Classical (Aristotelean or Boolean) logics provide a solid foundation for mathematical reasoning, but are limited in expressivity and necessarily incomplete. Effective understanding of logic in the modern world entails for the instructor and advanced students an understanding of the wider context. This chapter surveys standard extensions used in mathematical reasoning, artificial intelligence and cognitive science, and natural language reasoning and understanding, as well as inherent limitations on reasoning and computing. Initial technical extensions include equality of terms, integer arithmetic and quantification over sets and relations. To deal with natural reasoning, the chapter explores temporal and modal logics, fuzzy logic and probabilistic models, and relevance logic. Finally, the chapter considers limitations to logic and knowledge, via an overview of the fundamental results of Turing, Gödel, and others, and their connection to the state of mathematics, computing and science in the modern world.
Representations, Affordances, and Interactive Systems
