First-Order Fischer Servi Logic

We prove the completeness of a first-order analogue of the Fischer Servi logic $\mathsf{F}\mathsf{S}$ with respect to its expected birelational semantics. To this end we introduce the notion of the $\mathit{\text{trace}}\mathit{\text{model}}$ and, much like in a canonical model argument, prove a truth lemma. We conclude by examining a number of other first-order Fischer Servi logics, including the first-order analogue of $\mathsf{F}\mathsf{S}\mathsf{S}\mathsf{4}$, whose completeness can be similarly proved.