 For more information on second-order logic, see the entry on  second-order and higher-order logic,  as well as Church 1956 and Enderton 1972. Very recommended is Henkin 1996 as well as some of the papers in The Life and Work of Leon Henkin (see Manzano, Sain, &amp; Alonso 2014), in particular: “Changing a Semantics, Opportunism or courage?” (Andréka, van Benthem, Bezhanishvili and Németi 2014), “Henkin on Completeness” (Manzano 2014b), and “April the 19th” (Manzano 2014a).