Axiomatic classes of models in modal logics Evgeny Zolin Moscow - - PowerPoint PPT Presentation

Axiomatic classes of models in modal logics

Evgeny Zolin

Moscow State University

Workshop on Proof Theory, Modal Logic and Reflection Principles (Wormshop 2017) Steklov Mathematical Institute, Moscow October 20, 2017

Evgeny Zolin Axiomatic classes October 20, 2017 1 / 18

To worm up...

Evgeny Zolin Axiomatic classes October 20, 2017 2 / 18

To worm up...

Theorem (Keisler, 1961)

Let K be any class of first-order models (of a fixed signature). K is axiomatizable ⇐ ⇒ K is closed under ≡FO and ultraproducts Elementary equivalence: M ≡FO N means: for every formula A ∈ FO we have

• M |

= A ⇐ ⇒ N | = A

• Evgeny Zolin

Axiomatic classes October 20, 2017 2 / 18

To worm up...

Theorem (Keisler, 1961)

Let K be any class of first-order models (of a fixed signature). K is axiomatizable ⇐ ⇒ K is closed under ≡FO and ultraproducts K is finitely axiomatizable ⇐ ⇒ both K and its complement K are closed under ≡FO and ultraproducts Elementary equivalence: M ≡FO N means: for every formula A ∈ FO we have

• M |

= A ⇐ ⇒ N | = A

• Evgeny Zolin

Axiomatic classes October 20, 2017 2 / 18

To worm up...

Theorem (Keisler, 1961)

Let K be any class of first-order models (of a fixed signature). K is axiomatizable ⇐ ⇒ K is closed under ≡FO and ultraproducts K is finitely axiomatizable ⇐ ⇒ both K and its complement K are closed under ≡FO and ultraproducts Closure conditions Both K and K K K K is axiomatizable ≡FO uProd K is finitely axiomatizable ≡FO uProd uProd

Evgeny Zolin Axiomatic classes October 20, 2017 2 / 18

To worm up...

Theorem (Keisler, 1961)

Let K be any class of first-order models (of a fixed signature). K is axiomatizable ⇐ ⇒ K is closed under ≡FO and ultraproducts K is finitely axiomatizable ⇐ ⇒ both K and its complement K are closed under ≡FO and ultraproducts Closure conditions Both K and K K K K is axiomatizable ≡FO uProd K is finitely axiomatizable ≡FO uProd uProd Why non-symmetric?

Evgeny Zolin Axiomatic classes October 20, 2017 2 / 18

To worm up...

Theorem (Keisler, 1961)

Let K be any class of first-order models (of a fixed signature). K is axiomatizable ⇐ ⇒ K is closed under ≡FO and ultraproducts K is finitely axiomatizable ⇐ ⇒ both K and its complement K are closed under ≡FO and ultraproducts Closure conditions Both K and K K K K is axiomatizable ≡FO uProd K is finitely axiomatizable ≡FO uProd uProd Why non-symmetric? Because this is not the whole story!

Evgeny Zolin Axiomatic classes October 20, 2017 2 / 18

The 4 “species” of classes

• Definition. For a class of models K we write:

K ∈ L if K = Models(A), for some formula A ∈ FO.

Evgeny Zolin Axiomatic classes October 20, 2017 3 / 18

The 4 “species” of classes

• Definition. For a class of models K we write:

K ∈ L if K = Models(A), for some formula A ∈ FO. K ∈ ⩀L if K = Models(Γ), for some set of formulas Γ ⊆ FO.

Evgeny Zolin Axiomatic classes October 20, 2017 3 / 18

The 4 “species” of classes

• Definition. For a class of models K we write:

K ∈ L if K = Models(A), for some formula A ∈ FO. K ∈ ⩀L if K = Models(Γ), for some set of formulas Γ ⊆ FO. Equivalently: if K =

i∈I

Ki for some classes Ki ∈ L.

Evgeny Zolin Axiomatic classes October 20, 2017 3 / 18

The 4 “species” of classes

• Definition. For a class of models K we write:

K ∈ L if K = Models(A), for some formula A ∈ FO. K ∈ ⩀L if K = Models(Γ), for some set of formulas Γ ⊆ FO. Equivalently: if K =

i∈I

Ki for some classes Ki ∈ L. K ∈ ⊍L if K =

i∈I

Ki for some classes Ki ∈ L.

