Intuitionistic modalities in topology and...

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Intuitionistic modalities in topology and algebra Mamuka Jibladze Razmadze Mathematical Institute, Tbilisi TACL2011, Marseille 27.VII.2011

Transcript of Intuitionistic modalities in topology and...

Intuitionistic modalities in topology andalgebra

Mamuka Jibladze

Razmadze Mathematical Institute, Tbilisi

TACL2011, Marseille 27.VII.2011

L. Esakia, Quantification in intuitionistic logic with provabilitysmack. 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. Annalsof Pure and Applied Logic 103 (2000), 97–107

L. Esakia, Intuitionistic logic and modality via topology. Annalsof Pure and Applied Logic 127 (2004), 155–170

L. Esakia, The modalized Heyting calculus: a conservativemodal extension of the Intuitionistic Logic. Journal of AppliedNon-Classical Logics 16 (2006), 349–366.

Classical modal systems and topologyS4

The system S4 is normal and satisfiesp→ 3p,33p↔ 3p,3(p ∨ q)↔ 3p ∨3q.

Topological semantics (McKinsey & Tarski) —3 can be interpreted as the closure operator C of a topologicalspace X:

valuation v(ϕ) ⊆ X;

v(3ϕ) = C v(ϕ)

Classical modal systems and topologyS4

The system S4 is normal and satisfiesp→ 3p,33p↔ 3p,3(p ∨ q)↔ 3p ∨3q.

Topological semantics (McKinsey & Tarski) —3 can be interpreted as the closure operator C of a topologicalspace X:

valuation v(ϕ) ⊆ X;

v(3ϕ) = C v(ϕ)

Classical modal systems and topologyS4

The system S4 is normal and satisfiesp→ 3p,33p↔ 3p,3(p ∨ q)↔ 3p ∨3q.

Topological semantics (McKinsey & Tarski) —3 can be interpreted as the closure operator C of a topologicalspace X:

valuation v(ϕ) ⊆ X;

v(3ϕ) = C v(ϕ)

Classical modal systems and topologyS4

The system S4 is normal and satisfiesp→ 3p,33p↔ 3p,3(p ∨ q)↔ 3p ∨3q.

Topological semantics (McKinsey & Tarski) —3 can be interpreted as the closure operator C of a topologicalspace X:

valuation v(ϕ) ⊆ X;

v(3ϕ) = C v(ϕ)

Classical modal systems and topologyS4

Algebraic semantics —Closure algebra (B, c)

where B = (B,∧,∨,¬, 0, 1) is a Boolean algebra and c : B → Bsatisfies

c 0 = 0,b 6 c b,c c b = c b,c(b ∨ b′) = c b ∨ c b′.

Classical modal systems and topologyS4

Algebraic semantics —Closure algebra (B, c)

where B = (B,∧,∨,¬, 0, 1) is a Boolean algebra and c : B → Bsatisfies

c 0 = 0,b 6 c b,c c b = c b,c(b ∨ b′) = c b ∨ c b′.

Classical modal systems and topologyS4

Algebraic semantics —Closure algebra (B, c)

where B = (B,∧,∨,¬, 0, 1) is a Boolean algebra and c : B → Bsatisfies

c 0 = 0,b 6 c b,c c b = c b,c(b ∨ b′) = c b ∨ c b′.

Classical modal systems and topologyS4

Equivalently — using the dualinterior operator i = ¬ c¬

satisfying

i 1 = 1,i b 6 b,i i b = i b,i(b ∧ b′) = i b ∧ i b′.

These were considered by Rasiowa and Sikorski under thename of topological Boolean algebras; Blok used the terminterior algebra which is mostly used nowadays along withS4-algebra.

Classical modal systems and topologyS4

Equivalently — using the dualinterior operator i = ¬ c¬

satisfying

i 1 = 1,i b 6 b,i i b = i b,i(b ∧ b′) = i b ∧ i b′.

These were considered by Rasiowa and Sikorski under thename of topological Boolean algebras; Blok used the terminterior algebra which is mostly used nowadays along withS4-algebra.

Classical modal systems and topologyS4

Equivalently — using the dualinterior operator i = ¬ c¬

satisfying

i 1 = 1,i b 6 b,i i b = i b,i(b ∧ b′) = i b ∧ i b′.

These were considered by Rasiowa and Sikorski under thename of topological Boolean algebras; Blok used the terminterior algebra which is mostly used nowadays along withS4-algebra.

