Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011...
Transcript of Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011...
![Page 1: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/1.jpg)
TACL’2011 - Marseille, July 26, 2011.
Topological Semantics of Modal Logic
David Gabelaia
![Page 2: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/2.jpg)
Overview
• Personal story
• Three gracious ladies
• Completeness in C-semantics – Quasiorders as topologies
– Finite connected spaces are interior images of the real line
– Connected logics
• Completeness in d-semantics – Incompleteness
– Ordinal completeness of GL
– Completeness techniques for wK4 and K4.Grz
![Page 3: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/3.jpg)
![Page 4: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/4.jpg)
Motivations
• Gödel’s translation
– Bringing intuitionistic reasoning into theclassical setting.
• Tarski’s impetus towards “algebraization”
– Algebra of Topology, McKinsey and Tarski, 1944.
• Quine’s criticism
– Making Modal Logic meaningful in the rest of mathematics
![Page 5: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/5.jpg)
![Page 6: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/6.jpg)
TACL’2011 - Marseille, July 26, 2011.
Three Graces
Topological space (X,)
![Page 7: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/7.jpg)
TACL’2011 - Marseille, July 26, 2011.
Three Graces
Topological space (X,)
Closure Algebra
((X), C)
Derivative Algebra
((X), d)
Heyting Algebra
Op(X)
![Page 8: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/8.jpg)
TACL’2011 - Marseille, July 26, 2011.
Three Graces
Topological space (X,)
Closure Algebra
((X), C)
Derivative Algebra
((X), d)
Heyting Algebra
Op(X)
Hegemone
Cleta Delia
![Page 9: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/9.jpg)
TACL’2011 - Marseille, July 26, 2011.
The discourses of the Graces
• Hegemone talks about open subsets.
U(U V) V
![Page 10: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/10.jpg)
TACL’2011 - Marseille, July 26, 2011.
The discourses of the Graces
• Hegemone talks about open subsets.
U(U V) V
• Cleta can talk about everything Hegemone can:
IA I (-IA IB) IB
– and more:
• A CB subset B is “dense over” A
• CA CB = subsets A and B are “apart”
![Page 11: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/11.jpg)
TACL’2011 - Marseille, July 26, 2011.
The discourses of the Graces
• Hegemone talks about open subsets. U(U V) V
• Cleta can talk about everything Hegemone can: IA I (-IA IB) IB
– and more:
• A CB subset B is “dense over” A
• CA CB = subsets A and B are “apart”
• Delia can talk about everything Cleta can: A dA = CA
– and more:
A dA A is dense-in-itself (dii)
![Page 12: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/12.jpg)
TACL’2011 - Marseille, July 26, 2011.
Three Graces
Closure Algebra
((X), C)
Derivative Algebra
((X), d)
Heyting Algebra
Op(X)
Hegemone
Cleta Delia
![Page 13: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/13.jpg)
TACL’2011 - Marseille, July 26, 2011.
Three Graces
Closure Algebra
((X), C)
Derivative Algebra
((X), d)
Heyting Algebra
Op(X)
Hegemone
Cleta Delia
HC Heyting identities
![Page 14: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/14.jpg)
TACL’2011 - Marseille, July 26, 2011.
Three Graces
Closure Algebra
((X), C)
Derivative Algebra
((X), d)
Heyting Algebra
Op(X)
Hegemone
Cleta Delia
HC
Kuratowski Axioms
C =
C(AB) = CA CB
A CA
CCA = CA
Heyting identities
S4
![Page 15: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/15.jpg)
TACL’2011 - Marseille, July 26, 2011.
