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Logical matrix

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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.


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References

[a1] R. Wójcicki, "Theory of logical calculi" , Kluwer (1988)
How to Cite This Entry:
Logical matrix. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Logical_matrix&oldid=39749
This article was adapted from an original article by V.E. Plisko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article