Theory of Logical Calculi pp 189-251 | Cite as

# Logical Matrices

## 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).

## Keywords

Boolean Algebra Subdirect Product Heyting Algebra Modal Algebra Implication Algebra## Preview

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## References

- 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.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
Malcev [1970].Google Scholar**see**