On the reasoning of real-world agents: Toward a semantics for active logic

The current paper details a restricted semantics for active logic, a time-sensitive, contradictiontolerant logical reasoning formalism. Central to active logic are special rules controlling the inheritance of beliefs in general, and beliefs about the current time in particular, very tight controls on what can be derived from direct contradictions (P&¬P ), and mechanisms allowing an agent to represent and reason about its own beliefs and past reasoning. Using these ideas, we introduce a new definition of model and of logical consequence, as well as a new definition of soundness such that, when reasoning with consistent premises, all classically sound rules are sound for active logic. However, not everything that is classically sound remains sound in our sense, for by classical definitions, all rules with contradictory premises are vacuously sound, whereas in active logic not everything follows from a contradiction.