Artículos con la etiqueta ‘Lógica y computación’

Aristotle’s Logic Computed by Parametric Probability and Linear Optimization

Por • 4 jul, 2013 • Category: Crítica

A new computational method is provided to implement the system of deductive logic presented in Aristotle’s Prior Analytics. Each Aristotelian problem is interpreted as a parametric probability network in which the premises give constraints on probabilities relating the problem’s categorical terms (major, minor, and middle). The problem’s figure (schema) describes which specific probabilities are constrained, relative to those that are queried. Using numerical optimization methods, the minimum and maximum feasible values of certain queried probabilities are computed. These computed values determine whether a syllogism is present and if so, which precise conclusion has been deduced from the premises. This method of analysis prevents existential fallacies, and reveals new complementary patterns of syllogism that were previously unappreciated.

Initial Semantics for Reduction Rules

Por • 29 dic, 2012 • Category: Opinion

We give an algebraic characterization of the syntax and operational semantics of a class of simply–typed languages, such as the language PCF: we characterize simply–typed syntax with variable binding and equipped with reduction rules via a universal property, namely as the initial object of some category of models. For this purpose, we employ techniques developed in two previous works: in a first work we model syntactic translations between languages over different sets of types as initial morphisms in a category of models. In a second work we characterize untyped syntax with reduction rules as initial object in a category of models. In the present work, we show that the techniques used there are modular enough to be combined: we thus characterize simply–typed syntax with reduction rules as initial object in a category. The universal property yields an operator which allows to specify translations — that are semantically faithful by construction — between languages over possibly different sets of types.

La lógica matemática: una disciplina en busca de encuadre

Por • 5 feb, 2012 • Category: Opinion

We offer an analysis of the disciplinary transformations underwent by mathematical or symbolic logic since its emergence in the late 19th century. Examined are its origins as a hybrid of philosophy and mathematics, the maturity and institutionalisation attained under the label “logic and foundations,” a second wave of institutionalisation in the Postwar period, and the institutional developments since 1975 in connection with computer science and with the study of language and informatics. Although some “internal history” is discussed, the main focus is on the emergence, consolidation and convolutions of logic as a discipline, through various professional associations and journals, in centers such as Torino, Göttingen, Warsaw, Berkeley, Princeton, Carnegie Mellon, Stanford, and Amsterdam.

Resolving Gödel’s Incompleteness Myth: Polynomial Equations and Dynamical Systems for Algebraic Logic

Por • 26 dic, 2011 • Category: Crítica

A new computational method that uses polynomial equations and dynamical systems to evaluate logical propositions is introduced and applied to Goedel’s incompleteness theorems. The truth value of a logical formula subject to a set of axioms is computed from the solution to the corresponding system of polynomial equations. A reference by a formula to its own provability is shown to be a recurrence relation, which can be either interpreted as such to generate a discrete dynamical system, or interpreted in a static way to create an additional simultaneous equation. In this framework the truth values of logical formulas and other polynomial objectives have complex data structures: sets of elementary values, or dynamical systems that generate sets of infinite sequences of such solution-value sets. Besides the routine result that a formula has a definite elementary value, these data structures encode several exceptions: formulas that are ambiguous, unsatisfiable, unsteady, or contingent. These exceptions represent several semantically different types of undecidability; none causes any fundamental problem for mathematics. It is simple to calculate that Goedel’s formula, which asserts that it cannot be proven, is exceptional in specific ways: interpreted statically, the formula defines an inconsistent system of equations (thus it is called unsatisfiable); interpreted dynamically, it defines a dynamical system that has a periodic orbit and no fixed point (thus it is called unsteady). These exceptions are not catastrophic failures of logic; they are accurate mathematical descriptions of Goedel’s self-referential construction. Goedel’s analysis does not reveal any essential incompleteness in formal reasoning systems, nor any barrier to proving the consistency of such systems by ordinary mathematical means.

Computability, Gödel’s Incompleteness Theorem, and an inherent limit on the predictability of evolution

Por • 3 dic, 2011 • Category: Ambiente

The process of evolutionary diversification unfolds in a vast genotypic space of potential outcomes. During the past century there have been remarkable advances in the development of theory for this diversification, and the theory’s success rests, in part, on the scope of its applicability. A great deal of this theory focuses on a relatively small subset of the space of potential genotypes, chosen largely based on historical or contemporary patterns, and then predicts the evolutionary dynamics within this pre-defined set. To what extent can such an approach be pushed to a broader perspective that accounts for the potential open-endedness of evolutionary diversification? There have been a number of significant theoretical developments along these lines but the question of how far such theory can be pushed has not been addressed. Here a theorem is proven demonstrating that, because of the digital nature of inheritance, there are inherent limits on the kinds of questions that can be answered using such an approach. In particular, even in extremely simple evolutionary systems a complete theory accounting for the potential open-endedness of evolution is unattainable unless evolution is progressive. The theorem is closely related to G\»odel’s Incompleteness Theorem and to the Halting Problem from computability theory.