La palabra paradoja procede del griego (para y doxos) y significa etimológicamente “más allá de lo creíble”. Y ésta es probablemente su mejor definición. En general, una paradoja es una afirmación o razonamiento que nos lleva a una contradicción (real o aparente).

Today’s computing is told to be based on the classic paradigm, proposed by von Neumann, a three-quarter century ago. However, that paradigm was justified (for the timing relations of) vacuum tubes only. The technological development invalidated the classic paradigm (but not the model!) and led to catastrophic performance losses in computing systems, from operating gate level to large networks, including the neuromorphic ones. The paper reviews the critical points of the classic paradigm and scrutinizes the confusion made around it.

Al concebir las ciencias como conjuntos de conocimientos dotados de una mayor probabilidad que otros, en la línea del empirismo inglés, Russell reduce la gnoseología (la teoría de la ciencia) a epistemología (a teoría del conocimiento). Y, dentro de la república de las ciencias, otorga la hegemonía a la física y la psicología, encargadas de conocer el mundo objetivo y el mundo subjetivo respectivamente. No obstante, tras su conversión al monismo neutro de William James, Russell defenderá que lo físico y lo mental no son sino formas de denominar a una misma sustancia neutra, previa e intermedia, de modo que el dualismo ontológico es antes un dualismo epistemológico y gnoseológico (p. 56).

A formalism is proposed to describe entangled quantum histories, and their entanglement entropy. We define a history vector, living in a tensor space with basis elements corresponding to the allowed histories, i.e. histories with nonvanishing amplitudes. The amplitudes are the components of the history vector, and contain the dynamical information. Probabilities of measurement sequences, and resulting collapse, are given by generalized Born rules: they are all expressed by means of projections and scalar products involving the history vector. Entangled history states are introduced, and a history density matrix is defined in terms of ensembles of history vectors.

We present a computable algorithm that assigns probabilities to every logical statement in a given formal language, and refines those probabilities over time. For instance, if the language is Peano arithmetic, it assigns probabilities to all arithmetical statements, including claims about the twin prime conjecture, the outputs of long-running computations, and its own probabilities. We show that our algorithm, an instance of what we call a logical inductor, satisfies a number of intuitive desiderata, including: (1) it learns to predict patterns of truth and falsehood in logical statements, often long before having the resources to evaluate the statements, so long as the patterns can be written down in polynomial time; (2) it learns to use appropriate statistical summaries to predict sequences of statements whose truth values appear pseudorandom; and (3) it learns to have accurate beliefs about its own current beliefs, in a manner that avoids the standard paradoxes of self-reference.

Cien años después de la publicación de “On Denoting” son escasos los trabajos; que tratan de la relación entre la Teoría de las Descripciones Definidas (TDD); que; presentó Bertrand Russell en ese ensayo; y el Principio de Composicionalidad (PC)1; .; Ése es el territorio que explorará el presente trabajo con la vista puesta en la cuestión; de qué razones podría haber —de hecho; qué razones llevaron a Russell— a limitar la; aplicación de ese principio. Enfocadas así las cosas; la aceptación del PC emerge como una elección que ha guardar el justo equilibrio con otros tipos de desiderata: metafísicos; epistemológicos y lógicos.

Predicate Logic with Definitions (PLD or D-logic) is a modification of first-order logic intended mostly for practical formalization of mathematics. The main syntactic constructs of D-logic are terms, formulas and definitions. A definition is a definition of variables, a definition of constants, or a composite definition (D-logic has also abbreviation definitions called abbreviations). Definitions can be used inside terms and formulas. This possibility alleviates introducing new quantifier-like names. Composite definitions allow constructing new definitions from existing ones.

With the birth of quantum mechanics, the two disciplines that Hilbert proposed to axiomatize, probability and mechanics, became entangled and a new probabilistic model arose in addition to the classical one. Thus, to meet Hilbert’s challenge, an axiomatization should account deductively for the basic features of all three disciplines. This goal was achieved within the framework of quantum probability. The present paper surveys the quantum probabilistic axiomatization.

We begin with a context more general than set theory. The basic ingredients are essentially the object and functor primitives of category theory, and the logic is weak, requiring neither the Law of Excluded Middle nor quantification. Inside this we find «relaxed» set theory, which is much easier to use with full precision than traditional axiomatic theories. There is also an implementation of the Zermillo-Fraenkel-Choice axioms that is maximal in the sense that any other implementation uniquely embeds in it.

Contemporary research programs in fundamental physics appear to suggest that there could be two (physical) times—or none at all. This essay articulates these possibilities in the context of quantum gravity, and in particular of cosmological models developed in an approach called `loop quantum gravity’, and explains how they could nevertheless underwrite our manifestly temporal world. A proper interpretation of these models requires a negotiation of an atemporal and a temporal sense of the emergence of (space)time.