Towards Axiomatisation of Social Epistemic Logic

Abstract

This thesis is a theoretical development of epistemic logic to problems concerning the relationship between perception and knowledge. We closely follow the approach of Seligman, Liu and Girard’s “Logic in the Community” [55] which proposes a two-dimensional multi-agent epistemic logic, in which the model operator K (knows) is supplemented with a ‘social’ operators which allow reasoning about relations between agents. The logic also uses operators from hybrid logic, such as nominals n, which name agents, the perspective shifting operator @ₙ, which moves to agent n’s perspective, and the downarrow operator ↓ᵪ, which names the current agent a rigid name x. We review axiomatic and tableaux systems for this logic and propose a new axiomatisation and completeness proof, using the step-by-step method, first for the basic logic and then for the case of downarrow, which is more involved. While the framework is very general, we are specifically interested in a perceptual agent-oriented operator S (sees). Axioms for the interaction of seeing and knowing are explored. We then consider dynamic extensions of the basic logic with public announcement and “observational” announcement, in which information is given only to agents who can see the announcer. Various subtleties are discussed and connections are made to dynamic epistemic logic.