Active logic semantics for a single agent in a static world

For some time we have been developing, and have had significant practical success with, a time-sensitive, contradiction-tolerant logical reasoning engine called the active logic machine (ALMA). The current paper details a semantics for a general version of the underlying logical formalism, active logic. Central to active logic are special rules controlling the inheritance of beliefs in general (and of 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. Furthermore, inspired by the notion that until an agent notices that a set of beliefs is contradictory, that set seems consistent (and the agent therefore reasons with it as if it were consistent), we introduce an "apperception function" that represents an agent's limited awareness of its own beliefs, and serves to modify inconsistent belief sets so as to yield consistent sets. Using these ideas, we introduce a new definition of logical consequence in the context of active logic, as well as a new definition of soundness such that, when reasoning with consistent premises, all classically sound rules remain sound in our new sense. 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.