Difference between revisions of "Logical matrix"
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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/l/l060/l060740/l0607401.png" /></td> </tr></table> | <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/l/l060/l060740/l0607401.png" /></td> </tr></table> | ||
| − | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060740/l0607402.png" /> is a non-empty set; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060740/l0607403.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060740/l0607404.png" /> are binary | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060740/l0607402.png" /> is a non-empty set; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060740/l0607403.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060740/l0607404.png" /> are [[binary operation]]s; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060740/l0607405.png" /> is a [[unary operation]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060740/l0607406.png" />. Any formula of propositional logic, constructed from propositional variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060740/l0607407.png" /> by means of the logical connectives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060740/l0607408.png" />, can be regarded as an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060740/l0607409.png" />-place function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060740/l06074010.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060740/l06074011.png" /> are assumed to be variables with range of values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060740/l06074012.png" /> and the logical connectives are interpreted as the corresponding operations of the logical matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060740/l06074013.png" />. A formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060740/l06074014.png" /> is said to be generally valid in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060740/l06074015.png" /> if for any values of the variables in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060740/l06074016.png" /> the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060740/l06074017.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060740/l06074018.png" />. A logical matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060740/l06074019.png" /> is said to be characteristic for a propositional calculus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060740/l06074020.png" /> if the formulas that are generally valid in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060740/l06074021.png" /> are exactly those that are deducible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060740/l06074022.png" />. An example of a logical matrix is the system |
<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/l/l060/l060740/l06074023.png" /></td> </tr></table> | <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/l/l060/l060740/l06074023.png" /></td> </tr></table> | ||
Revision as of 19:13, 13 November 2016
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.
Comments
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
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