Evgeny Zolin Axiomatic classes October 20, 2017 3 / 18

The 4 “species” of classes

• Definition. For a class of models K we write:

K ∈ L if K = Models(A), for some formula A ∈ FO. K ∈ ⩀L if K = Models(Γ), for some set of formulas Γ ⊆ FO. Equivalently: if K =

i∈I

Ki for some classes Ki ∈ L. K ∈ ⊍L if K =

i∈I

Ki for some classes Ki ∈ L. K ∈ ⊍⩀L if K =

i∈I

• j∈Ji

Ki,j for some classes Ki,j ∈ L.

Evgeny Zolin Axiomatic classes October 20, 2017 3 / 18

The 4 “species” of classes

• Definition. For a class of models K we write:

K ∈ L if K = Models(A), for some formula A ∈ FO. K ∈ ⩀L if K = Models(Γ), for some set of formulas Γ ⊆ FO. Equivalently: if K =

i∈I

Ki for some classes Ki ∈ L. K ∈ ⊍L if K =

i∈I

Ki for some classes Ki ∈ L. K ∈ ⊍⩀L if K =

i∈I

• j∈Ji

Ki,j for some classes Ki,j ∈ L. A somewhat “oldish” terminology: K ∈ L — an elementary class of models (finitely axiomatizable) K ∈ ⩀L — a ∆-elementary class of models (axiomatizable) K ∈ ⊍L — a Σ-elementary class of models (co-axiomatizable?) K ∈ ⊍⩀L — a Σ∆-elementary class of models

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The hierarchy of the 4 species of classes

⊍⩀L = ⩀⊍L ⩀L ⊍L

L

Evgeny Zolin Axiomatic classes October 20, 2017 4 / 18

The hierarchy of the 4 species of classes

⊍⩀L = ⩀⊍L ⩀L ⊍L

L

In L: the classes of all groups (rings, fields) In ⩀L: infinite groups (rings, fields), algebraically closed fields In ⊍L: finite groups (rings, fields) In ⊍⩀L: infinite fields of characteristic p > 0; infinite finitely dimensional vector spaces Not even in ⊍⩀L: well-ordered sets, periodic groups, simple groups

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First-order language | Criteria for the 4 species

Theorem (Keisler, 1961)

Both K K K ∈ ⊍⩀L ≡FO K ∈ ⊍L ≡FO uProd K ∈ ⩀L ≡FO uProd K ∈ L ≡FO uProd uProd Legend: uProd = ultraproduct

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First-order language | Criteria for the 4 species

Theorem (Keisler, 1961) (Keisler, 1961; Shelah, 1971)

Both K K K ∈ ⊍⩀L ≡FO K ∈ ⊍L ≡FO uProd K ∈ ⩀L ≡FO uProd K ∈ L ≡FO uProd uProd Both K K ∼ = uPow uPow ∼ = uPow uProd ∼ = uProd uPow ∼ = uProd uProd Legend: uProd = ultraproduct uPow = ultrapower

Evgeny Zolin Axiomatic classes October 20, 2017 5 / 18

First-order language | Criteria for the 4 species

Theorem (Keisler, 1961) (Keisler, 1961; Shelah, 1971)

Both K K K ∈ ⊍⩀L ≡FO K ∈ ⊍L ≡FO uProd K ∈ ⩀L ≡FO uProd K ∈ L ≡FO uProd uProd Both K K ∼ = uPow uPow ∼ = uPow uProd ∼ = uProd uPow ∼ = uProd uProd Legend: uProd = ultraproduct uPow = ultrapower Main reason for the symmetry in the tables: M | = A ⇐ ⇒ M | = ¬A

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Modal language | Kripke semantics

Formulas: pi | ¬A | (A ∧ B) | (A ∨ B) | (A → B) | A

Evgeny Zolin Axiomatic classes October 20, 2017 6 / 18

Modal language | Kripke semantics

Formulas: pi | ¬A | (A ∧ B) | (A ∨ B) | (A → B) | A Kripke semantics: Kripke model: M = (W , R, V ), where W = ∅ — a nonempty set of worlds R ⊆ W × W — a accessibility relation between worlds V (pi) ⊆ W — a valuation of variables

