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Logical Matrices

  • Ryszard Wójcicki
Part of the Synthese Library book series (SYLI, volume 199)

Abstract

A (logical) matrix for a propositional language S is a couple M = (A, D) where A is an algebra similar to S and D is a subset of the set A of elements of A. If A = S the matrix is called Lindenbaum. If the only congruence in A consistent with D (matrix congruence) is identity, the matrix is called simple. A valuation h for S in M is a homomorphism h from S into A. Given a class K of matrices for S, we define αK (X) iff Xα is satisfied in all (A, D) ∈ K by all valuations h in (A, D) i.e. haD whenever hX ∈ D. If C = K , C is said to be determined by K. ∠K is defined so that ∠K = K (ø).

Keywords

Boolean Algebra Subdirect Product Heyting Algebra Modal Algebra Implication Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    This interpretation is implicit in Lukasiewicz and Tarski [1930]. In an explicit form it was developed and examined by many students of many-valued logics. It should be mentioned that quite recently its adequacy has been put in doubt, see 4.3.0.Google Scholar
  2. 2.
    In an explicit manner the idea of using logical matrices to define consequence operations was stated in Loś and Suszko [19581Google Scholar
  3. 6.
    For this notion as well as many others discussed in this note see Malcev [1970].Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1988

Authors and Affiliations

  • Ryszard Wójcicki
    • 1
  1. 1.Section of Logic, Institute of Philosophy and SociologyPolish Academy of SciencesPoland

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