable, The number of axioms proper to the Robinson system is finite and The logical calculation of the Robinson system is equal to the calculation of the first-order logic.

The following note has on purpose to introduce interested students to logicism. Our objective is not to show any new interpretation or thesis about logicism or its rebirth between the 60s and 80s of the last century. What we will do is systematically show the evolution of logicism from Frege to Russell-Whitehead, with greater emphasis on this latest development, and approach some problems that arise within that movement, for example: The logical paradoxes and the principle of intuitive comprehension, the impredicative definitions and the semantic paradoxes, the Ramified Hierarchy and real numbers, the axiom of reducibility and the impossibility of reducing mathematics to logic, etc.

The turbulent times we live in, product of the COVID-19 pandemic, creating an ideal breeding ground for the proliferation of myths and conspiracy theories. These myths and theories are often popular on social media, defended by irresponsible politicians and published by the media in search of ratings or likes. But none of these meet the theoretical virtues that a good scientific theory should enjoy, in particular, none of these conspiracy theories about the disease caused by the SARS CoV-2 virus has the attribute of being simple. Our purpose in this article is to illuminate how it is possible that simplicity, within the framework of theoretical virtues, can function as an epistemic criterion to reveal the irrational and, consequently, unscientific nature of myths and conspiracy theories.

Hacia finales del siglo XIX se llevó a cabo una gran revolución conceptual y metodológica en la matemática. En tal revolución se empezaron a emplear conceptos, métodos y técnicas que dejaban de lado la antigua forma de hacer matemática, propia del siglo XVIII y principios del siglo XIX, y a su vez proponían un Hacer abstracto, es decir, una forma abstracta de ocuparse del ente matemático. Pero no sólo se trataba de un cambio metodológico, sino que la pregunta por los fundamentos se vuelve cada vez más importante y trae consigo interrogantes de carácter filosófico, como es el caso de la inquietud por la naturaleza del objeto matemático (la interrogante ontológica), y la posibilidad de conocimiento de dicho objeto (la interrogante epistemológica). Nuestro interés en este artículo es mostrar cómo la filosofía que respalda las investigaciones matemáticas de Cantor trata de dar respuestas a las interrogantes ontológicas y epistemológicas. Para ello hemos tratado de ofrecer un contexto histórico-conceptual que gira en torno a la pregunta por los fundamentos, y dentro de dicho contexto hemos señalado como se presenta el Platonismo absoluto de Cantor An approach to Cantor´s absolute PlatonismA great conceptual and methodological revolution in mathematics was carried out by the end of nineteen century. In that revolution people began to use concepts, methods and techniques which set aside the old way of doing mathematics, typical of the eighteenth and early nineteenth century, and in turn they proposed an Abstract Make, i.e., an abstract form of dealing with the mathematical entity. But it was not only a methodological change, but the question of the foundations is becoming increasingly important which arises more philosophical questions, such as the concern about the nature of the mathematical object -the ontological question- and the possibility of knowledge of this object -the epistemological question. Our interest in this article is to show how the philosophy behind Cantor's mathematical research is intended to answer the ontological and epistemological questions. For it, this paper tries to provide a conceptual and historical context, which is revolving around the question of the foundations, and within this context, it is noted as the Absolute Platonism of Cantor.