Bimodal bilattice logic Igor Sedlr Institute of Computer Science, - PowerPoint PPT Presentation

Bimodal bilattice logic Igor Sedlár Institute of Computer Science, Czech Academy of Sciences, Prague TACL 2017, Prague 26 June 2017

Outline 1. Preliminaries and aims Lattice-valued modal logics The Dunn–Belnap bilattice A modal Dunn–Belnap logic and the aim of the talk 2. Bimodal bilattice logic Motivating the second modality Some properties of the bimodal logic 3. Completeness A standard argument Bits of a general theory 4. Conclusion

Lattice-valued modal logics Defined in terms of A -valued Kripke models for a lattice A , M = ⟨ W, R, e ⟩ • W is a non-empty set • R is a function from W × W to A • e is a function from Fm 0 × W to A The value of ✷ ϕ at w is defined in terms of the lattice-order infimum of values related to ϕ . M is called crisp if the range of R is the { 0 , 1 } -subalgebra of A . In crisp models, we have ¯ e ( ✷ ϕ, w ) = inf { ¯ e ( ϕ, u ) ; Rwu } 1 / 14

Lattice-valued modal logics Defined in terms of A -valued Kripke models for a lattice A , M = ⟨ W, R, e ⟩ • W is a non-empty set • R is a function from W × W to A • e is a function from Fm 0 × W to A The value of ✷ ϕ at w is defined in terms of the lattice-order infimum of values related to ϕ . M is called crisp if the range of R is the { 0 , 1 } -subalgebra of A . In crisp models, we have ¯ e ( ✷ ϕ, w ) = inf { ¯ e ( ϕ, u ) ; Rwu } A bilattice is, roughly, an algebra with two lattice orders . The literature on bilattice-valued modal logics (Odintsov and Wansing, 2010; Rivieccio et al., 2017) considers languages where only one of the orders corresponds to a modal operator. 1 / 14

Lattice-valued modal logics Defined in terms of A -valued Kripke models for a lattice A , M = ⟨ W, R, e ⟩ • W is a non-empty set • R is a function from W × W to A • e is a function from Fm 0 × W to A The value of ✷ ϕ at w is defined in terms of the lattice-order infimum of values related to ϕ . M is called crisp if the range of R is the { 0 , 1 } -subalgebra of A . In crisp models, we have ¯ e ( ✷ ϕ, w ) = inf { ¯ e ( ϕ, u ) ; Rwu } A bilattice is, roughly, an algebra with two lattice orders . The literature on bilattice-valued modal logics (Odintsov and Wansing, 2010; Rivieccio et al., 2017) considers languages where only one of the orders corresponds to a modal operator. So what happens if we add a second one?? 1 / 14

The Dunn–Belnap bilattice Dunn (1966), Belnap (1977a, 1977b) d e t s i g n a t e d n truth b f information 2 / 14

The Dunn–Belnap bilattice Dunn (1966), Belnap (1977a, 1977b) d e ( ϕ ∧ ψ ) = inf ≤ t { e ( ϕ ) , e ( ψ ) } e t s i g n a t e e ( ϕ ∨ ψ ) = sup ≤ t { e ( ϕ ) , e ( ψ ) } d n truth b  if e ( ϕ ) = f t   e ( ¬ ϕ ) = if e ( ϕ ) = t f  f e ( ϕ ) otherwise  information 2 / 14

The Dunn–Belnap bilattice Dunn (1966), Belnap (1977a, 1977b) d e ( ϕ ∧ ψ ) = inf ≤ t { e ( ϕ ) , e ( ψ ) } e t s i g n a t e e ( ϕ ∨ ψ ) = sup ≤ t { e ( ϕ ) , e ( ψ ) } d n truth b  if e ( ϕ ) = f t   e ( ¬ ϕ ) = if e ( ϕ ) = t f  f e ( ϕ ) otherwise  information Arieli and Avron (1996), BL { if e ( ϕ, w ) ̸∈ D t e ( ϕ ⊃ ψ ) = e ( ψ ) otherwise e ( ϕ ⊃ ψ ) ∈ D iff ( e ( ϕ ) ∈ D = ⇒ e ( ψ ) ∈ D ) . 2 / 14

