Source: Notre Dame J. Formal Logic Volume 47, Number 4
(2006), 525-540.
We outline the Gödel-McKinsey-Tarski Theorem on embedding of Intuitionistic Propositional Logic Int into modal logic
S4 and further developments which led to the Generalized
Embedding Theorem. The latter in turn opened a full-scale comparative
exploration of lattices of the (normal) extensions of modal propositional
logic S4, provability logic GL, proof-intuitionistic
logic KM, and others, including Int. The present paper is
a contribution to this part of the research originated from the
Gödel-McKinsey-Tarski Theorem. In particular, we show that the lattice ExtInt of intermediate logics is likely to be the only constructing block with which ExtS4, the lattice of the extensions of S4, can be formed. We, however, advise the reader that our exposition is different from the historical lines along which some of the results discussed below came to light. Part 1, presented here, deals mostly with structural issues of extensions of logics, where algebraic semantics, though underlying this approach, is used merely occasionally. Part 2 will be devoted to algebraic analysis of the Embedding Theorem.
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