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Modal logics based on the derivative operation in topological spaces Philippe Balbiani, Levan Uridia Toulouse, Madrid Outline 1. Modal logic and topology: basic results 2. Modal logic and topology: further direction 3. The derivative operation in topological spaces 4. Iterating the derivative operation 5. On the modal logic of iterated derivative Modal logic and topology: basic results A topological space is a pair F = (X , τ ) where I I X is a nonempty set τ is a set of subsets of X satisfying the following conditions: I I I ∅ ∈ τ and X ∈ τ if A, B ∈ τ then A ∩ B ∈ τ if (Ai )i is a collection of subsets of X such that Ai ∈ τ for S every i then i Ai ∈ τ Let F = (X , τ ) be a topological space I F is called an Alexandroff space if it satisfies the following condition: I if (Ai )i is a collection of subsets of X such that Ai ∈ τ for T every i then i Ai ∈ τ Modal logic and topology: basic results Let F = (X , τ ) be a topological space I The elements of τ are called open sets I For A ⊆ X , a point x ∈ X is called a co-limit point of A iff there exists an open neighborhood B of x such that B ⊆ A ∪ {x} I For A ⊆ X , t(A) will denote the set of co-limit points of A and In(A) will denote A ∩ t(A) I Remark that In(A) is the greatest open set contained in A (called the interior of A) Modal logic and topology: basic results Let F = (X , τ ) be a topological space I The complements of open sets are called closed sets I For A ⊆ X , a point x ∈ X is called a limit point of A iff for all open neighborhoods B of x, A ∩ (B \ {x}) is nonempty I For A ⊆ X , d(A) will denote the set of limit points of A and Cl(A) will denote A ∪ d(A) I Remark that Cl(A) is the least closed set containing A (called the closure of A) Modal logic and topology: basic results Let F = (X , τ ) be a topological space I t(A ∩ B) = t(A) ∩ t(B) and d(A ∪ B) = d(A) ∪ d(B) I A ∩ t(A) ⊆ t(t(A)) and d(d(A)) ⊆ A ∪ d(A) I t(X \ A) = X \ d(A) and d(X \ A) = X \ t(A) Modal logic and topology: basic results Let F = (X , τ ) be a topological space I In(X ) = X and Cl(∅) = ∅ I In(A ∩ B) = In(A) ∩ In(B) and Cl(A ∪ B) = Cl(A) ∪ Cl(B) I In(A) = In(In(A)) and Cl(Cl(A)) = Cl(A) I In(A) ⊆ A and A ⊆ Cl(A) I In(X \ A) = X \ Cl(A) and Cl(X \ A) = X \ In(A) Modal logic and topology: basic results Let F = (X , τ ) be a topological space (separation axioms) I F is called a T0 -space iff for each pair of different points there exists an open set containing one and not containing the other I F is called a Td -space iff for each x ∈ X there exists an open neighborhood A of x such that {x} is closed in A I F is called a T1 -space iff for each pair of different points there exists an open set containing exactly one of the points I F is called a T2 -space iff for each pair x, y ∈ X of different points there exists disjoint open neighborhoods of x and y Modal logic and topology: basic results Syntax of modal logic I I φ ::= p | ⊥ | ¬φ | (φ ∨ ψ) | φ Abbreviations I I I Topological interpretation I I Standard definitions for the remaining Boolean operations ♦φ ::= ¬¬φ The interpretation of φ is the interior of φ’s interpretation As a result, I The interpretation of ♦φ is the closure of φ’s interpretation Modal logic and topology: basic results Topological semantics for modal logic I Let M = (X , τ, V ) where I I I Truth of a modal formula φ at a point x in X is defined as follows: I I I F = (X , τ ) is a topological space V is a valuation function: p 7−→ V (p) ⊆ X M, x |= p iff x ∈ V (p) M, x |= φ iff there exists O ∈ τ such that x ∈ O and for all y ∈ O, M, y |= φ As a result, I M, x |= ♦φ iff for all O ∈ τ , if x ∈ O then there exists y ∈ O such that M, y |= φ Modal logic and topology: basic results Topological semantics