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”
Ordinary modal logic
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)
Ordinary modal logic
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”
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)
Contingency and non-contingency
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
Contingency and non-contingency
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)
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 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.
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 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.
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 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.
Contingency and non-contingency
Cresswell (1988)New primitiveI ∆ϕ : “it is non-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 contingent that 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)
Contingency and non-contingency
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′.
Contingency and non-contingency
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
Contingency and non-contingency
Cresswell (1988)QuestionI Find normal modal logics L such that T 6⊆ L, Verum 6⊆ L
and � is ∆-definable in L
Contingency and non-contingency
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
Contingency and non-contingency
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.
Contingency and non-contingency
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))))
Contingency and non-contingency
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.
Contingency and non-contingency
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.
Contingency and non-contingency
Humberstone (1995)
Some principles for non-contingency
I ∆⊥I ∆p → ∆¬pI ∆p ∧∆q → ∆(p ∨ q)
I ∆p1 ∧ . . . ∧∆pn → ∆](p1, . . . ,pn)
Contingency and non-contingency
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}
Contingency and non-contingency
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)
Contingency and non-contingency
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
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 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.
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 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)
Axiomatization (in the class of all frames)
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)
Contingency and non-contingency
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
Truth Lemma : for all ϕ, for all x ∈Wc
I Mc , x |= ϕ iff ϕ ∈ x
Contingency and non-contingency
Fan et al. (2015)
Axiomatization (in the class of all frames)
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)
Contingency and non-contingency
Fan et al. (2015)
Axiomatization (in the class of all frames)
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 Euclidean frames)I additional axiom : ∇p → ∆(∇p ∨ q)
I reference: Zolin (1999)
Contingency and non-contingency
Fan et al. (2015)
Axiomatization (in the class of all frames)
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 symmetric frames)I additional axiom : p → ∆(∆p ∧∆(p → q)→ ∆q ∨ r)
I reference: Fan et al. (2015)
Contingency and non-contingency
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
I Mc , x |= ϕ iff ϕ ∈ x
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)
Axiomatization (in the class of all frames)
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
Truth Lemma : for all ϕ, for all x ∈Wc
I Mc , x |= ϕ iff ϕ ∈ x
Wrongly believing
Steinsvold (2011)New primitiveI Wϕ : “wrongly believing that ϕ”
Truth condition in model M = (W ,R,V )
I M, s |= Wϕ iff ∀t ∈W (sRt ⇒ M, t |= ϕ) & M, s 6|= ϕ
Axiomatization (in the class of all frames)
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
I 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 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
Essence, accident and strong non-contingency
Fan (2015)New primitiveI Nϕ : “it is strongly non-contingent that ϕ”
Truth condition in model M = (W ,R,V )
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”
Essence, accident and strong non-contingency
Fan (2015)A family of modal connectivesI �, ∆, ◦ and N
Truth condition in model M = (W ,R,V )
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
Essence, accident and strong non-contingency
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)
Essence, accident and strong non-contingency
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.
Essence, accident and strong non-contingency
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.
Essence, accident and strong non-contingency
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)
Axiomatization (in the class of all frames)
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)
Essence, accident and strong non-contingency
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
Truth Lemma : for all ϕ, for all x ∈Wc
I Mc , x |= ϕ iff ϕ ∈ 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.
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