A logical system generally consists of a language and a set of sentences of , called provable in . is defined inductively, as being the smallest set of sentences of which contains a given set of -sentences and closed under certain specified operations. The elements of are called the logical axioms of .
|||E. Mendelson, "Introduction to mathematical logic" , v. Nostrand (1964)|
|||J.R. Shoenfield, "Mathematical logic" , Addison-Wesley (1967)|
The phrase "logical axiom" is often more specifically used to distinguish those axioms, in a formal theory, which are concerned with securing the meaning of the logical connectives and quantifiers (cf. Logical calculus), as opposed to the "non-logical axioms" which are the standing hypotheses about the interpretation of the particular function and predicate symbols in the language in which the theory is formulated (cf. Logico-mathematical calculus).
Logical axiom. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Logical_axiom&oldid=15648