Some Philosophical Aspects of Reasoning about Knowledge

The first part of the talk will highlight the way in which assumptions widespread in the literature on reasoning about knowledge generate the traditional philosophical problem of scepticism. The problematic assumptions to be discussed here do not include logical omniscience, but are equally serious. My particular concern will be with the so-called Brouwerian axiom: p $ K~Kp (if p is false, an agent knows that she does not know p). This axiom follows from the 'negative introspection' axiom ~Kp $ K~Kp and the 'truth' axiom Kp $ p. It corresponds to the condition that the accessibility relation for knowledge be symmetric, and holds on the partition model of knowledge. Let s be a situation in which an agent gains some knowledge p of the external world (Kp), believes that she does so (BKp) and has consistent beliefs (so ~B-Kp). Let s* be a situation in which p is false (-p) but the input from the external world to the agent is the same as in s, so that the agent has the same beliefs in s* as in s. Thus ~B~Kp also holds in s*. Given that knowledge implies belief, it follows that -K~Kp also holds in s*, and therefore that the Brouwerian axiom fails in s*. A defender of the Brouwerian axiom must deny that there can be such pairs of situations s and s*, but this is tantamount to denying that there can be knowledge of the external world. In fact the outlook for the axiom is even worse, since a similar argument shows it to imply that an agent who has consistent beliefs and believes only when he believes that he knows (Bp $ BKp) has no false beliefs: but obviously one cannot guarantee the truth of one's beliefs about the external world merely by satisfying conditions of internal coherence. Computational problems associated with the axiom will also be briefly mentioned (some connected with G0ders second incompleteness theorem and some with agents with limited ability to survey the contents of their own memories). Since the truth axiom is basic to the meaning of 'knowledge' , and negative introspection implies the Brouwerian axiom given truthfulness, the failure of the Brouwerian axiom implies the failure of negative introspection ('positive introspection', Kp $ KKp, is problematic too but will not be discussed in this talk).