Classical modal systems and topologywK4

Closure/interior is one among many ways to define the topologyof 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.

Classical modal systems and topologywK4

Closure/interior is one among many ways to define the topologyof 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.

Classical modal systems and topologywK4

Closure/interior is one among many ways to define the topologyof 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.

Classical modal systems and topologywK4

Closure/interior is one among many ways to define the topologyof 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.

Classical modal systems and topologywK4

Half of the time, it is more useful to work with the dualcoderivative operator given by τA := X − δ(X −A);

one has τA = IA ∪ Isol(X −A),

where I is the interior, IA = X −C(X −A) and IsolS denotesthe set of isolated points of S ⊆ X (with respect to the topologyinduced from X).

This τ satisfies the dual identities

τX = X,τ(A ∩B) = τA ∩ τB,A ∩ τA ⊆ ττA.

Classical modal systems and topologywK4

Half of the time, it is more useful to work with the dualcoderivative operator given by τA := X − δ(X −A);

one has τA = IA ∪ Isol(X −A),

where I is the interior, IA = X −C(X −A) and IsolS denotesthe set of isolated points of S ⊆ X (with respect to the topologyinduced from X).

This τ satisfies the dual identities

τX = X,τ(A ∩B) = τA ∩ τB,A ∩ τA ⊆ ττA.

Classical modal systems and topologywK4

Half of the time, it is more useful to work with the dualcoderivative operator given by τA := X − δ(X −A);

one has τA = IA ∪ Isol(X −A),

where I is the interior, IA = X −C(X −A) and IsolS denotesthe set of isolated points of S ⊆ X (with respect to the topologyinduced from X).

This τ satisfies the dual identities

τX = X,τ(A ∩B) = τA ∩ τB,A ∩ τA ⊆ ττA.

Classical modal systems and topologywK4

Half of the time, it is more useful to work with the dualcoderivative operator given by τA := X − δ(X −A);

one has τA = IA ∪ Isol(X −A),

where I is the interior, IA = X −C(X −A) and IsolS denotesthe set of isolated points of S ⊆ X (with respect to the topologyinduced from X).

This τ satisfies the dual identities

τX = X,τ(A ∩B) = τA ∩ τB,A ∩ τA ⊆ ττA.

Classical modal systems and topologywK4

If one wants to have a modal system with topological semanticsinterpreting 3 as δ, and 2 as τ , one thus arrives at a normalsystem with an additional axiom

(p&2p)→ 22p,

or equivalently

33p→ (p ∨3p).

This is wK4, or weak K4.

Classical modal systems and topologywK4

If one wants to have a modal system with topological semanticsinterpreting 3 as δ, and 2 as τ , one thus arrives at a normalsystem with an additional axiom

(p&2p)→ 22p,

or equivalently

33p→ (p ∨3p).

This is wK4, or weak K4.

Classical modal systems and topologywK4

If one wants to have a modal system with topological semanticsinterpreting 3 as δ, and 2 as τ , one thus arrives at a normalsystem with an additional axiom

(p&2p)→ 22p,

or equivalently

33p→ (p ∨3p).

This is wK4, or weak K4.

Classical modal systems and topologywK4

If one wants to have a modal system with topological semanticsinterpreting 3 as δ, and 2 as τ , one thus arrives at a normalsystem with an additional axiom

(p&2p)→ 22p,

or equivalently

33p→ (p ∨3p).

This is wK4, or weak K4.

Classical modal systems and topologywK4

For the algebraic semantics one has derivative algebras (B, δ)— Boolean algebras with an operator δ satisfying

δ 0 = 0,δ(b ∨ b′) = δ b ∨ δ b′,δ δ b 6 b ∨ δ b.

Or, one might define them in terms of the dual coderivativeoperator τ = ¬ δ ¬ with axioms

τ 1 = 1,τ (b ∧ b′) = τ b ∧ τ b′,b ∧ τ b 6 τ τ b.

Classical modal systems and topologywK4

For the algebraic semantics one has derivative algebras (B, δ)— Boolean algebras with an operator δ satisfying

δ 0 = 0,δ(b ∨ b′) = δ b ∨ δ b′,δ δ b 6 b ∨ δ b.

Or, one might define them in terms of the dual coderivativeoperator τ = ¬ δ ¬ with axioms

τ 1 = 1,τ (b ∧ b′) = τ b ∧ τ b′,b ∧ τ b 6 τ τ b.

The intuitionistic sideHC

Let us now consider the intuitionistic counterparts of thesesystems.

