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(ϕ) ⊆ X;

v(3ϕ) = C v(ϕ)

Classical modal systems and topology S4

The system S4 is normal and satisﬁes p → 3p, 33p ↔ 3p, 3(p ∨ q) ↔ 3p ∨ 3q. valuation v(ϕ) ⊆ X;

v(3ϕ) = C v(ϕ)

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(3ϕ) = C v(ϕ)

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(ϕ) ⊆ 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(ϕ) ⊆ X;

v(3ϕ) = C v(ϕ) 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 := {x ∈ X | every neighborhood of x meets A \{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. 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 := {x ∈ X | every neighborhood of x meets A \{x}}. 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 := {x ∈ X | every neighborhood of x meets A \{x}}.

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 := {x ∈ X | every neighborhood of x meets A \{x}}.

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

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) = {i b | b ∈ B} = Fix(i) = {h ∈ B | i h = h} is a sublattice of B; it is a Heyting algebra with respect to the implication

The intuitionistic side HC

Let us now consider the intuitionistic counterparts of these systems. it is a Heyting algebra with respect to the implication

The intuitionistic side HC

Let us now consider the intuitionistic counterparts of these systems. For a closure algebra (B, c), the subset H = i(B) = {i b | b ∈ B} = Fix(i) = {h ∈ B | i h = h} is a sublattice of B; and is thus an algebraic model of the Heyting propositional Calculus HC. Obviously, i itself disappears from sight in H; also in general c does not leave any manageable “trace” on H.

The intuitionistic side HC

Let us now consider the intuitionistic counterparts of these systems. For a closure algebra (B, c), the subset H = i(B) = {i b | b ∈ B} = Fix(i) = {h ∈ B | i h = h} is a sublattice of B; it is a Heyting algebra with respect to the implication

h −→ h0 := i(h → h0) H Obviously, i itself disappears from sight in H; also in general c does not leave any manageable “trace” on H.

The intuitionistic side HC

h −→ h0 := i(h → h0) H and is thus an algebraic model of the Heyting propositional Calculus HC. also in general c does not leave any manageable “trace” on H.

The intuitionistic side HC

h −→ h0 := i(h → h0) H and is thus an algebraic model of the Heyting propositional Calculus HC. Obviously, i itself disappears from sight in H; The intuitionistic side HC

h −→ h0 := i(h → h0) H and is thus an algebraic model of the Heyting propositional Calculus HC. Obviously, i itself disappears from sight in H; also in general c does not leave any manageable “trace” on H. Note that when (B, c) = (P(X), C) for a topological space X, the corresponding H is the Heyting algebra O(X) of all open sets of X. In terms of the coderivative, a subset U ⊆ X is open if and only if U ⊆ τU. More generally, for a (co)derivative algebra (B, τ ) one gets the Heyting algebra H := {h ∈ B | h 6 τ h}. And now, τ restricts to H in a nontrivial way: since τ is obviously monotone, h 6 τ h implies τ h 6 τ τ h, i. e. τ |H ⊆ H.

The intuitionistic side mHC

The situation is more interesting with the (co)derivative semantics. In terms of the coderivative, a subset U ⊆ X is open if and only if U ⊆ τU. More generally, for a (co)derivative algebra (B, τ ) one gets the Heyting algebra H := {h ∈ B | h 6 τ h}. And now, τ restricts to H in a nontrivial way: since τ is obviously monotone, h 6 τ h implies τ h 6 τ τ h, i. e. τ |H ⊆ H.

The intuitionistic side mHC

The situation is more interesting with the (co)derivative semantics. Note that when (B, c) = (P(X), C) for a topological space X, the corresponding H is the Heyting algebra O(X) of all open sets of X. More generally, for a (co)derivative algebra (B, τ ) one gets the Heyting algebra H := {h ∈ B | h 6 τ h}. And now, τ restricts to H in a nontrivial way: since τ is obviously monotone, h 6 τ h implies τ h 6 τ τ h, i. e. τ |H ⊆ H.

The intuitionistic side mHC

The situation is more interesting with the (co)derivative semantics. Note that when (B, c) = (P(X), C) for a topological space X, the corresponding H is the Heyting algebra O(X) of all open sets of X. In terms of the coderivative, a subset U ⊆ X is open if and only if U ⊆ τU. And now, τ restricts to H in a nontrivial way: since τ is obviously monotone, h 6 τ h implies τ h 6 τ τ h, i. e. τ |H ⊆ H.

