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Intuitionistic Modalities in Topology and Algebra

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Intuitionistic Modalities in Topology and Algebra Intuitionistic modalities in topology and algebra Mamuka Jibladze Razmadze Mathematical Institute, Tbilisi TACL2011, Marseille 27.VII.2011 L. Esakia, Quantification in intuitionistic logic with provability smack. Bull. Sect. Logic 27 (1998), 26-28. L. Esakia, Synopsis of Fronton theory. Logical Investigations 7 (2000), 137-147 (in Russian). L. Esakia, M. Jibladze, D. Pataraia, Scattered toposes. Annals of Pure and Applied Logic 103 (2000), 97–107 L. Esakia, Intuitionistic logic and modality via topology. Annals of Pure and Applied Logic 127 (2004), 155–170 L. Esakia, The modalized Heyting calculus: a conservative modal extension of the Intuitionistic Logic. Journal of Applied Non-Classical Logics 16 (2006), 349–366. Topological semantics (McKinsey & Tarski) — 3 can be interpreted as the closure operator C of a topological space X: valuation v(') ⊆ X; v(3') = C v(') Classical modal systems and topology S4 The system S4 is normal and satisfies p ! 3p, 33p $ 3p, 3(p _ q) $ 3p _ 3q. valuation v(') ⊆ X; v(3') = C v(') Classical modal systems and topology S4 The system S4 is normal and satisfies p ! 3p, 33p $ 3p, 3(p _ q) $ 3p _ 3q. Topological semantics (McKinsey & Tarski) — 3 can be interpreted as the closure operator C of a topological space X: v(3') = C v(') Classical modal systems and topology S4 The system S4 is normal and satisfies p ! 3p, 33p $ 3p, 3(p _ q) $ 3p _ 3q. Topological semantics (McKinsey & Tarski) — 3 can be interpreted as the closure operator C of a topological space X: valuation v(') ⊆ X; Classical modal systems and topology S4 The system S4 is normal and satisfies p ! 3p, 33p $ 3p, 3(p _ q) $ 3p _ 3q. Topological semantics (McKinsey & Tarski) — 3 can be interpreted as the closure operator C of a topological space X: valuation v(') ⊆ X; v(3') = C v(') where B = (B; ^; _; :; 0; 1) is a Boolean algebra and c : B ! B satisfies c 0 = 0, b 6 c b, c c b = c b, c(b _ b0) = c b _ c b0. Classical modal systems and topology S4 Algebraic semantics — Closure algebra (B; c) c 0 = 0, b 6 c b, c c b = c b, c(b _ b0) = c b _ c b0. Classical modal systems and topology S4 Algebraic semantics — Closure algebra (B; c) where B = (B; ^; _; :; 0; 1) is a Boolean algebra and c : B ! B satisfies Classical modal systems and topology S4 Algebraic semantics — Closure algebra (B; c) where B = (B; ^; _; :; 0; 1) is a Boolean algebra and c : B ! B satisfies c 0 = 0, b 6 c b, c c b = c b, c(b _ b0) = c b _ c b0. satisfying i 1 = 1, i b 6 b, i i b = i b, i(b ^ b0) = i b ^ i b0. These were considered by Rasiowa and Sikorski under the name of topological Boolean algebras; Blok used the term interior algebra which is mostly used nowadays along with S4-algebra. Classical modal systems and topology S4 Equivalently — using the dual interior operator i = : c : These were considered by Rasiowa and Sikorski under the name of topological Boolean algebras; Blok used the term interior algebra which is mostly used nowadays along with S4-algebra. Classical modal systems and topology S4 Equivalently — using the dual interior operator i = : c : satisfying i 1 = 1, i b 6 b, i i b = i b, i(b ^ b0) = i b ^ i b0. Classical modal systems and topology S4 Equivalently — using the dual interior operator i = : c : satisfying i 1 = 1, i b 6 b, i i b = i b, i(b ^ b0) = i b ^ i b0. These were considered by Rasiowa and Sikorski under the name of topological Boolean algebras; Blok used the term interior algebra which is mostly used nowadays along with S4-algebra. Every topology has its derivative operator — for A ⊆ X, δA := fx 2 X j every neighborhood of x meets A n fxgg. For every topology, the corresponding δ satisfies δ? = ?, δ(A [ B) = δA [ δB, δδA ⊆ A [ δA. Classical modal systems and topology wK4 Closure/interior is one among many ways to define the topology of a space X. For every topology, the corresponding δ satisfies δ? = ?, δ(A [ B) = δA [ δB, δδA ⊆ A [ δA. Classical modal systems and topology wK4 Closure/interior is one among many ways to define the topology of a space X. Every topology has its derivative operator — for A ⊆ X, δA := fx 2 X j every neighborhood of x meets A n fxgg. δ? = ?, δ(A [ B) = δA [ δB, δδA ⊆ A [ δA. Classical modal systems and topology wK4 Closure/interior is one among many ways to define the topology of a space X. Every topology has its derivative operator — for A ⊆ X, δA := fx 2 X j every neighborhood of x meets A n fxgg. For every topology, the corresponding δ satisfies Classical modal systems and topology wK4 Closure/interior is one among many ways to define the topology of a space X. Every topology has its derivative operator — for A ⊆ X, δA := fx 2 X j every neighborhood of x meets A n fxgg. For every topology, the corresponding δ satisfies δ? = ?, δ(A [ B) = δA [ δB, δδA ⊆ A [ δA. one has τA = I A [ Isol(X − A), where I is the interior, I A = X − C(X − A) and Isol S denotes the set of isolated points of S ⊆ X (with respect to the topology induced from X). This τ satisfies the dual identities τX = X, τ(A \ B) = τA \ τB, A \ τA ⊆ ττA. Classical modal systems and topology wK4 Half of the time, it is more useful to work with the dual coderivative operator given by τA := X − δ(X − A); where I is the interior, I A = X − C(X − A) and Isol S denotes the set of isolated points of S ⊆ X (with respect to the topology induced from X). This τ satisfies the dual identities τX = X, τ(A \ B) = τA \ τB, A \ τA ⊆ ττA. Classical modal systems and topology wK4 Half of the time, it is more useful to work with the dual coderivative operator given by τA := X − δ(X − A); one has τA = I A [ Isol(X − A), This τ satisfies the dual identities τX = X, τ(A \ B) = τA \ τB, A \ τA ⊆ ττA. Classical modal systems and topology wK4 Half of the time, it is more useful to work with the dual coderivative operator given by τA := X − δ(X − A); one has τA = I A [ Isol(X − A), where I is the interior, I A = X − C(X − A) and Isol S denotes the set of isolated points of S ⊆ X (with respect to the topology induced from X). Classical modal systems and topology wK4 Half of the time, it is more useful to work with the dual coderivative operator given by τA := X − δ(X − A); one has τA = I A [ Isol(X − A), where I is the interior, I A = X − C(X − A) and Isol S denotes the set of isolated points of S ⊆ X (with respect to the topology induced from X). This τ satisfies the dual identities τX = X, τ(A \ B) = τA \ τB, A \ τA ⊆ ττA. (p&2p) ! 22p; or equivalently 33p ! (p _ 3p): This is wK4, or weak K4. Classical modal systems and topology wK4 If one wants to have a modal system with topological semantics interpreting 3 as δ, and 2 as τ, one thus arrives at a normal system with an additional axiom or equivalently 33p ! (p _ 3p): This is wK4, or weak K4. Classical modal systems and topology wK4 If one wants to have a modal system with topological semantics interpreting 3 as δ, and 2 as τ, one thus arrives at a normal system with an additional axiom (p&2p) ! 22p; This is wK4, or weak K4. Classical modal systems and topology wK4 If one wants to have a modal system with topological semantics interpreting 3 as δ, and 2 as τ, one thus arrives at a normal system with an additional axiom (p&2p) ! 22p; or equivalently 33p ! (p _ 3p): Classical modal systems and topology wK4 If one wants to have a modal system with topological semantics interpreting 3 as δ, and 2 as τ, one thus arrives at a normal system with an additional axiom (p&2p) ! 22p; or equivalently 33p ! (p _ 3p): This is wK4, or weak K4. Or, one might define them in terms of the dual coderivative operator τ = : δ : with axioms τ 1 = 1, τ (b ^ b0) = τ b ^ τ b0, b ^ τ b 6 τ τ b. Classical modal systems and topology wK4 For the algebraic semantics one has derivative algebras (B; δ) — Boolean algebras with an operator δ satisfying δ 0 = 0, δ(b _ b0) = δ b _ δ b0, δ δ b 6 b _ δ b. Classical modal systems and topology wK4 For the algebraic semantics one has derivative algebras (B; δ) — Boolean algebras with an operator δ satisfying δ 0 = 0, δ(b _ b0) = δ b _ δ b0, δ δ b 6 b _ δ b. Or, one might define them in terms of the dual coderivative operator τ = : δ : with axioms τ 1 = 1, τ (b ^ b0) = τ b ^ τ b0, b ^ τ b 6 τ τ b. For a closure algebra (B; c), the subset H = i(B) = fi b j b 2 Bg = Fix(i) = fh 2 B j i h = hg is a sublattice of B; it is a Heyting algebra with respect to the implication h −! h0 := i(h ! h0) H and is thus an algebraic model of the Heyting propositional Calculus HC.
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