Three Graces
Closure Algebra
((X), C)
Derivative Algebra
((X), d)
Heyting Algebra
Op(X)
Hegemone
Cleta Delia
HC
Kuratowski Axioms
C =
C(AB) = CA CB
A CA
CCA = CA
d =
d(AB) = dA dB
ddA A dA
Heyting identities
S4 wK4
![Page 16: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/16.jpg)
Graceful translations
S4
HC
S4.Grz
Gődel Translation
―Box‖ everything
![Page 17: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/17.jpg)
Graceful translations
S4 wK4
K4.Grz HC
S4.Grz
GL
Gődel Translation Splitting Translation
―Box‖ everything Split Boxes
![Page 18: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/18.jpg)
TACL’2011 - Marseille, July 26, 2011.
Syntax and Semantics
Structures Formulas
Log Str ()
K Str Log (K)
Str
Log
![Page 19: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/19.jpg)
TACL’2011 - Marseille, July 26, 2011.
Syntax and Semantics
Log Str ()
K Str Log (K)
K is definable, if
K = Str Log (K)
is complete, if
= Log Str ()
Structures Formulas
Str
Log
![Page 20: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/20.jpg)
TACL’2011 - Marseille, July 26, 2011.
Completeness for Hegemone
• Heyting Calculus (HC) is complete wrt the class of all topological spaces
[Tarski 1938]
• HC is also complete wrt the class of finite topological spaces
• HC is also complete wrt the class of finite partial orders
• Is there an intermediate logic that is topologically incomplete? (Kuznetsov’s Problem).
![Page 21: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/21.jpg)
TACL’2011 - Marseille, July 26, 2011.
Completeness for Cleta
Kuratowski Axioms Axioms of modal S4 C = 0 = 0
C(AB) = CA CB (p q) = p q
A CA p p = 1
CCA = CA p = p
So S4 is definitely valid on all topological spaces
(soundness). How do we know that nothing extra
goes through (completeness)?
![Page 22: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/22.jpg)
TACL’2011 - Marseille, July 26, 2011.
Kripke semantics for S4
• Quasiorders are reflexive-transitive frames.
• Just partial orders with clusters.
• S4 is the logic of all quasiorders.
• Indeed, finite tree-like
quasiorders suffice to
generate S4 (unravelling).
![Page 23: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/23.jpg)
Intermezzo: Gödel Translation (quasi)orderly
p-morphism of “pinching” clusters
HC |- iff S4 |- Tr()
![Page 24: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/24.jpg)
Intermezzo: Gödel Translation (quasi)orderly
Quasiorders Partial orders
“pinching” clusters
embedding
?
![Page 25: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/25.jpg)
Quasiorder as a partially ordered
sum of clusters
External skeleton
(partial order)
![Page 26: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/26.jpg)
Quasiorder as a partially ordered
sum of clusters
External skeleton
(partial order) Internal worlds
(clusters)
![Page 27: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/27.jpg)
Quasiorder as a partially ordered
sum of clusters
External skeleton
(partial order) Internal worlds
(clusters)
![Page 28: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/28.jpg)
Quasiorder as a partially ordered
sum of clusters
External skeleton
(partial order) Internal worlds
(clusters)
The sum
![Page 29: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/29.jpg)
Partially ordered sums
A partial order P (Skeleton)
1
2 3
![Page 30: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/30.jpg)
Partially ordered sums
A partial order P (Skeleton)
Family of frames (Fi)iP
indexed by P
(Components)
1
2 3
F2
F1
F3
![Page 31: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/31.jpg)
Partially ordered sums
A partial order P (Skeleton)
Family of frames (Fi)iP
indexed by P
(Components)
1
2 3
F2
F1
F3
P-ordered sum of (Fi)
P Fi
![Page 32: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/32.jpg)
Partially ordered sums
A partial order P (Skeleton)
Family of frames (Fi)iP
indexed by P
(Components)
1
2 3
F2
F1
F3
P-ordered sum of (Fi)
P Fi
![Page 33: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/33.jpg)
Partially ordered sums
A partial order P (Skeleton)
Family of frames (Fi)iP
indexed by P
(Components)
1
2 3
F2
F1
F3
P-ordered sum of (Fi)
P Fi
internal
(component)
![Page 34: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/34.jpg)
Partially ordered sums
A partial order P (Skeleton)
Family of frames (Fi)iP
indexed by P
(Components)
1
2 3
F2
F1
F3
P-ordered sum of (Fi)
P Fi
external
(skeleton)
![Page 35: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/35.jpg)
Partially ordered sums
A partial order P (Skeleton)
Family of frames (Fi)iP
indexed by P
(Components)
1
2 3
F2
F1
F3
P-ordered sum of (Fi)
P Fi
![Page 36: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/36.jpg)
Quasiorders as topologies
• Topology is generated by upwards closed sets.
