On an axiom of predicate calculus from ``Gödel's incompleteness theorems'' by R. Smullyan (Q1584141)

The author discusses one error found in the book by \textit{R.~Smullyan} [Gödel's incompleteness theorems, Oxford University Press, New York (1992; Zbl 0787.03003)], p. 58, Theorem~B. The axioms \(L_1\) -- \(L_7\), \(N_1\) -- \(N_9\), \(N_{12}\) from this book need to be refined. In this paper it is claimed that the error found may be corrected by formulating axiom \(L_7\) in the following way [similar to \textit{D.~Kalish} and \textit{R.~Montague}, Arch. Math. Log. Grundlagenforsch. 7, 81-101 (1965; Zbl 0166.00105)] \[ (t_1=t_2\supset(X_1t_1X_2\supset X_1t_2X_2)), \] where \(t_1\) and \(t_2\) are terms and \(X_1t_1X_2\) is an atomic formula.