# Logical axiom

A logical system $ S $
generally consists of a language $ L $
and a set $ T $
of sentences of $ L $,
called provable in $ S $.
$ T $
is defined inductively, as being the smallest set of sentences of $ L $
which contains a given set $ A $
of $ L $-
sentences and closed under certain specified operations. The elements of $ A $
are called the logical axioms of $ S $.

#### References

[1] | E. Mendelson, "Introduction to mathematical logic" , v. Nostrand (1964) |

[2] | J.R. Shoenfield, "Mathematical logic" , Addison-Wesley (1967) |

#### Comments

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).

**How to Cite This Entry:**

Logical axiom.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Logical_axiom&oldid=47708