For a closure algebra (B, c), the subsetH = i(B) = i b | b ∈ B = Fix(i) = h ∈ B | ih = h is asublattice of B;

it is a Heyting algebra with respect to the implication

h −→H

h′ := i(h→ h′)

and is thus an algebraic model of the Heyting propositionalCalculus HC.

Obviously, i itself disappears from sight in H;

also in general c does not leave any manageable “trace” on H.

The intuitionistic sideHC

Let us now consider the intuitionistic counterparts of thesesystems.

For a closure algebra (B, c), the subsetH = i(B) = i b | b ∈ B = Fix(i) = h ∈ B | ih = h is asublattice of B;

it is a Heyting algebra with respect to the implication

h −→H

h′ := i(h→ h′)

and is thus an algebraic model of the Heyting propositionalCalculus HC.

Obviously, i itself disappears from sight in H;

also in general c does not leave any manageable “trace” on H.

The intuitionistic sideHC

Let us now consider the intuitionistic counterparts of thesesystems.

For a closure algebra (B, c), the subsetH = i(B) = i b | b ∈ B = Fix(i) = h ∈ B | ih = h is asublattice of B;

it is a Heyting algebra with respect to the implication

h −→H

h′ := i(h→ h′)

and is thus an algebraic model of the Heyting propositionalCalculus HC.

Obviously, i itself disappears from sight in H;

also in general c does not leave any manageable “trace” on H.

The intuitionistic sideHC

Let us now consider the intuitionistic counterparts of thesesystems.

For a closure algebra (B, c), the subsetH = i(B) = i b | b ∈ B = Fix(i) = h ∈ B | ih = h is asublattice of B;

it is a Heyting algebra with respect to the implication

h −→H

h′ := i(h→ h′)

and is thus an algebraic model of the Heyting propositionalCalculus HC.

Obviously, i itself disappears from sight in H;

also in general c does not leave any manageable “trace” on H.

The intuitionistic sideHC

Let us now consider the intuitionistic counterparts of thesesystems.

For a closure algebra (B, c), the subsetH = i(B) = i b | b ∈ B = Fix(i) = h ∈ B | ih = h is asublattice of B;

it is a Heyting algebra with respect to the implication

h −→H

h′ := i(h→ h′)

and is thus an algebraic model of the Heyting propositionalCalculus HC.

Obviously, i itself disappears from sight in H;

also in general c does not leave any manageable “trace” on H.

The intuitionistic sideHC

Let us now consider the intuitionistic counterparts of thesesystems.

For a closure algebra (B, c), the subsetH = i(B) = i b | b ∈ B = Fix(i) = h ∈ B | ih = h is asublattice of B;

it is a Heyting algebra with respect to the implication

h −→H

h′ := i(h→ h′)

and is thus an algebraic model of the Heyting propositionalCalculus HC.

Obviously, i itself disappears from sight in H;

also in general c does not leave any manageable “trace” on H.

The intuitionistic sidemHC

The situation is more interesting with the (co)derivativesemantics.

Note that when (B, c) = (P(X),C) for a topological space X,the corresponding H is the Heyting algebra O(X) of all opensets of X.

In terms of the coderivative, a subset U ⊆ X is open if and onlyif U ⊆ τU .

More generally, for a (co)derivative algebra (B, τ ) one gets theHeyting 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 sidemHC

The situation is more interesting with the (co)derivativesemantics.

Note that when (B, c) = (P(X),C) for a topological space X,the corresponding H is the Heyting algebra O(X) of all opensets of X.

In terms of the coderivative, a subset U ⊆ X is open if and onlyif U ⊆ τU .

More generally, for a (co)derivative algebra (B, τ ) one gets theHeyting 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 sidemHC

The situation is more interesting with the (co)derivativesemantics.

Note that when (B, c) = (P(X),C) for a topological space X,the corresponding H is the Heyting algebra O(X) of all opensets of X.

In terms of the coderivative, a subset U ⊆ X is open if and onlyif U ⊆ τU .

More generally, for a (co)derivative algebra (B, τ ) one gets theHeyting 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 sidemHC

The situation is more interesting with the (co)derivativesemantics.

Note that when (B, c) = (P(X),C) for a topological space X,the corresponding H is the Heyting algebra O(X) of all opensets of X.

In terms of the coderivative, a subset U ⊆ X is open if and onlyif U ⊆ τU .

