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 ) ϕ ◮ generalization : � 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.