![Page 37: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/37.jpg)
Quasiorders as topologies
• Topology is generated by upwards closed sets.
![Page 38: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/38.jpg)
Quasiorders as topologies
• Topology is generated by upwards closed sets.
![Page 39: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/39.jpg)
Quasiorders as topologies
• Topology is generated by upwards closed sets.
![Page 40: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/40.jpg)
C-completeness via Kripke
completeness
Quasiorders
Topological spaces
All spaces for a
logic L
A complete class of Kripke countermodels for a logic L
![Page 41: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/41.jpg)
C-completeness via Kripke
completeness
• Any Kripke complete logic above S4 is topologically
complete.
• There exist topologically complete logics that are
not Kripke complete [Gerson 1975]
– Even above S4.Grz [Shehtman 1998]
• Stronger completeness result by McKinsey and Tarski
(1944):
– S4 is complete wrt any metric separable dense-in-itself
space.
– In particular, LogC(R) = S4.
![Page 42: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/42.jpg)
LogC(R) = S4: Insights
R Finite Quasitrees
Following: G. Bezhanishvili, M. Gehrke. Completeness of S4 with
respect to the real line: revisited, Annals of Pure and Applied Logic, 131
(2005), pp. 287—301.
Interior (open continuous)
maps
![Page 43: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/43.jpg)
TACL’2011 - Marseille, July 26, 2011.
Mapping R onto finite connected quasiorders
... R
...
![Page 44: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/44.jpg)
TACL’2011 - Marseille, July 26, 2011.
Mapping R onto finite connected quasiorders
... R
... ( 0 )
![Page 45: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/45.jpg)
TACL’2011 - Marseille, July 26, 2011.
Mapping R onto finite connected quasiorders
... R
...
![Page 46: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/46.jpg)
TACL’2011 - Marseille, July 26, 2011.
Mapping R onto finite connected quasiorders
... R
... ... ( ( ( ) ) ) ) ( -1 -1/2 -1/4
-1/8
0
![Page 47: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/47.jpg)
TACL’2011 - Marseille, July 26, 2011.
Mapping R onto finite connected quasiorders
... R
...
![Page 48: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/48.jpg)
TACL’2011 - Marseille, July 26, 2011.
Mapping R onto finite connected quasiorders
... R
... ) ( 0
Problems: What if clusters are present?
What if the 3-fork is taken instead of the 2-fork??
![Page 49: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/49.jpg)
Cantor Space
0 1
![Page 50: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/50.jpg)
Cantor Space
0 1 1/3 2/3
![Page 51: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/51.jpg)
Cantor Space
0 1 1/3 2/3
![Page 52: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/52.jpg)
Cantor Space
0 1 1/3 2/3
![Page 53: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/53.jpg)
Cantor Space
0 1 1/3 2/3
![Page 54: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/54.jpg)
Cantor Space
0 1 1/3 2/3
![Page 55: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/55.jpg)
Cantor Space
0 1 1/3 2/3
![Page 56: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/56.jpg)
Cantor Space
![Page 57: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/57.jpg)
Cantor Space
![Page 58: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/58.jpg)
Cantor Space
![Page 59: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/59.jpg)
Cantor Space
It’s fractal-like
![Page 60: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/60.jpg)
Cantor Space
In the limit – Cantor set.