More generally, for a (co)derivative algebra (B, τ ) one gets theHeyting 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 sidemHC

The situation is more interesting with the (co)derivativesemantics.

Note that when (B, c) = (P(X),C) for a topological space X,the corresponding H is the Heyting algebra O(X) of all opensets of X.

In terms of the coderivative, a subset U ⊆ X is open if and onlyif U ⊆ τU .

More generally, for a (co)derivative algebra (B, τ ) one gets theHeyting 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 sidemHC

The situation is more interesting with the (co)derivativesemantics.

Note that when (B, c) = (P(X),C) for a topological space X,the corresponding H is the Heyting algebra O(X) of all opensets of X.

In terms of the coderivative, a subset U ⊆ X is open if and onlyif U ⊆ τU .

More generally, for a (co)derivative algebra (B, τ ) one gets theHeyting 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 sidemHC

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

Then obviouslyτ 1 = 1,τ (h ∧ h′) = τ h ∧ τ h′

on H.

The intuitionistic sidemHC

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

Then obviouslyτ 1 = 1,τ (h ∧ h′) = τ h ∧ τ h′

on H.

The intuitionistic sidemHC

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

τ(U) =∧

V ∪ (V −−−→O(X)

U) | V ∈ O(X)

.

In particular, one has

τ(U) 6 V ∪(V −−−→

O(X)U

)for any U, V ∈ O(X).

The intuitionistic sidemHC

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

τ(U) =∧

V ∪ (V −−−→O(X)

U) | V ∈ O(X)

.

In particular, one has

τ(U) 6 V ∪(V −−−→

O(X)U

)for any U, V ∈ O(X).

The intuitionistic sidemHC

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

τ(U) =∧

V ∪ (V −−−→O(X)

U) | V ∈ O(X)

.

In particular, one has

τ(U) 6 V ∪(V −−−→

O(X)U

)for any U, V ∈ O(X).

The intuitionistic sidemHC

We thus arrive at an intuitionistic modal system mHC, withtopological semantics given by valuations via open sets of aspace and the modality 2 interpreted as the coderivativerestricted to open sets.

It is given by adding to the axioms of HC the axioms2(p→ q)→ (2p→ 2q),p→ 2p,2p→ (q ∨ (q → p)).

NB The second axiom might look sort of “wrong way” for amodal logician, as for multiplicative operators implicationusually goes in the opposite direction.However in intuitionistic modal systems such things happen.E. g. for those familiar with nuclei — a nucleus is an inflationarymultiplicative idempotent operator.

The intuitionistic sidemHC

We thus arrive at an intuitionistic modal system mHC, withtopological semantics given by valuations via open sets of aspace and the modality 2 interpreted as the coderivativerestricted 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 amodal logician, as for multiplicative operators implicationusually goes in the opposite direction.However in intuitionistic modal systems such things happen.E. g. for those familiar with nuclei — a nucleus is an inflationarymultiplicative idempotent operator.

The intuitionistic sidemHC

We thus arrive at an intuitionistic modal system mHC, withtopological semantics given by valuations via open sets of aspace and the modality 2 interpreted as the coderivativerestricted 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 amodal logician, as for multiplicative operators implicationusually goes in the opposite direction.

However in intuitionistic modal systems such things happen.E. g. for those familiar with nuclei — a nucleus is an inflationarymultiplicative idempotent operator.

The intuitionistic sidemHC

We thus arrive at an intuitionistic modal system mHC, withtopological semantics given by valuations via open sets of aspace and the modality 2 interpreted as the coderivativerestricted 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 amodal logician, as for multiplicative operators implicationusually goes in the opposite direction.However in intuitionistic modal systems such things happen.E. g. for those familiar with nuclei — a nucleus is an inflationarymultiplicative idempotent operator.

The intuitionistic sidemHC

Algebraic models of mHC are thus of the form (H, τ ) where His a Heyting algebra and τ : H → H satisfies

h 6 τ h,τ (h ∧ h′) = τ h ∧ τ h′,τ h 6 h′ ∨ (h′ → h).

Back to the classicsK4.Grz

Moreover one has

Theorem. For every mHC-algebra (H, τ ) there exists acoderivative algebra (B(H), τ ) such thatH = h ∈ B(H) | h 6 τ h and τ |H = τ .

