Disjunction and Existence Properties in Inquisitive Logic Gianluca - PowerPoint PPT Presentation

Disjunction and Existence Properties in Inquisitive Logic Gianluca Grilletti June 30, 2017 Institute for Logic, Language and Computation (ILLC), 1 Amsterdam, the Netherlands

Motivating example: hospital protocol • A disease gives rise to two symptoms S 1 and S 2 . • S 1 is much worse than S 2 . • Depending on which symptoms the patients show, they have to be put in quarantine. 2

Motivating example: hospital protocol • A disease gives rise to two symptoms S 1 and S 2 . • S 1 is much worse than S 2 . • Depending on which symptoms the patients show, they have to be put in quarantine. Protocol • Patient x shows S 1 ⇒ x in quarantine. • Everyone shows S 2 ⇒ Everyone in quarantine. • Otherwise, no quarantine. 2

Q 1 ∶ Wether x shows S 1 Q 2 ∶ Wether everyone shows S 2 determine Q 3 ∶ Wether x is in quarantine 3

Q 1 ∶ Wether x shows S 1 Q 2 ∶ Wether everyone shows S 2 determine Q 3 ∶ Wether x is in quarantine Observation: Q 1 , Q 2 and Q 3 are questions . Question Q 3 depends on questions Q 1 and Q 2 . 3

How can we represent dependency between questions in a logical framework? Question Q 3 depends on questions Q 1 and Q 2 . 4

Logic and Questions In FOL (classical first-order logic) a formula is determined by its associated truth-value in any context ⇒ a FOL formula represents a statement . Questions do not have an associated truth-value ⇒ questions are not (directly) representable in FOL . The aim of the logic InqBQ (inquisitive first-order logic) is to • extend FOL to represent questions as formulas; • extend FOL entailment to capture dependency between questions. 5

InqBQ : Adding Questions to FOL Disjunction Property Existence Property 6

InqBQ : Adding Questions to FOL

Syntax of InqBQ : introducing questions φ ∶∶= �∣ [ t 1 = t 2 ]∣ R ( t 1 ,...,t n )∣ φ ∧ φ ∣ φ → φ ∣ ∀ x.φ ∣ φ ∣ ∃ x.φ φ ⩾ shorthands ¬ φ ∶= φ → � φ ∨ ψ ∶= ¬ ( ¬ φ ∧ ¬ ψ ) ∃ x.φ ∶= ¬∀ x. ¬ φ 7

Syntax of InqBQ : introducing questions φ ∶∶= � ∣ [ t 1 = t 2 ]∣ R ( t 1 ,...,t n )∣ φ ∧ φ ∣ φ → φ ∣ ∀ x.φ ∣ φ ∣ ∃ x.φ φ ⩾ shorthands ¬ φ ∶= φ → � φ ∨ ψ ∶= ¬ ( ¬ φ ∧ ¬ ψ ) ∃ x.φ ∶= ¬∀ x. ¬ φ A formula is called FOL or classical if it does not contain the symbols and ∃ . ⩾ FOL formulas are denoted with α , β , . . . 7

Intuition FOL formulas represent statements . ( c = d ) ∨ ( c ≠ d ) ≡ “ c is equal to d or not” ∃ x. [ x = c ] ≡ “There is an element equal to c ” The operator introduces alternative questions . ⩾ ( c = d ) ( c ≠ d ) ≡ “Is c equal to d or not?” ⩾ The operator ∃ introduces existential questions . ∃ x. [ x = c ] ≡ “Which is an element equal to c ?” 8

Some notations Fix a signature Σ = { f i ,R j } i ∈ I,j ∈ J . Definition ( FOL structure) M = ⟨ D , f i , R j , ∼ ⟩ i ∈ I,j ∈ J where • f i ∶ D ar ( f i ) → D is the interpretation of f i ; • R j ⊆ D ar ( R j ) is the interpretation of R j ; • [∼] ⊆ D 2 is an equivalence relation and a congruence with respect to { f i , R j } i ∈ I,j ∈ J . 9

M = ⟨ D , f i , R j , ∼ ⟩ i ∈ I,j ∈ J Definition (Skeleton) Given M a FOL structure, define Sk ( M ) = ⟨ D, f i ⟩ i ∈ I i.e., leaving out relations and equality. 10

Models of InqBQ : representing information Definition (Information structure) M = ⟨ M w ∣ w ∈ W M ⟩ where the M w are classical structures sharing the same skeleton . We will call W M the set of worlds of the structure. 11