Some properties of DB d e t s i “Classical negation” g n a t If ∼ ϕ := ϕ ⊃ f , then e ( ∼ ϕ ) ∈ D iff e d e ( ϕ ) ̸∈ D ; but, for example, not always n b truth e ( ∼∼ ϕ ) = e ( ϕ ) . f  iff { ϕ, ∼¬ ϕ } ∈ D t information    f iff {∼ ϕ, ¬ ϕ } ∈ D  e ( ϕ ) = Expressing truth values iff { ϕ, ¬ ϕ } ∈ D b    iff {∼ ϕ, ∼¬ ϕ } ∈ D  n Filters Both D and { x ; ∼¬ x ∈ D } are prime filters wrt the truth order; Both D and { x ; ¬ x ∈ D } is a prime filter wrt the info order 3 / 14

A modal Dunn–Belnap logic d e t s i g n Odintsov and Wansing (2010), BK a t e d Language {∧ , ∨ , ¬ , ⊃ , f , ✷ } , n b truth DB -valued crisp Kripke models; and e ( ✷ ϕ, w ) = inf ≤ t { e ( ϕ, w ′ ) ; Rww ′ } f information 4 / 14

A modal Dunn–Belnap logic d e t s i g n Odintsov and Wansing (2010), BK a t e d Language {∧ , ∨ , ¬ , ⊃ , f , ✷ } , n b truth DB -valued crisp Kripke models; and e ( ✷ ϕ, w ) = inf ≤ t { e ( ϕ, w ′ ) ; Rww ′ } f information Think of the states in a DB -valued crisp model as possibly incomplete and in- consistent bodies of information within a network (graph). For example, agents in a social network, interconnected databases etc. A modal logic over such models expresses properties of and represents reasoning about such “infor- mation networks”. 4 / 14

A modal Dunn–Belnap logic d e t s i g n Odintsov and Wansing (2010), BK a t e d Language {∧ , ∨ , ¬ , ⊃ , f , ✷ } , n b truth DB -valued crisp Kripke models; and e ( ✷ ϕ, w ) = inf ≤ t { e ( ϕ, w ′ ) ; Rww ′ } f information Think of the states in a DB -valued crisp model as possibly incomplete and in- consistent bodies of information within a network (graph). For example, agents in a social network, interconnected databases etc. A modal logic over such models expresses properties of and represents reasoning about such “infor- mation networks”. Example: “Hereditarity” p ⊃ ✷ p and ¬ p ⊃ ✷ ¬ p . 4 / 14

A modal Dunn–Belnap logic d e t s i g n Odintsov and Wansing (2010), BK a t e d Language {∧ , ∨ , ¬ , ⊃ , f , ✷ } , n b truth DB -valued crisp Kripke models; and e ( ✷ ϕ, w ) = inf ≤ t { e ( ϕ, w ′ ) ; Rww ′ } f information Think of the states in a DB -valued crisp model as possibly incomplete and in- consistent bodies of information within a network (graph). For example, agents in a social network, interconnected databases etc. A modal logic over such models expresses properties of and represents reasoning about such “infor- mation networks”. Example: “Hereditarity” p ⊃ ✷ p and ¬ p ⊃ ✷ ¬ p . The story invites to consider an information-order-based modality as well! 4 / 14

Outline 1. Preliminaries and aims Lattice-valued modal logics The Dunn–Belnap bilattice A modal Dunn–Belnap logic and the aim of the talk 2. Bimodal bilattice logic Motivating the second modality Some properties of the bimodal logic 3. Completeness A standard argument Bits of a general theory 4. Conclusion