Here B(H) is the free Boolean extension of H, so that everyelement of B(H) is a finite meet of elements of the form ¬h′ ∨ hfor some h, h′ ∈ H. One defines

τ (¬h′ ∨ h) := h′ −→H

h

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

Back to the classicsK4.Grz

Moreover one has

Theorem. For every mHC-algebra (H, τ ) there exists acoderivative algebra (B(H), τ ) such thatH = h ∈ B(H) | h 6 τ h and τ |H = τ .

Here B(H) is the free Boolean extension of H, so that everyelement of B(H) is a finite meet of elements of the form ¬h′ ∨ hfor some h, h′ ∈ H. One defines

τ (¬h′ ∨ h) := h′ −→H

h

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

Back to the classicsK4.Grz

It turns out that actually the coderivative algebras of the aboveform (B(H), τ ) land in a proper subvariety: they all areK4-algebras, i. e. satisfy τ b 6 τ τ b; moreover they satisfy theidentity

τ (τ (b→ τ b)→ b) = τ b.

This enabled Esakia to construct a translation # of mHC intothe 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 classicsK4.Grz

It turns out that actually the coderivative algebras of the aboveform (B(H), τ ) land in a proper subvariety: they all areK4-algebras, i. e. satisfy τ b 6 τ τ b; moreover they satisfy theidentity

τ (τ (b→ τ b)→ b) = τ b.

This enabled Esakia to construct a translation # of mHC intothe 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 classicsK4.Grz

It turns out that actually the coderivative algebras of the aboveform (B(H), τ ) land in a proper subvariety: they all areK4-algebras, i. e. satisfy τ b 6 τ τ b; moreover they satisfy theidentity

τ (τ (b→ τ b)→ b) = τ b.

This enabled Esakia to construct a translation # of mHC intothe 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 classicsK4.Grz

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

This result may be viewed as an analog/generalization of theKuznetsov-Muravitsky theorem.

Back to the classicsK4.Grz

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

This result may be viewed as an analog/generalization of theKuznetsov-Muravitsky theorem.

Canonical choices of the modalityKM

The Kuznetsov-Muravitsky calculus KM may be defined as theresult 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 allextensions of KM is isomorphic to the lattice of all normalextensions of GL.

From the point of view of topological/algebraic semantics, thissystem is interesting in that in its models, the coderivativeoperator is in fact uniquely determined.

Canonical choices of the modalityKM

The Kuznetsov-Muravitsky calculus KM may be defined as theresult 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 allextensions of KM is isomorphic to the lattice of all normalextensions of GL.

From the point of view of topological/algebraic semantics, thissystem is interesting in that in its models, the coderivativeoperator is in fact uniquely determined.

Canonical choices of the modalityKM

The Kuznetsov-Muravitsky calculus KM may be defined as theresult 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 allextensions of KM is isomorphic to the lattice of all normalextensions of GL.

From the point of view of topological/algebraic semantics, thissystem is interesting in that in its models, the coderivativeoperator is in fact uniquely determined.

Canonical choices of the modalityKM

The Kuznetsov-Muravitsky calculus KM may be defined as theresult 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 allextensions of KM is isomorphic to the lattice of all normalextensions of GL.

From the point of view of topological/algebraic semantics, thissystem is interesting in that in its models, the coderivativeoperator is in fact uniquely determined.

Canonical choices of the modalityKM

For H of the form O(X) for a space X, there is a canonicalchoice of the “correct” τ .

However for general Heyting algebrasH there might be several operators τ satisfying the aboveidentities. For example, τ h ≡ h would always do.Whereas if (H, τ ) happens to be a model of KM, i. e. τ h→ his equal to h for all h ∈ H, then in addition toτ h 6 h′ ∨ (h′ → h), also τ h itself is of the form h′ ∨ (h′ → h) forsome h′ (in fact for h′ = τ h).Thus τ h is the smallest element of the seth′ ∨ (h′ → h) | h′ ∈ H, and this property makes it uniquelydetermined.An algebra of the form (O(X), τ) is a model of KM iff the spaceX is scattered (every nonempty subspace has an isolatedpoint).

Canonical choices of the modalityKM

For H of the form O(X) for a space X, there is a canonicalchoice of the “correct” τ . However for general Heyting algebrasH there might be several operators τ satisfying the aboveidentities.

For example, τ h ≡ h would always do.Whereas if (H, τ ) happens to be a model of KM, i. e. τ h→ his equal to h for all h ∈ H, then in addition toτ h 6 h′ ∨ (h′ → h), also τ h itself is of the form h′ ∨ (h′ → h) forsome h′ (in fact for h′ = τ h).Thus τ h is the smallest element of the seth′ ∨ (h′ → h) | h′ ∈ H, and this property makes it uniquelydetermined.An algebra of the form (O(X), τ) is a model of KM iff the spaceX is scattered (every nonempty subspace has an isolatedpoint).

