About contingency and ignorance - ruhr-uni-bochum.de fileHans van Ditmarsch and Jie Fan. Introduction Modal logic Study of principles of reasoning involving Inecessity Ipossibility

About contingency and ignorance

Philippe Balbiani

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

with a little help fromHans van Ditmarsch and Jie Fan

Introduction

Modal logicStudy of principles of reasoning involvingI necessityI possibilityI impossibilityI unnecessityI non-contingencyI contingency

Introduction

In modal logicA proposition is non-contingent iffI it is necessarily true or it is necessarily false

A proposition is contingent iffI it is possibly true and it is possibly false

Introduction

In a doxastic contextA proposition is non-contingent iffI you are opinionated as to whether the proposition is true

A proposition is contingent iffI you are agnostic about the value of the proposition

Introduction

In an epistemic contextA proposition is non-contingent iffI you know whether the proposition is true

A proposition is contingent iffI you are ignorant about the truth value of the proposition

Introduction

For exampleIn agent communication languagesI an agent will reply she is unable to answer a query if she is

ignorant about the value of the information she is beingasked

In communication protocolsI a desirable property of the interaction is that the state of

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

IntroductionReferences in modal logicAbout non-contingency and contingencyI Montgomery and Routley (1966, 1968)I Cresswell (1988)I Humberstone (1995)I Kuhn (1995)I Zolin (1999)

References in a doxastic or epistemic contextAbout ignoranceI Moses et al. (1986)I Orłowska (1989)I Demri (1997)I Van der Hoek and Lomuscio (2004)I Steinsvold (2008, 2011)

Introduction

Our aim todayWe willI study the literature on contingency logicI study the literature on the logic of ignoranceI bridge the gap between the two literaturesI give an overview of the known axiomatizationsI attack the difficulties of some completeness proofs

Ordinary modal logic

SyntaxFormulasI ϕ ::= p | ⊥ | ¬ϕ | (ϕ ∨ ψ) | �ϕ

AbbreviationsI (ϕ ∧ ψ) for ¬(¬ϕ ∨ ¬ψ), etcI ♦ϕ for ¬�¬ϕ

ReadingsI �ϕ : “ϕ is necessarily true”I ♦ϕ : “ϕ is possibly true”


Relational semanticsFrames : F = (W ,R) whereI W 6= ∅I R ⊆W ×W

Models : M = (W ,R,V ) whereI V : p 7→ V (p) ⊆W

Truth conditionsI M, s |= p iff s ∈ V (p)

I M, s 6|= ⊥, etcI M, s |= �ϕ iff ∀t ∈W (sRt ⇒ M, t |= ϕ)

I M, s |= ♦ϕ iff ∃t ∈W (sRt & M, t |= ϕ)


Axiomatization/completeness

Minimal normal logic K

I tautologies, modus ponensI �(p → q)→ (�p → �q)

I generalization : ϕ�ϕ

ExtensionsI D : ♦>I T : �p → pI B : p → �♦pI 4 : �p → ��pI 5 : ♦p → �♦p

Contingency and non-contingency

Montgomery and Routley (1966, 1968)New primitiveI ∇ϕ : “it is contingent that ϕ”I ϕ ::= p | ⊥ | ¬ϕ | (ϕ ∨ ψ) | ∇ϕ

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

I M, s |= ∇ϕ iff∃t ∈W (sRt & M, t |= ϕ) & ∃u ∈W (sRu & M,u 6|= ϕ)

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


I M, s |= ∆ϕ iff∀t ∈W (sRt ⇒ M, t |= ϕ) ∨ ∀u ∈W (sRu ⇒ M,u 6|= ϕ)


Segerberg (1982)In the class of all framesI ∇ϕ is equivalent to ♦ϕ ∧ ♦¬ϕI ∆ϕ is equivalent to �ϕ ∨�¬ϕ

In the class of all reflexive framesI �ϕ is equivalent to ϕ ∧∆ϕ

I ♦ϕ is equivalent to ϕ ∨∇ϕ


Montgomery and Routley (1966, 1968)

