Logical axiom

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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 $.


[1] E. Mendelson, "Introduction to mathematical logic" , v. Nostrand (1964)
[2] 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).

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Logical axiom. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.E. Plisko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article