Editing Formal system (section)

==== Formal semantics of logical system ====
{{main|Semantics of logic|Interpretation (logic)|Model theory}}

A ''logical system'' is a deductive system (most commonly [[First-order logic|first order logic]]) together with additional [[non-logical axioms]]. According to [[model theory]], a logical system may be given [[interpretation (logic)|interpretation]]s which describe whether a given [[Structure (mathematical logic)|structure]] - the mapping of formulas to a particular meaning - satisfies a well-formed formula. A structure that satisfies all the axioms of the formal system is known as a [[Model (model theory)|model]] of the logical system. 

A logical system is:
*[[Soundness|Sound]], if each well-formed formula that can be inferred from the axioms is satisfied by every model of the logical system.
*[[Completeness (logic)#Semantic completeness|Semantically complete]], if each well-formed formula that is satisfied by every model of the logical system can be inferred from the axioms.

An example of a logical system is [[Peano arithmetic]]. The standard model of arithmetic sets the [[domain of discourse]] to be the [[nonnegative integer]]s and gives the symbols their usual meaning.<ref>{{cite book |last1=Kaye |first1=Richard |title=Models of Peano arithmetic |date=1991 |publisher=Clarendon Press |location=Oxford |isbn=9780198532132 |page=10 |chapter=1. The Standard Model}}</ref> There are also [[non-standard model of arithmetic|non-standard models of arithmetic]].