Defining Knowledge in Terms of Belief: The Modal Logic Perspective

Abstract

The question of whether knowledge is definable in terms of belief, which has played an important role in epistemology for the last 50 years, is studied here in the framework of epistemic and doxastic logics. Three notions of definability are considered: explicit definability, implicit definability, and reducibility, where explicit definability is equivalent to the combination of implicit definability and reducibility. It is shown that if knowledge satisfies any set of axioms contained in S5, then it cannot be explicitly defined in terms of belief. S5 knowledge can be implicitly defined by belief, but not reduced to it. On the other hand, S4.4 knowledge and weaker notions of knowledge cannot be implicitly defined by belief, but can be reduced to it by defining knowledge as true belief. It is also shown that S5 knowledge cannot be reduced to belief and justification, provided that there are no axioms that involve both belief and justification.

Principles of Knowledge, Belief and Conditional Belief

“…This result is in contrast with Theorem 4.8 in (Halpern et al, 2009a), from which it follows that the belief modality cannot be explicitly defined in the logic (S4.x) K + (KD45) B + {KB1,KB2}, and so on for any x ∈ {.2, .3, .3.2, .4}. We see here that the increase in expressivity due to the addition of the interaction axiom KB3 plays an important role in bridging the gap between belief and knowledge.…”

“…As the following proposition shows, knowledge can be defined in terms of belief if the logic of knowledge is S4.4, but not if the logic of knowledge is S4 and S4.x, where x ranges over {.2, .3, .3.2}. This result can be contrasted with Theorem 4.1 in (Halpern et al, 2009a), from which it follows that the knowledge modality cannot be explicitly defined in the logic (S4.4) K + (KD45) B + {KB1,KB2}. We see once again that the increase in expressivity due to the addition of the interaction axiom KB3 plays an important role in bridging the gap between belief and knowledge.…”

“…Roughly speaking, this means that the agent could believe a proposition which turns out by chance to be true, although this belief cannot qualify as knowledge if one considers the whole epistemic context. An explanation of this notion of "epistemic luck" in logical terms is proposed in (Halpern et al, 2009a) (but also see (Prichard, 2004)). …”

“…This study is subsequently applied to epistemic logic in (Halpern et al, 2009a). Three notions of definability emerge from this work: explicit definability, implicit definability and reducibility.…”

“…We need to add interaction axioms. In (Halpern et al, 2009a), only the interaction axioms KB1 and KB2 suggested by Hintikka (Hintikka, 1962) are considered. In this section, we will also add the interaction axiom KB3, suggested by Lenzen (Lenzen, 1978), since this axiom is characteristic of the notion of strong belief, as we explained in Section 4.1.…”