Canonical choices of the modalityKM

For H of the form O(X) for a space X, there is a canonicalchoice of the “correct” τ . However for general Heyting algebrasH there might be several operators τ satisfying the aboveidentities. For example, τ h ≡ h would always do.

Whereas if (H, τ ) happens to be a model of KM, i. e. τ h→ his equal to h for all h ∈ H, then in addition toτ h 6 h′ ∨ (h′ → h), also τ h itself is of the form h′ ∨ (h′ → h) forsome h′ (in fact for h′ = τ h).Thus τ h is the smallest element of the seth′ ∨ (h′ → h) | h′ ∈ H, and this property makes it uniquelydetermined.An algebra of the form (O(X), τ) is a model of KM iff the spaceX is scattered (every nonempty subspace has an isolatedpoint).

Canonical choices of the modalityKM

For H of the form O(X) for a space X, there is a canonicalchoice of the “correct” τ . However for general Heyting algebrasH there might be several operators τ satisfying the aboveidentities. For example, τ h ≡ h would always do.Whereas if (H, τ ) happens to be a model of KM, i. e. τ h→ his equal to h for all h ∈ H, then in addition toτ h 6 h′ ∨ (h′ → h), also τ h itself is of the form h′ ∨ (h′ → h) forsome h′ (in fact for h′ = τ h).

Thus τ h is the smallest element of the seth′ ∨ (h′ → h) | h′ ∈ H, and this property makes it uniquelydetermined.An algebra of the form (O(X), τ) is a model of KM iff the spaceX is scattered (every nonempty subspace has an isolatedpoint).

Canonical choices of the modalityKM

For H of the form O(X) for a space X, there is a canonicalchoice of the “correct” τ . However for general Heyting algebrasH there might be several operators τ satisfying the aboveidentities. For example, τ h ≡ h would always do.Whereas if (H, τ ) happens to be a model of KM, i. e. τ h→ his equal to h for all h ∈ H, then in addition toτ h 6 h′ ∨ (h′ → h), also τ h itself is of the form h′ ∨ (h′ → h) forsome h′ (in fact for h′ = τ h).Thus τ h is the smallest element of the seth′ ∨ (h′ → h) | h′ ∈ H, and this property makes it uniquelydetermined.

An algebra of the form (O(X), τ) is a model of KM iff the spaceX is scattered (every nonempty subspace has an isolatedpoint).

Canonical choices of the modalityKM

For H of the form O(X) for a space X, there is a canonicalchoice of the “correct” τ . However for general Heyting algebrasH there might be several operators τ satisfying the aboveidentities. For example, τ h ≡ h would always do.Whereas if (H, τ ) happens to be a model of KM, i. e. τ h→ his equal to h for all h ∈ H, then in addition toτ h 6 h′ ∨ (h′ → h), also τ h itself is of the form h′ ∨ (h′ → h) forsome h′ (in fact for h′ = τ h).Thus τ h is the smallest element of the seth′ ∨ (h′ → h) | h′ ∈ H, and this property makes it uniquelydetermined.An algebra of the form (O(X), τ) is a model of KM iff the spaceX is scattered (every nonempty subspace has an isolatedpoint).

Canonical choices of the modalityQHC

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

Note that whenever H is a complete Heyting algebra, it comeswith the “correct” τ , viz.

τ h :=∧

h′ ∨ (h′ → h) | h′ ∈ H.

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

Canonical choices of the modalityQHC

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

Note that whenever H is a complete Heyting algebra, it comeswith the “correct” τ , viz.

τ h :=∧

h′ ∨ (h′ → h) | h′ ∈ H.

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

Canonical choices of the modalityQHC

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

Note that whenever H is a complete Heyting algebra, it comeswith the “correct” τ , viz.

τ h :=∧

h′ ∨ (h′ → h) | h′ ∈ H.

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

Canonical choices of the modalityQHC

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 resultof 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 ofmHC.

Canonical choices of the modalityQHC

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 resultof 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 ofmHC.

Canonical choices of the modalityQHC

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 resultof 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 ofmHC.

Canonical choices of the modalityQ+HC

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

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

∀p((ϕ→ ∀pϕ)→ ∀pϕ)→ ∀pϕ.