The intuitionistic side mHC

The situation is more interesting with the (co)derivative semantics. Note that when (B, c) = (P(X), C) for a topological space X, the corresponding H is the Heyting algebra O(X) of all open sets of X. In terms of the coderivative, a subset U ⊆ X is open if and only if U ⊆ τU. More generally, for a (co)derivative algebra (B, τ ) one gets the Heyting algebra H := {h ∈ B | h 6 τ h}. since τ is obviously monotone, h 6 τ h implies τ h 6 τ τ h, i. e. τ |H ⊆ H.

The intuitionistic side mHC

The situation is more interesting with the (co)derivative semantics. Note that when (B, c) = (P(X), C) for a topological space X, the corresponding H is the Heyting algebra O(X) of all open sets of X. In terms of the coderivative, a subset U ⊆ X is open if and only if U ⊆ τU. More generally, for a (co)derivative algebra (B, τ ) one gets the Heyting algebra H := {h ∈ B | h 6 τ h}. And now, τ restricts to H in a nontrivial way: The intuitionistic side mHC

The situation is more interesting with the (co)derivative semantics. Note that when (B, c) = (P(X), C) for a topological space X, the corresponding H is the Heyting algebra O(X) of all open sets of X. In terms of the coderivative, a subset U ⊆ X is open if and only if U ⊆ τU. More generally, for a (co)derivative algebra (B, τ ) one gets the Heyting algebra H := {h ∈ B | h 6 τ h}. And now, τ restricts to H in a nontrivial way: since τ is obviously monotone, h 6 τ h implies τ h 6 τ τ h, i. e. τ |H ⊆ H. Then obviously τ 1 = 1, τ (h ∧ h0) = τ h ∧ τ h0 on H.

The intuitionistic side mHC

Thus, each coderivative algebra (B, τ ) gives rise to a Heyting algebra H = {h ∈ B | h 6 τ h} equipped with an operator τ = τ |H : H → H. The intuitionistic side mHC

Thus, each coderivative algebra (B, τ ) gives rise to a Heyting algebra H = {h ∈ B | h 6 τ h} equipped with an operator τ = τ |H : H → H.

Then obviously τ 1 = 1, τ (h ∧ h0) = τ h ∧ τ h0 on H. ^ τ(U) = V ∪ (V −−−→ U) | V ∈ O(X) . O(X)

In particular, one has

τ(U) 6 V ∪ V −−−→ U O(X)

for any U, V ∈ O(X).

The intuitionistic side mHC

Now if (B, τ ) = (P(X), τ) for a topological space X, the corresponding operator τ = τ|O(X) : O(X) → O(X) can be deﬁned entirely in terms of O(X) as follows: In particular, one has

τ(U) 6 V ∪ V −−−→ U O(X)

for any U, V ∈ O(X).

The intuitionistic side mHC

Now if (B, τ ) = (P(X), τ) for a topological space X, the corresponding operator τ = τ|O(X) : O(X) → O(X) can be deﬁned entirely in terms of O(X) as follows:

^ τ(U) = V ∪ (V −−−→ U) | V ∈ O(X) . O(X) The intuitionistic side mHC

Now if (B, τ ) = (P(X), τ) for a topological space X, the corresponding operator τ = τ|O(X) : O(X) → O(X) can be deﬁned entirely in terms of O(X) as follows:

^ τ(U) = V ∪ (V −−−→ U) | V ∈ O(X) . O(X)

In particular, one has

τ(U) 6 V ∪ V −−−→ U O(X)

for any U, V ∈ O(X). It is given by adding to the axioms of HC the axioms 2(p → q) → (2p → 2q), p → 2p, 2p → (q ∨ (q → p)). NB The second axiom might look sort of “wrong way” for a modal logician, as for multiplicative operators implication usually goes in the opposite direction. However in intuitionistic modal systems such things happen. E. g. for those familiar with nuclei — a nucleus is an inﬂationary multiplicative idempotent operator.

The intuitionistic side mHC

We thus arrive at an intuitionistic modal system mHC, with topological semantics given by valuations via open sets of a space and the modality 2 interpreted as the coderivative restricted to open sets. NB The second axiom might look sort of “wrong way” for a modal logician, as for multiplicative operators implication usually goes in the opposite direction. However in intuitionistic modal systems such things happen. E. g. for those familiar with nuclei — a nucleus is an inﬂationary multiplicative idempotent operator.