![Page 61: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/61.jpg)
(0,1) mapped onto the fork
0 1
![Page 62: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/62.jpg)
(0,1) mapped onto the fork
0 1 1/3 2/3
![Page 63: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/63.jpg)
(0,1) mapped onto the fork
0 1 1/3 2/3
![Page 64: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/64.jpg)
(0,1) mapped onto the fork
0 1 1/3 2/3
![Page 65: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/65.jpg)
(0,1) mapped onto the fork
0 1 1/3 2/3
![Page 66: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/66.jpg)
(0,1) mapped onto the fork
0 1 1/3 2/3
![Page 67: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/67.jpg)
(0,1) mapped onto the fork
0 1 1/3 2/3
![Page 68: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/68.jpg)
(0,1) mapped onto the fork
![Page 69: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/69.jpg)
(0,1) mapped onto the fork
![Page 70: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/70.jpg)
Problems solved
• It is straightforward to generalize this procedure to a
3-fork and, indeed, to any n-fork.
• Clusters are no problem:
– the Cantor set can be decomposed into infinitely many
disjoint subsets which are dense in it.
– Similarly, an open interval (and thus, any open subset of
the reals) can be decomposed into infinitely many disjoint,
dense in it subsets.
• How about increasing the depth?
![Page 71: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/71.jpg)
Iterating the procedure
![Page 72: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/72.jpg)
Iterating the procedure
![Page 73: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/73.jpg)
Iterating the procedure
0 1 1/3 2/3
![Page 74: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/74.jpg)
Iterating the procedure
0 1 1/3 2/3
![Page 75: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/75.jpg)
Iterating the procedure
0 1 1/3 2/3
![Page 76: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/76.jpg)
Connected logics
• What more can a modal logic say about the topology of R in
C-semantics?
• Consider the closure algebra R+ = ((R), C). Which modal
logics can be generated by subalgebras of R+?
Answer: Any connected modal logic above S4 with fmp. [G. Bezhanishvili, Gabelaia 2010]
• More questions like this – e.g. what about homomorphic
images? What about logics without fmp?
• Recently Philip Kremer has shown strong completeness of S4
wrt the real line!
![Page 77: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/77.jpg)
Story of Delia
• d-completeness doesn’t straightforwardly follow from Kripke completeness.
• Incompleteness theorems.
• Extensions allow automatic transfer of d-completeness of GL.
• Completeness of GL wrt ordinals.
• Completeness of wK4
• Completeness of K4.Grz
• Some other recent results.
![Page 78: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/78.jpg)
Story of Delia (d-semantics)
wK4 – weak K4
wK4-frames are weakly transitive.
Tbilisi-Munich-Marseille is a transit flight,
Tbilisi-Munich-Tbilisi is not really a transit flight.
Axioms for derivation Axioms of wK4 d = 0 = 0
d(AB) = dA dB (p q) = p q
ddA A dA p p p
![Page 79: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/79.jpg)
Story of Delia (d-semantics)
wK4 – weak K4
wK4-frames are weakly transitive.
Axioms for derivation Axioms of wK4 d = 0 = 0
d(AB) = dA dB (p q) = p q
ddA A dA p p p
![Page 80: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/80.jpg)
Story of Delia (d-semantics)
wK4 – weak K4
wK4-frames are weakly transitive.
Tbilisi-Munich-Marseille is a transit flight,
Axioms for derivation Axioms of wK4 d = 0 = 0
d(AB) = dA dB (p q) = p q
ddA A dA p p p
![Page 81: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/81.jpg)
Story of Delia (d-semantics)
wK4 – weak K4
wK4-frames are weakly transitive.
Tbilisi-Munich-Marseille is a transit flight,
Tbilisi-Munich-Tbilisi is not really a transit flight.
Axioms for derivation Axioms of wK4 d = 0 = 0
d(AB) = dA dB (p q) = p q
ddA A dA p p p
![Page 82: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/82.jpg)
wK4-frames
xyz(xRy yRz xz xRz)
• Weak quasiorders (delete any reflexive arrows in a quasiorder).
