On semantics of first-order justification logic with binding modalities

We introduce the first order logic of proofs $FOL{P}^{\mathrm{\square }}$ in the joint language combining justification terms and binding modalities. The main issue is Kripke–style semantics for this logic. We describe models for $FOL{P}^{\mathrm{\square }}$ in terms of valuations of individual variables instead of introducing constants to the language. This approach requires a new format of the evidence function. This allows us to assign semantic meaning to formulas that contain free variables. The main results are soundness and completeness of $FOL{P}^{\mathrm{\square }}$ with respect to the described semantics.