About contingency and ignorance Philippe Balbiani Institut de - - PowerPoint PPT Presentation

About contingency and ignorance

Philippe Balbiani

Institut de recherche en informatique de Toulouse CNRS — Toulouse University, France

with a little help from Hans van Ditmarsch and Jie Fan

Introduction

Modal logic

Study of principles of reasoning involving

◮ necessity ◮ possibility ◮ impossibility ◮ unnecessity ◮ non-contingency ◮ contingency

Introduction

In modal logic

A proposition is non-contingent iff

◮ it is necessarily true or it is necessarily false

A proposition is contingent iff

◮ it is possibly true and it is possibly false

Introduction

In a doxastic context

A proposition is non-contingent iff

◮ you are opinionated as to whether the proposition is true

A proposition is contingent iff

◮ you are agnostic about the value of the proposition

Introduction

In an epistemic context

A proposition is non-contingent iff

◮ you know whether the proposition is true

A proposition is contingent iff

◮ you are ignorant about the truth value of the proposition

Introduction

For example

In agent communication languages

◮ an agent will reply she is unable to answer a query if she is

ignorant about the value of the information she is being asked In communication protocols

◮ a desirable property of the interaction is that the state of

ignorance of the intruder with respect to the content of the messages is preserved

Introduction

References in modal logic

About non-contingency and contingency

◮ Montgomery and Routley (1966, 1968) ◮ Cresswell (1988) ◮ Humberstone (1995) ◮ Kuhn (1995) ◮ Zolin (1999)

References in a doxastic or epistemic context

About ignorance

◮ Moses et al. (1986) ◮ Orłowska (1989) ◮ Demri (1997) ◮ Van der Hoek and Lomuscio (2004) ◮ Steinsvold (2008, 2011)

Introduction

Our aim today

We will

◮ study the literature on contingency logic ◮ study the literature on the logic of ignorance ◮ bridge the gap between the two literatures ◮ give an overview of the known axiomatizations ◮ attack the difficulties of some completeness proofs

Ordinary modal logic

Syntax

Formulas

◮ ϕ ::= p | ⊥ | ¬ϕ | (ϕ ∨ ψ) | ϕ

Abbreviations

◮ (ϕ ∧ ψ) for ¬(¬ϕ ∨ ¬ψ), etc ◮ ♦ϕ for ¬¬ϕ

Readings

◮ ϕ : “ϕ is necessarily true” ◮ ♦ϕ : “ϕ is possibly true”

Ordinary modal logic

Relational semantics

Frames : F = (W, R) where

◮ W = ∅ ◮ R ⊆ W × W

Models : M = (W, R, V) where

◮ V : p → V(p) ⊆ W

Truth conditions

◮ M, s |

= p iff s ∈ V(p)

◮ M, s |

= ⊥, etc

◮ M, s |

= ϕ iff ∀t ∈ W (sRt ⇒ M, t | = ϕ)

◮ M, s |

= ♦ϕ iff ∃t ∈ W (sRt & M, t | = ϕ)

Ordinary modal logic

Axiomatization/completeness

Minimal normal logic K

◮ tautologies, modus ponens ◮ (p → q) → (p → q) ◮ generalization :

ϕ ϕ

Extensions

◮ D : ♦⊤ ◮ T : p → p ◮ B : p → ♦p ◮ 4 : p → p ◮ 5 : ♦p → ♦p

Contingency and non-contingency

Montgomery and Routley (1966, 1968)

New primitive

◮ ∇ϕ : “it is contingent that ϕ” ◮ ϕ ::= p | ⊥ | ¬ϕ | (ϕ ∨ ψ) | ∇ϕ

Truth condition in model M = (W, R, V)

◮ M, s |

= ∇ϕ iff ∃t ∈ W (sRt & M, t | = ϕ) & ∃u ∈ W (sRu & M, u | = ϕ) Abbreviation

◮ ∆ϕ for ¬∇ϕ : “it is non-contingent that ϕ”

Truth condition in model M = (W, R, V)