Axiomatization (in the class of all reflexive frames)

I tautologies, modus ponensI ∆p ↔ ∆¬pI p → (∆(p → q)→ (∆p → ∆q))

Iϕ

∆ϕ

Axiomatization (in the class of all reflexive transitive frames)I additional axiom : ∆p → ∆∆p

Axiomatization (in the class of all partitions)I additional axiom : ∆∆p

Axiomatization (in the class of all frames)I open problem (1966)


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 suchthat M, s 6|= p → (∆(p → q)→ (∆p → ∆q)).

2. Hence, M, s |= p, M, s |= ∆(p → q), M, s |= ∆p andM, s 6|= ∆q.

3. Let t ,u ∈W be such that sRt , sRu, M, t |= q and M,u 6|= 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 6|= q, therefore M,u 6|= p.6. Since R is reflexive, M, s |= ∆p and sRu, therefore

M, s |= p iff M,u |= p : a contradiction.


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 ands ∈W be such that M, s 6|= ∆p → ∆∆p.

2. Hence, M, s |= ∆p and M, s 6|= ∆∆p.3. Let t ,u ∈W be such that sRt , sRu, M, t |= ∆p and

M,u 6|= ∆p.4. Let v ,w ∈W be such that uRv , uRw , M, v |= p and

M,w 6|= 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.


Montgomery and Routley (1966, 1968)Validity of ∆∆p in partitions

1. Let M = (W ,R,V ) where R is reflexive, symmetric andtransitive and s ∈W be such that M, s 6|= ∆∆p.

2. Let t ,u ∈W be such that sRt , sRu, M, t |= ∆p andM,u 6|= ∆p.

3. Let v ,w ∈W be such that uRv , uRw , M, v |= p andM,w 6|= 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 : acontradiction.


Cresswell (1988)New primitiveI ∆ϕ : “it is non-contingent that ϕ”I ϕ ::= p | ⊥ | ¬ϕ | (ϕ ∨ ψ) | ∆ϕ


I M, s |= ∆ϕ iff∀t ∈W (sRt ⇒ M, t |= ϕ) ∨ ∀u ∈W (sRu ⇒ M,u 6|= ϕ)

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


I M, s |= ∇ϕ iff∃t ∈W (sRt & M, t |= ϕ) & ∃u ∈W (sRu & M,u 6|= ϕ)


Cresswell (1988)� is ∆-definable in the normal modal logic L iffI there exists a formula ϕ(p) in L(⊥,¬,∨,∆) such that

I �p ↔ ϕ?(p) ∈ L

where ϕ?(p) is obtained from ϕ(p) by iteratively replacingthe subformulas of the form ∆ψ by the correspondingformulas �ψ ∨�¬ψ

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

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


Cresswell (1988)For example, � is ∆-definable in the normal modal logicT = K +�p → p seeing thatI �p ↔ p ∧ (�p ∨�¬p) ∈ T

Another example, � is ∆-definable in the normal modal logicVerum = K +�⊥ seeing thatI �p ↔ > ∈ Verum


Cresswell (1988)QuestionI Find normal modal logics L such that T 6⊆ L, Verum 6⊆ L

and � is ∆-definable in L


Cresswell (1988)A resultI 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.

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

that T 6⊆ L, Verum 6⊆ L and ♦> ∈ L


Cresswell (1988)Other results

1. Let L be a normal modal logic such that ♦> ∈ L. If thecanonical model of L contains an irreflexive s ∈W withexactly one successor then � is not ∆-definable in L.

2. Let L be a normal modal logic such that ♦> ∈ L andF = (W ,R) be an L-frame. If there exists an irreflexives ∈W such that for all t ∈W , t 6= s, sR+t and tRs then �is not ∆-definable in L.

3. Let L be a normal modal logic such that ♦> ∈ L andF = (W ,R) be an L-frame. If there exists an irreflexives ∈W such that for all t ∈W , t 6= s, sR+t and tRs then �is not ∆-definable in L.


