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''second-order variable''
 
''second-order variable''
  
 
A variable whose values can be predicates (cf. [[Predicate|Predicate]]). In the formal structure of an axiomatic system, predicate variables differ from individual variables (cf. [[Individual variable|Individual variable]]) by the fact that formulas may be substituted for them. Thus, in second-order predicate calculus, if in the axiom
 
A variable whose values can be predicates (cf. [[Predicate|Predicate]]). In the formal structure of an axiomatic system, predicate variables differ from individual variables (cf. [[Individual variable|Individual variable]]) by the fact that formulas may be substituted for them. Thus, in second-order predicate calculus, if in the axiom
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074380/p0743801.png" /></td> </tr></table>
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$$
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\forall x  \phi ( x)  \rightarrow  \phi ( t),
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$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074380/p0743802.png" /> is a predicate variable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074380/p0743803.png" />-place predicates, then any formula with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074380/p0743804.png" /> distinguished variables may be taken for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074380/p0743805.png" />. Here the result of substituting a formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074380/p0743806.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074380/p0743807.png" /> distinguished variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074380/p0743808.png" /> for the predicate variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074380/p0743809.png" /> in the atomic formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074380/p07438010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074380/p07438011.png" /> are individual constants, is the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074380/p07438012.png" /> obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074380/p07438013.png" /> by simultaneously replacing the free occurrences of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074380/p07438014.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074380/p07438015.png" />, respectively.
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$  x $
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is a predicate variable for $  n $-
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place predicates, then any formula with $  n $
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distinguished variables may be taken for $  t $.  
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Here the result of substituting a formula $  t $
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with $  n $
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distinguished variables $  z _ {1} \dots z _ {n} $
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for the predicate variable $  x $
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in the atomic formula $  x ( y _ {1} \dots y _ {n} ) $,  
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where $  y _ {1} \dots y _ {n} $
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are individual constants, is the formula $  t ( y _ {1} | z _ {1} \dots y _ {n} | z _ {n} ) $
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obtained from $  t $
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by simultaneously replacing the free occurrences of $  z _ {1} \dots z _ {n} $
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by $  y _ {1} \dots y _ {n} $,  
 +
respectively.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Church,  "Introduction to mathematical logic" , '''1''' , Princeton Univ. Press  (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Takeuti,  "Proof theory" , North-Holland  (1987)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Church,  "Introduction to mathematical logic" , '''1''' , Princeton Univ. Press  (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Takeuti,  "Proof theory" , North-Holland  (1987)</TD></TR></table>

Latest revision as of 08:07, 6 June 2020


second-order variable

A variable whose values can be predicates (cf. Predicate). In the formal structure of an axiomatic system, predicate variables differ from individual variables (cf. Individual variable) by the fact that formulas may be substituted for them. Thus, in second-order predicate calculus, if in the axiom

$$ \forall x \phi ( x) \rightarrow \phi ( t), $$

$ x $ is a predicate variable for $ n $- place predicates, then any formula with $ n $ distinguished variables may be taken for $ t $. Here the result of substituting a formula $ t $ with $ n $ distinguished variables $ z _ {1} \dots z _ {n} $ for the predicate variable $ x $ in the atomic formula $ x ( y _ {1} \dots y _ {n} ) $, where $ y _ {1} \dots y _ {n} $ are individual constants, is the formula $ t ( y _ {1} | z _ {1} \dots y _ {n} | z _ {n} ) $ obtained from $ t $ by simultaneously replacing the free occurrences of $ z _ {1} \dots z _ {n} $ by $ y _ {1} \dots y _ {n} $, respectively.

References

[1] A. Church, "Introduction to mathematical logic" , 1 , Princeton Univ. Press (1956)
[2] G. Takeuti, "Proof theory" , North-Holland (1987)
How to Cite This Entry:
Predicate variable. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Predicate_variable&oldid=48278
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article