View source for Logical matrix

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A system

$$ 
\mathfrak M  =  \langle  M ;  D , \& , \lor , \supset , \neg \rangle ,
$$

where  $  M $
is a non-empty set;  $  D \subseteq M $; 
$  \& , \lor , \supset $
are [[binary operation]]s; and  $  \neg $
is a [[unary operation]] on  $  M $. 
Any formula of propositional logic, constructed from propositional variables  $  p _ {1} \dots p _ {n} $
by means of the logical connectives  $  \& , \lor , \supset , \neg $, 
can be regarded as an  $  n $-
place function on  $  M $
if  $  p _ {1} \dots p _ {n} $
are assumed to be variables with range of values  $  M $
and the logical connectives are interpreted as the corresponding operations of the logical matrix  $  \mathfrak M $. 
A formula  $  \mathfrak A $
is said to be generally valid in  $  \mathfrak M $
if for any values of the variables in  $  M $
the value of  $  \mathfrak A $
belongs to  $  D $. 
A logical matrix  $  \mathfrak M $
is said to be characteristic for a propositional calculus  $  K $
if the formulas that are generally valid in  $  \mathfrak M $
are exactly those that are deducible in  $  K $. 
An example of a logical matrix is the system

$$ 
\langle  \{ 0 , 1 \} ;  \{ 1 \} , \& , \lor , \supset , \neg \rangle ,
$$

where

$$ 
x \& y  =  \min \{ x , y \} ,
$$

$$ 
x \lor y  =  \max \{ x , y \} ,
$$

$$ 
x \supset y  =  \max \{ 1 - x , y \} ,
$$

$$ 
\neg x  =  1 - x .
$$

This logical matrix is characteristic for the classical propositional calculus. t logic','../p/p110060.htm','Set theory','../s/s084750.htm','Syntax','../s/s091900.htm','Undecidability','../u/u095140.htm','Unsolvability','../u/u095800.htm','ZFC','../z/z130100.htm')" style="background-color:yellow;">K. Gödel proved that it is impossible to construct a logical matrix with a finite set  $  M $
that is characteristic for the intuitionistic propositional calculus.

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Wójcicki,   "Theory of logical calculi" , Kluwer  (1988)</TD></TR></table>