Modal Logics of Stone Spaces

Interpreting modal diamond as the closure of a topological space, we axiomatize the modal logic of each metrizable Stone space and of each extremally disconnected Stone space. As a corollary, we obtain that S4.1 is the modal logic of the Pelczynski compactification of the natural numbers and S4.2 is the modal logic of the Gleason cover of the Cantor space. As another corollary, we obtain an axiomatization of the intermediate logic of each metrizable Stone space and of each extremally disconnected Stone space. In particular, we obtain that the intuitionistic logic is the logic of the Pelczynski compactification of the natural numbers and the logic of weak excluded middle is the logic of the Gleason cover of the Cantor space.

[1]  Johan van Benthem,et al.  Reasoning About Space: The Modal Way , 2003, J. Log. Comput..

[2]  Sabine Koppelberg,et al.  Handbook of Boolean Algebras , 1989 .

[3]  Guram Bezhanishvili,et al.  Scattered, Hausdorff-reducible, and hereditarily irresolvable spaces , 2003 .

[4]  Guram Bezhanishvili,et al.  Scattered and hereditarily irresolvable spaces in modal logic , 2010, Arch. Math. Log..

[5]  Edwin Hewitt,et al.  A problem of set-theoretic topology , 1943 .

[6]  M. de Rijke,et al.  Modal Logic , 2001, Cambridge Tracts in Theoretical Computer Science.

[7]  Countably Complementable,et al.  LINEAR ORDERINGS , 2006 .

[8]  G. Bezhanishvili,et al.  The modal logic of β(N) , 2009 .

[9]  W. Wistar Comfort,et al.  The Theory of Ultrafilters , 1974 .

[10]  Zbigniew Semadeni,et al.  Banach spaces of continuous functions , 1971 .

[11]  Johan van Benthem,et al.  Modal Logics of Space , 2007, Handbook of Spatial Logics.

[12]  Johan van Benthem,et al.  Euclidean Hierarchy in Modal Logic , 2003, Stud Logica.

[13]  Michael Zakharyaschev,et al.  Modal Logic , 1997, Oxford logic guides.

[14]  Johan van Benthem,et al.  Handbook of Spatial Logics , 2007 .

[15]  Guram Bezhanishvili,et al.  The modal logic of $${\beta(\mathbb{N})}$$ , 2009, Arch. Math. Log..

[16]  David Gabelaia,et al.  Modal definability in topology , 2001 .

[17]  M. Stone,et al.  Algebraic characterizations of special Boolean rings , 1937 .

[18]  A. Chagrov,et al.  Modal Logic (Oxford Logic Guides, vol. 35) , 1997 .

[19]  Guram Bezhanishvili,et al.  Completeness of S4 with respect to the real line: revisited , 2005, Ann. Pure Appl. Log..

[20]  A. Tarski,et al.  The Algebra of Topology , 1944 .

[21]  M. Stone.  The theory of representations for Boolean algebras , 1936 .