for modal logic I Let M = (X , τ, V ) where I I I Truth of a modal formula φ at a point x in X is defined as follows: I I I F = (X , τ ) is a topological space V is a valuation function: p 7−→ V (p) ⊆ X M, x |= p iff x ∈ V (p) M, x |= φ iff x ∈ In({y ∈ X : M, y |= φ} As a result, I M, x |= ♦φ iff x ∈ Cl({y ∈ X : M, y |= φ} Modal logic and topology: basic results Neighborhood semantics for modal logic I Let M = (W , N, V ) where I I I Truth of a modal formula φ at a possible world x in W is defined as follows: I I I (W , N) is a neighborhood structure V is a valuation function: p 7−→ V (p) ⊆ X M, x |= p iff x ∈ V (p) M, x |= φ iff there exists O ⊆ W such that O ∈ N(x) and for all y ∈ O, M, y |= φ As a result, I M, x |= ♦φ iff for all O ⊆ W , if O ∈ N(x) then there exists y ∈ O such that M, y |= φ Modal logic and topology: basic results Let F = (W , R) be an S4-relational structure I A subset A of W is called an upset of F iff I I A subset A of W is called a downset of F iff I I x ∈ A and xRy imply y ∈ A x ∈ A and xR −1 y imply y ∈ A Remark that I I if A is an upset then W \ A is a downset if A is a downset then W \ A is an upset Modal logic and topology: basic results For a given S4-relational structure F = (W , R), we define the topology τR on W by declaring I the upsets of F to be open I the downsets of F to be closed Remark that I I the topological space ts(F) = (W , τR ) is an Alexandroff space the least neighborhood of x ∈ W is I I the interior of a set A ⊆ W is I I R(x) = {y ∈ W : xRy } {x ∈ W : R(x) ⊆ A} the closure of a set A ⊆ W is I {x ∈ W : R(x) ∩ A 6= ∅} Modal logic and topology: basic results For a given topological space F = (X , τ ), we define the binary relation Rτ on W by I xRτ y iff x ∈ Cl({y }) Remark that I the relational structure rs(F) = (W , Rτ ) is an S4-structure I (W , Rτ ) is a partial order iff F is T0 Modal logic and topology: basic results Let F = (W , R) be an S4-relational structure, ts(F) = (W , τR ) be the associated topological space and rs(ts(F)) = (W , RτR ) be the associated relational structure. Then I RτR = R Let F = (W , τ ) be a topological space, rs(F) = (W , Rτ ) be the associated relational structure and ts(rs(F)) = (W , τRτ ) be the associated topological space. Then I τ ⊆ τRτ I τ = τRτ iff F is Alexandroff Modal logic and topology: basic results Topo-bisimulation: Let M = (X , τ, V ) and M0 = (X 0 , τ 0 , V 0 ) be topo-models I A topo-bisimulation between M and M0 is a binary relation Z ⊆ X × X 0 such that if xZx 0 then I I I I for all Boolean variables p, x ∈ V (p) iff x 0 ∈ V 0 (p) for all O ∈ τ , if x ∈ O then there exists O 0 ∈ τ 0 such that x 0 ∈ O 0 and for all y 0 ∈ O 0 , there exists y ∈ O such that yZy 0 for all O 0 ∈ τ 0 , if x 0 ∈ O 0 then there exists O ∈ τ such that x ∈ O and for all y ∈ O, there exists y 0 ∈ O 0 such that yZy 0 x and x 0 are topo-bisimilar iff there exists a topo-bisimulation Z between M and M0 such that xZx 0 Modal logic and topology: basic results Theorem (Aiello and van Benthem): Let M = (X , τ, V ) and M0 = (X 0 , τ 0 , V 0 ) be topo-models. If x and x 0 are topo-bisimilar then I for all formulas φ, M, x |= φ iff M0 , x 0 |= φ. Theorem (Aiello and van Benthem): Let M = (X , τ, V ) and M0 = (X 0 , τ 0 , V 0 ) be finite topo-models. If for all formulas φ, M, x |= φ iff M0 , x 0 |= φ then I x and x 0 are topo-bisimilar. Modal logic and topology: basic results Topological semantics for modal logic I Let M = (X , τ, V ) be a topological model I I I I We say that φ is true in M iff φ is true at every point in M Let F = (X , τ ) be a topological space We say that φ is valid in F iff φ is true in every model based on F Let C be a class of topological spaces I We say that φ is valid in C iff φ is valid in every topological