◮ M, s |

= ∆ϕ iff ∀t ∈ W (sRt ⇒ M, t | = ϕ) ∨ ∀u ∈ W (sRu ⇒ M, u | = ϕ)

Contingency and non-contingency

Segerberg (1982)

In the class of all frames

◮ ∇ϕ is equivalent to ♦ϕ ∧ ♦¬ϕ ◮ ∆ϕ is equivalent to ϕ ∨ ¬ϕ

In the class of all reflexive frames

◮ ϕ is equivalent to ϕ ∧ ∆ϕ ◮ ♦ϕ is equivalent to ϕ ∨ ∇ϕ

Contingency and non-contingency

Montgomery and Routley (1966, 1968)

Axiomatization (in the class of all reflexive frames)

◮ tautologies, modus ponens ◮ ∆p ↔ ∆¬p ◮ p → (∆(p → q) → (∆p → ∆q)) ◮

ϕ ∆ϕ

Axiomatization (in the class of all reflexive transitive frames)

◮ additional axiom : ∆p → ∆∆p

Axiomatization (in the class of all partitions)

◮ additional axiom : ∆∆p

Axiomatization (in the class of all frames)

◮ open problem (1966)

Contingency and non-contingency

Montgomery and Routley (1966, 1968)

Validity of p → (∆(p → q) → (∆p → ∆q)) in reflexive frames

• 1. Let M = (W, R, V) where R is reflexive and s ∈ W be such

that M, s | = p → (∆(p → q) → (∆p → ∆q)).

• 2. Hence, M, s |

= p, M, s | = ∆(p → q), M, s | = ∆p and M, s | = ∆q.

• 3. Let t, u ∈ W be such that sRt, sRu, M, t |

= q and M, u | = q.

• 4. Since M, s |

= ∆(p → q), therefore M, t | = p → q iff M, u | = p → q.

• 5. Since M, t |

= q and M, u | = q, therefore M, u | = p.

• 6. Since R is reflexive, M, s |

= ∆p and sRu, therefore M, s | = p iff M, u | = p : a contradiction.

Contingency and non-contingency

Montgomery and Routley (1966, 1968)

Validity of ∆p → ∆∆p in reflexive transitive frames

• 1. Let M = (W, R, V) where R is reflexive and transitive and

s ∈ W be such that M, s | = ∆p → ∆∆p.

• 2. Hence, M, s |

= ∆p and M, s | = ∆∆p.

• 3. Let t, u ∈ W be such that sRt, sRu, M, t |

= ∆p and M, u | = ∆p.

• 4. Let v, w ∈ W be such that uRv, uRw, M, v |

= p and M, w | = p.

• 5. Since R is transitive and sRu, therefore sRv and sRw.
• 6. Since M, s |

= ∆p, therefore M, v | = p iff M, w | = p : a contradiction.

Contingency and non-contingency

Montgomery and Routley (1966, 1968)

Validity of ∆∆p in partitions

• 1. Let M = (W, R, V) where R is reflexive, symmetric and

transitive and s ∈ W be such that M, s | = ∆∆p.

• 2. Let t, u ∈ W be such that sRt, sRu, M, t |

= ∆p and M, u | = ∆p.

• 3. Let v, w ∈ W be such that uRv, uRw, M, v |

= p and M, w | = p.

• 4. Since R is symmetric and transitive, sRt and sRu,

therefore tRv and tRw.

• 5. Since M, t |

= ∆p, therefore M, v | = p iff M, w | = p : a contradiction.

Contingency and non-contingency

Cresswell (1988)

New primitive

◮ ∆ϕ : “it is non-contingent that ϕ” ◮ ϕ ::= p | ⊥ | ¬ϕ | (ϕ ∨ ψ) | ∆ϕ

Truth condition in model M = (W, R, V)

◮ M, s |

= ∆ϕ iff ∀t ∈ W (sRt ⇒ M, t | = ϕ) ∨ ∀u ∈ W (sRu ⇒ M, u | = ϕ) Abbreviation

◮ ∇ϕ for ¬∆ϕ : “it is contingent that ϕ”