Cresswell (1988)A natural questionI is there a normal modal logic L such that T 6⊆ L,

Verum 6⊆ L, ♦> ∈ L and � is ∆-definable in L ?Cresswell’s answerI yes

I K +�p ↔ (∆p ∧ (p ↔ ∆∆p))?

I that is to sayI K +�p ↔ ((�p ∨�¬p) ∧ (p ↔

(�(�p ∨�¬p) ∨�¬(�p ∨�¬p))))


Expressivity and definabilitySome properties

1. ∆ and � are equally expressive on the class of all reflexiveframes.

2. ∆ is strictly less expressive than � onthe class of all frames,the class of all serial frames,the class of all transitive frames,the class of all Euclidean frames andthe class of all symmetric frames.


Expressivity and definabilitySome properties

1. The class of all frames and the class of all serial framesvalidate the same ∆-formulas.

2. The frame properties of reflexivity, seriality, transitivity,Euclideanity and symmetry are not ∆-definable.


Humberstone (1995)

Some principles for non-contingency

I ∆⊥I ∆p → ∆¬pI ∆p ∧∆q → ∆(p ∨ q)

I ∆p1 ∧ . . . ∧∆pn → ∆](p1, . . . ,pn)


Humberstone (1995)

Some principles for non-contingency

I¬ϕ→ψ0, ϕ→ψ1∆ϕ→∆ψ0∨∆ψ1

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

∆ϕ∧∆ϕ′→∆ψ00∨∆ψ01∨∆ψ10∨∆ψ11

I{ϕε11 ∧...∧ϕ

εnn →ψε1...εn : ε1...εn is a tuple of n bits}

∆ϕ1∧...∧∆ϕn→∨{∆ψε1...εn : ε1...εn is a tuple of n bits}


Humberstone (1995)

Axiomatization (in the class of all frames)

I tautologies, modus ponensI ∆¬p → ∆p

I{ϕε11 ∧...∧ϕ

εnn →ψε1...εn : ε1...εn is a tuple of n bits}

∆ϕ1∧...∧∆ϕn→∨{∆ψε1...εn : ε1...εn is a tuple of n bits}

The following formula is derivableI ∆p1 ∧ . . . ∧∆pn → ∆](p1, . . . ,pn)

Finite axiomatization (in the class of all frames)I open problem (1995)

Axiomatization (in the class of all transitive frames)I open problem (1995)


Humberstone (1995)Canonical model : Mc = (Wc ,Rc ,Vc)

I Wc : set of all maximal consistent sets of formulasI λc : x ∈Wc 7→ λc(x) is a set of formulas such that for all

formulas ϕ, if ϕ is a consequence of λC(x) then ∆ϕ ∈ xI Rc : xRcy iff λc(x) ⊆ yI Vc : x ∈ Vc(p) iff p ∈ x

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

I Mc , x |= ϕ iff ϕ ∈ x


Kuhn (1995)Validity of ∆p → ∆(p → q) ∨∆(r → p) in arbitrary frames

1. Let M = (W ,R,V ) and s ∈W be such thatM, s 6|= ∆p → ∆(p → q) ∨∆(r → p).

2. Hence, M, s |= ∆p, M, s 6|= ∆(p → q) andM, s 6|= ∆(r → p).

3. Let t ,u, v ,w ∈W be such that sRt , sRu, sRv , sRw ,M, t |= p → q, M,u 6|= p → q, M, v |= r → p andM,w 6|= r → p.

4. Since M, s |= ∆p, therefore M,u |= p iff M,w |= p.5. Since M,u 6|= p → q and M,w 6|= r → p, therefore

M,u |= p and M,w 6|= p : a contradiction.


