A Note on Fixed Points in Justification Logics and the Surprise Test Paradox

Abstract: 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 Smory`nski'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 G\"odel-L\"ob provability logic.