A system
where 
is a non-empty set; 
; 
are binary operations; and 
is a unary operation on 
. Any formula of propositional logic, constructed from propositional variables 
by means of the logical connectives 
, can be regarded as an 
-place function on 
if 
are assumed to be variables with range of values 
and the logical connectives are interpreted as the corresponding operations of the logical matrix 
. A formula 
is said to be generally valid in 
if for any values of the variables in 
the value of 
belongs to 
. A logical matrix 
is said to be characteristic for a propositional calculus 
if the formulas that are generally valid in 
are exactly those that are deducible in 
. An example of a logical matrix is the system
where
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 
that is characteristic for the intuitionistic propositional calculus.
References
| [a1] | R. Wójcicki, "Theory of logical calculi" , Kluwer (1988) |
How to Cite This Entry:
Logical matrix. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Logical_matrix&oldid=16307
This article was adapted from an original article by V.E. Plisko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article