• Partially ordered sums of weak clusters
clusters with irreflexive points:
![Page 83: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/83.jpg)
Delia is capricious (d-incompleteness)
• S4 is an extension of wK4 (add reflexivity axiom)
• S4 has no d-models whatsoever!!
• S4 is incomplete in d-semantics.
Reason: The relation induced by d is always irreflexive:
xd[x]
![Page 84: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/84.jpg)
Caprice exemplified
Topological non-topological
![Page 85: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/85.jpg)
How capricious is Delia?
Definition: Weak partial orders are obtained from
partial orders by deleting (some) reflexive arrows.
• For any class of weak partial orders of depth n, if
there is a root-reflexive frame in this class with the
depth exactly n, then the logic of this class is
d-incomplete.
![Page 86: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/86.jpg)
Gracious Delia
• Kripke completeness implies d-completeness for extensions of
GL.
• GL is the logic of finite irreflexive trees.
• In d-semantics, GL defines the class of scattered topologies [Esakia 1981]
• GL is d-complete wrt to the class of ordinals.
• GL is the d-logic of . [Abashidze 1988, Blass 1990]
![Page 87: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/87.jpg)
Finite irreflexive trees recursively
• Irreflexive point is an i-tree.
• Irreflexive n-fork is an i-tree.
• Tree sum of i-trees is an i-tree.
![Page 88: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/88.jpg)
Finite irreflexive trees recursively
• Irreflexive point is an i-tree.
• Irreflexive n-fork is an i-tree.
• Tree sum of i-trees is an i-tree.
What is a tree sum?
![Page 89: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/89.jpg)
Finite irreflexive trees recursively
• Irreflexive point is an i-tree.
• Irreflexive n-fork is an i-tree.
• Tree sum of i-trees is an i-tree.
What is a tree sum?
Similar to the ordered sum, but only leaves of a tree
can be “blown up” (e.g. substituted by other
trees).
![Page 90: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/90.jpg)
Tree sum exemplified
![Page 91: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/91.jpg)
Tree sum exemplified
![Page 92: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/92.jpg)
Tree sum exemplified
![Page 93: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/93.jpg)
Tree sum exemplified
![Page 94: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/94.jpg)
Tree sum exemplified
![Page 95: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/95.jpg)
Tree sum exemplified
![Page 96: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/96.jpg)
d-maps
• f: X Y is a d-map iff: – f is open
– f is continuous
– f is pointwise discrete
• d-maps preserve d-validity of modal formulas – so they anti-preserve (reflect) satisfiability.
• One can show that each finite i-tree is an image of an ordinal via a d-map.
• This gives ordinal completeness of GL.
![Page 97: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/97.jpg)
Mapping ordinals to i-trees
… 0 3 2 1
[0] [1]
[r]
![Page 98: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/98.jpg)
Mapping ordinals to i-trees
… 0 3 2 1
[0] [1]
[0]
[r]
![Page 99: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/99.jpg)
Mapping ordinals to i-trees
… 0 3 2 1
[0] [1]
[0] [1]
[r]
![Page 100: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/100.jpg)
Mapping ordinals to i-trees
… 0 3 2 1
[0] [1]
[0] [0] [1]
[r]
![Page 101: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/101.jpg)
Mapping ordinals to i-trees
… 0 3 2 1
[r]
[0] [1]
[0] [0] [1] [1]
[r]
![Page 102: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/102.jpg)
Mapping ordinals to i-trees
![Page 103: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/103.jpg)
Mapping ordinals to i-trees
![Page 104: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/104.jpg)
Mapping ordinals to i-trees
![Page 105: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/105.jpg)
Mapping ordinals to i-trees
…
![Page 106: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/106.jpg)
Mapping ordinals to i-trees
… …
![Page 107: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/107.jpg)
Mapping ordinals to i-trees
… …
![Page 108: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/108.jpg)
Mapping ordinals to i-trees
… …
… …
![Page 109: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/109.jpg)
Mapping ordinals to i-trees
… … …
![Page 110: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/110.jpg)
Mapping ordinals to i-trees
… … …
0 +1 2 2
![Page 111: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/111.jpg)
Ordinals recursively
• 0 is an ordinal
• + 1 is an ordinal
• ordinal sums of ordinals are ordinals
![Page 112: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/112.jpg)
Ordinals recursively
• 0 is an ordinal
• + 1 is an ordinal
• ordinal sums of ordinals are ordinals
What is an ordinal sum?