space of C Modal logic and topology: basic results Topological interpretation of some valid formulas I > ↔ >: the whole space is open I ♦⊥ ↔ ⊥: the empty set is closed I (φ ∧ ψ) ↔ φ ∧ ψ: open sets are closed under finite intersections I ♦(φ ∨ ψ) ↔ ♦φ ∨ ♦ψ: closed sets are closed under finite unions I φ ↔ φ: the interior operator is idempotent I ♦φ ↔ ♦♦φ: the closure operator is idempotent I φ → φ: the interior of any set is contained in the set I φ → ♦φ: the closure of any set contains the set Modal logic and topology: basic results Several standard modal logics and their axiomatizations I K is axiomatized by (K ) (p → q) → (p → q) (N) from φ, infer φ I T is axiomatized by K + (T ) p → p I K 4 is axiomatized by K + (4) p → p I wK 4 is axiomatized by K + (w4) p ∧ p → p I S4 is axiomatized by K + (T ) + (4) I S5 is axiomatized by K + (T ) + (4) + (B) p → ♦p Modal logic and topology: basic results Proposition: Let Top be the class of all topological spaces. Then I (K ) is valid in Top, I (N) preserves validity in Top, I (T ) is valid in Top, I (4) is valid in Top, i.e. the modal logic S4 is sound with respect to interpreting as interior. Modal logic and topology: basic results The canonical topo-model of S4 is the triple Mc = (Xc , τc , Vc ) where I Xc is the set of all maximal S4-consistent sets of formulas I τc is the set generated by arbitrary unions of basic sets, i.e. sets of the form Oφ = {x ∈ Xc : φ ∈ x} I Vc : p 7−→ Vc (p) ⊆ Xc is the valuation function defined by x ∈ Vc (p) iff p ∈ x Truth lemma: For all formulas φ, for all x ∈ Xc , I Mc , x |= φ iff φ ∈ x Modal logic and topology: basic results Theorem (completeness): I S4 is the logic of the class of all topological spaces Theorem (McKinsey and Tarski): I S4 is complete for any dense-in-itself metric separable space. Theorem (completeness in special spaces): I S4 is the logic of the rational line I S4 is the logic of the real line Modal logic and topology: basic results Thus, such topological properties as I being dense-in-itself I being metric I being separable are not definable in the basic modal language Topological definability: Let C be a class of topological spaces I C is topologically definable iff there exists a set Γ of modal formulas such that for each topological space F = (X , τ ), we have I F ∈ C iff F |= Γ Modal logic and topology: basic results Topological completeness of S4 tells us that I the class Top of all topological spaces is topo-definable by the formula > Proposition (Gabelaia): I Neither compactness nor connectedness is topo-definable I None of the separation axioms T0 , Td , T1 and T2 is topo-definable Proposition (Gabelaia): Let C be a class of topological spaces closed under formation of Alexandroff extensions I Then C is modally definable iff it is closed under taking open subspaces, interior images, topological sums and it reflects Alexandroff extensions Modal logic and topology: basic results Let F = (X , τ ) be a topological space. Then I F |= p → p iff F is discrete I F |= p → ♦p iff every closed subset of X is open I F |= ♦p → ♦p iff F is extremally disconnected I F |= ♦p → ♦p iff the set of all dense subsets of X is a filter Modal logic and topology: further direction Let F = (X , τ ) be a topological space I Co-limit points I I I I For A ⊆ X , a point x ∈ X is a co-limit point of A iff there exists an open neighborhood B of x such that B ⊆ A ∪ {x} For A ⊆ X , t(A) is the set of co-limit points of A Remind that In(A) = A ∩ t(A) Limit points I I I For A ⊆ X , a point x ∈ X is a limit point of A iff for all open neighborhoods B of x, A ∩ (B \ {x}) is nonempty For A ⊆ X , d(A) is the set of limit points of A Remind that Cl(A) = A ∪ d(A) Modal logic and topology: further direction Syntax of modal logic