Journal of Symbolic Logic

The Lattice of Modal Logics: An Algebraic Investigation

W. J. Blok

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Modal logics are studied in their algebraic disguise of varieties of so-called modal algebras. This enables us to apply strong results of a universal algebraic nature, notably those obtained by B. Jonsson. It is shown that the degree of incompleteness with respect to Kripke semantics of any modal logic containing the axiom $\square p \rightarrow p$ or containing an axiom of the form $\square^mp \leftrightarrow\square^{m + 1}p$ for some natural number $m$ is $2^{\aleph_0}$. Furthermore, we show that there exists an immediate predecessor of classical logic (axiomatized by $p \leftrightarrow \square p$) which is not characterized by any finite algebra. The existence of modal logics having $2^{\aleph_0}$ immediate predecessors is established. In contrast with these results we prove that the lattice of extensions of S4 behaves much better: a logic extending S4 is characterized by a finite algebra iff it has finitely many extensions and any such logic has only finitely many immediate predecessors, all of which are characterized by a finite algebra.

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J. Symbolic Logic, Volume 45, Issue 2 (1980), 221-236.

First available in Project Euclid: 6 July 2007

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Blok, W. J. The Lattice of Modal Logics: An Algebraic Investigation. J. Symbolic Logic 45 (1980), no. 2, 221--236.

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