Evgeny Zolin Axiomatic classes October 20, 2017 6 / 18

Modal language | Kripke semantics

Formulas: pi | ¬A | (A ∧ B) | (A ∨ B) | (A → B) | A Kripke semantics: Kripke model: M = (W , R, V ), where W = ∅ — a nonempty set of worlds R ⊆ W × W — a accessibility relation between worlds V (pi) ⊆ W — a valuation of variables Truth of a formula is defined in a pointed model (M, x): M, x | = pi ⇌ x ∈ V (pi) M, x | = ¬A ⇌ M, x | = A M, x | = A ∧ B ⇌ M, x | = A and M, x | = B M, x | = A ∨ B ⇌ M, x | = A

• r

M, x | = B M, x | = A → B ⇌ M, x | = A ⇒ M, x | = B M, x | = A ⇌ for every y ∈ W (x R y ⇒ M, y | = A) Truth of a formula in a model: M | = A ⇌ ∀x ∈ W M, x | = A.

Evgeny Zolin Axiomatic classes October 20, 2017 6 / 18

Modal language | Relations between (pointed) models

Modal equivalence of two (pointed) Kripke models M ≡ML N ⇌ for every formula A ∈ ML: M | = A ⇐ ⇒ N | = A

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Modal language | Relations between (pointed) models

Modal equivalence of two (pointed) Kripke models M ≡ML N ⇌ for every formula A ∈ ML: M | = A ⇐ ⇒ N | = A Bisimulation between two pointed Kripke models M, a ≃ N, b —

• respects the valuation of variables
• every step in M is “simulated” by some step in N
• every step in N is “simulated” by some step in M

Evgeny Zolin Axiomatic classes October 20, 2017 7 / 18

Modal language | Relations between (pointed) models

Modal equivalence of two (pointed) Kripke models M ≡ML N ⇌ for every formula A ∈ ML: M | = A ⇐ ⇒ N | = A Bisimulation between two pointed Kripke models M, a ≃ N, b —

• respects the valuation of variables
• every step in M is “simulated” by some step in N
• every step in N is “simulated” by some step in M

Global bisimulation between Kripke models M :≃: N — bisimulation that covers the whole models M and N

Evgeny Zolin Axiomatic classes October 20, 2017 7 / 18

Compact and saturated classes of structures

Let K be a class of pointed Kripke models: (M, a)

Definition

K is called (modally) compact if, for every set of modal formulas Γ ⊆ ML, every finite subset ∆ ⊆ Γ is satisfiable in the class K = ⇒ Γ is satisfiable in the class K

Evgeny Zolin Axiomatic classes October 20, 2017 8 / 18

Compact and saturated classes of structures

Let K be a class of pointed Kripke models: (M, a)

Definition

K is called (modally) saturated if for every (M, a) ∈ K there is (M♯, a♯) ∈ K such that: 1) (M, a) ≡ML (M♯, a♯); 2) the model M♯ is modally saturated.

Evgeny Zolin Axiomatic classes October 20, 2017 8 / 18

Compact and saturated classes of structures

Let K be a class of pointed Kripke models: (M, a)

Definition

K is called (modally) saturated if for every (M, a) ∈ K there is (M♯, a♯) ∈ K such that: 1) (M, a) ≡ML (M♯, a♯); 2) the model M♯ is modally saturated.

Fact 1.

For arbitrary models M, N: (M, a) ≃ (N, b) = ⇒ (M, a) ≡ML (N, b)

Fact 2.

For saturated models M, N: (M, a) ≃ (N, b) ⇐ ⇒ (M, a) ≡ML (N, b)

Evgeny Zolin Axiomatic classes October 20, 2017 8 / 18

Definability: classes of pointed models: abstract formulation

Theorem (Z, 2017)

Both K K K ∈ ⊍⩀L ≡ML K ∈ ⊍L ≡ML compact K ∈ ⩀L ≡ML compact K ∈ L ≡ML compact compact

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Definability: classes of pointed models: abstract formulation

Theorem (Z, 2017)

Both K K K ∈ ⊍⩀L ≡ML K ∈ ⊍L ≡ML compact K ∈ ⩀L ≡ML compact K ∈ L ≡ML compact compact Can we replace the “linguistic” relation of modal equivalence ≡ML with the “structural” relation of bisimulation ≃? Yes, at the price of adding saturatedness of K and K.

