On tautologies in $\omega^+$-valued logic

The connection is established in this paper between the tautologies of the $\omega^+$-valued predicate logic studied in [1] and the tautologies of m-valued logic for various $m<\omega$. As a consequence it is proven that the set of tautologies of $\omega^+$-valued predicate logic is an forallexist-set. An algorithm is constructed which, for any arbitrary formula of $\omega^+$-valued logic, recognizes whether or not that formula is an $\omega^+$-valued tautology; one axiomatization is proposed for the $\omega^+$-valued propositional logic.