Hybrid Languages and Temporal Logic

Hybridization is a method invented by Arthur Prior for extending the expressive power of modal languages. Although developed in interesting ways by Robert Bull, and by the Sofia school (notably, George Gargov, Valentin Goranko, Solomon Passy and Tinko Tinchev) the method remains little known. In our view this has deprived temporal logic of a valuable tool. The aim of the paper is to explain why hybridization is useful in temporal logic. We make two major points, the first technical, the second conceptual. First, we show that hybridization gives rise to well-behaved logics that exhibit an interesting synergy between modal and classical ideas. This synergy, obvious for hybrid languages with full first-order expressive strength, is demonstrated for a weaker local language capable of defining the Until operator, we provide a minimal axiomatization, and show that in a wide range of temporally interesting cases extended completeness results can be obtained automatically. Second, we argue that the idea of sorted atomic symbols which underpins the hybrid enterprise can be developed further. To illustrate this, we discuss the advantages and disadvantages of a simple hybrid language which can quantify over paths

La lógica híbrida como extensión de las lógicas modal y temporal

Developed by Arthur Prior, Temporal Logic allows to represent temporal information on a logical system using modal (temporal) operators such as P, F, H or G, whose intuitive meaning is “it was sometime in the Past...”, “it will be sometime in the Future...”, “it Has always been in the past...” and “it will always Going to be in the future...” respectively. Valuation of formulae built from these operators are carried out on Kripke semantics, so Modal Logic and Temporal Logic are consequently related. In fact, Temporal Logic is an extension of Modal one. Even when both logics mechanisms are able to formalize modal-temporal information with some accuracy, they suffer from a lack of expressiveness which Hybrid Logic can solve. Indeed, one of the problems of Modal Logic consists in its incapacity of naming specific points inside a model. As Temporal Logic is based on it, it cannot make such a thing neither. But First-Order Logic does can by means of constants and equality relation. Hybrid Logic, which results from combining Modal Logic and First-Order Logic, may solve this shortcoming. The main aim of this paper is to explain how Hybrid Logic emanates from Modal and Temporal ones in order to show what it adds to both logics with regard to information representation, why it is more expressive than them and what relation it maintains with the First-Order Correspondence Language.La lógica temporal fue creada por Arthur Prior para representar información temporal en un sistema lógico mediante operadores modales-temporales como P, F, H o G. Intuitivamente tales operadores pueden entenderse respectivamente como “fue alguna vez en el pasado...”, “será alguna vez en el futuro...”, “ha sido siempre en el pasado...” y “será siempre en el futuro...”. La evaluación de las fórmulas construidas a partir de ellos se lleva a cabo en semánticas kripkeanas y, de este modo, la lógica modal y la temporal están relacionadas. Sin embargo, aunque sus mecanismos permiten formalizar la información modal-temporal con cierta precisión, ambas lógicas adolecen de un problema de expresividad que la lógica híbrida es capaz de solventar. En efecto, uno de los problemas de la lógica modal reside en su incapacidad para nombrar puntos concretos dentro de un modelo. La lógica temporal, al basarse en ella, tampoco puede hacerlo. Pero la lógica de primer orden sí es capaz gracias a las constantes y a la relación de identidad. La lógica híbrida, que resulta de combinar la lógica modal con la lógica de primer orden, sería una solución a este problema. El principal objetivo de este artículo consiste en explicar el origen de la lógica híbrida a partir de la modal-temporal para mostrar qué añade a ambos sistemas en la representación de información, porqué es más expresiva que ellos y qué relación guarda con el lenguaje de correspondencia de la lógica de primer orden

Hybrid type theory: a quartet in four movements

This paper sings a song -a song created by bringing together the work of four great names in the history of logic: Hans Reichenbach, Arthur Prior, Richard Montague, and Leon Henkin. Although the work of the first three of these authors have previously been combined, adding the ideas of Leon Henkin is the addition required to make the combination work at the logical level. But the present paper does not focus on the underlying technicalities (these can be found in Areces, Blackburn, Huertas, and Manzano [to appear]) rather it focusses on the underlying instruments, and the way they work together. We hope the reader will be tempted to sing along

To Teach Modal Logic: An Opinionated Survey

I aim to promote an alternative agenda for teaching modal logic chiefly inspired by the relationships between modal logic and philosophy. The guiding idea for this proposal is a reappraisal of the interest of modal logic in philosophy, which do not stem mainly from mathematical issues, but which is motivated by central problems of philosophy and language. I will point out some themes to start elaborating a guide for a more comprehensive approach to teach modal logic, and consider the contributions of dual-process theories in cognitive science, in order to explore a pedagogical framework for the proposed point of view.Comment: Proceedings of the Fourth International Conference on Tools for Teaching Logic (TTL2015), Rennes, France, June 9-12, 2015. Editors: M. Antonia Huertas, Jo\~ao Marcos, Mar\'ia Manzano, Sophie Pinchinat, Fran\c{c}ois Schwarzentrube

La lógica híbrida como extensión de la lógica temporal

El objetivo de este trabajo consiste en explicar el origen de la lógica híbrida a partir de la modal/ temporal para mostrar qué añade a ambos sistemas en la representación de información, porqué es más potente que ellos y qué relación guarda con el lenguaje de correspondencia de la lógica de primer orden. La lógica temporal permite la representación de información temporal en un sistema lógico. Su creador fue Arthur Prior, cuya propuesta se basa en definir operadores temporales para representar enunciados como “fue alguna vez en el pasado p”, “será alguna vez en el futuro p”, “ha sido siempre en el pasado p” o “será siempre en el futuro p”. La evaluación de tales enunciados se lleva a cabo en semánticas kripkeanas. Lógica temporal y lógica modal en consecuencia están relacionadas. Sin embargo, la primera no tiene la capacidad de nombrar puntos concretos dentro de un modelo, por ejemplo. La lógica temporal tampoco puede hacerlo al fundamentarse en ella. Pero la lógica de primer orden sí puede mediante las constantes y la relación de identidad. La lógica híbrida es el resultado de combinar la lógica modal con la lógica de primer orden para realizar tal cosa.Abstract. The aim of this paper is to explain the origin of Hybrid Logic from modal/temporal one to show how much it contributes to the representation of formulae, why it is stronger than them and which relation holds with first order correspondence language. Temporal Logic allows the representation of temporal information in logical systems. Its origin goes back to Arthur Prior´s works, whose proposal consists in defining temporal operators which can be applied to propositions to represent sentences such as “p was sometime in the past”, “p will be sometime in the future p”, “p has always been in the past” or “p will always be in the future”. The evaluation of such a sentences is carrying out on Kripke semantics. Temporal Logic and Modal Logic are consequently related. However Modal Logic is not able to name points inside models, for instance. And as Temporal Logic is based on it, it cannot make such a thing neither. First Order Logic on the contrary does can through constants and equality relation. Hybrid Logic is the result of combining Modal Logic and First Order Logic to make that thing

Design of strapdown gyroscopes for a dynamic environment Semiannual report, Dec. 1967 - May 1968

Systems analysis, design, and operating characteristics of strapdown gyroscopes for dynamic environmen