Evgeny Zolin Axiomatic classes October 20, 2017 9 / 18

Definability: classes of pointed models: abstract formulation

Theorem (Z, 2017)

Both K K K ∈ ⊍⩀L ≡ML K ∈ ⊍L ≡ML compact K ∈ ⩀L ≡ML compact K ∈ L ≡ML compact compact

Theorem (Z, 2017)

Both K K K ∈ ⊍⩀L ≃ saturated saturated K ∈ ⊍L ≃ saturated compact saturated K ∈ ⩀L ≃ compact saturated saturated K ∈ L ≃ compact saturated compact saturated

Evgeny Zolin Axiomatic classes October 20, 2017 9 / 18

Definability: classes of pointed models: concrete results

Theorem (Basic ML; de Rijke, 1993)

Both K K K ∈ ⊍⩀L ≡ML K ∈ ⊍L ≡ML uProd K ∈ ⩀L ≡ML uProd K ∈ L ≡ML uProd uProd

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Definability: classes of pointed models: concrete results

Theorem (Basic ML; de Rijke, 1993) (Tense language)

Both K K K ∈ ⊍⩀L ≡ML K ∈ ⊍L ≡ML uProd K ∈ ⩀L ≡ML uProd K ∈ L ≡ML uProd uProd Both K K ≡ML.t ≡ML.t uProd ≡ML.t uProd ≡ML.t uProd uProd

Evgeny Zolin Axiomatic classes October 20, 2017 10 / 18

Definability: classes of pointed models: concrete results

Theorem (Basic ML; de Rijke, 1993) (Tense language)

Both K K K ∈ ⊍⩀L ≡ML K ∈ ⊍L ≡ML uProd K ∈ ⩀L ≡ML uProd K ∈ L ≡ML uProd uProd Both K K ≡ML.t ≡ML.t uProd ≡ML.t uProd ≡ML.t uProd uProd

(ML with universal modality)

Both K K K ∈ ⊍⩀L ≡ML∀ K ∈ ⊍L ≡ML∀ uProd K ∈ ⩀L ≡ML∀ uProd K ∈ L ≡ML∀ uProd uProd

Evgeny Zolin Axiomatic classes October 20, 2017 10 / 18

Definability: classes of pointed models: concrete results

Theorem (Basic ML; de Rijke, 1993) (Tense language)

Both K K K ∈ ⊍⩀L ≡ML K ∈ ⊍L ≡ML uProd K ∈ ⩀L ≡ML uProd K ∈ L ≡ML uProd uProd Both K K ≡ML.t ≡ML.t uProd ≡ML.t uProd ≡ML.t uProd uProd

(ML with universal modality) (Graded ML, de Rijke, 2000)

Both K K K ∈ ⊍⩀L ≡ML∀ K ∈ ⊍L ≡ML∀ uProd K ∈ ⩀L ≡ML∀ uProd K ∈ L ≡ML∀ uProd uProd Both K K ≡MLG ≡MLG uProd ≡MLG uProd ≡MLG uProd uProd

Evgeny Zolin Axiomatic classes October 20, 2017 10 / 18

Definability: classes of pointed models: concrete

Theorem (Basic ML; de Rijke, 1993) (Tense language)

Both K K K ∈ ⊍⩀L ≃ uPow uPow K ∈ ⊍L ≃ uPow uProd K ∈ ⩀L ≃ uProd uPow K ∈ L ≃ uProd uProd Both K K ≃t uPow uPow ≃t uPow uProd ≃t uProd uPow ≃t uProd uProd

(ML with universal modality) (Graded ML, de Rijke, 2000)

Both K K K ∈ ⊍⩀L :≃: uPow uPow K ∈ ⊍L :≃: uPow uProd K ∈ ⩀L :≃: uProd uPow K ∈ L :≃: uProd uProd Both K K ≃G uPow uPow ≃G uPow uProd ≃G uProd uPow ≃G uProd uProd