Canonical choices of the modalityQ+HC

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

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

∀p((ϕ→ ∀pϕ)→ ∀pϕ)→ ∀pϕ.

Canonical choices of the modalityKripke-Joyal semantics

Naturally defined propositional quantifiers are readily availablein the topos semantics.

If a Heyting algebra H happens to be the algebra of allsubobjects of some object in an elementary topos, it comesequipped with such quantifiers.

In particular, on the subobject classifier Ω one has thecorresponding operator τ : Ω→ Ω given, in the Kripke-Joyalsemantics, by

τ(u) = ∀p(p ∨ (p→ u)).

Thus τ classifies the Higgs subobject µ ∈ Ω | > ∪ ↓µ = Ωof Ω.

Canonical choices of the modalityKripke-Joyal semantics

Naturally defined propositional quantifiers are readily availablein the topos semantics.

If a Heyting algebra H happens to be the algebra of allsubobjects of some object in an elementary topos, it comesequipped with such quantifiers.

In particular, on the subobject classifier Ω one has thecorresponding operator τ : Ω→ Ω given, in the Kripke-Joyalsemantics, by

τ(u) = ∀p(p ∨ (p→ u)).

Thus τ classifies the Higgs subobject µ ∈ Ω | > ∪ ↓µ = Ωof Ω.

Canonical choices of the modalityKripke-Joyal semantics

Naturally defined propositional quantifiers are readily availablein the topos semantics.

If a Heyting algebra H happens to be the algebra of allsubobjects of some object in an elementary topos, it comesequipped with such quantifiers.

In particular, on the subobject classifier Ω one has thecorresponding operator τ : Ω→ Ω given, in the Kripke-Joyalsemantics, by

τ(u) = ∀p(p ∨ (p→ u)).

Thus τ classifies the Higgs subobject µ ∈ Ω | > ∪ ↓µ = Ωof Ω.

Canonical choices of the modalityKripke-Joyal semantics

Naturally defined propositional quantifiers are readily availablein the topos semantics.

If a Heyting algebra H happens to be the algebra of allsubobjects of some object in an elementary topos, it comesequipped with such quantifiers.

In particular, on the subobject classifier Ω one has thecorresponding operator τ : Ω→ Ω given, in the Kripke-Joyalsemantics, by

τ(u) = ∀p(p ∨ (p→ u)).

Thus τ classifies the Higgs subobject µ ∈ Ω | > ∪ ↓µ = Ωof Ω.

Canonical choices of the modalityScattered toposes

Call a topos scattered if this τ satisfies

(τ(u)→ u)→ u.

Note that this identity may be viewed as a kind of inductionprinciple: the Higgs object contains the top together with itsimmediate predecessor, if any.

Then the identity says that if we want to prove some statementu, we might as well assume that it is (either the top or) theimmediate predecessor of the top.

There are non-Boolean scattered toposes — e. g. the sheaveson any scattered space form a scattered topos.

Canonical choices of the modalityScattered toposes

Call a topos scattered if this τ satisfies

(τ(u)→ u)→ u.

Note that this identity may be viewed as a kind of inductionprinciple

: the Higgs object contains the top together with itsimmediate predecessor, if any.

Then the identity says that if we want to prove some statementu, we might as well assume that it is (either the top or) theimmediate predecessor of the top.

There are non-Boolean scattered toposes — e. g. the sheaveson any scattered space form a scattered topos.

Canonical choices of the modalityScattered toposes

Call a topos scattered if this τ satisfies

(τ(u)→ u)→ u.

Note that this identity may be viewed as a kind of inductionprinciple: the Higgs object contains the top together with itsimmediate predecessor, if any.

Then the identity says that if we want to prove some statementu, we might as well assume that it is (either the top or) theimmediate predecessor of the top.

There are non-Boolean scattered toposes — e. g. the sheaveson any scattered space form a scattered topos.

Canonical choices of the modalityScattered toposes

Call a topos scattered if this τ satisfies

(τ(u)→ u)→ u.

Note that this identity may be viewed as a kind of inductionprinciple: the Higgs object contains the top together with itsimmediate predecessor, if any.

Then the identity says that if we want to prove some statementu, we might as well assume that it is (either the top or) theimmediate predecessor of the top.

There are non-Boolean scattered toposes — e. g. the sheaveson any scattered space form a scattered topos.

Canonical choices of the modalityScattered toposes

Call a topos scattered if this τ satisfies

(τ(u)→ u)→ u.

