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Difference between revisions of "Minimal functional calculus"

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''minimal predicate calculus''
 
''minimal predicate calculus''
  
 
The calculus of predicates given by all axiom schemes of the [[Minimal propositional calculus|minimal propositional calculus]] and by the usual quantifier axiom schemes and deduction rules, that is,
 
The calculus of predicates given by all axiom schemes of the [[Minimal propositional calculus|minimal propositional calculus]] and by the usual quantifier axiom schemes and deduction rules, that is,
  
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<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/m/m063/m063820/m0638201.png" /></td> </tr></table>
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$$\forall xA(x)\supset A(t),\quad A(t)\supset\exists xA(x)$$
  
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(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063820/m0638202.png" /> an arbitrary term), [[Modus ponens|modus ponens]] and
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($t$ an arbitrary term), [[Modus ponens|modus ponens]] and
  
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<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/m/m063/m063820/m0638203.png" /></td> </tr></table>
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$$\frac{C\supset A(a)}{C\supset\forall xA(x)},\quad\frac{A(a)\supset C}{\exists xA(x)\supset C}$$
  
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(provided the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063820/m0638204.png" /> does not occur in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063820/m0638205.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063820/m0638206.png" />).
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(provided the variable $a$ does not occur in $A(x)$ and $C$).
  
 
====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></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Church,  "Introduction to mathematical logic" , '''1''' , Princeton Univ. Press  (1956)</TD></TR></table>

Latest revision as of 08:11, 12 August 2014

minimal predicate calculus

The calculus of predicates given by all axiom schemes of the minimal propositional calculus and by the usual quantifier axiom schemes and deduction rules, that is,

$$\forall xA(x)\supset A(t),\quad A(t)\supset\exists xA(x)$$

($t$ an arbitrary term), modus ponens and

$$\frac{C\supset A(a)}{C\supset\forall xA(x)},\quad\frac{A(a)\supset C}{\exists xA(x)\supset C}$$

(provided the variable $a$ does not occur in $A(x)$ and $C$).

References

[1] A. Church, "Introduction to mathematical logic" , 1 , Princeton Univ. Press (1956)
How to Cite This Entry:
Minimal functional calculus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Minimal_functional_calculus&oldid=12860
This article was adapted from an original article by S.K. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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