Truth condition in model M = (W, R, V)

◮ M, s |

= ∇ϕ iff ∃t ∈ W (sRt & M, t | = ϕ) & ∃u ∈ W (sRu & M, u | = ϕ)

Contingency and non-contingency

Cresswell (1988)

is ∆-definable in the normal modal logic L iff

◮ there exists a formula ϕ(p) in L(⊥, ¬, ∨, ∆) such that

◮ p ↔ ϕ⋆(p) ∈ L

where ϕ⋆(p) is obtained from ϕ(p) by iteratively replacing the subformulas of the form ∆ψ by the corresponding formulas ψ ∨ ¬ψ Obviously

◮ Let L, L′ be normal modal logics such that L ⊆ L′. If is

∆-definable in L then is ∆-definable in L′.

Contingency and non-contingency

Cresswell (1988)

For example, is ∆-definable in the normal modal logic T = K + p → p seeing that

◮ p ↔ p ∧ (p ∨ ¬p) ∈ T

Another example, is ∆-definable in the normal modal logic Verum = K + ⊥ seeing that

◮ p ↔ ⊤ ∈ Verum

Contingency and non-contingency

Cresswell (1988)

Question

◮ Find normal modal logics L such that T ⊆ L, Verum ⊆ L

and is ∆-definable in L

Contingency and non-contingency

Cresswell (1988)

A result

◮ Let L be a normal modal logic. If the canonical model of L

contains a dead end and a non-dead end then is not ∆-definable in L. Therefore

◮ it is only needed to consider normal modal logics L such

that T ⊆ L, Verum ⊆ L and ♦⊤ ∈ L

Contingency and non-contingency

Cresswell (1988)

Other results

• 1. Let L be a normal modal logic such that ♦⊤ ∈ L. If the

canonical model of L contains an irreflexive s ∈ W with exactly one successor then is not ∆-definable in L.

• 2. Let L be a normal modal logic such that ♦⊤ ∈ L and

F = (W, R) be an L-frame. If there exists an irreflexive s ∈ W such that for all t ∈ W, t = s, sR+t and t ¯ Rs then is not ∆-definable in L.

• 3. Let L be a normal modal logic such that ♦⊤ ∈ L and

F = (W, R) be an L-frame. If there exists an irreflexive s ∈ W such that for all t ∈ W, t = s, sR+t and tRs then is not ∆-definable in L.

Contingency and non-contingency

Cresswell (1988)

A natural question

◮ is there a normal modal logic L such that T ⊆ L,

Verum ⊆ L, ♦⊤ ∈ L and is ∆-definable in L ? Cresswell’s answer

◮ yes

◮ K + p ↔ (∆p ∧ (p ↔ ∆∆p))⋆

◮ that is to say

◮ K + p ↔ ((p ∨ ¬p) ∧ (p ↔

((p ∨ ¬p) ∨ ¬(p ∨ ¬p))))

Contingency and non-contingency

Expressivity and definability

Some properties

• 1. ∆ and are equally expressive on the class of all reflexive

frames.

• 2. ∆ is strictly less expressive than on

the class of all frames, the class of all serial frames, the class of all transitive frames, the class of all Euclidean frames and the class of all symmetric frames.

Contingency and non-contingency

Expressivity and definability

Some properties

• 1. The class of all frames and the class of all serial frames

validate the same ∆-formulas.

• 2. The frame properties of reflexivity, seriality, transitivity,

Euclideanity and symmetry are not ∆-definable.