The intuitionistic side mHC

We thus arrive at an intuitionistic modal system mHC, with topological semantics given by valuations via open sets of a space and the modality 2 interpreted as the coderivative restricted to open sets. It is given by adding to the axioms of HC the axioms 2(p → q) → (2p → 2q), p → 2p, 2p → (q ∨ (q → p)). However in intuitionistic modal systems such things happen. E. g. for those familiar with nuclei — a nucleus is an inﬂationary multiplicative idempotent operator.

The intuitionistic side mHC

We thus arrive at an intuitionistic modal system mHC, with topological semantics given by valuations via open sets of a space and the modality 2 interpreted as the coderivative restricted to open sets. It is given by adding to the axioms of HC the axioms 2(p → q) → (2p → 2q), p → 2p, 2p → (q ∨ (q → p)). NB The second axiom might look sort of “wrong way” for a modal logician, as for multiplicative operators implication usually goes in the opposite direction. The intuitionistic side mHC

We thus arrive at an intuitionistic modal system mHC, with topological semantics given by valuations via open sets of a space and the modality 2 interpreted as the coderivative restricted to open sets. It is given by adding to the axioms of HC the axioms 2(p → q) → (2p → 2q), p → 2p, 2p → (q ∨ (q → p)). NB The second axiom might look sort of “wrong way” for a modal logician, as for multiplicative operators implication usually goes in the opposite direction. However in intuitionistic modal systems such things happen. E. g. for those familiar with nuclei — a nucleus is an inﬂationary multiplicative idempotent operator. The intuitionistic side mHC

Algebraic models of mHC are thus of the form (H, τ ) where H is a Heyting algebra and τ : H → H satisfies h 6 τ h, τ (h ∧ h0) = τ h ∧ τ h0, 0 0 τ h 6 h ∨ (h → h). Here B(H) is the free Boolean extension of H, so that every element of B(H) is a finite meet of elements of the form ¬h0 ∨ h for some h, h0 ∈ H. One defines

τ (¬h0 ∨ h) := h0 −→ h H

and then extends to the whole B(H) by multiplicativity (correctness must be ensured).

Back to the classics K4.Grz

Moreover one has Theorem. For every mHC-algebra (H, τ ) there exists a coderivative algebra (B(H), τ ) such that H = {h ∈ B(H) | h 6 τ h} and τ |H = τ . Back to the classics K4.Grz

Moreover one has Theorem. For every mHC-algebra (H, τ ) there exists a coderivative algebra (B(H), τ ) such that H = {h ∈ B(H) | h 6 τ h} and τ |H = τ . Here B(H) is the free Boolean extension of H, so that every element of B(H) is a ﬁnite meet of elements of the form ¬h0 ∨ h for some h, h0 ∈ H. One deﬁnes

τ (¬h0 ∨ h) := h0 −→ h H

and then extends to the whole B(H) by multiplicativity (correctness must be ensured). This enabled Esakia to construct a translation # of mHC into the corresponding modal system K4.Grz. His # is deﬁned on propositional variables p by #p := p ∧ 2p, commutes with ∧, ∨, ⊥, 2 and moreover

#(ϕ → ψ) := (#ϕ → #ψ) ∧ 2(#ϕ → #ψ).

Back to the classics K4.Grz

It turns out that actually the coderivative algebras of the above form (B(H), τ ) land in a proper subvariety: they all are K4-algebras, i. e. satisfy τ b 6 τ τ b; moreover they satisfy the identity τ (τ (b → τ b) → b) = τ b. His # is deﬁned on propositional variables p by #p := p ∧ 2p, commutes with ∧, ∨, ⊥, 2 and moreover

#(ϕ → ψ) := (#ϕ → #ψ) ∧ 2(#ϕ → #ψ).

Back to the classics K4.Grz

This enabled Esakia to construct a translation # of mHC into the corresponding modal system K4.Grz. Back to the classics K4.Grz

This enabled Esakia to construct a translation # of mHC into the corresponding modal system K4.Grz. His # is deﬁned on propositional variables p by #p := p ∧ 2p, commutes with ∧, ∨, ⊥, 2 and moreover

#(ϕ → ψ) := (#ϕ → #ψ) ∧ 2(#ϕ → #ψ). This result may be viewed as an analog/generalization of the Kuznetsov-Muravitsky theorem.