Evgeny Zolin Axiomatic classes October 20, 2017 11 / 18

Modal language: “purely modal” operations on models

Ultra-extension of a Kripke model M = (W , R, V ) is a Kripke model Mue = (W ue, Rue, V ue), where worlds: W ue — all ultrafilters over the set W ; relation: α Rue β ⇌ ∀X ⊆ W ( ◇X ∈ α ⇐ X ∈ β ) ⇔ ∀X ⊆ W ( X ∈ α ⇒ X ∈ β ) valuation: α | = pi ⇌ V (pi) ∈ α

Evgeny Zolin Axiomatic classes October 20, 2017 12 / 18

Modal language: “purely modal” operations on models

Ultra-extension of a Kripke model M = (W , R, V ) is a Kripke model Mue = (W ue, Rue, V ue), where worlds: W ue — all ultrafilters over the set W ; relation: α Rue β ⇌ ∀X ⊆ W ( ◇X ∈ α ⇐ X ∈ β ) ⇔ ∀X ⊆ W ( X ∈ α ⇒ X ∈ β ) valuation: α | = pi ⇌ V (pi) ∈ α

• Observation. 1) M ≡ML Mue;

2) the model Mue is modally saturated.

Evgeny Zolin Axiomatic classes October 20, 2017 12 / 18

Modal language: “purely modal” operations on models

Ultra-extension of a Kripke model M = (W , R, V ) is a Kripke model Mue = (W ue, Rue, V ue), where worlds: W ue — all ultrafilters over the set W ; relation: α Rue β ⇌ ∀X ⊆ W ( ◇X ∈ α ⇐ X ∈ β ) ⇔ ∀X ⊆ W ( X ∈ α ⇒ X ∈ β ) valuation: α | = pi ⇌ V (pi) ∈ α

• Observation. 1) M ≡ML Mue;

2) the model Mue is modally saturated. Ultra-union of a family of pointed Kripke models (Mi, ai)i∈I M =

• ( ⊎

i∈I Mi)ue, α

• , all co-finite subsets of { ai, i | i ∈ I } are in α.
• Observation. Ultra-union behaves like the ultra-product.

Evgeny Zolin Axiomatic classes October 20, 2017 12 / 18

Definability: classes of pointed models: “pure” criteria

Theorem (Yde Venema, 1999)

Both K K K ∈ ⊍⩀L ≡ML K ∈ ⊍L ≡ML

⊎ue

K ∈ ⩀L ≡ML

⊎ue

K ∈ L ≡ML

⊎ue ⊎ue

Evgeny Zolin Axiomatic classes October 20, 2017 13 / 18

Definability: classes of pointed models: “pure” criteria

Theorem (Yde Venema, 1999)

Both K K K ∈ ⊍⩀L ≃

ue ue

K ∈ ⊍L ≃

ue ⊎ue

K ∈ ⩀L ≃

⊎ue ue

K ∈ L ≃

⊎ue ⊎ue

Evgeny Zolin Axiomatic classes October 20, 2017 13 / 18

Definability: classes of pointed models: “pure” criteria

Theorem (Yde Venema, 1999)

Both K K K ∈ ⊍⩀L ≃

ue ue

K ∈ ⊍L ≃

ue ⊎ue

K ∈ ⩀L ≃

⊎ue ue

K ∈ L ≃

⊎ue ⊎ue Theorem (Tense language) (exercise)

Both K K K ∈ ⊍⩀L ≡ML.t K ∈ ⊍L ≡ML.t

⊎ue

K ∈ ⩀L ≡ML.t

⊎ue

K ∈ L ≡ML.t

⊎ue ⊎ue

Evgeny Zolin Axiomatic classes October 20, 2017 13 / 18

Definability: classes of pointed models: “pure” criteria

Theorem (Yde Venema, 1999)

Both K K K ∈ ⊍⩀L ≃

ue ue

K ∈ ⊍L ≃

ue ⊎ue

K ∈ ⩀L ≃

⊎ue ue

K ∈ L ≃

⊎ue ⊎ue Theorem (Tense language) (exercise)

Both K K K ∈ ⊍⩀L ≃t

ue ue

K ∈ ⊍L ≃t

ue ⊎ue

K ∈ ⩀L ≃t

⊎ue ue

K ∈ L ≃t

⊎ue ⊎ue

Evgeny Zolin Axiomatic classes October 20, 2017 13 / 18

Now what about classes of models?

