Ricardo Oscar Rodriguez

El sistema de demostración KE es una variante de los tableaux analíticos pero es computacionalmente más eficiente que estos [2]. Este sistema fue extendido a la lógica intuicionista proposicional en [8]. Además de su eficiencia y a diferencia de los resolvedores SAT, el sistema KE intuicionista no requiere transformaciones a formas normales y es modular en tanto que se puede extender, simple y naturalmente, a una familia amplia de lógicas que admiten semánticas relacionales. En este trabajo exploramos parte de esta modularidad, extendiendo KE a una familia de lógicas modales intuicionistas, las cuales tienen aplicaciones relevantes en representación del conocimiento y razonamiento [3], [4], [10].

The tableau-like system KE is generalized to intuitionistic propositional logic by means of labeled signed formulas and constraints between labels, mimicking the relational semantics. To improve on proof-search and exploiting the meaning of negation, we further endow the system with free-variables. The resulting system enjoys the subformula property and terminates, either with a proof or a finite countermodel, without any extra mechanism. Proof and countermodel search is guided by generalizations of traditional reasoning patterns.

The development of new intelligent technologies for military use is already yielding a new stage in the arms race that could have serious implications for peace and security across the globe. Governments, international organizations, civil society, and the scientific community face the challenge of promoting legal, ethical, and moral parameters for the development of Artificial Intelligence for military purposes. This article explores a series of concepts that are relevant to the design and development of autonomous weapons and proposes a series of general (political and ethical) and technical considerations that may lead to effective lines of action in the matter.

Gödel modal logics can be seen as extensions of intutionistic modal logics with the prelinearity axiom. In this paper we focus on the algebraic and relational semantics for Gödel modal logics that leverages on the duality between finite Gödel algebras and finite forests, i.e. finite posets whose principal downsets are totally ordered. We consider different subvarieties of the basic variety ????????????\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {GAO}$$\end{document} of Gödel algebras with two modal operators (GAOs for short) and their corresponding classes of forest frames, either with one or two accessibility relations. These relational structures can be considered as prelinear versions of the usual relational semantics of intuitionistic modal logic. More precisely we consider two main extensions of finite Gödel algebras with operators: the one obtained by adding Dunn axioms, typically studied in the fragment of positive classical (and intuitionistic) logic, and the one determined by adding Fischer Servi axioms. We present Jónsson-Tarski like representation theorems for the different types of finite GAOs considered in the paper.

Alchourrón devoted his last years to the analysis of the notion of defeasible conditionalization. He developed a formal system capturing the essentials of this notion. His definition of the defeasible conditional is given in terms of strict implication operator and a modal operator f which is interpreted as a revision function at the language level. In this paper, we will point out that this underlying revision function is more general than the well known AGM revision [4]. In addition, we will give a complete characterization of that more general kind of revision and will show how permits to unify models of revision given by other authors. Alchourrón dedicó sus últimos años al análisis de la noción de condicionalización derrotable. Desarrolló un sistema formal que captura los aspectos esenciales de esta noción. Su definición de condicional revisable está dada en términos del operador de implicación estricta y un operador modal f el cual es interpretado en el nivel del lenguaje como una función de revisión. En este trabajo señalamos que la función de revisión subyacente es más general que la bien conocida función de revisión de AGM. Además, presentaremos una caracterización completa de una clase de revisión más general y mostraremos cómo nuestro modelo permite unificar los modelos de revisión propuestos por otros autores

In this paper we provide a simplified, possibilistic semantics for the logics K45, i.e. a many-valued counterpart of the classical modal logic K45 over the [0, 1]-valued Gödel fuzzy logic \. More precisely, we characterize K45 as the set of valid formulae of the class of possibilistic Gödel frames \, where W is a non-empty set of worlds and \ is a possibility distribution on W. We provide decidability results as well. Moreover, we show that all the results also apply to the extension of K45 with the axiom, provided that we restrict ourselves to normalised Gödel Kripke frames, i.e. frames \ where \ satisfies the normalisation condition \ = 1\).

In this paper we consider the modal logic with both $$\Box $$ and $$\Diamond $$ arising from Kripke models with a crisp accessibility and whose propositions are valued over the standard Gödel algebra $$[0,1]_G$$. We provide an axiomatic system extending the one from Caicedo and Rodriguez (J Logic Comput 25(1):37–55, 2015) for models with a valued accessibility with Dunn axiom from positive modal logics, and show it is strongly complete with respect to the intended semantics. The axiomatizations of the most usual frame restrictions are given too. We also prove that in the studied logic it is not possible to get $$\Diamond $$ as an abbreviation of $$\Box $$, nor vice-versa, showing that indeed the axiomatic system we present does not coincide with any of the mono-modal fragments previously axiomatized in the literature.