Note that this identity may be viewed as a kind of inductionprinciple: the Higgs object contains the top together with itsimmediate predecessor, if any.

Then the identity says that if we want to prove some statementu, we might as well assume that it is (either the top or) theimmediate predecessor of the top.

There are non-Boolean scattered toposes — e. g. the sheaveson any scattered space form a scattered topos.

Canonical choices of the modalityScattered toposes

Theorem. For an elementary topos E , the following areequivalent:

(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 logictHC

The temporal Heyting Calculus tHC results from adding tomHC one more modal operator 3, with additional axioms

p→ 23p;32p→ p;3(p ∨ q)→ (3p ∨3q);3⊥ → ⊥

and an additional rulep→ q

3p→ 3q.

Temporal intuitionistic logictHC

In the algebraic semantics, this corresponds to having a leftadjoint to τ : H → H.

That is, an operator τ: H → H such that h 6 τ h′ if and only if

τh 6 h′.

If such an adjoint exists on a coderivative algebra (P(X), τ )corresponding to a space X, then one can show that thetopology on X is the Alexandroff topology for some preorder onX (namely, for the specialization preorder).

For (O(X), τ ) existence of the adjoint is less stringent. Thereare “almost Alexandroff” spaces with this property which arenot Alexandroff.

Temporal intuitionistic logictHC

In the algebraic semantics, this corresponds to having a leftadjoint to τ : H → H.

That is, an operator τ: H → H such that h 6 τ h′ if and only if

τh 6 h′.

If such an adjoint exists on a coderivative algebra (P(X), τ )corresponding to a space X, then one can show that thetopology on X is the Alexandroff topology for some preorder onX (namely, for the specialization preorder).

For (O(X), τ ) existence of the adjoint is less stringent. Thereare “almost Alexandroff” spaces with this property which arenot Alexandroff.

Temporal intuitionistic logictHC

In the algebraic semantics, this corresponds to having a leftadjoint to τ : H → H.

That is, an operator τ: H → H such that h 6 τ h′ if and only if

τh 6 h′.

If such an adjoint exists on a coderivative algebra (P(X), τ )corresponding to a space X, then one can show that thetopology on X is the Alexandroff topology for some preorder onX (namely, for the specialization preorder).

For (O(X), τ ) existence of the adjoint is less stringent. Thereare “almost Alexandroff” spaces with this property which arenot Alexandroff.

Temporal intuitionistic logictHC

In the algebraic semantics, this corresponds to having a leftadjoint to τ : H → H.

That is, an operator τ: H → H such that h 6 τ h′ if and only if

τh 6 h′.

If such an adjoint exists on a coderivative algebra (P(X), τ )corresponding to a space X, then one can show that thetopology on X is the Alexandroff topology for some preorder onX (namely, for the specialization preorder).

For (O(X), τ ) existence of the adjoint is less stringent. Thereare “almost Alexandroff” spaces with this property which arenot Alexandroff.

Temporal intuitionistic logictHC

A simple example: the real line R with the topology for whichthe only open sets are the open rays (r,∞) for −∞ 6 r 6∞.

More generally, if H is a complete bi-Heyting algebra then itscanonical coderivative operator τH given by

τH(h) =∧

h′ ∨ (h′ → h) | h′ ∈ H

has a left adjoint τH , 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 isgiven by > = a0 > a1 > a2 > a3 > ... andb0 > b1 > b2 > b3 > ... > ⊥, with an > bn for each n. It is thealgebra of open sets of a space, so has a canonicalcoderivative operator τ . The left adjoint τto τ is given by

τan = τbn = bn+1.

Temporal intuitionistic logictHC

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

τH(h) =∧

h′ ∨ (h′ → h) | h′ ∈ H

has a left adjoint τH , 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 isgiven by > = a0 > a1 > a2 > a3 > ... andb0 > b1 > b2 > b3 > ... > ⊥, with an > bn for each n. It is thealgebra of open sets of a space, so has a canonicalcoderivative operator τ . The left adjoint τto τ is given by

τan = τbn = bn+1.

Temporal intuitionistic logictHC

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

τH(h) =∧

h′ ∨ (h′ → h) | h′ ∈ H

has a left adjoint τH , 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 isgiven by > = a0 > a1 > a2 > a3 > ... andb0 > b1 > b2 > b3 > ... > ⊥, with an > bn for each n. It is thealgebra of open sets of a space, so has a canonicalcoderivative operator τ . The left adjoint τto τ is given by

τan = τbn = bn+1.