Contingency and non-contingency

Humberstone (1995)

Some principles for non-contingency

◮ ∆⊥ ◮ ∆p → ∆¬p ◮ ∆p ∧ ∆q → ∆(p ∨ q) ◮ ∆p1 ∧ . . . ∧ ∆pn → ∆♯(p1, . . . , pn)

Contingency and non-contingency

Humberstone (1995)

Some principles for non-contingency

◮ ¬ϕ→ψ0, ϕ→ψ1

∆ϕ→∆ψ0∨∆ψ1

◮ ¬ϕ∧¬ϕ′→ψ00, ¬ϕ∧ϕ′→ψ01, ϕ∧¬ϕ′→ψ10, ϕ∧ϕ′→ψ11

∆ϕ∧∆ϕ′→∆ψ00∨∆ψ01∨∆ψ10∨∆ψ11

◮

{ϕ

ǫ1 1 ∧...∧ϕǫn n →ψǫ1...ǫn : ǫ1...ǫn is a tuple of n bits}

∆ϕ1∧...∧∆ϕn→{∆ψǫ1...ǫn : ǫ1...ǫn is a tuple of n bits}

Contingency and non-contingency

Humberstone (1995)

Axiomatization (in the class of all frames)

◮ tautologies, modus ponens ◮ ∆¬p → ∆p ◮

{ϕ

ǫ1 1 ∧...∧ϕǫn n →ψǫ1...ǫn : ǫ1...ǫn is a tuple of n bits}

∆ϕ1∧...∧∆ϕn→{∆ψǫ1...ǫn : ǫ1...ǫn is a tuple of n bits}

The following formula is derivable

◮ ∆p1 ∧ . . . ∧ ∆pn → ∆♯(p1, . . . , pn)

Finite axiomatization (in the class of all frames)

◮ open problem (1995)

Axiomatization (in the class of all transitive frames)

◮ open problem (1995)

Contingency and non-contingency

Humberstone (1995)

Canonical model : Mc = (Wc, Rc, Vc)

◮ Wc : set of all maximal consistent sets of formulas ◮ λc : x ∈ Wc → λc(x) is a set of formulas such that for all

formulas ϕ, if ϕ is a consequence of λC(x) then ∆ϕ ∈ x

◮ Rc : xRcy iff λc(x) ⊆ y ◮ Vc : x ∈ Vc(p) iff p ∈ x

Truth Lemma : for all ϕ, for all x ∈ Wc

◮ Mc, x |

= ϕ iff ϕ ∈ x

Contingency and non-contingency

Kuhn (1995)

Validity of ∆p → ∆(p → q) ∨ ∆(r → p) in arbitrary frames

• 1. Let M = (W, R, V) and s ∈ W be such that

M, s | = ∆p → ∆(p → q) ∨ ∆(r → p).

• 2. Hence, M, s |

= ∆p, M, s | = ∆(p → q) and M, s | = ∆(r → p).

• 3. Let t, u, v, w ∈ W be such that sRt, sRu, sRv, sRw,

M, t | = p → q, M, u | = p → q, M, v | = r → p and M, w | = r → p.

• 4. Since M, s |

= ∆p, therefore M, u | = p iff M, w | = p.

• 5. Since M, u |

= p → q and M, w | = r → p, therefore M, u | = p and M, w | = p : a contradiction.

Contingency and non-contingency

Kuhn (1995)

Validity of ∆p → ∆(∆p ∨ q) in transitive frames

• 1. Let M = (W, R, V) where R is transitive and s ∈ W be

such that M, s | = ∆p → ∆(∆p ∨ q).

• 2. Hence, M, s |

= ∆p and M, s | = ∆(∆p ∨ q).

• 3. Let t, u ∈ W be such that sRt, sRu, M, t |

= ∆p ∨ q and M, u | = ∆p ∨ q.

• 4. Thus, M, u |

= ∆p and M, u | = q.

• 5. Let v, w ∈ W be such that uRv, uRw, M, v |

= p and M, w | = p.

• 6. Since R is transitive and sRu, therefore sRv and sRw.
• 7. Since M, s |

= ∆p, therefore M, v | = p iff M, w | = p : a contradiction.