Kuhn (1995)Validity of ∆p → ∆(∆p ∨ q) in transitive frames

1. Let M = (W ,R,V ) where R is transitive and s ∈W besuch that M, s 6|= ∆p → ∆(∆p ∨ q).

2. Hence, M, s |= ∆p and M, s 6|= ∆(∆p ∨ q).3. Let t ,u ∈W be such that sRt , sRu, M, t |= ∆p ∨ q and

M,u 6|= ∆p ∨ q.4. Thus, M,u 6|= ∆p and M,u 6|= q.5. Let v ,w ∈W be such that uRv , uRw , M, v |= p and

M,w 6|= 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-contingencyKuhn (1995)


I tautologies, modus ponensI ∆¬p → ∆pI ∆p ∧∆q → ∆(p ∧ q)

I ∆p → ∆(p → q) ∨∆(r → p)

Iϕ

∆ϕ

Iϕ↔ψ

∆ϕ↔∆ψ

Axiomatization (in the class of all transitive frames)I additional axiom : ∆p → ∆(∆p ∨ q)

Axiomatization (in the class of all symmetric frames)I open problem (1995)


Kuhn (1995)Canonical model : Mc = (Wc ,Rc ,Vc)

I Wc : set of all maximal consistent sets of formulasI λc : x ∈Wc 7→ λc(x) = {ϕ : for all ψ, ∆(ϕ ∨ ψ) ∈ x}I Rc : xRcy iff λc(x) ⊆ yI Vc : x ∈ Vc(p) iff p ∈ x




Fan et al. (2015)


I tautologies, modus ponensI ∆¬p → ∆pI ∆(p ∨ q) ∧∆(p ∨ ¬q)→ ∆pI ∆p → ∆(p → q) ∨∆(r → p)

Iϕ

∆ϕ

Iϕ↔ψ

∆ϕ↔∆ψ

Axiomatization (in the class of all transitive frames)I additional axiom : ∆p → ∆(∆p ∨ q)

I references: Kuhn (1995), Zolin (1999)


Fan et al. (2015)



Iϕ

∆ϕ

Iϕ↔ψ

∆ϕ↔∆ψ

Axiomatization (in the class of all Euclidean frames)I additional axiom : ∇p → ∆(∇p ∨ q)

I reference: Zolin (1999)


Fan et al. (2015)



Iϕ

∆ϕ

Iϕ↔ψ

∆ϕ↔∆ψ

Axiomatization (in the class of all symmetric frames)I additional axiom : p → ∆(∆p ∧∆(p → q)→ ∆q ∨ r)

I reference: Fan et al. (2015)


Fan et al. (2015)Almost definability of � by means of ∆

I (�ϕ↔ ∆ϕ ∧∆(ϕ ∨ ψ)) ∨∆ψ

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

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

I ∆ψ 6∈ xI for all ϕ, if ∆ϕ ∈ x and ∆(ϕ ∨ ψ) ∈ x then ϕ ∈ y

I Vc : x ∈ Vc(p) iff p ∈ xTruth Lemma : for all ϕ, for all x ∈Wc


Ignorance

Van der Hoek and Lomuscio (2004)FormulasI ϕ ::= p | ⊥ | ¬ϕ | (ϕ ∨ ψ) | Iϕ

ReadingI Iϕ : “the agent is ignorant about ϕ”

Models : M = (W ,R,V ) whereI W 6= ∅I R ⊆W ×WI V : p 7→ V (p) ⊆W

Truth conditionsI M, s |= p iff s ∈ V (p)

I M, s 6|= ⊥, etcI M, s |= Iϕ iff∃t ∈W (sRt & M, t |= ϕ) & ∃u ∈W (sRu & M,u 6|= ϕ)

Ignorance

Van der Hoek and Lomuscio (2004)Some principles for ignoranceI Ip ↔ I¬pI I(p ∧ q)→ Ip ∨ IqI I(p ∧ r) ∧ ¬Ir ∧ I(q ∧ (r → s)) ∧ ¬I(r → s)→ ¬Is ∧ I(p ∧ s)

I ¬Ip ∧ Iq → I(p ∧ q) ∨ I(¬p ∧ q)

