A b s t r a c t. Let ${\bf N}_n = \lg \{1,2,\dots,n\}, =, 1,2,\dots,n \rg$ be the relational system of cardinality $n$ with identity and with designated elements $1,2,\dots,n$. In the paper we consider the predicate logic $L({\bf N}_n)$ determined by the relational system ${\bf N}_n$ (cf. [1]).