Justifications, Ontology, and Conservativity

An ontologically transparent semantics for justifications that interprets justifications as sets of formulas they justify has been recently presented by Artemov. However, this semantics of modular models has only been studied for the case of the basic justification logic J, corresponding to the modal logic K. It has been left open how to extend and relate modular models to the already existing symbolic and epistemic semantics for justification logics with additional axioms, in particular, for logics of knowledge with factive justifications. We introduce modular models for extensions of J with any combination of the axioms (jd), (jt), (j4), (j5), and (jb), which are the explicit counterparts of standard modal axioms. After establishing soundness and completeness results, we examine the relationship of modular models to more traditional symbolic and epistemic models. This comparison yields several new semantics, including symbolic models for logics of belief with negative introspection (j5) and models for logics with the axiom (jb). Besides pure justification logics, we also consider logics with both justifications and a belief/knowledge modal operator of the same strength. In particular, we use modular models to study the conditions under which the addition of such an operator to a justification logic yields a conservative extension.