Back to the classics K4.Grz

Theorem. mHC ` ϕ iff K4.Grz ` #ϕ. Moreover, the lattice of all extensions of mHC is isomorphic to the lattice of all normal extensions of K4.Grz. Back to the classics K4.Grz

Theorem. mHC ` ϕ iff K4.Grz ` #ϕ. Moreover, the lattice of all extensions of mHC is isomorphic to the lattice of all normal extensions of K4.Grz.

This result may be viewed as an analog/generalization of the Kuznetsov-Muravitsky theorem. This system relates to GL in the same way as mHC to K4.Grz: the Kuznetsov-Muravitsky theorem states that the lattice of all extensions of KM is isomorphic to the lattice of all normal extensions of GL. From the point of view of topological/algebraic semantics, this system is interesting in that in its models, the coderivative operator is in fact uniquely determined.

Canonical choices of the modality KM

The Kuznetsov-Muravitsky calculus KM may be deﬁned as the result of adding to mHC the axiom

(2p → p) → p. : the Kuznetsov-Muravitsky theorem states that the lattice of all extensions of KM is isomorphic to the lattice of all normal extensions of GL. From the point of view of topological/algebraic semantics, this system is interesting in that in its models, the coderivative operator is in fact uniquely determined.

Canonical choices of the modality KM

The Kuznetsov-Muravitsky calculus KM may be deﬁned as the result of adding to mHC the axiom

(2p → p) → p.

This system relates to GL in the same way as mHC to K4.Grz From the point of view of topological/algebraic semantics, this system is interesting in that in its models, the coderivative operator is in fact uniquely determined.

Canonical choices of the modality KM

The Kuznetsov-Muravitsky calculus KM may be deﬁned as the result of adding to mHC the axiom

(2p → p) → p.

The Kuznetsov-Muravitsky calculus KM may be deﬁned as the result of adding to mHC the axiom

(2p → p) → p.

This system relates to GL in the same way as mHC to K4.Grz: the Kuznetsov-Muravitsky theorem states that the lattice of all extensions of KM is isomorphic to the lattice of all normal extensions of GL. From the point of view of topological/algebraic semantics, this system is interesting in that in its models, the coderivative operator is in fact uniquely determined. However for general Heyting algebras H there might be several operators τ satisfying the above identities. For example, τ h ≡ h would always do. Whereas if (H, τ ) happens to be a model of KM, i. e. τ h → h is equal to h for all h ∈ H, then in addition to 0 0 0 0 τ h 6 h ∨ (h → h), also τ h itself is of the form h ∨ (h → h) for some h0 (in fact for h0 = τ h). Thus τ h is the smallest element of the set {h0 ∨ (h0 → h) | h0 ∈ H}, and this property makes it uniquely determined. An algebra of the form (O(X), τ) is a model of KM iff the space X is scattered (every nonempty subspace has an isolated point).

Canonical choices of the modality KM

For H of the form O(X) for a space X, there is a canonical choice of the “correct” τ . For example, τ h ≡ h would always do. Whereas if (H, τ ) happens to be a model of KM, i. e. τ h → h is equal to h for all h ∈ H, then in addition to 0 0 0 0 τ h 6 h ∨ (h → h), also τ h itself is of the form h ∨ (h → h) for some h0 (in fact for h0 = τ h). Thus τ h is the smallest element of the set {h0 ∨ (h0 → h) | h0 ∈ H}, and this property makes it uniquely determined. An algebra of the form (O(X), τ) is a model of KM iff the space X is scattered (every nonempty subspace has an isolated point).

Canonical choices of the modality KM

For H of the form O(X) for a space X, there is a canonical choice of the “correct” τ . However for general Heyting algebras H there might be several operators τ satisfying the above identities. Whereas if (H, τ ) happens to be a model of KM, i. e. τ h → h is equal to h for all h ∈ H, then in addition to 0 0 0 0 τ h 6 h ∨ (h → h), also τ h itself is of the form h ∨ (h → h) for some h0 (in fact for h0 = τ h). Thus τ h is the smallest element of the set {h0 ∨ (h0 → h) | h0 ∈ H}, and this property makes it uniquely determined. An algebra of the form (O(X), τ) is a model of KM iff the space X is scattered (every nonempty subspace has an isolated point).

