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Difference between revisions of "Implicative normal form"

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A [[Propositional form|propositional form]] of the type
 
A [[Propositional form|propositional form]] of the type
  
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$$C_1\supset(C_2\supset\ldots(C_n\supset\bot)\ldots),$$
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$$C_1\supset(C_2\supset\dotsb(C_n\supset\bot)\cdots),$$
  
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where all the $C_i$, $i=1,\ldots,n$, have the form
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where all the $C_i$, $i=1,\dotsc,n$, have the form
  
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$$C_{i1}\supset(C_{i2}\supset\ldots(C_{im_i}\supset\bot)\ldots).$$
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$$C_{i1}\supset(C_{i2}\supset\dotsb(C_{im_i}\supset\bot)\cdots).$$
  
−
Here, each $C_{ij}$ ($i=1,\ldots,m$; $j=1,\ldots,m_i$) is either a variable or the negation of a variable, and $\bot$ is the logical symbol denoting falsehood. For each propositional formula $A$ one can construct an implicative normal form $B$ classically equivalent to it and containing the same variables as $A$. Such a $B$ is called an implicative normal form of $A$.
+
Here, each $C_{ij}$ ($i=1,\dotsc,m$; $j=1,\dotsc,m_i$) is either a variable or the negation of a variable, and $\bot$ is the logical symbol denoting falsehood. For each propositional formula $A$ one can construct an implicative normal form $B$ classically equivalent to it and containing the same variables as $A$. Such a $B$ is called an implicative normal form of $A$.
  
 
====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 13:03, 14 February 2020

A propositional form of the type

$$C_1\supset(C_2\supset\dotsb(C_n\supset\bot)\cdots),$$

where all the $C_i$, $i=1,\dotsc,n$, have the form

$$C_{i1}\supset(C_{i2}\supset\dotsb(C_{im_i}\supset\bot)\cdots).$$

Here, each $C_{ij}$ ($i=1,\dotsc,m$; $j=1,\dotsc,m_i$) is either a variable or the negation of a variable, and $\bot$ is the logical symbol denoting falsehood. For each propositional formula $A$ one can construct an implicative normal form $B$ classically equivalent to it and containing the same variables as $A$. Such a $B$ is called an implicative normal form of $A$.

References

[1] A. Church, "Introduction to mathematical logic" , 1 , Princeton Univ. Press (1956)
How to Cite This Entry:
Implicative normal form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Implicative_normal_form&oldid=44609
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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