Theorem (Lines 1 and 3: Z, 2017; Lines 2 and 4: open questions)

Both K K K ∈ ⊍⩀L ≡ML ֒ →

⊎ compact

• K ∈ ⊍L

≡ML ֒ →

⊎ compact

? K ∈ ⩀L ≡ML ֒ → ⊎ compact

⊎ compact

• K ∈

L ≡ML ֒ → ⊎ compact

⊎ compact

?

Evgeny Zolin Axiomatic classes October 20, 2017 14 / 18

Now what about classes of models?

Theorem (Lines 1 and 3: Z, 2017; Lines 2 and 4: open questions)

Both K K K ∈ ⊍⩀L ≡ML ֒ →

⊎ compact

• K ∈ ⊍L

≡ML ֒ →

⊎ compact

? K ∈ ⩀L ≡ML ֒ → ⊎ compact

⊎ compact

• K ∈

L ≡ML ֒ → ⊎ compact

⊎ compact

? Now we get rid of ≡ML in favour of bisimulation ≃.

Evgeny Zolin Axiomatic classes October 20, 2017 14 / 18

Now what about classes of models?

Theorem (Lines 1 and 3: Z, 2017; Lines 2 and 4: open questions)

Both K K K ∈ ⊍⩀L ≡ML ֒ →

⊎ compact

• K ∈ ⊍L

≡ML ֒ →

⊎ compact

? K ∈ ⩀L ≡ML ֒ → ⊎ compact

⊎ compact

• K ∈

L ≡ML ֒ → ⊎ compact

⊎ compact

?

Theorem (Lines 1 and 3: Z, 2017; Lines 2 and 4: open questions)

Both K K K ∈ ⊍⩀L :≃: ֒ → saturated

⊎ compact saturated

• K ∈ ⊍L

:≃: ֒ → saturated

⊎ compact saturated

? K ∈ ⩀L :≃: ֒ → ⊎ compact saturated

⊎ compact saturated

• K ∈

L :≃: ֒ → ⊎ compact saturated

⊎ compact saturated

?

Evgeny Zolin Axiomatic classes October 20, 2017 14 / 18

Definability: classes of models: basic modal language

Theorem (Lines 3-4: de Rijke, Sturm, 2001; Lines 1–2: Z, 2017)

Both K K K ∈ ⊍⩀L ≡ML ֒ → K ∈ ⊍L ≡ML ֒ → uProd K ∈ ⩀L ≡ML ֒ → ⊎ uProd K ∈ L ≡ML ֒ → ⊎ uProd uProd

Evgeny Zolin Axiomatic classes October 20, 2017 15 / 18

Definability: classes of models: basic modal language

Theorem (Lines 3-4: de Rijke, Sturm, 2001; Lines 1–2: Z, 2017)

Both K K K ∈ ⊍⩀L ≡ML ֒ → K ∈ ⊍L ≡ML ֒ → uProd K ∈ ⩀L ≡ML ֒ → ⊎ uProd K ∈ L ≡ML ֒ → ⊎ uProd uProd Both K K K ∈ ⊍⩀L :≃: ֒ → uPow uPow K ∈ ⊍L :≃: ֒ → uPow uProd K ∈ ⩀L :≃: ֒ → ⊎ uProd uPow K ∈ L :≃: ֒ → ⊎ uProd uProd

Evgeny Zolin Axiomatic classes October 20, 2017 15 / 18

Definability: classes of models: “pure” criteria

Theorem (Line 3: Yde Venema, 1999; Line 1: Z, 2017)

Both K K K ∈ ⊍⩀L ≡ML ֒ → K ∈ ⊍L ? K ∈ ⩀L ≡ML ֒ → ⊎ ue K ∈ L ? Both K K :≃: ֒ →

ue ue

? :≃: ֒ → ⊎ ue

ue

?

Evgeny Zolin Axiomatic classes October 20, 2017 16 / 18

Definability: classes of models: ML with universal modality

We can “internalize” negation: M | = ϕ ⇐ ⇒ M | = ¬[∀]ϕ. Hence, the symmetry is again with us!

