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 α ∈ K ⊨ (X) iff X ⊢ α is satisfied in all (A, D) ∈ K by all valuations h in (A, D) i.e. ha ∈ D whenever hX ∈ D. If C = K ⊨, C is said to be determined by K. ∠K is defined so that ∠K = K ⊨(ø).
KeywordsBoolean Algebra Subdirect Product Heyting Algebra Modal Algebra Implication Algebra
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- 1.This interpretation is implicit in Lukasiewicz and Tarski . 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.In an explicit manner the idea of using logical matrices to define consequence operations was stated in Loś and Suszko [19581Google Scholar
- 6.For this notion as well as many others discussed in this note see Malcev .Google Scholar