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Difference between revisions of "Logical axiom"

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A logical system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060680/l0606801.png" /> generally consists of a language <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060680/l0606802.png" /> and a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060680/l0606803.png" /> of sentences of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060680/l0606804.png" />, called provable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060680/l0606805.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060680/l0606806.png" /> is defined inductively, as being the smallest set of sentences of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060680/l0606807.png" /> which contains a given set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060680/l0606808.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060680/l0606809.png" />-sentences and closed under certain specified operations. The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060680/l06068010.png" /> are called the logical axioms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060680/l06068011.png" />.
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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>
 
 
  
 
====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
This article was adapted from an original article by V.E. Plisko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article