Canonical choices of the modality KM

For H of the form O(X) for a space X, there is a canonical choice of the “correct” τ . However for general Heyting algebras H there might be several operators τ satisfying the above identities. For example, τ h ≡ h would always do. Thus τ h is the smallest element of the set {h0 ∨ (h0 → h) | h0 ∈ H}, and this property makes it uniquely determined. An algebra of the form (O(X), τ) is a model of KM iff the space X is scattered (every nonempty subspace has an isolated point).

Canonical choices of the modality KM

For H of the form O(X) for a space X, there is a canonical choice of the “correct” τ . However for general Heyting algebras H there might be several operators τ satisfying the above identities. For example, τ h ≡ h would always do. Whereas if (H, τ ) happens to be a model of KM, i. e. τ h → h is equal to h for all h ∈ H, then in addition to 0 0 0 0 τ h 6 h ∨ (h → h), also τ h itself is of the form h ∨ (h → h) for some h0 (in fact for h0 = τ h). Thus τ h is the smallest element of the set {h0 ∨ (h0 → h) | h0 ∈ H}, and this property makes it uniquely determined. Canonical choices of the modality KM

For H of the form O(X) for a space X, there is a canonical choice of the “correct” τ . However for general Heyting algebras H there might be several operators τ satisfying the above identities. For example, τ h ≡ h would always do. Whereas if (H, τ ) happens to be a model of KM, i. e. τ h → h is equal to h for all h ∈ H, then in addition to 0 0 0 0 τ h 6 h ∨ (h → h), also τ h itself is of the form h ∨ (h → h) for some h0 (in fact for h0 = τ h). Thus τ h is the smallest element of the set {h0 ∨ (h0 → h) | h0 ∈ H}, and this property makes it uniquely determined. An algebra of the form (O(X), τ) is a model of KM iff the space X is scattered (every nonempty subspace has an isolated point). Note that whenever H is a complete Heyting algebra, it comes with the “correct” τ , viz. ^ τ h := h0 ∨ (h0 → h) | h0 ∈ H .

One way to do this syntactically is to enrich the language with propositional quantiﬁers.

Canonical choices of the modality QHC

Another way to ensure the preferred choice of τ is to enrich the syntax. One way to do this syntactically is to enrich the language with propositional quantiﬁers.

Canonical choices of the modality QHC

Another way to ensure the preferred choice of τ is to enrich the syntax.

Note that whenever H is a complete Heyting algebra, it comes with the “correct” τ , viz. ^ τ h := h0 ∨ (h0 → h) | h0 ∈ H . Canonical choices of the modality QHC

Another way to ensure the preferred choice of τ is to enrich the syntax.

Note that whenever H is a complete Heyting algebra, it comes with the “correct” τ , viz. ^ τ h := h0 ∨ (h0 → h) | h0 ∈ H .

One way to do this syntactically is to enrich the language with propositional quantiﬁers. In QHC, one has axioms (∀pϕ) → ϕ|ψ←p and inference rules ψ → ϕ ψ → ∀pϕ

whenever p does not occur freely in ψ (here ϕ|ψ←p is the result of substituting ψ for p everywhere in ϕ). It is then easy to see that the modality 2 given by

2ϕ := ∀p(p ∨ (p → ϕ)),

for any p which does not occur freely in ϕ, satisﬁes all axioms of mHC.

Canonical choices of the modality QHC

The calculus QHC is obtained from HC by adding operators ∀p, one for each propositional variable p. It is then easy to see that the modality 2 given by

2ϕ := ∀p(p ∨ (p → ϕ)),

for any p which does not occur freely in ϕ, satisﬁes all axioms of mHC.

Canonical choices of the modality QHC

The calculus QHC is obtained from HC by adding operators ∀p, one for each propositional variable p.

In QHC, one has axioms (∀pϕ) → ϕ|ψ←p and inference rules ψ → ϕ ψ → ∀pϕ

whenever p does not occur freely in ψ (here ϕ|ψ←p is the result of substituting ψ for p everywhere in ϕ). Canonical choices of the modality QHC

The calculus QHC is obtained from HC by adding operators ∀p, one for each propositional variable p.

