knower paradox

After introducing Fitch's paradox of knowability and the knower paradox, the paper critically discusses the dialetheist unified solution to both problems that Beall and Priest have proposed. It is first argued that the dialetheist approach to the knower paradox can withstand the main objections against it, these being that the approach entails an understanding of negation that is intolerably weak (allowing one to stay in agreement with something that one negates) and that it commits dialetheists to jointly accept and reject the same thing. The lesson of the knower paradox, according to dialetheism, is that human knowledge is inconsistent. The paper also argues that this inconsistency has not been shown by dialetheists to be wide enough in its scope to justify their approach to Fitch's problem. The connection between the two problems is superficial and therefore the proposed unified solution fails.

In this note we study the effect of adding fixed points to justification logics. We introduce two extensions of justification logics: extensions by fixed point (or diagonal) operators, and extensions by least fixed points. The former is a justification version of Smorynski's Diagonalization Operator Logic, and the latter is a justification version of Kozen's modal $\mu$-calculus. We also introduce fixed point extensions of Fitting's quantified logic of proofs, and formalize the Knower Paradox and the Surprise Test Paradox in these extensions. By interpreting a surprise statement as a statement for which there is no justification, we give a solution to the self-reference version of the Surprise Test Paradox in quantified logic of proofs. We also give formalizations of the Surprise Test Paradox in timed modal epistemic logics, and in Gödel-Löb provability logic.