I I φ ::= p | ⊥ | ¬φ | (φ ∨ ψ) | φ Abbreviations I I I Topological interpretation I I Standard definitions for the remaining Boolean operations ♦φ ::= ¬¬φ The interpretation of φ is the set of co-limit points of φ’s interpretation As a result, I The interpretation of ♦φ is the set of limit points of φ’s interpretation Modal logic and topology: further direction Topological semantics for modal logic I Let M = (X , τ, V ) where I I I Truth of a modal formula φ at a point x in X is defined as follows: I I I (X , τ ) is a topological space V is a valuation function: p 7−→ V (p) ⊆ X M, x |= p iff x ∈ V (p) M, x |= φ iff there exists O ∈ τ such that x ∈ O and for all y ∈ O, if x 6= y then M, y |= φ As a result, I M, x |= ♦φ iff for all O ∈ τ , if x ∈ O then there exists y ∈ O such that x 6= y and M, y |= φ Modal logic and topology: further direction Topological semantics for modal logic I Let M = (X , τ, V ) where I I I Truth of a modal formula φ at a point x in X is defined as follows: I I I (X , τ ) is a topological space V is a valuation function: p 7−→ V (p) ⊆ X M, x |= p iff x ∈ V (p) M, x |= φ iff x ∈ t({y ∈ X : M, y |= φ}) As a result, I M, x |= ♦φ iff x ∈ d({y ∈ X : M, y |= φ}) Modal logic and topology: further direction Topological semantics for modal logic I Let M = (X , τ, V ) be a topological model I I I I We say that φ is true in M iff φ is true at every point in M Let F = (X , τ ) be a topological space We say that φ is valid in F iff φ is true in every model based on F Let C be a class of topological spaces I We say that φ is valid in C iff φ is valid in every topological space of C Modal logic and topology: further direction Proposition: Let Top be the class of all topological spaces. Then I (K ) is valid in Top, I (N) preserves validity in Top, I (w4) is valid in Top, i.e. the modal logic wK 4 is sound with respect to interpreting as set of co-limit points. Modal logic and topology: further direction For a given wK 4-relational structure F = (W , R), we define the topology τR on W by declaring I F = (W , R) to be the reflexive closure of F I F = (W , R) to be the irreflexive fragment of F I dR to be the limit operator in F Remark that I I the topological space ts(F) = (W , τR ) is an Alexandroff space the least neighborhood of x ∈ W is I I the interior of a set A ⊆ W is I I R(x) = {y ∈ W : xRy} {x ∈ W : R(x) ⊆ A} the closure of a set A ⊆ W is I {x ∈ W : R(x) ∩ A 6= ∅} Modal logic and topology: further direction For a given topological space F = (X , τ ), we define the binary relation Rτ on W by I xRτ y iff x ∈ dτ ({y}) Remark that I the relational structure rs(F) = (W , Rτ ) is an irreflexive wK 4-structure Modal logic and topology: further direction Lemma: 1. If F = (W , R) is a wK 4-frame then RτR ⊆ R. 2. If F = (W , R) is an irreflexive wK 4-frame then RτR = R. 3. If F = (X , τ ) is a topological space then Rτ−1 (A) ⊆ d(A). 4. If F = (X , τ ) is an Alexandroff space then Rτ−1 (A) = d(A). Corollary: For a nonempty set X , there is a 1-1 correspondence between I Alexandroff topologies on X I reflexive and transitive binary relations on X I irreflexive and weakly transitive binary relations on X Modal logic and topology: further direction Theorem (Esakia): 1. wK 4 is the logic of all topological spaces. 2. wK 4 is the logic of all finite topological spaces. 