Logical matrix

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 operations; 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.

References

[a1]	R. Wójcicki, "Theory of logical calculi" , Kluwer (1988)