Models of InqBQ : representing information Definition (Information structure) M = ⟨ M w ∣ w ∈ W M ⟩ where the M w are classical structures sharing the same skeleton . We will call W M the set of worlds of the structure. ⋯ w 0 w 1 ⋯ Example of a simple model in the signature { f ( 1 ) } . 11

Models of InqBQ : representing information Definition (Information structure) M = ⟨ M w ∣ w ∈ W M ⟩ where the M w are classical structures sharing the same skeleton . We will call W M the set of worlds of the structure. ⋯ w 0 w 1 ⋯ Example of a simple model in the signature { f ( 1 ) } . The arrow represents f . 11

Models of InqBQ : representing information Definition (Information structure) M = ⟨ M w ∣ w ∈ W M ⟩ where the M w are classical structures sharing the same skeleton . We will call W M the set of worlds of the structure. ⋯ w 0 w 1 ⋯ Example of a simple model in the signature { f ( 1 ) } . The arrow represents f . The colours represent equality. 11

w 0 w 1 ⋯ ⋯ 12

World w 0 w 1 ⋯ ⋯ encoded by Truth-condition World 12

Info state World w 0 w 1 ⋯ ⋯ encoded by Truth-condition World Information encoded by Info State 12

Semantics of InqBQ : supporting relation M ↝ info structure s ↝ info state g ↝ assignment M ,s ⊧ g φ M ,s ⊧ g ⊥ ⇐ ⇒ s = ∅ ∀ w ∈ s. [ g ( t 1 ) ∼ M M ,s ⊧ g [ t 1 = t 2 ] ⇐ ⇒ w g ( t 2 )] ∀ w ∈ s. [ R M w ( g ( t 1 ) ,...,g ( t n ))] M ,s ⊧ g R ( t 1 ,...,t n ) ⇐ ⇒ M ,s ⊧ g φ ∧ ψ ⇐ ⇒ M ,s ⊧ g φ and M ,s ⊧ g ψ M ,s ⊧ g φ → ψ ⇐ ⇒ ∀ t ⊆ s. [M ,t ⊧ g φ ⇒ M ,t ⊧ g ψ ] M ,s ⊧ g ∀ x.φ ⇐ ⇒ ∀ d ∈ D M . M ,s ⊧ g [ x ↦ d ] φ M ,s ⊧ g φ ⇐ ⇒ M ,s ⊧ g φ or M ,s ⊧ g ψ ψ ⩾ ∃ d ∈ D M . M ,s ⊧ g [ x ↦ d ] φ M ,s ⊧ g ∃ x.φ ⇐ ⇒ 13

M ,s ⊧ g [ t 1 = t 2 ] ⇐ ⇒ ∀ w ∈ s. [ g ( t 1 ) ∼ M w g ( t 2 )] c = d ≅ “ c is equal to d ” w 0 w 1 { w 0 ,w 1 } ⊆ ⊇ { w 0 } { w 1 } c c ⊇ ⊆ ∅ d d 14

M ,s ⊧ g [ t 1 = t 2 ] ⇐ ⇒ ∀ w ∈ s. [ g ( t 1 ) ∼ M w g ( t 2 )] c = d ≅ “ c is equal to d ” w 0 w 1 { w 0 ,w 1 } ⊆ ⊇ { w 0 } { w 1 } c c ⊇ ⊆ ∅ d d 14

M ,s ⊧ g [ t 1 = t 2 ] ⇐ ⇒ ∀ w ∈ s. [ g ( t 1 ) ∼ M w g ( t 2 )] c = d ≅ “ c is equal to d ” w 0 w 1 { w 0 ,w 1 } ⊆ ⊇ { w 0 } { w 1 } c c ⊇ ⊆ ∅ d d 14

M ,s ⊧ g [ t 1 = t 2 ] ⇐ ⇒ ∀ w ∈ s. [ g ( t 1 ) ∼ M w g ( t 2 )] c = d ≅ “ c is equal to d ” w 0 w 1 { w 0 ,w 1 } ⊆ ⊇ { w 0 } { w 1 } c c ⊇ ⊆ ∅ d d Fact 1: The info states that support a FOL formula form a principal ideal (truth-conditionality). An alternative way to state this: s ⊧ α iff ∀ w ∈ s. { w } ⊧ α . 14