3. wK 4 has the effective finite model property with respect to the class of all topological spaces. The derivative operation in topological spaces Let F = (X , τ ) be a topological space I Co-limit points I I I I Limit points I I I I For A ⊆ X , a point x ∈ X is a co-limit point of A iff there exists an open neighborhood B of x such that B ⊆ A ∪ {x} For A ⊆ X , t(A) is the set of co-limit points of A Remind that In(A) = A ∩ t(A) For A ⊆ X , a point x ∈ X is a limit point of A iff for all open neighborhoods B of x, A ∩ (B \ {x}) is nonempty For A ⊆ X , d(A) is the set of limit points of A Remind that Cl(A) = A ∪ d(A) Remind that I I t(X \ A) = X \ d(A) d(X \ A) = X \ t(A) The derivative operation in topological spaces In words I x belongs to the set of co-limit points of A iff some open set B around x is contained in A ∪ {x} I x belongs to the set of limit points of A iff every open set B around x intersects A \ {x} This set d(A) of limit points of A has many other names I derivative I derived set I Cantor-Bendixson derivative I set of limit points I set of accumulation points I set of cluster points The derivative operation in topological spaces Example: Let IR be the reals with the usual topology and 1 A = {m : m ≥ 1}. Then I d(A) = {0} I d(d(A)) = ∅ Example: Let IR be the reals with the usual topology and 1 B = {m + n1 : m, n ≥ 1}. Then I d(B) = A ∪ {0} I d(d(B)) = {0} I d(d(d(B))) = ∅ The derivative operation in topological spaces Example: Let F = (X , τ ) be a topological space such that I X is an infinite set I there exists an infinite subset D of X such that τ is the family of subsets A ⊆ W such that A = ∅ or D \ A is finite Then I d(A) = ∅ if D ∩ A is finite I d(A) = A if D ∩ A is infinite The derivative operation in topological spaces Example: If F = (W , R) is an S4-frame then its (Alexandroff) topology is defined as follows: I the opens are the R-closed sets Moreover, I d(A) = {x ∈ W : there exists y ∈ A such that x 6= y and xRy} The derivative operation in topological spaces Proposition: Concerning the derivative operation d on a topological space F = (X , τ ), 1. If every set of the form d({x}) is open then F is a T1 -space. 2. d(d(A)) ⊆ A ∪ d(A). 3. If F is a Td -space then d(d(A)) ⊆ d(A). A dense-in-itself topological space F = (X , τ ) is a DSO space iff I every derived set d(A) is open The derivative operation in topological spaces Topological semantics for modal logic I Let M = (X , τ, V ) where I I I Truth of a modal formula φ at a point x in X is defined as follows: I I I (X , τ ) is a topological space V is a valuation function: p 7−→ V (p) ⊆ X M, x |= p iff x ∈ V (p) M, x |= φ iff x ∈ t({y ∈ X : M, y |= φ}) As a result, I M, x |= ♦φ iff x ∈ d({y ∈ X : M, y |= φ}) The derivative operation in topological spaces Theorem: Let F = (X , τ ) be a topological space. Then I F |= p → ♦p iff F is dense-in-itself I F |= p → p iff F is a Td -space I F |= ♦p → ♦p iff all derived sets are open Theorem: The logic KD45 is sound and complete for DSO topological models. Iterating the derivative operation Let F = (X , τ ) be a topological space I Limit points I I I I I For A ⊆ X , a point x ∈ X is a limit point of A iff for all open neighborhoods B of x, A ∩ (B \ {x}) is nonempty For A ⊆ X , dτ (A) is the set of limit points of A Remind that Clτ (A) = A ∪ dτ (A) In general, dτ (dτ (A)) ⊆ A ∪ dτ (A) If F is a Td -space, dτ (dτ (A)) ⊆ dτ (A) Sometimes, (X , τ ) will be denoted by (X , dτ ) or by (X , d) Iterating the derivative operation Let F = (X , d) be a topological space I Iterated limit points I Let Ld be the set of all functions d 0 : 2X 7−→ 2X such that I I Let ≤d be the binary relation on Ld defined by I I for all A ⊆ X , d 0 (A) ⊆ d(A) d 0 ≤d d 00 iff for all A ⊆ X , d 0 (A) ⊆ d 00 (A) Obviously, (Ld , ≤d ) is a complete lattice Iterating the derivative operation Let F = (X , d) be a topological space I Iterated limit points I If F is a Td -space, let θ be the following operator: I I I I Ld −→ Ld d 0 7−→ θ(d 0 ) = d ◦ d 0 Obviously, θ is a ≤d -preserving operator Thus, I d∗ = W {d 0 ∈ Ld : d 0 ≤d θ(d 0 )} is the greatest fixpoint of θ Iterating the derivative operation Proposition: Let F = (X , d) be a topological space. If F is a Td -space then for all A ⊆ X , 1. d ∗ (A) ⊆ d(A). 