In QHC, one has axioms (∀pϕ) → ϕ|ψ←p and inference rules ψ → ϕ ψ → ∀pϕ

whenever p does not occur freely in ψ (here ϕ|ψ←p is the result of substituting ψ for p everywhere in ϕ). It is then easy to see that the modality 2 given by

2ϕ := ∀p(p ∨ (p → ϕ)),

for any p which does not occur freely in ϕ, satisﬁes all axioms of mHC. The corresponding system Q+HC is given by adding to QHC the Casari schema

∀p((ϕ → ∀pϕ) → ∀pϕ) → ∀pϕ.

Canonical choices of the modality Q+HC

The natural question arises — which conditions on ∀p would ensure KM for this 2? Canonical choices of the modality Q+HC

The natural question arises — which conditions on ∀p would ensure KM for this 2?

The corresponding system Q+HC is given by adding to QHC the Casari schema

∀p((ϕ → ∀pϕ) → ∀pϕ) → ∀pϕ. If a Heyting algebra H happens to be the algebra of all subobjects of some object in an elementary topos, it comes equipped with such quantiﬁers. In particular, on the subobject classiﬁer Ω one has the corresponding operator τ :Ω → Ω given, in the Kripke-Joyal semantics, by τ(u) = ∀p(p ∨ (p → u)).

Thus τ classiﬁes the Higgs subobject {µ ∈ Ω | {>} ∪ ↓µ = Ω} of Ω.

Canonical choices of the modality Kripke-Joyal semantics

Naturally defined propositional quantifiers are readily available in the topos semantics. In particular, on the subobject classifier Ω one has the corresponding operator τ :Ω → Ω given, in the Kripke-Joyal semantics, by τ(u) = ∀p(p ∨ (p → u)).

Thus τ classiﬁes the Higgs subobject {µ ∈ Ω | {>} ∪ ↓µ = Ω} of Ω.

Canonical choices of the modality Kripke-Joyal semantics

Naturally defined propositional quantifiers are readily available in the topos semantics. If a Heyting algebra H happens to be the algebra of all subobjects of some object in an elementary topos, it comes equipped with such quantifiers. In particular, on the subobject classifier Ω one has the corresponding operator τ :Ω → Ω given, in the Kripke-Joyal semantics, by τ(u) = ∀p(p ∨ (p → u)). Canonical choices of the modality Kripke-Joyal semantics

Thus τ classiﬁes the Higgs subobject {µ ∈ Ω | {>} ∪ ↓µ = Ω} of Ω. Note that this identity may be viewed as a kind of induction principle: the Higgs object contains the top together with its immediate predecessor, if any. Then the identity says that if we want to prove some statement u, we might as well assume that it is (either the top or) the immediate predecessor of the top. There are non-Boolean scattered toposes — e. g. the sheaves on any scattered space form a scattered topos.

Canonical choices of the modality Scattered toposes

Call a topos scattered if this τ satisﬁes

(τ(u) → u) → u. : the Higgs object contains the top together with its immediate predecessor, if any. Then the identity says that if we want to prove some statement u, we might as well assume that it is (either the top or) the immediate predecessor of the top. There are non-Boolean scattered toposes — e. g. the sheaves on any scattered space form a scattered topos.

Canonical choices of the modality Scattered toposes

Call a topos scattered if this τ satisﬁes

(τ(u) → u) → u.

Note that this identity may be viewed as a kind of induction principle Then the identity says that if we want to prove some statement u, we might as well assume that it is (either the top or) the immediate predecessor of the top. There are non-Boolean scattered toposes — e. g. the sheaves on any scattered space form a scattered topos.

Canonical choices of the modality Scattered toposes

Call a topos scattered if this τ satisﬁes

(τ(u) → u) → u.

Note that this identity may be viewed as a kind of induction principle: the Higgs object contains the top together with its immediate predecessor, if any. There are non-Boolean scattered toposes — e. g. the sheaves on any scattered space form a scattered topos.

Canonical choices of the modality Scattered toposes

Call a topos scattered if this τ satisﬁes

(τ(u) → u) → u.

Note that this identity may be viewed as a kind of induction principle: the Higgs object contains the top together with its immediate predecessor, if any. Then the identity says that if we want to prove some statement u, we might as well assume that it is (either the top or) the immediate predecessor of the top. Canonical choices of the modality Scattered toposes

Call a topos scattered if this τ satisﬁes

(τ(u) → u) → u.

