Logika, wiedza i czas. Problemy i metody temporalno-logicznej reprezentacji wiedzy

Abstract

This monograph discusses the problems of implementing formaI logic apparatus to represent and investigate knowledge undergoing changes in time. By 'formal logic apparatus' we understand systems which are a combination of epistemic modal logic systems applied for the description of knowledge and temporal logic systems which enable the expression of the temporaI context. Therefore, we implement various systems of epistemic temporal logic to describe knowledge changing in time. The monograph is divided into seven chapters. Indicated in the title, the concept of knowledge is the subject of the First Chapter. We provide examples of designations of knowledge and point out basic differences in understanding knowledge in selected disciplines. In the case of a classical understanding of knowledge, we refer to Gettier's arguments which indicate the limitations of such reasoning. However, the first chapter does not consist of a full exposition of the theory of knowledge. It should rather be treated in categories signaling the main ideas and problems connected with understanding and defining knowledge. In the Second Chapter we deal with the formalization of the concept of knowledge. We mainly focus on questions connected with modal logies, in particular with the formaI apparatus of modal epistemic logies. Therefore, we discuss basic systems of epistemic logic. We supply a basic notional apparatus of modal epistemic logic concerning knowledge of a singular cognitive subject and the notional apparatus connected with issues of knowledge of groups of cognitive subjects. Formal languages considered here are appropriate for the description of static knowledge which is not subject to changes. In the Third Chapter basic systems of temporaI logic, the superstructure over classical sentence calculus, are called to mind and basic properties of these systems are discussed. These systems are employed for the temporalization of logical systems utilizing the method described in Chapter Four, as well as to build on their basis epistemic systems of temporal logic discussed in Chapter Five. In Chapter Four the problem of methods of temporalizing logical systems is considered. Two basic methods of temporalization are discussed: internal temporalization and outer temporalization. Due to its possibilities for implementation we concentrate on the method of outer temporalization. We describe the Finger-Gabbay method and indicate conditions which should be met so that when joining two logical systems using the mentioned method, the newly created logical system retains the metaproperties of the component systems. We have in mind here properties such as consistency, fullness, and decidability. We also discuss the generalization of the Finger-Gabbay method proposed by Finger and Weiss. In the Fifth Chapter we discuss selected systems of epistemic temporaI logic. We begin the review of selected systems of epistemic temporal logic by presenting systems whose languages possess lower expressive power. We discuss two systems of epistemic temporaI logic of linear time created as a result of the temporalization of system S5 using the method described in Chapter Four. We analyze classes of systems whose semantics is based on the interpreted systems and we investigate properties of epistemic temporaI logic of alternating time. Furthermore, in this Chapter, we provide exampIes of implementations of the discussed systems in philosophy, computer science, and game theory. In Chapter Six we discuss the possibilities of analyzing knowledge by means of the sentence language of temporal logic, superstructure over intuitive logic. We prove that intuitive temporal logic may be treated as epistemic temporaI logic. We analyze the system of minimaI intuitive temporaI logic suggested by EwaId. We prove that proposed axioms create a structure of dependent axioms. We suggest our own, reduced axiomatic system. We also suggest our own version of semantics for minimaI intuitive temporal logic. We examine basic properties of this system and indicate basic differences between the constructed system and its classical counterpart. In the Seventh Chapter we consider issues connected with temporal-epistemic formalization of algorithmic knowledge. This monograph may be placed at the intersection of philosophy and artificial intelligence. Philosophy attempts to render various aspects of knowledge and formalize them by the means of the logic apparatus. Artificial intelligence, on the other hand, makes efforts to implement these elements in artificial systems, in particular in systems based on information technology.