Two systems of epistemic logic

Abstract

Drawing on the results of chapters 2 to 4, two non-normal, multimodal axiomatic systems for both knowledge (k) and being in a position to know (K) are introduced—an idealized system and a weaker, more realistic system. Both share important theorems governing the complex operators ‘¬K¬K’ and ‘¬K¬K’, whose availability will be of crucial importance in later chapters. Unlike the realistic system, the idealized system requires subjects to be logically omniscient and must therefore ultimately be rejected in favour of the realistic system. A semantic characterization of the idealized system is devised that shows it to be sound and allows us to invalidate principles we previously found unacceptable for independent reasons. Since the realistic system is weaker, this result implies that it too has these features. Both systems imply that each of ⌜¬K¬Kφ‎⌝, ⌜K¬Kφ‎⌝, and ⌜¬K¬Kφ‎⌝ encodes a luminous condition. The scenario of the unmarked clock presents a prima facie case against this implication. It is shown that the relevant anti-luminosity argument presupposes the principle that being in a position to know (K) distributes across provable conditionals—a principle that has been shown to be deeply problematic and that the realistic system is designed to flout.