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 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).
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References
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.
In an explicit manner the idea of using logical matrices to define consequence operations was stated in Loś and Suszko [19581
For this notion as well as many others discussed in this note see Malcev [1970].
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© 1988 Springer Science+Business Media Dordrecht
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Wójcicki, R. (1988). Logical Matrices. In: Theory of Logical Calculi. Synthese Library, vol 199. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-6942-2_4
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DOI: https://doi.org/10.1007/978-94-015-6942-2_4
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