![Page 113: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/113.jpg)
Ordinals recursively
• 0 is an ordinal
• + 1 is an ordinal
• ordinal sums of ordinals are ordinals
What is an ordinal sum?
Roughly: take an ordinal, take it’s isolated points and plug in
other spaces in place of them.
In the sum, a set is open if:
(a) It’s trace on the original ordinal is open (externally).
(b) it’s intersection with each plugged space is open (internally)
![Page 114: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/114.jpg)
d-morphisms
f: X F is a d-morphism if:
(a) f: X F+ is an interior map.
(b) f is i-discrete (preimages of irreflexive
points are discrete)
(c) f is r-dense (preimages of reflexive
points are dense-in-itself)
• d-morphisms preserve validity.
• We use d-morphisms to obtain d-completeness from
Kripke completeness.
![Page 115: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/115.jpg)
d-completeness for wK4
![Page 116: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/116.jpg)
d-completeness for wK4
![Page 117: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/117.jpg)
d-completeness for wK4
![Page 118: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/118.jpg)
d-completeness for wK4
d-morphism
Recipe: Substitute each reflexive point with a two-point irreflexive cluster.
![Page 119: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/119.jpg)
d-completeness for K4.Grz
• K4.Grz doesn’t admit two-point clusters at all.
• Kripke models for K4.Grz are weak partial orders.
• Finite weak trees suffice.
• How to build a K4.Grz-space that maps d-
morphically onto a given finite weak tree?
• Toy (but key) example: single reflexive point
![Page 120: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/120.jpg)
El’kin space
• A set E, together with a free ultrafilter U.
• nonempty OA is open iff OU
• E is dense-in-itself
• E is a K4.Grz-space (no subset can be
decomposed into two disjoint dense in it sets)
Pictorial representation:
![Page 121: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/121.jpg)
El’kin space
• A set E, together with a free ultrafilter U.
• nonempty OA is open iff OU
• E is dense-in-itself
• E is a K4.Grz-space (no subset can be
decomposed into two disjoint dense in it sets)
Pictorial representation:
d-morphism
![Page 122: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/122.jpg)
Building K4.Grz-space preimages
![Page 123: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/123.jpg)
Building K4.Grz-space preimages
![Page 124: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/124.jpg)
Building K4.Grz-space preimages
![Page 125: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/125.jpg)
Building K4.Grz-space preimages
![Page 126: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/126.jpg)
Building K4.Grz-space preimages
Recipe: Substitute each reflexive point with a copy of Elkin’s Space.
d-morphism
![Page 127: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/127.jpg)
Topo-sums of spaces
A space X (Skeleton)
![Page 128: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/128.jpg)
Topo-sums of spaces
A space X (Skeleton)
Family of spaces (Yi)iX
indexed by X
(Components)
Y2
Y1
Y3
![Page 129: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/129.jpg)
Topo-sums of spaces
A space X (Skeleton)
Family of spaces (Yi)iX
indexed by X
(Components)
Y2
Y1
Y3
X-ordered sum of (Yi)
Y = X Yi
![Page 130: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/130.jpg)
Topo-sums of spaces
A space X (Skeleton)
Family of spaces (Yi)iX
indexed by X
(Components)
Y2
Y1
Y3
X-ordered sum of (Yi)
Y = X Yi
A set UY is open iff it’s trace on the skeleton is open
and its traces on all the components are open.