Contingency and non-contingency

Kuhn (1995)

Axiomatization (in the class of all frames)

◮ tautologies, modus ponens ◮ ∆¬p → ∆p ◮ ∆p ∧ ∆q → ∆(p ∧ q) ◮ ∆p → ∆(p → q) ∨ ∆(r → p) ◮

ϕ ∆ϕ

◮

ϕ↔ψ ∆ϕ↔∆ψ

Axiomatization (in the class of all transitive frames)

◮ additional axiom : ∆p → ∆(∆p ∨ q)

Axiomatization (in the class of all symmetric frames)

◮ open problem (1995)

Contingency and non-contingency

Kuhn (1995)

Canonical model : Mc = (Wc, Rc, Vc)

◮ Wc : set of all maximal consistent sets of formulas ◮ λc : x ∈ Wc → λc(x) = {ϕ : for all ψ, ∆(ϕ ∨ ψ) ∈ x} ◮ Rc : xRcy iff λc(x) ⊆ y ◮ Vc : x ∈ Vc(p) iff p ∈ x

Truth Lemma : for all ϕ, for all x ∈ Wc

◮ Mc, x |

= ϕ iff ϕ ∈ x

Contingency and non-contingency

Fan et al. (2015)

Axiomatization (in the class of all frames)

◮ tautologies, modus ponens ◮ ∆¬p → ∆p ◮ ∆(p ∨ q) ∧ ∆(p ∨ ¬q) → ∆p ◮ ∆p → ∆(p → q) ∨ ∆(r → p) ◮

ϕ ∆ϕ

◮

ϕ↔ψ ∆ϕ↔∆ψ

Axiomatization (in the class of all transitive frames)

◮ additional axiom : ∆p → ∆(∆p ∨ q) ◮ references: Kuhn (1995), Zolin (1999)

Contingency and non-contingency

Fan et al. (2015)

Axiomatization (in the class of all frames)

◮ tautologies, modus ponens ◮ ∆¬p → ∆p ◮ ∆(p ∨ q) ∧ ∆(p ∨ ¬q) → ∆p ◮ ∆p → ∆(p → q) ∨ ∆(r → p) ◮

ϕ ∆ϕ

◮

ϕ↔ψ ∆ϕ↔∆ψ

Axiomatization (in the class of all Euclidean frames)

◮ additional axiom : ∇p → ∆(∇p ∨ q) ◮ reference: Zolin (1999)

Contingency and non-contingency

Fan et al. (2015)

Axiomatization (in the class of all frames)

◮ tautologies, modus ponens ◮ ∆¬p → ∆p ◮ ∆(p ∨ q) ∧ ∆(p ∨ ¬q) → ∆p ◮ ∆p → ∆(p → q) ∨ ∆(r → p) ◮

ϕ ∆ϕ

◮

ϕ↔ψ ∆ϕ↔∆ψ

Axiomatization (in the class of all symmetric frames)

◮ additional axiom : p → ∆(∆p ∧ ∆(p → q) → ∆q ∨ r) ◮ reference: Fan et al. (2015)

Contingency and non-contingency

Fan et al. (2015)

Almost definability of by means of ∆

◮ (ϕ ↔ ∆ϕ ∧ ∆(ϕ ∨ ψ)) ∨ ∆ψ

Canonical model : Mc = (Wc, Rc, Vc)

◮ Wc : set of all maximal consistent sets of formulas ◮ Rc : xRcy iff there exists ψ such that

◮ ∆ψ ∈ x ◮ for all ϕ, if ∆ϕ ∈ x and ∆(ϕ ∨ ψ) ∈ x then ϕ ∈ y

◮ Vc : x ∈ Vc(p) iff p ∈ x

Truth Lemma : for all ϕ, for all x ∈ Wc

◮ Mc, x |

= ϕ iff ϕ ∈ x

Ignorance

Van der Hoek and Lomuscio (2004)

Formulas

◮ ϕ ::= p | ⊥ | ¬ϕ | (ϕ ∨ ψ) | Iϕ

Reading

◮ Iϕ : “the agent is ignorant about ϕ”

Models : M = (W, R, V) where

◮ W = ∅ ◮ R ⊆ W × W ◮ V : p → V(p) ⊆ W

Truth conditions

◮ M, s |

= p iff s ∈ V(p)

◮ M, s |

= ⊥, etc

◮ M, s |

= Iϕ iff ∃t ∈ W (sRt & M, t | = ϕ) & ∃u ∈ W (sRu & M, u | = ϕ)

