Basics of Second-Order Predicate Logic

The article presents the basics of second-order predicate logic (SOL). The need for a symbolic representation of the general quantifier is pointed out. A distinction is made between the first-order predicate logic (FOL) and the second-order predicate logic (predicates of predicates, relations of relations). The syntax and semantics of the second-order predicate logic are introduced. Logical and non-logical designators and operators, terms, rules for forming logical formulas, status of variables, and rules for variable substitution are introduced. Reference is made to Henkin's semantics of controlled predicates, and an axiomatic system of second-order predicate logic. Russell's analogy for the axiom of choice and methods of proving the validity of the deduction for second-order predicate logic are given.