I ϕ¬Iϕ

I ϕIψ→I(ϕ∧ψ)

Ignorance

Van der Hoek and Lomuscio (2004)


I tautologies, modus ponensI Ip ↔ I¬pI I(p ∧ q)→ Ip ∨ IqI I(p ∧ r) ∧ ¬Ir ∧ I(q ∧ (r → s)) ∧ ¬I(r → s)→¬Is ∧ I(p ∧ s)

I ¬Ip ∧ Iq → I(p ∧ q) ∨ I(¬p ∧ q)

Iϕ¬Iϕ

Iϕ

Iψ→I(ϕ∧ψ)

Ignorance

Van der Hoek and Lomuscio (2004)Canonical model : Mc = (Wc ,Rc ,Vc)

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

I for all ϕ, if ¬Iϕ ∈ x and I(ϕ ∧ ψ) ∈ x then ϕ ∈ yI Vc : x ∈ Vc(p) iff p ∈ x



Wrongly believing

Steinsvold (2011)New primitiveI Wϕ : “wrongly believing that ϕ”


I M, s |= Wϕ iff ∀t ∈W (sRt ⇒ M, t |= ϕ) & M, s 6|= ϕ


I tautologies, modus ponensI Wp → ¬pI Wp ∧Wq →W (p ∧ q)

Iϕ→ψ

Wϕ∧¬ψ→Wψ

Axiomatization (in the class of all transitive frames)I open problem (2011)

Being wrong

Steinsvold (2011)Canonical model : Mc = (Wc ,Rc ,Vc)

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

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

I Vc : x ∈ Vc(p) iff p ∈ xTruth Lemma : for all ϕ, for all x ∈Wc


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 iffI it is true but not necessarily true

A proposition is essential iffI if it is true then it is necessarily true

In an epistemic contextA proposition is accidental iffI it is true but you do not know that

A proposition is essential iffI if it is true then you know that


Fan (2015)New primitiveI Nϕ : “it is strongly non-contingent that ϕ”


I M, s |= Nϕ iff either ∀t ∈W (sRt ⇒ M, t |= ϕ) & M, s |= ϕ,or ∀t ∈W (sRt ⇒ M, t 6|= ϕ) & M, s 6|= ϕ

AbbreviationI Hϕ for ¬Nϕ

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

necessarily”I Hϕ : “no matter whether ϕ is true or false, it could have

been otherwise”


Fan (2015)A family of modal connectivesI �, ∆, ◦ and N


I M, s |= �ϕ iff ∀t ∈W (sRt ⇒ M, t |= ϕ)

I M, s |= ∆ϕ iff either ∀t ∈W (sRt ⇒ M, t |= ϕ), or∀t ∈W (sRt ⇒ M, t 6|= ϕ)

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

I M, s |= Nϕ iff either ∀t ∈W (sRt ⇒ M, t |= ϕ) & M, s |= ϕ,or ∀t ∈W (sRt ⇒ M, t 6|= ϕ) & M, s 6|= ϕ


Fan (2015)ValiditiesI ∆ϕ↔ �ϕ ∨�¬ϕ ∆ϕ : “ϕ is non-contingent”I ◦ϕ↔ (ϕ→ �ϕ) ◦ϕ : “ϕ is essential”I Nϕ↔ ◦ϕ ∧ ◦¬ϕ Nϕ : “ϕ is strongly non-contingent”I Nϕ↔ (ϕ→ �ϕ) ∧ (¬ϕ→ �¬ϕ)


Fan (2015)

I N and � are equally expressive on the class of all reflexiveframes.

I N is strictly less expressive than � onthe class of all frames,the class of all serial frames,the class of all transitive frames,the class of all Euclidean frames andthe class of all symmetric frames.


Fan (2015)

I ∆ and N are equally expressive on the class of all reflexiveframes.

I ∆ is strictly less expressive than N onthe class of all frames,the class of all serial frames,the class of all transitive frames,the class of all Euclidean frames andthe class of all symmetric frames.