Ignorance

Van der Hoek and Lomuscio (2004)

Some principles for ignorance

◮ Ip ↔ I¬p ◮ I(p ∧ q) → Ip ∨ Iq ◮ I(p ∧ r) ∧ ¬Ir ∧ I(q ∧ (r → s)) ∧ ¬I(r → s) → ¬Is ∧ I(p ∧ s) ◮ ¬Ip ∧ Iq → I(p ∧ q) ∨ I(¬p ∧ q) ◮ ϕ ¬Iϕ ◮ ϕ Iψ→I(ϕ∧ψ)

Ignorance

Van der Hoek and Lomuscio (2004)

Axiomatization (in the class of all frames)

◮ tautologies, modus ponens ◮ Ip ↔ I¬p ◮ I(p ∧ q) → Ip ∨ Iq ◮ I(p ∧ r) ∧ ¬Ir ∧ I(q ∧ (r → s)) ∧ ¬I(r → s) →

¬Is ∧ I(p ∧ s)

◮ ¬Ip ∧ Iq → I(p ∧ q) ∨ I(¬p ∧ q) ◮

ϕ ¬Iϕ

◮

ϕ Iψ→I(ϕ∧ψ)

Ignorance

Van der Hoek and Lomuscio (2004)

Canonical model : Mc = (Wc, Rc, Vc)

◮ Wc : set of all maximal consistent sets of formulas ◮ Rc : xRcy iff there exists Iψ ∈ x such that

◮ for all ϕ, if ¬Iϕ ∈ x and I(ϕ ∧ ψ) ∈ x then ϕ ∈ y

◮ Vc : x ∈ Vc(p) iff p ∈ x

Truth Lemma : for all ϕ, for all x ∈ Wc

◮ Mc, x |

= ϕ iff ϕ ∈ x

Wrongly believing

Steinsvold (2011)

New primitive

◮ Wϕ : “wrongly believing that ϕ”

Truth condition in model M = (W, R, V)

◮ M, s |

= Wϕ iff ∀t ∈ W (sRt ⇒ M, t | = ϕ) & M, s | = ϕ Axiomatization (in the class of all frames)

◮ tautologies, modus ponens ◮ Wp → ¬p ◮ Wp ∧ Wq → W(p ∧ q) ◮

ϕ→ψ Wϕ∧¬ψ→Wψ

Axiomatization (in the class of all transitive frames)

◮ open problem (2011)

Being wrong

Steinsvold (2011)

Canonical model : Mc = (Wc, Rc, Vc)

◮ Wc : set of all maximal consistent sets of formulas ◮ wrong worlds : x ∈ Wc such that Wψ ∈ x for some ψ ◮ Rc : xRcy iff one of the following conditions holds

◮ x is wrong and for all ϕ, if Wϕ ∈ x then ϕ ∈ y ◮ x is not wrong and x = y

◮ Vc : x ∈ Vc(p) iff p ∈ x

Truth Lemma : for all ϕ, for all x ∈ Wc

◮ Mc, x |

= ϕ iff ϕ ∈ x

Being wrong

Steinsvold (2011)

Some properties

• 1. The frame property of seriality is W-defined by ¬W⊥.
• 2. The frame property of post-reflexivity is W-defined by

Wp → W(p ∧ ¬Wq).

• 3. The frame properties of transitivity, Euclideanity, symmetry,

weak connectedness, weak directedness, determinism, narcissism and weak narcissism are not W-definable.

Essence, accident and strong non-contingency

Fine (1994, 1995, 2000)

A proposition is accidental iff

◮ it is true but not necessarily true

A proposition is essential iff

◮ if it is true then it is necessarily true

In an epistemic context

A proposition is accidental iff

◮ it is true but you do not know that

A proposition is essential iff

◮ if it is true then you know that

Essence, accident and strong non-contingency

Fan (2015)

New primitive

◮ ϕ : “it is strongly non-contingent that ϕ”

Truth condition in model M = (W, R, V)

