A number of significant results concerning the first-order functional or predicate calculus (with identity) date from a paper published in 1915 by Leopold Löwenheim (1878–1957), a mathematician of Schröder's school. In this paper, "Über Möglichkeiten im Relativkalkül" (Mathematische Annalen 76 [1915]: 447–470), Löwenheim showed how the problem of deciding the validity of formulas in this calculus reduces to the problem of determining the validity of formulas in which only two-place predicate letters occur. Since (from the point of view of decidability) such formulas are accordingly no less general than arbitrary formulas of the calculus, we know from a later result, by Alonzo Church, that the decision problem for this class is unsolvable. However, Löwenheim was able to provide a decision procedure for a more restricted class of formulas, those in which only one-place predicate letters occur. He also showed that no...