2. d(d(A)) ⊆ d(A). 3. d(d ∗ (A)) ⊆ d ∗ (A). 4. d ∗ (d(A)) ⊆ d ∗ (A). 5. d ∗ (d ∗ (A)) ⊆ d ∗ (A). 6. d ∗ (A) ⊆ d(d ∗ (A)). Iterating the derivative operation Let F = (W , R) be a relational structure I Iterated relation I Let LR be the set of all binary relations R 0 on W such that I I Let ≤R be the binary relation on LR defined by I I R0 ⊆ R R 0 ≤R R 00 iff R 0 ⊆ R 00 Obviously, (LR , ≤R ) is a complete lattice Iterating the derivative operation Let F = (W , R) be a relational structure I Iterated relation I If F is a K 4-relational structure, let f be the following operator: I I I I LR −→ LR R 0 7−→ f (R 0 ) = R ◦ R 0 Obviously, f is a ≤R -preserving operator Thus, I R∗ = W {R 0 ∈ LR : R 0 ≤R f (R 0 )} is the greatest fixpoint of f Iterating the derivative operation Proposition: Let F = (W , R) be a relational structure. If F is a K 4-relational structure then for all x, y ∈ W , 1. If xR ∗ y then xRy . 2. If x(R ◦ R)y then xRy. 3. If x(R ◦ R ∗ )y then xR ∗ y . 4. If x(R ∗ ◦ R)y then xR ∗ y . 5. If x(R ∗ ◦ R ∗ )y then xR ∗ y. 6. If xR ∗ y then x(R ◦ R ∗ )y . Iterating the derivative operation For a given Td topological space F = (X , d), we define the binary relation Rd on W by I xRd y iff x ∈ d({y}) Remark that in all cases I Rd∗ = Rd ∗ For a given K 4-relational structure F = (W , R), we define the function dR : 2W 7−→ 2W by I dR (A) = R −1 (A) Remark that in some cases I dR∗ 6= dR ∗ Iterating the derivative operation An iterated relational structure is a triple F = (W , R, S) where I W is a nonempty set I R is a transitive binary relation on W S is the greatest fixpoint of the following operator: I I I LR −→ LR S 7−→ f (S) = R ◦ S where LR is the complete lattice of the set of all binary relations on W contained in R Sometimes, S will be denoted by R ∗ Iterating the derivative operation Example: Let F = (W , R) where W = IN. Then I R∗ = ∅ Example: Let F = (W , R) where W = IN ∪ {ω}. Then I R ∗ = IN × {ω} Example: Let F = (W , R) where W = IN ⊕ IN0 . Then I R ∗ = {(m, m0 ): m ∈ IN and m0 ∈ IN0 } Iterating the derivative operation Proposition: Let F = (W , R, S) be an iterated relational structure. Then 1. S ⊆ R. 2. R ◦ R ⊆ R. 3. R ◦ S ⊆ S. 4. S ◦ R ⊆ S. 5. S ◦ S ⊆ S. 6. S ⊆ R ◦ S. Iterating the derivative operation Proposition: There exists no first-order formula φ(u, v ) in Lan(=, R) with free variables u, v such that for all iterated relational structures F = (W , R, S) and for all possible worlds x, y in W , I xSy iff F |= φ[x, y] Iterating the derivative operation Proof: 1. Assume such φ(u, v ) exists and let r be its quantifier rank. 2. Let F1 = (ZZ, <) and F2 = (ZZ ⊕ ZZ0 , < ⊕ <0 ). 3. Let x1 = −2r and y1 = +2r . Remark that not x1 S1 y1 . 4. Let x2 = 0 and y2 = 00 . Remark that x2 S2 y2 . 5. Lemma: The Duplicator wins the r -move Ehrenfeucht-Fraı̈ssé game on (F1 , x1 , y1 ) and (F2 , x2 , y2 ). 6. By 5, for all first-order formulas ϕ(u, v ) of quantifier rank r in Lan(=, R) with free variables u, v , F1 |= ϕ[x1 , y1 ] iff F2 |= ϕ[x2 , y2 ]. 7. By 6, F1 |= φ[x1 , y1 ] iff F2 |= φ[x2 , y2 ]. 8. By 1, x1 S1 y1 iff x2 S2 y2 : a contradiction with 3 and 4. Iterating the derivative operation Proposition: There exists no first-order sentence φ in Lan(=, R, S) such that for all relational structures F = (W , R, S), I F is an iterated relational structure iff F |= φ On the modal logic of iterated derivative Syntax of modal logic I I φ ::= p | ⊥ | ¬φ | (φ ∨ ψ) | φ | ∗ φ Abbreviations I I I I Topological interpretation I I I Standard definitions for the remaining Boolean operations ♦φ ::= ¬¬φ ♦∗ φ ::= ¬∗ ¬φ The interpretation of φ is the set of co-limit points of φ’s interpretation The interpretation of ∗ φ is the set of