◮ M, s |

= ϕ iff either ∀t ∈ W (sRt ⇒ M, t | = ϕ) & M, s | = ϕ,

• r ∀t ∈ W (sRt ⇒ M, t |

= ϕ) & M, s | = ϕ Abbreviation

◮ ϕ for ¬ϕ

Readings

◮ ϕ : “no matter whether ϕ is true or false, it does it

necessarily”

◮ ϕ : “no matter whether ϕ is true or false, it could have

been otherwise”

Essence, accident and strong non-contingency

Fan (2015)

A family of modal connectives

◮ , ∆, ◦ and

Truth condition in model M = (W, R, V)

◮ M, s |

= ϕ iff ∀t ∈ W (sRt ⇒ M, t | = ϕ)

◮ M, s |

= ∆ϕ iff either ∀t ∈ W (sRt ⇒ M, t | = ϕ), or ∀t ∈ W (sRt ⇒ M, t | = ϕ)

◮ M, s |

= ◦ϕ iff if M, s | = ϕ then ∀t ∈ W (sRt ⇒ M, t | = ϕ)

◮ M, s |

= ϕ iff either ∀t ∈ W (sRt ⇒ M, t | = ϕ) & M, s | = ϕ,

• r ∀t ∈ W (sRt ⇒ M, t |

= ϕ) & M, s | = ϕ

Essence, accident and strong non-contingency

Fan (2015)

Validities

◮ ∆ϕ ↔ ϕ ∨ ¬ϕ

∆ϕ : “ϕ is non-contingent”

◮ ◦ϕ ↔ (ϕ → ϕ)

• ϕ : “ϕ is essential”

◮ ϕ ↔ ◦ϕ ∧ ◦¬ϕ

ϕ : “ϕ is strongly non-contingent”

◮ ϕ ↔ (ϕ → ϕ) ∧ (¬ϕ → ¬ϕ)

Essence, accident and strong non-contingency

Fan (2015)

◮ and are equally expressive on the class of all reflexive

frames.

◮ is strictly less expressive than on

the class of all frames, the class of all serial frames, the class of all transitive frames, the class of all Euclidean frames and the class of all symmetric frames.

Essence, accident and strong non-contingency

Fan (2015)

◮ ∆ and are equally expressive on the class of all reflexive

frames.

◮ ∆ is strictly less expressive than on

the class of all frames, the class of all serial frames, the class of all transitive frames, the class of all Euclidean frames and the class of all symmetric frames.

Essence, accident and strong non-contingency

Fan (2015)

◮ The frame properties of reflexivity, seriality, transitivity and

Euclideanity are not -definable.

◮ The frame property of symmetry is -definable by

p → (p → p)

◮ The frame property of weak narcissism is -definable by

p

Essence, accident and strong non-contingency

Fan (2015)

Axiomatization (in the class of all frames)

◮ tautologies, modus ponens ◮ p ↔ ¬p ◮ p ∧ q → (p ∧ q) ◮ ⊤ ◮

ϕ→ψ ϕ∧ϕ→ψ

Axiomatization (in the class of all transitive frames)

◮ additional axiom : p → p

Axiomatization (in the class of all symmetric frames)

◮ additional axiom : p → (p → p)

Axiomatization (in the class of all Euclidean frames)

◮ open problem (2015)

Essence, accident and strong non-contingency

Fan (2015)

Canonical model : Mc = (Wc, Rc, Vc)

◮ Wc : set of all maximal consistent sets of formulas ◮ Rc : xRcy iff for all ϕ, if ϕ ∈ x and ϕ ∈ x then ϕ ∈ y ◮ Vc : x ∈ Vc(p) iff p ∈ x

Truth Lemma : for all ϕ, for all x ∈ Wc

◮ Mc, x |

= ϕ iff ϕ ∈ x

Directions for further research

Correspondence theory

◮ prove an equivalent of the Sahlqvist’s theorems

Complexity of the validity problem

◮ find the lower bound and the upper bound of the

complexity of the validity problem Expressivity and succinctness

◮ compare the expressivity and succinctness of ∆, , I, W,

and ◦ Multimodal version

◮ add group operators for knowing-whether

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