In 2015 Dag Prawitz proposed an Ecumenical system where classical and intuitionistic logic could coexist in peace. The classical logician and the intuitionistic logician would share the universal quantifier, conjunction, negation and the constant for the absurd, but they would each have their own existential quantifier, disjunction and implication, with different meanings. Prawitz’ main idea is that these different meanings are given by a semantical framework that can be accepted by both parties. The aim of the present paper is [1] to prove the normalization theorem for the propositional fragment NEp of Prawitz’ ecumenical system, and [2] to show that NEp is sound and complete with respect to a Kripke-style semantics for the language of NEp.

Las investigaciones que Carlos Alchourrón desarrolló en la Filosofía del Derecho se vincularon, desde inicios de la década de 1980, con problemáticas críticas de la Inteligencia Artificial. Su contribución a la construcción de una lógica de las normas se conectó rápidamente con la deducción automática y los Sistemas Expertos Jurídicos. Su preocupación por la cuestión de los conflictos de obligaciones que pueden plantearse en un sistema normativo cuando el juez se enfrenta a la necesidad de emitir un veredicto considerando ciertos hechos, se vinculó, en sus trabajos, con la noción más general de "condicionales derrotables" y con la formalización del Razonamiento No Monótono. Pero su contribución de mayor impacto fue aquella derivada de la búsqueda de una solución al problema de los resultados múltiples que surgen de la derogación de una norma en un sistema jurídico, problema que puede ser visto, en un contexto epistémico, como el de la contracción de un conjunto de creencias. Su enfoque constructivo, que desarrolló con David Makinson, convergió con el axiomático elaborado por Peter Gärdenfors, dando lugar al surgimiento de la teoría de los Cambios Racionales de Creencias, más conocida como AGM, teoría que fue adoptada en IA como un modelo standard para la especificación de las actualizaciones de Bases de Conocimiento, operaciones que, habitualmente, involucran la introducción de una nueva creencia que puede ser inconsistente con las anteriores. ¿Por qué AGM adquirió rápidamente carta de ciudadanía en IA, al punto de que su influencia, medida según la cantidad de menciones bibliográficas del artículo fundacional de 1985, se ha duplicado en la última década? Mostraremos en este trabajo que la teoría apareció en un momento de crisis de la IA y que el alto nivel de abstracción del modelo de AGM era lo que muchos investigadores de la disciplina estaban buscando. El presente artículo, luego de describir la génesis de AGM, realiza un análisis histórico de las circunstancias en las cuales la teoría se introdujo en la IA y una evaluación cuantitativa y cualitativa de su impacto en dicha disciplina. The main lines of research that Carlos Alchourrón developed in the Philosophy of Law, were connected, from the beginnings of the 80' s, to some critical problems in the Artificial Intelligence. His contributions to the construction of a logic of norms were rapidly linked to automatic deduction and the Legal Expert Systems. His concerns for the conflict of obligations that could arise in a normative system when a judge has to issue a ruling taking some particular facts into account were associated in his work with the more general notion of "defeasible conditionals" and with the formalization of Non Monotonic Reasoning. But his most relevant contribution was the one derived from the search of a solution to the problem of multiple outcomes resulting from the derogation of a norm in a legal corpus, which may be considered -within an epistemic framework- analogous to the contraction of a set of beliefs. The constructive approach he worked out with David Makinson converged with Peter Gärdenfors' axiomatic, giving rise to the formulation of the theory of Rational Changes of Belief, which was adopted in AI as a model for the specification of the updates of Knowledge Bases as it usually involves a new belief that may be inconsistent with the old ones.

We prove strong completeness of the □-version and the ◊-version of a Gödel modal logic based on Kripke models where propositions at each world and the accessibility relation are both infinitely valued in the standard Gödel algebra [0,1]. Some asymmetries are revealed: validity in the first logic is reducible to the class of frames having two-valued accessibility relation and this logic does not enjoy the finite model property, while validity in the second logic requires truly fuzzy accessibility relations and this logic has the finite model property. Analogues of the classical modal systems D, T, S4 and S5 are considered also, and the completeness results are extended to languages enriched with a discrete well ordered set of truth constants.