In logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory.
First-order logic uses only variables that range over individuals (elements of the domain of discourse); second-order logic has these variables as well as additional variables that range over sets of individuals. For example, the second-order sentence says that for every set P of individuals and every individual x, either x is in P or it is not (this is the principle of bivalence). Second-order logic also includes variables quantifying over functions, and other variables as explained in the section Syntax below. Both first-order and second-order logic use the idea of a domain of discourse (often called simply the "domain" or the "universe"). The domain is a set of individual elements which can be quantified over.
Read more about Second-order Logic: Syntax and Fragments, Semantics, Expressive Power, Deductive Systems, Non-reducibility To First-order Logic, Metalogical Results, History and Disputed Value, Relation To Computational Complexity
Famous quotes containing the word logic:
“...some sort of false logic has crept into our schools, for the people whom I have seen doing housework or cooking know nothing of botany or chemistry, and the people who know botany and chemistry do not cook or sweep. The conclusion seems to be, if one knows chemistry she must not cook or do housework.”
—Ellen Henrietta Swallow Richards (1842–1911)