Fan (2015)

I The frame properties of reflexivity, seriality, transitivity andEuclideanity are not N-definable.

I The frame property of symmetry is N-definable byp → N(Np → p)

I The frame property of weak narcissism is N-definable byNp

Essence, accident and strong non-contingencyFan (2015)


I tautologies, modus ponensI Np ↔ N¬pI Np ∧ Nq → N(p ∧ q)

I N>I

ϕ→ψNϕ∧ϕ→Nψ

Axiomatization (in the class of all transitive frames)I additional axiom : Np → NNp

Axiomatization (in the class of all symmetric frames)I additional axiom : p → N(Np → p)

Axiomatization (in the class of all Euclidean frames)I open problem (2015)


Fan (2015)Canonical model : Mc = (Wc ,Rc ,Vc)

I Wc : set of all maximal consistent sets of formulasI Rc : xRcy iff for all ϕ, if Nϕ ∈ x and ϕ ∈ x then ϕ ∈ yI Vc : x ∈ Vc(p) iff p ∈ x



Directions for further research

Correspondence theoryI prove an equivalent of the Sahlqvist’s theorems

Complexity of the validity problemI find the lower bound and the upper bound of the

complexity of the validity problemExpressivity and succinctnessI compare the expressivity and succinctness of ∆, �, I, W ,N and ◦

Multimodal versionI add group operators for knowing-whether

Bibliography

I Cresswell, M. Necessity and contingency. Studia Logic 47(1988) 145–149.

I Demri, S. A completeness proof for a logic with analternative necessity operator. Studia Logic 58 (1997)99–112.

I Fan, J. Logics of strong noncontingency.arXiv:1505.03950v1 [cs.LO].

I Fan, J., Wang, Y., van Ditmarsch, H. Contingency andknowing whether. The Review of Symbolic Logic 8 (2015)75–107.

I Fine, K. Essence and modality. Philosophical Perspectives8 (1994) 1–16.

I Fine, K. The logic of essence. Journal of PhilosophicalLogic 24 (1995) 241–273.

Bibliography

I Fine, K. Semantics for the logic of essence. Journal ofPhilosophical Logic 29 (2000) 543–584.

I Van der Hoek, W., Lomuscio, A. A logic for ignorance.Electronic Notes in Theoretical Computer Science 85(2004) http://www.elsevier.nl/locate/entcs/volume85.html.

I Humberstone, I. The logic of non-contingency. Notre DameJournal of Formal Logic 36 (1995) 214–229.

I Humberstone, I. The modal logic of agreement andnoncontingency. Notre Dame Journal of Formal Logic 43(2002) 95–127.

I Kuhn, S. Minimal non-contingency logic. Notre DameJournal of Formal Logic 36 (1995) 230–234.

I Montgomery, H., Routley, R. Contingency andnon-contingency bases for normal modal logics. Logique &Analyse 9 (1966) 318–328.

Bibliography

I Montgomery, H., Routley, R. Non-contingency axioms forS4 and S5. Logique & Analyse 11 (1968) 422–428.

I Moses, Y., Dolev, D., Halpern, J. Cheating husbands andother stories: a case study in knowledge, action, andcommunication. Distributed Computing 1 (1986) 167–176.

I Orłowska, E. Logic for reasoning about knowledge.Zeitschrift fur mathematische Logik und Grundlagen derMathematik 35 (1989) 559–568.

I Steinsvold, C. A note on logics of ignorance and borders.Notre Dame Journal of Formal Logic 49 (2008) 385–392.

I Steinsvold, C. Being wrong: logics for false belief. NotreDame Journal of Formal Logic 52 (2011) 245–253.

I Zolin, E. Completeness and definability in the logic ofnoncontingency. Notre Dame Journal of Formal Logic 40(1999) 533–548

Documents

About contingency and ignorance - ruhr-uni-bochum.de fileHans van Ditmarsch and Jie Fan. Introduction Modal logic Study of principles of reasoning involving Inecessity Ipossibility