Evgeny Zolin Axiomatic classes October 20, 2017 17 / 18

Definability: classes of models: ML with universal modality

We can “internalize” negation: M | = ϕ ⇐ ⇒ M | = ¬[∀]ϕ. Hence, the symmetry is again with us!

Theorem (ML with universal modality; Z, 2017)

Both K K K ∈ ⊍⩀L ≡ML∀ K ∈ ⊍L ≡ML∀ uProd K ∈ ⩀L ≡ML∀ uProd K ∈ L ≡ML∀ uProd uProd Both K K :≃: uPow uPow :≃: uPow uProd :≃: uProd uPow :≃: uProd uProd

Evgeny Zolin Axiomatic classes October 20, 2017 17 / 18

Definability: classes of models: ML with universal modality

We can “internalize” negation: M | = ϕ ⇐ ⇒ M | = ¬[∀]ϕ. Hence, the symmetry is again with us!

Theorem (ML with universal modality; Z, 2017)

Both K K K ∈ ⊍⩀L ≡ML∀ K ∈ ⊍L ≡ML∀ uProd K ∈ ⩀L ≡ML∀ uProd K ∈ L ≡ML∀ uProd uProd Both K K :≃: uPow uPow :≃: uPow uProd :≃: uProd uPow :≃: uProd uProd

Question

What are “purely modal” criteria for the ML with universal modality?

Evgeny Zolin Axiomatic classes October 20, 2017 17 / 18

Further directions

Criteria for other semantics of the basic modal language:

neighbourhood semantics topological semantics algebraic semantics

Evgeny Zolin Axiomatic classes October 20, 2017 18 / 18

Further directions

Criteria for other semantics of the basic modal language:

neighbourhood semantics topological semantics algebraic semantics

Criteria for other languages:

intuitionistic propositional language

Evgeny Zolin Axiomatic classes October 20, 2017 18 / 18

Further directions

Criteria for other semantics of the basic modal language:

neighbourhood semantics topological semantics algebraic semantics

Criteria for other languages:

intuitionistic propositional language new modalities: inequality [=], transitive closure modality ⊞, nominals

Evgeny Zolin Axiomatic classes October 20, 2017 18 / 18

Further directions

Criteria for other semantics of the basic modal language:

neighbourhood semantics topological semantics algebraic semantics

Criteria for other languages:

intuitionistic propositional language new modalities: inequality [=], transitive closure modality ⊞, nominals infinitary modal language (for any set Φ of formulas, Φ is a formula): – L: classes of models definable by a single formula – ⩀L: classes of models definable by a class (not set!) of formulas

Evgeny Zolin Axiomatic classes October 20, 2017 18 / 18

Further directions

Criteria for other semantics of the basic modal language:

neighbourhood semantics topological semantics algebraic semantics

Criteria for other languages:

intuitionistic propositional language new modalities: inequality [=], transitive closure modality ⊞, nominals infinitary modal language (for any set Φ of formulas, Φ is a formula): – L: classes of models definable by a single formula – ⩀L: classes of models definable by a class (not set!) of formulas modal and intuitionistic predicate languages

Evgeny Zolin Axiomatic classes October 20, 2017 18 / 18

Further directions

Criteria for other semantics of the basic modal language:

neighbourhood semantics topological semantics algebraic semantics

Criteria for other languages:

intuitionistic propositional language new modalities: inequality [=], transitive closure modality ⊞, nominals infinitary modal language (for any set Φ of formulas, Φ is a formula): – L: classes of models definable by a single formula – ⩀L: classes of models definable by a class (not set!) of formulas modal and intuitionistic predicate languages

Will the results survive if we go to finite models?

Evgeny Zolin Axiomatic classes October 20, 2017 18 / 18

Further directions

Criteria for other semantics of the basic modal language:

neighbourhood semantics topological semantics algebraic semantics

Criteria for other languages:

intuitionistic propositional language new modalities: inequality [=], transitive closure modality ⊞, nominals infinitary modal language (for any set Φ of formulas, Φ is a formula): – L: classes of models definable by a single formula – ⩀L: classes of models definable by a class (not set!) of formulas modal and intuitionistic predicate languages

Will the results survive if we go to finite models? Thank you!

Evgeny Zolin Axiomatic classes October 20, 2017 18 / 18