The true belief components of Plato's tripartite definition of knowledge as justified true belief are represented in formal epistemology by modal logic and its possible worlds semantics. At the same time, the justification component of Plato's definition did not have a formal representation. This

We introduce an extension of the propositional logic of single-conclusion proofs by the second order variables denoting the reference constructors of the type “the formula which is proved by x. ” The resulting Logic of Proofs with References, FLPref, is shown to be decidable, and to enjoy soundness and completeness with respect to the intended provability semantics. We show that FLPref provides a complete test of admissibility of inference rules in a sound extension of arithmetic. Key words: proof theory, explicit modal logic, single conclusion logic of proofs, proof term, reference, unification, admissible inference rule. 1

An issue of an epistemic logic with justification has been discussed since the early 1990s. Such a logic, along with the usual knowledge operator ✷F (F is known), should contain assertions t:F (t is a justification for F), which gives a more nuanced and realistic model of knowledge. In this paper, we build two systems of epistemic logic with justification: the minimal one—S4LP—which is an extension of the basic epistemic logic S4 by an appropriate calculus of justification corresponding to the logic of proofs LP, and S4LPN—which is S4LP augmented by the explicit negative introspection principle ¬(t:F) → ✷¬(t:F). Epistemic semantics for both systems are suggested. Completeness and specific properties of S4LP and S4LPN, reflecting the explicit character of those systems, are established. 1