M ,s ⊧ g φ ⇐ ⇒ M ,s ⊧ g φ or M ,s ⊧ g ψ ψ ⩾ [ c = d ] [ c ≠ d ] ≡ “Is c equal to d ?” ⩾ w 0 w 1 { w 0 ,w 1 } ⊆ ⊇ { w 0 } { w 1 } c c ⊇ ⊆ ∅ d d 15

M ,s ⊧ g φ ⇐ ⇒ M ,s ⊧ g φ or M ,s ⊧ g ψ ψ ⩾ [ c = d ] [ c ≠ d ] ≡ “Is c equal to d ?” ⩾ w 0 w 1 { w 0 ,w 1 } ⊆ ⊇ { w 0 } { w 1 } c c ⊇ ⊆ ∅ d d 15

M ,s ⊧ g φ ⇐ ⇒ M ,s ⊧ g φ or M ,s ⊧ g ψ ψ ⩾ [ c = d ] [ c ≠ d ] ≡ “Is c equal to d ?” ⩾ w 0 w 1 { w 0 ,w 1 } ⊆ ⊇ { w 0 } { w 1 } c c ⊇ ⊆ ∅ d d 15

M ,s ⊧ g φ ⇐ ⇒ M ,s ⊧ g φ or M ,s ⊧ g ψ ψ ⩾ [ c = d ] [ c ≠ d ] ≡ “Is c equal to d ?” ⩾ w 0 w 1 { w 0 ,w 1 } ⊆ ⊇ { w 0 } { w 1 } c c ⊇ ⊆ ∅ d d Fact 2: The info states that support a formula form an ideal , but in general not principal (Persistency). Uniform substitution does not hold! Fact 3: φ is truth-conditional iff is equivalent to a FOL formula. 15

M ,s ⊧ g ∃ x.φ ⇐ ⇒ ∃ d ∈ D M . M ,s ⊧ g [ x ↦ d ] ∃ x. [ f ( x ) = x ] ≡ “Which is a fixed point of f ?” { w 0 ,w 1 } w 0 w 1 ⊆ ⊇ { w 0 } { w 1 } c c ⊇ ⊆ ∅ d d 16

Some insight. . . Information structures as Kripke models w 0 w 1 w 2 17

Some insight. . . Information structures as Kripke models { w 0 ,w 1 ,w 2 } { w 0 ,w 1 } { w 0 ,w 2 } { w 1 ,w 2 } w 0 w 1 w 2 17

Some insight. . . Information structures as Kripke models { w 0 ,w 1 ,w 2 } { w 0 ,w 1 } { w 0 ,w 2 } { w 1 ,w 2 } w 0 w 1 w 2 • Frame = ⟨P( W ) ∖ {∅} , ⊇⟩ 17

Some insight. . . Information structures as Kripke models { w 0 ,w 1 ,w 2 } { w 0 ,w 1 } { w 0 ,w 2 } { w 1 ,w 2 } w 0 w 1 w 2 • Frame = ⟨P( W ) ∖ {∅} , ⊇⟩ • Constant domain D M . 17

Some insight. . . Information structures as Kripke models { w 0 ,w 1 ,w 2 } { w 0 ,w 1 } { w 0 ,w 2 } { w 1 ,w 2 } w 0 w 1 w 2 • Frame = ⟨P( W ) ∖ {∅} , ⊇⟩ • Constant domain D M . A } ↓ for A atomic • � A � g = { w ∣ M w ⊧ FOL g 17

Some insight. . . Information structures as Kripke models { w 0 ,w 1 ,w 2 } { w 0 ,w 1 } { w 0 ,w 2 } { w 1 ,w 2 } w 0 w 1 w 2 • Frame = ⟨P( W ) ∖ {∅} , ⊇⟩ • Constant domain D M . A } ↓ for A atomic • � A � g = { w ∣ M w ⊧ FOL g Fact: InqBQ is the logic of a class of Kripke models . 17

The Main Result: DP and EP in InqBQ Theorem (Disjunction and Existence Property) Consider Γ a FOL theory. Then • If Γ ⊧ φ ψ then Γ ⊧ φ or Γ ⊧ ψ . ⩾ • If Γ ⊧ ∃ x.φ ( x ) then Γ ⊧ φ ( t ) for some term t . Corollary If Γ ⊧ ∀ x ∃ ! y.φ ( x,y ) (i.e., φ defines a function), then there exists a term t such that Γ ⊧ ∀ x.φ ( x,t ) . 18