Logical equations in monadic logic

A logical formula $F(\mathbf X,\mathbf P)$ can be treated as an equation to be satisfied by the solutions $\mathbf X_0(\mathbf P)$ for the predicates $\mathbf X$ with the expressions $\mathbf P$ as parameters (if there are such solutions). J. McCarthy [8] considers the parameterization of the solutions, gives the general solution in the case of propositional logic and states the problem for other logics. We find the general solution for the formulas in the first-order language with monadic predicates and equality. The solutions are obtained via quantifier elimination and parametrized by $\epsilon$-terms. Bibl. – 10 titles.