In Guram Bezhanishvili, Giovanna D'Agostino, George Metcalfe & Thomas Studer (eds.), Advances in Modal Logic 12, proceedings of the 12th conference on "Advances in Modal Logic," held in Bern, Switzerland, August 27-31, 2018. pp. 219-235 (2018)

Yifeng Ding
University of California, Berkeley
Scroggs's theorem on the extensions of S5 is an early landmark in the modern mathematical studies of modal logics. From it, we know that the lattice of normal extensions of S5 is isomorphic to the inverse order of the natural numbers with infinity and that all extensions of S5 are in fact normal. In this paper, we consider extending Scroggs's theorem to modal logics with propositional quantifiers governed by the axioms and rules analogous to the usual ones for ordinary quantifiers. We call them Π-logics. Taking S5Π, the smallest normal Π-logic extending S5, as the natural counterpart to S5 in Scroggs's theorem, we show that all normal Π-logics extending S5Π are complete with respect to their complete simple S5 algebras, that they form a lattice that is isomorphic to the lattice of the open sets of the disjoint union of two copies of the one-point compactification of N, that they have arbitrarily high Turing-degrees, and that there are non-normal Π-logics extending S5Π.
Keywords Propositional quantifiers  Scroggs’s theorem  lattice of modal logics  algebraic semantics
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Propositional Quantifiers in Modal Logic.Kit Fine - 1970 - Theoria 36 (3):336-346.
On Modal Logic with Propositional Quantifiers.R. A. Bull - 1969 - Journal of Symbolic Logic 34 (2):257-263.
Algebraic Logic.Paul Richard Halmos - 1962 - New York, NY, USA: Chelsea.
For Some Proposition and so Many Possible Worlds.Kit Fine - 1969 - Dissertation, University of Warwick

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