Logic of Awareness for Nested Knowledge

Abstract

Reasoning abilities of human beings are limited. Logics that treat logical inference for human knowledge should reflect these limited abilities. Logic of awareness is one of those logics. In the logic, what an agent with a limited reasoning ability actually knows at a given moment (explicit knowledge) is distinguished from the ideal knowledge that an agent obtains by performing all possible inferences with what she already knows (implicit knowledge). This paper proposes a logic for explicit knowledge. In particular, we focus more on nested explicit knowledge, which means another agent's knowledge that an agent actually knows at a given moment. We develope a new formalization of two ideas and propose Kripke-style semantics. The first idea is the effect on an agent's reasoning ability by a state of an agent's awareness. We incorporate a relation on possible worlds called an indistinguishable relation to represent ignorance due to lack of awareness. The second idea is a state of each agent's awareness in the other agent's mind. We incorporate a non-empty finite sequence of agents called \textit{a chain of belief for awareness}. Our logic is called Awareness Logic with Partitions and Chains (ALPC). Employing an example, we show how nested explicit knowledge is formalized with our logic. Thereafter, we propose the proof system and prove the completeness. Finally, we discuss directions for extending and applying our logic and conclude. Our logic offers a foundation for a formal representation of human knowledge. We expect that the logic can be applied to computer science and game theory by describing and analyzing strategic behavior in a game and practical agent communication.