We apply a paraconsistent logic to reason about fractions.

Desde la llegada de Juan David García Bacca a Venezuela, en 1947, invitado a refundar la Facultad de Filosofía, junto con otros compañeros de exilio, brota en él un prolongado interés por estudiar y dar a conocer, a través de diversos artículos y libros, la obra de los principales filósofos escolásticos de la época de la Colonia española en el ámbito de Venezuela y Colombia (Nueva Granada). En otro escrito nos hemos ocupado de estudiar el conjunto de los autores escolásticos investigados por García Bacca’, y aquí nos vamos a referir exclusivamente al polifacético escritor de origen venezolano, pero afincado en Chile, Andrés Bello. Si en el siglo XVII, época de predominio de la filosofía escolástica, el autor más importante será Alfonso Briceño, en el que se da una curiosa complementariedad con Andrés Bello (puesto que había nacido en Chile, y ejerció su labor magisterial y apostólica primero en Perú y luego en Venezuela), en el siglo XVIII, época de decadencia de la escolástica y momento en que el conjunto de las tierras que componían la Corona española se estaban abriendo a las nuevas corrientes filosóficas modernas, el personaje más significativo en el terreno filosófico será sin duda Andrés Bello’.

Mathematical experts couldn’t find any algorithm of obtaining the exact answer for one mass NP-complete task during four decades after emergence of the concept “class NP”. It is logical to assume that perfectly developed traditional and/or developed experimentally approaches to creation of algorithms and will not provide considerable progress in this question. Therefore, despite of the fact what the final answer to “P ? NP” dilemma will be, it is essential now to look for other ideologies of algorithms functioning of the solution of similar tasks on discrete structures. “Metaedroalgorithm” can be one of such nonconventional, but very perspective approaches.

Entre 1854, año en el que Riemann presentó en la Universidad de Göttingen su disertación de habilitación titulada Über die Hypothesen, welche der Geometrie zu Grunde liegen, precisamente el mismo año de la publicación en Londres del libro de Boole An Investigation of the Laws of Thought on Which Are Founded the Mathematical Theories of Logic and Probabilities, y 1931, año en el que se publicó el artículo de Gödel Über formal unentscheidbare Sätze der Principia Matemathica und verwandter Systeme, Europa, y muy particularmente Alemania, vivió una de las más grandiosas aventuras del espíritu humano. Una de esas revoluciones espirituales que cambian radicalmente la forma de ver y entender el mundo de los humanos y que afectará a generaciones enteras. Sus ecos y consecuencias inmediatas, tanto conceptuales como prácticos, durarían décadas y llegan a nuestros días, dando lugar a debates que aún no están cerrados y arrojando nueva luz sobre pensadores que creíamos haber entendido y que ahora podemos comprender de otra manera1 . La Lógica y las Matemáticas experimentaron un desarrollo sin precedentes en sus planteamientos y métodos, y, junto con una reflexión sobre sus fundamentos y sus relaciones mutuas, se crearon nuevas teorías y nuevas disciplinas matemáticas; y siendo éstas de alguna forma el lenguaje de las ciencias de la naturaleza, esto tuvo consecuencias inmediatas en todas ellas y en la tecnología

The main object of study in this collection is the status of a theory, ascribed to Frege, popularized by Russell and widely accepted by the last two generations of teachers in logic. The theory discriminates four meanings of the term “being”. Frege, this theory claims, discovered that “is” designates (i) Existence: “God is”; (ii) Predication: The Copula in “John is ill”; (iii) Identity: “The morning star is the evening star”; and (i’v) Class inclusion: generic implication in “If something is a horse, it is an animal”. However, when historians look at classical Greek philosophy, they discover that neither Greek language, nor the expressed convictions of great thinkers such as Plato and Aristotle agree completely or even partially with the Frege-Russellian orthodoxy.

We compute minimal solutions of the rational interpolation problem in terms of different notions of degrees associated to these functions. In all the cases, the rational interpolating functions with smallest degree can be computed via the Extended Euclidean Algorithm and syzygies of polynomials. As a by-product, we describe the minimal degree in a mu-basis of a polynomial planar parametrization in terms of a “critical” degree arising in the EEA.

” Kant’s classification of mathematical truth as synthetic a priori has given rise to a very considerable literature on the subject. We will limit ourselves here to discussing a recent attempt to vindicate Kant in terms of certain results in contemporary first order logic. In a series of letures delivered at Oxford in the early sixties, J. Hintikka has proposed formal explications of the Kantian distinction between analytic and synthetic truths and arguments. Entitled “An Analysis of Analyticity”, “Are Logical Truths Tautologies? “, “Kant Vindicated”, and “Kant and the Tradition of Analysis”, these lectures now form a central part of Hintikka’s book Logic, Language Games and Infom1ation : Kantian Themes in the Philosophy of Logic(Oxford, 1973); henceforth references to Hintikka’s presentation will simply be given by the appropriate page number in the book. We shall first give an overview of Hintikka’s arguments, then examine then criticallly, and finally argue for Church’s Theorem as a superior formal vindication of Kant’s position.

La logique classique n’a réussi à systématiser, d’une manière satisfaisante, ni un groupe suffisamment vaste de propositions logiquement vraies, ni un ensemble suffisamment étendu de procédés d’inférence pouvant être appliqués au cours d’un raisonnement déductif. Cet échec est dû au fait qu’on se sert, dans la logique classique, de l’opérateur de l’implication matérielle, lequel ne correspond que d’une façon approximative à l’expression {< si, alors >}.

We explore the issue of spacetime emergence in quantum gravity, by articulating several levels at which this can be intended. These levels correspond to the reconstruction moves that are needed to recover the classical and continuum notion of space and time, which are progressively lost in a progressively deeper sense in the more fundamental quantum gravity description. They can also be understood as successive steps in a process of widening of the perspective, revealing new details and new questions at each step. Each level carries indeed new technical issues and opportunities, and raises new conceptual issues. This deepens the scope of the debate on the nature of spacetime, both philosophically and physically.

There are human beings whose intellectual power exceeds that of ordinary men. In my life, in my personal experience, there were three such men, and one of them was Andrei Nikolaevich Kolmogorov. I was lucky enough to be his immediate pupil. He invited me to be his pupil at the third year of my being student at the Moscow University. This talk is my tribute, my homage to my great teacher. Andrei Nikolaevich Kolmogorov was born on April 25, 1903. He graduated from Moscow University in 1925, finished his post-graduate education at the same University in 1929, and since then without any interruption worked at Moscow University till his death on October 20, 1987, at the age 84-. Kolmogorov was not only one of the greatest mathematicians ofthe twentieth century. By the width of his scientific interests and results he reminds one of the titans of the Renaissance. Indeed, he made prominent contributions to various fields from the theory of shooting to the theory of versification, from hydrodynamics to set theory. In this talk I should like to expound his contributions to mathematical logic.