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Algebraic logic, where does it stand today? (English) Zbl 1111.03053

From the abstract: This is a survey paper on algebraic logic, but not so-called abstract algebraic logic (AAL). The paper focuses on cylindric algebras, which were proposed to prove the completeness theorem of first-order logic by algebraic methods. The author relates the algebraic notion of neat embedding to the metalogical ones of provability, interpolation and omitting types in variants of first-order logic. Furthermore he relates the algebraic notion of atom-canonicity for a class of Boolean algebras with operators to the metalogical one of omitting types for the corresponding logic. In the paper 8 open problems are presented to be solved. These problems are central ones in the field of algebraic logic.

MSC:

03Gxx Algebraic logic
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
03G15 Cylindric and polyadic algebras; relation algebras
03B10 Classical first-order logic
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