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Logical equations in monadic logic
Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part XI, Tome 358 (2008), pp. 251-270 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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G. Mints; T. Hoshi. Logical equations in monadic logic. Zapiski Nauchnykh Seminarov POMI, Studies in constructive mathematics and mathematical logic. Part XI, Tome 358 (2008), pp. 251-270. http://geodesic.mathdoc.fr/item/ZNSL_2008_358_a12/

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