![Page 131: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/131.jpg)
Some results
• d-completeness of some extensions of K4.Grz “with a
provability smack” [Bezhanishvili, Esakia, Gabelaia 2010]
• d-logics of maximal, submaximal, nodec spaces. [Bezhanishvili, Esakia, Gabelaia, Studia 2005]
• d-logic of Stone spaces is K4. [Bezhanishvili, Esakia, Gabelaia, RSL 2010]
• d-logic of Spectral spaces. [Bezhanishvili, Esakia, Gabelaia 2011]
• d-definability of T0 separation axiom. [Bezhanishvili, Esakia, Gabelaia 2011]
• d-completeness of the GLP. [Beklemishev, Gabelaia 201?]
![Page 132: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/132.jpg)
![Page 133: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/133.jpg)
TACL’2011 - Marseille, July 26, 2011.
Quasiorders as topologies
• Interior is the largest open contained in a set.
![Page 134: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/134.jpg)
TACL’2011 - Marseille, July 26, 2011.
Quasiorders as topologies
• Interior is the largest open contained in a set.
![Page 135: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/135.jpg)
TACL’2011 - Marseille, July 26, 2011.
Quasiorders as topologies
• Interior is the largest open contained in a set.
![Page 136: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/136.jpg)
TACL’2011 - Marseille, July 26, 2011.
Quasiorders as topologies
• Closure takes all the points below.
![Page 137: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/137.jpg)
TACL’2011 - Marseille, July 26, 2011.
Quasiorders as topologies
• Closure takes all the points below.
![Page 138: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/138.jpg)
TACL’2011 - Marseille, July 26, 2011.
Interior fields of sets
Some examples of Interior Fields of Sets in R and their logics:
– B(Op(R)) Boolean combinations of opens S4
– C (R) Finite unions of convex sets S4.Grz
– C (OD(R)) Boolean comb. of open dense subsets S4.Grz.2
– B(C(R)) Countable unions of convex sets Log( )
– All subsets of R with small boundary S4.1
– Nowhere dense and interior dense subsets of R S4.1.2
Question: Which logics arise in this way from R?
![Page 139: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/139.jpg)
TACL’2011 - Marseille, July 26, 2011.
Theorem
Suppose L is an extension of S4 with fmp. Then the
following conditions are equivalent:
(1) L arises from a subalgebra of R+.
(2) L is the logic of a path-connected quasiorder.
(3) L is the logic of a connected space.
(4) L is a logic of a connected Closure Algebra.
Corollary: All logics extending S4.1 with the finite model
property arise from a subalgebra of R+.
![Page 140: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/140.jpg)
TACL’2011 - Marseille, July 26, 2011.
Glueing the finite frames
Suppose L admits the frame:
Then L also admits the frame:
n
n
![Page 141: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/141.jpg)
TACL’2011 - Marseille, July 26, 2011.
Glueing the finite frames
n m
![Page 142: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/142.jpg)
TACL’2011 - Marseille, July 26, 2011.
Glueing the finite frames
n
m
m
n
![Page 143: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/143.jpg)
TACL’2011 - Marseille, July 26, 2011.
Glueing the finite frames
n
m
m
n
![Page 144: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/144.jpg)
TACL’2011 - Marseille, July 26, 2011.
Glueing the finite frames
n m
![Page 145: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/145.jpg)
TACL’2011 - Marseille, July 26, 2011.
Glueing all finite frames
F1 F2 F3
. . .
![Page 146: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/146.jpg)
TACL’2011 - Marseille, July 26, 2011.
Glueing all finite frames
F1 F2 F3
. . .
![Page 147: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/147.jpg)
TACL’2011 - Marseille, July 26, 2011.
Glueing all finite frames
F1 F2 F3
. . .
![Page 148: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/148.jpg)
TACL’2011 - Marseille, July 26, 2011.
Glueing interior maps
F1 F2 F3
. . .
... Q
...
