Difference between revisions of "Logical axiom"
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| + | $#C+1 = 11 : ~/encyclopedia/old_files/data/L060/L.0600680 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 $. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Mendelson, "Introduction to mathematical logic" , v. Nostrand (1964)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.R. Shoenfield, "Mathematical logic" , Addison-Wesley (1967)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Mendelson, "Introduction to mathematical logic" , v. Nostrand (1964)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.R. Shoenfield, "Mathematical logic" , Addison-Wesley (1967)</TD></TR></table> | ||
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====Comments==== | ====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|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|Logico-mathematical calculus]]). | 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|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|Logico-mathematical calculus]]). | ||
Latest revision as of 04:11, 6 June 2020
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=15648
Logical axiom. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Logical_axiom&oldid=15648
This article was adapted from an original article by V.E. Plisko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article