The information box – motivation d e t s i g n BBK extends the language of BK by a new a t e d modality ✷ i with the semantic clause n truth b e ( ✷ i ϕ, w ) = inf ≤ i { e ( ϕ, w ′ ) ; Rww ′ } f information Sources. Graphs represent “sources of information”; the value of ✷ p is the value that can be assigned to p after considering all the sources (i.e. the info on which all the sources agree). Supervaluations. Graphs represent possibly incomplete or inconsistent val- uations; ✷ p is the “supervalue” of p , i.e. the “least” value on which all the ac- cessible “supervaluations” agree (cf. p ⊃ ✷ p and ¬ p ⊃ ✷ ¬ p ). 5 / 14

Some properties of BBK ✷ ϕ ⊃ ⊂ ✷ i ϕ is valid, but ¬ ✷ ϕ ⊃ ⊂ ¬ ✷ i ϕ is not. In fact, ¬ ✷ i ϕ ⊃ ⊂ ✷ i ¬ ϕ is valid. ∧ Γ ⊃ ϕ ∧ ✷ i Γ ⊃ ✷ i ϕ preserves validity. Note: If n is added to the language, then the information modality is definable ✷ i ϕ := ( n ∧ ¬ ✷ ¬ ϕ ) ∨ ✷ ϕ 6 / 14

Outline 1. Preliminaries and aims Lattice-valued modal logics The Dunn–Belnap bilattice A modal Dunn–Belnap logic and the aim of the talk 2. Bimodal bilattice logic Motivating the second modality Some properties of the bimodal logic 3. Completeness A standard argument Bits of a general theory 4. Conclusion

The axiom system BBK Implication axioms Modal “filter” axioms ϕ ⊃ ( ψ ⊃ ϕ ) ✷ ∼¬ ϕ ⊃ ⊂ ∼¬ ✷ ϕ ( ϕ ⊃ ( ψ ⊃ χ )) ⊃ (( ϕ ⊃ ψ ) ⊃ ( ϕ ⊃ χ )) ✷ i ¬ ϕ ⊃ ⊂ ¬ ✷ i ϕ (( ϕ ⊃ ψ ) ⊃ ϕ ) ⊃ ϕ Lattice axioms Normality rules ( ϕ ∧ ψ ) ⊃ ϕ and ( ϕ ∧ ψ ) ⊃ ψ ∧ Γ ⊃ ϕ ϕ ⊃ ( ϕ ∨ ψ ) and ψ ⊃ ( ϕ ∨ ψ ) ∧ ✷ Γ ⊃ ✷ ϕ ϕ ⊃ ( ψ ⊃ ϕ ∧ ψ ) ( ϕ ⊃ χ ) ⊃ (( ψ ⊃ χ ) ⊃ ( ϕ ∨ ψ ⊃ χ )) f ⊃ ϕ ∧ Γ ⊃ ϕ Negation axioms ∧ ✷ i Γ ⊃ ✷ i ϕ Γ ⊆ ω Fm ϕ ⊃ ⊂ ¬¬ ϕ ϕ ⊃ ¬ f ¬ ( ϕ ∧ ψ ) ⊃ ⊂ ( ¬ ϕ ∨ ¬ ψ ) Modus ponens ¬ ( ϕ ∨ ψ ) ⊃ ⊂ ( ¬ ϕ ∧ ¬ ψ ) ϕ ϕ ⊃ ψ ¬ ( ϕ ⊃ ψ ) ⊃ ⊂ ( ϕ ∧ ¬ ψ ) ψ 7 / 14

Completeness (Prime theories and extension) A nontrivial prime theory is any set of formulas Γ such that ( Γ ⊢ ϕ := Γ ′ ⊆ ω Γ , provable ∧ Γ ′ ⊃ ϕ ) • Γ ∈ ϕ iff Γ ⊢ ϕ • Γ ̸ = Fm • ϕ ∨ ψ ∈ Γ iff ϕ ∈ Γ or ψ ∈ Γ A pair of arbitrary sets of formulas ⟨ Γ , ∆ ⟩ is an independent pair iff there are no finite Γ ′ ⊆ Γ , ∆ ′ ⊆ ∆ where Γ ′ ⊃ ∧ ∨ ∆ ′ . ⊢ 8 / 14