iterated co-limit points of φ’s interpretation As a result, I I The interpretation of ♦φ is the set of limit points of φ’s interpretation The interpretation of ♦∗ φ is the set of iterated limit points of φ’s interpretation On the modal logic of iterated derivative Relational semantics for modal logic I Let M = (W , R, S, V ) where I I I Truth of a modal formula φ at a possible world x in W is defined as follows: I I I I (W , R, S) is an iterated relational structure V is a valuation function: p 7−→ V (p) ⊆ X M, x |= p iff x ∈ V (p) M, x |= φ iff R(x) ⊆ {y ∈ X : M, y |= φ} M, x |= ∗ φ iff S(x) ⊆ {y ∈ X : M, y |= φ} As a result, I I M, x |= ♦φ iff R(x) ∩ {y ∈ X : M, y |= φ} = 6 ∅ M, x |= ♦∗ φ iff S(x) ∩ {y ∈ X : M, y |= φ} = 6 ∅ On the modal logic of iterated derivative Relational semantics for modal logic I Let M = (W , R, S, V ) be an iterated relational model I I Let F = (W , R, S) be an iterated relational structure I I We say that φ is true in M iff φ is true at every possible world in M We say that φ is valid in F iff φ is true in every model based on F Let C be a class of iterated relational structures I We say that φ is valid in C iff φ is valid in every iterated relational structure of C On the modal logic of iterated derivative Proposition: There exists no modal formula φ in Lan() such that for all iterated relational structures F = (W , R, S), I F |= ∗ p ↔ φ On the modal logic of iterated derivative Proof: 1. Assume such φ exists. 2. Let F = (IN, <, ∅). Remark that F 6|= ¬∗ ⊥. 3. Let F 0 = (W 0 , R 0 , S 0 ) where (W 0 , R 0 ) is the ultrafilter extension of (IN, <) and S 0 = R 0 ∗ . Remark that F 0 |= ¬∗ ⊥. 4. By 1, F 0 |= ¬φ(p := ⊥). 5. Thus, F |= ¬φ(p := ⊥). 6. By 1, F |= ¬∗ ⊥: a contradiction with 2. On the modal logic of iterated derivative Proposition: The modal logic of the class of all iterated relational structures is the least normal logic Log(, ∗ ) containing the following axioms: (Ax1 ) p → ∗ p (Ax2 ) p → p (Ax3 ) ∗ p → ∗ p (Ax4 ) ∗ p → ∗ p (Ax5 ) ∗ p → ∗ ∗ p (Ax6 ) ∗ p → ∗ p Proof: Step-by-step construction. On the modal logic of iterated derivative Proposition: Log(, ∗ ) has not the finite model property with respect to the class of all irreflexive iterated relational structures. Proof: 1. Take φ = ♦∗ >. 2. Remark that φ is valid in the irreflexive iterated relational structure (IR, <, <). 3. For all finite irreflexive iterated relational structures (W , R, S), S = ∅. 4. Thus, φ is satisfiable in no finite irreflexive iterated relational structure. On the modal logic of iterated derivative Open problems: 1. Philosophical interpretation of ∗ in terms of beliefs ? 2. Finite model property of Log(, ∗ ) with respect to the class of all relational structures validating it ? 3. Modal definability in Lan(, ∗ ) of the class of all iterated relational structures ? 4. Decidability/complexity of the membership problem in Log(, ∗ ) ? 5. What is the modal logic of ∗ alone ? K 4 ? 6. Do the class of all Td -spaces of the form (X , d, d ∗ ) and the class of all K 4-relational structures of the form (W , R, R ∗ ) validate the same formulas in Lan(, ∗ ) ? Bibliography I Van Benthem, J., Bezhanishvili, G.: Modal logics of space. In Aiello, M., Pratt-Hartmann, I., van Benthem, J. (Editeurs) : Handbook of Spatial Logics. Springer (2007) 217–298. I Bezhanishvili, G., Esakia, L., Gabelaia, D.: Some results on modal axiomatization and definability for topological spaces. Studia Logica 81 (2005) 325–355. I Parikh, R., Moss, L., Steinsvold, C.: Topology and epistemic logic. In Aiello, M., Pratt-Hartmann, I., van Benthem, J. (Editeurs) : Handbook of Spatial Logics. Springer (2007) 299–341. I Shehtman, V.: Derived sets in Euclidean spaces and modal logic. ITLI Prepublication Series X-90-05 (1990).