![Page 149: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/149.jpg)
TACL’2011 - Marseille, July 26, 2011.
Glueing interior maps
F1 F2 F3
. . .
... Q
... ... ) ( ) ( ) ( ) ( ) (
![Page 150: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/150.jpg)
TACL’2011 - Marseille, July 26, 2011.
Glueing interior maps
F1 F2 F3
. . .
... Q
... ... ) ) ( ) ( ) ( ) ( (
...
![Page 151: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/151.jpg)
TACL’2011 - Marseille, July 26, 2011.
Going from algebras to topologies
Each closure algebra (A, ) is isomorphic to a subalgebra of X+ for some topological space (X,).
[McKinsey&Tarski, 1944]
Each closure algebra (A, ) is isomorphic to a subalgebra of ((X), R-1) for some quasiorder (X,R).
[Jonsson&Tarski, 1951]
X is a set of Ultrafilters of A and (X, R) is a Stone space of A.
[Bezhanishvili, Mines, Morandi, 2006 ]
![Page 152: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/152.jpg)
TACL’2011 - Marseille, July 26, 2011.
Mapping R onto finite connected quasiorders
... R
...
![Page 153: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/153.jpg)
TACL’2011 - Marseille, July 26, 2011.
Mapping R onto finite connected quasiorders
... R
-/2 2 0 -1/2 1
Q I
...
![Page 154: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/154.jpg)
TACL’2011 - Marseille, July 26, 2011.
Mapping R onto finite connected quasiorders
... R
...
![Page 155: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/155.jpg)
TACL’2011 - Marseille, July 26, 2011.
Mapping R onto finite connected quasiorders
... R
... ( 0 )
![Page 156: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/156.jpg)
TACL’2011 - Marseille, July 26, 2011.
Mapping R onto finite connected quasiorders
... R
...
![Page 157: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/157.jpg)
TACL’2011 - Marseille, July 26, 2011.
Mapping R onto finite connected quasiorders
... R
... ... ( ( ( ) ) ) ) ( -1 -1/2 -1/4
-1/8
0
![Page 158: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/158.jpg)
TACL’2011 - Marseille, July 26, 2011.
Mapping R onto finite connected quasiorders
... R
...
![Page 159: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/159.jpg)
TACL’2011 - Marseille, July 26, 2011.
Mapping R onto finite connected quasiorders
... R
... ) ( 0
![Page 160: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/160.jpg)
TACL’2011 - Marseille, July 26, 2011.
Mapping R onto finite connected quasiorders
... R
... ) ( 0
We can use this map to falsify formulas on R.
![Page 161: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/161.jpg)
TACL’2011 - Marseille, July 26, 2011.
Interior fields of sets
R
Q I
... ... B = {, I, Q, R}
Q = I = ,
R = R.
(B,) – Interior Algebra
![Page 162: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/162.jpg)
TACL’2011 - Marseille, July 26, 2011.
Interior fields of sets
B = {, I, Q, R}
Q = I = ,
R = R.
(B,) – Interior Algebra
pp is valid on B, but not on R.
In R: A(R).(CA ICA)
In B: A B.(CA ICA)
R
Q I
... ...
![Page 163: Topological Semantics of Modal Logiclogica.dmi.unisa.it/tacl/tacl2011/slides/109.pdf · TACL’2011 - Marseille, July 26, 2011. Topological Semantics of Modal Logic David Gabelaia](https://reader033.fdocuments.in/reader033/viewer/2022042401/5f0fd2a17e708231d4460e75/html5/thumbnails/163.jpg)
TACL’2011 - Marseille, July 26, 2011.
Interior fields of sets
B = {, I, Q, R}
Q = I = ,
R = R.
(B,) – Interior Algebra
pp is valid on B, but not on R.
In R: A(R).(CA ICA)
In B: A B.(CA ICA)
Interior field of sets is a Boolean algebra of subsets which is
closed under operators of interior and closure.
R
Q I
... ...