Knowledge, proof and the Knower

Abstract

The Knower Paradox demonstrates that any theory T which 1) extends Robinson arithmetic Q, 2) includes a predicate K(x) intended to formalize "the formula with godel number x is known by agent i," and 3) contains certain elementary epistemic principles involving K(x) is inconsistent. The purpose of this paper is to show how this paradox may be redeveloped within a system of quantified explicit modal logic in the tradition of Artemov [4] and Fitting [10], [11] which we argue allows for a more faithful formulation of some of the epistemic principles on which it is based. Along the way, we isolate a principle -- the so-called Uniform Barcan Formula [UBF] -- which we show is required to derive an explicit counterpart of the axiom U (i.e. K(⌜K(⌝φ⌍) → φ⌍)) which was used in the original formulation of the Paradox. We argue that since there are independent epistemic reasons to be suspicious of UBF, the paradox may be resolved by abandoning this principle (and thereby U as well).