On some classes of formulas in S5 which are pre-complete relative to existential expressibility

Existential expressibility for all k-valued functions was proposed by A. V. Kuznetsov and later was investigated in more details by S. S. Marchenkov. In the present paper, we consider existential expressibility in the case of formulas defined by a logical calculus and find out some conditions for a system of formulas to be closed relative to existential expressibility. As a consequence, it has been established some pre-complete as to existential expressibility classes of formulas in some finite extensions of the paraconsistent modal logic S5.