TR-2007015: Justification Logics and Conservative Extensions

Several justification logics have evolved, starting with the logic LP, [2]. These can be thought of as explicit versions of modal logics, or logics of knowledge or belief in which the unanalyzed necessity operator has been replaced with a family of explicit justification terms. Modal logics come in various strengths. For their corresponding justification logics, differing strength is reflected in different vocabularies. What we show here is that for justification logics corresponding to modal logics extending T, extensions are actually conservative. Our method of proof is very simple, and general enough to also handle several justification logics not directly corresponding to modal logics. Our methods do not, however, allow us to prove comparable results for justification logics corresponding to modal logics that do not extend T. That is, we are able to handle explicit logics of knowledge, but not explicit logics of belief. This remains open.