Note that this identity may be viewed as a kind of induction principle: the Higgs object contains the top together with its immediate predecessor, if any. Then the identity says that if we want to prove some statement u, we might as well assume that it is (either the top or) the immediate predecessor of the top. There are non-Boolean scattered toposes — e. g. the sheaves on any scattered space form a scattered topos. Canonical choices of the modality Scattered toposes

Theorem. For an elementary topos E , the following are equivalent: (i) E is scattered, i. e. (τp → p) → p holds in E ;

(ii) The Casari schema ∀p((ϕ → ∀pϕ) → ∀pϕ) → ∀pϕ holds in E ;

(iii) (∀x¬¬ϕ(x)) → ¬¬∀xϕ(x) holds in every closed subtopos of E . Temporal intuitionistic logic tHC

The temporal Heyting Calculus tHC results from adding to mHC one more modal operator 3, with additional axioms p → 23p; 32p → p; 3(p ∨ q) → (3p ∨ 3q); 3⊥ → ⊥ and an additional rule p → q . 3p → 3q τ 0

That is, an operator : H → H such that h 6 τ h if and only if τ 0 h 6 h . If such an adjoint exists on a coderivative algebra (P(X), τ ) corresponding to a space X, then one can show that the topology on X is the Alexandroff topology for some preorder on X (namely, for the specialization preorder).

For (O(X), τ ) existence of the adjoint is less stringent. There are “almost Alexandroff” spaces with this property which are not Alexandroff.

Temporal intuitionistic logic tHC

In the algebraic semantics, this corresponds to having a left adjoint to τ : H → H. If such an adjoint exists on a coderivative algebra (P(X), τ ) corresponding to a space X, then one can show that the topology on X is the Alexandroff topology for some preorder on X (namely, for the specialization preorder).

For (O(X), τ ) existence of the adjoint is less stringent. There are “almost Alexandroff” spaces with this property which are not Alexandroff.

Temporal intuitionistic logic tHC

In the algebraic semantics, this corresponds to having a left

adjoint to τ : H → H. τ 0

That is, an operator : H → H such that h 6 τ h if and only if τ 0 h 6 h . For (O(X), τ ) existence of the adjoint is less stringent. There are “almost Alexandroff” spaces with this property which are not Alexandroff.

Temporal intuitionistic logic tHC

In the algebraic semantics, this corresponds to having a left

adjoint to τ : H → H. τ 0

In the algebraic semantics, this corresponds to having a left

adjoint to τ : H → H. τ 0

For (O(X), τ ) existence of the adjoint is less stringent. There are “almost Alexandroff” spaces with this property which are not Alexandroff. More generally, if H is a complete bi-Heyting algebra then its canonical coderivative operator τ H given by

^ 0 0 0 τ H (h) = h ∨ (h → h) | h ∈ H

◦ has a left adjoint τ H◦ , where H is H with the order reversed. There are still more general spaces with this property. The (probably) simplest Heyting algebra which is not bi-Heyting is given by > = a0 > a1 > a2 > a3 > ... and b0 > b1 > b2 > b3 > ... > ⊥, with an > bn for each n. It is the

algebra of open sets of a space, so has a canonical

coderivative operator τ . The left adjoint τ to τ is given by

τ τ an = bn = bn+1.

Temporal intuitionistic logic tHC

A simple example: the real line R with the topology for which the only open sets are the open rays (r, ∞) for −∞ 6 r 6 ∞. There are still more general spaces with this property. The (probably) simplest Heyting algebra which is not bi-Heyting is given by > = a0 > a1 > a2 > a3 > ... and b0 > b1 > b2 > b3 > ... > ⊥, with an > bn for each n. It is the

algebra of open sets of a space, so has a canonical

coderivative operator τ . The left adjoint τ to τ is given by

τ τ an = bn = bn+1.

Temporal intuitionistic logic tHC

A simple example: the real line R with the topology for which the only open sets are the open rays (r, ∞) for −∞ 6 r 6 ∞. More generally, if H is a complete bi-Heyting algebra then its canonical coderivative operator τ H given by

^ 0 0 0 τ H (h) = h ∨ (h → h) | h ∈ H

◦ has a left adjoint τ H◦ , where H is H with the order reversed. Temporal intuitionistic logic tHC

^ 0 0 0 τ H (h) = h ∨ (h → h) | h ∈ H

algebra of open sets of a space, so has a canonical

coderivative operator τ . The left adjoint τ to τ is given by

τ τ an = bn = bn+1.