Knowledge Representation: Modalities, Conditionals, and Nonmonotonic Reasoning

Abstract

The aim of the present chapter is to overview three important tools for knowledge representation that are strongly interrelated. All three can be traced back to a fundamental limitation of classical logic: its connectives are truth-functional, which does not allow to reason about some concepts such as modalities and “if-then” relationships between propositions. To witness, most of the students in an introductory course on logic have a hard time to accept that the implication “if A then B” should be identified with “A is false or B is true”. Indeed, such an identification leads to validities that are rather counter-intuitive, such as “B implies A implies B” or “A implies B, or B implies A”. In introductory courses it is often omitted that the above interpretation of the so-called material implication was subject of much concern among scholars in the past. Their work led to the development of several families of formalisms that will be presented in this chapter: modal logics, conditional logics, and nonmonotonic formalisms. The next three sections detail the definitions of each of these: the modal logics K and S5, the conditional logics due to Stalnaker and Lewis, and the preferential and rational nonmonotonic reasoning formalisms. We then study the relationship between conditional logics and dynamic epistemic logics. The latter are a family of modal logics that got popular recently. We show that they can be viewed as particular logics of indicative conditionals: they are in the Stalnaker family and violate all of Lewis’s principles.