Editing Second-order logic (section)

==Deductive systems==

A [[deductive system]] for a logic is a set of [[inference rules]] and logical axioms that determine which sequences of formulas constitute valid proofs. Several deductive systems can be used for second-order logic, although none can be complete for the standard semantics (see below). Each of these systems is [[soundness|sound]], which means any sentence they can be used to prove is logically valid in the appropriate semantics.

The weakest deductive system that can be used consists of a standard deductive system for first-order logic (such as [[natural deduction]]) augmented with substitution rules for second-order terms.{{efn|Such a system is used without comment by {{harvtxt|Hinman|2005}}.}} This deductive system is commonly used in the study of [[second-order arithmetic]].

The deductive systems considered by {{harvtxt|Shapiro|2000}} and {{harvtxt|Henkin|1950}} add to the augmented first-order deductive scheme both comprehension axioms and choice axioms. These axioms are sound for standard second-order semantics. They are sound for Henkin semantics restricted to Henkin models satisfying the comprehension and choice axioms.{{efn|These are the models originally studied by {{harvtxt|Henkin|1950}}.}}