Some extensions of the Belnap-Dunn logic

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1 Some extensions of the Belnap-Dunn logic Umberto Rivieccio Università di Genova November 11th, 2010 Barcelona U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

2 The Belnap-Dunn logic Introduction The Belnap-Dunn four-valued logic (a.k.a. first degree entailment) is a well-known system related to relevant and paraconsistent logics, widely known and applied in computer science. In recent years it has also been studied algebraically (Font, Pynko) and in connection with more general structures such as bilattices and generalized Kleene logics (Fitting, Arieli & Avron, Shramko & Wansing). U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

3 The Belnap-Dunn logic Semantic definition of B (Dunn 1966, Belnap 1977) The Belnap-Dunn logic (that we denote by B) is the logic determined by the matrix xm 4, tt, Juy, where M 4 is the four-element diamond De Morgan lattice shown below. K t J f t f K J f t K J U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

4 The Belnap-Dunn logic Miscellaneous facts (Pynko 1995, Font 1997) B is: a theoremless (a.k.a. purely inferential ) logic paraconsistent, in the sense that ϕ ^ ϕ * B ψ relevant, in the sense that if ϕ ( B ψ, then varpϕq X varpψq H the logic of the lattice order of De Morgan lattices, i.e. Γ ( B ϕ iff DMLat ( Γ ϕ finitely axiomatized by Hilbert- or Gentzen-style calculi U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

5 The Belnap-Dunn logic Miscellaneous facts (Pynko 1995, Font 1997) Moreover B is non-protoalgebraic, self-extensional and non-fregean B has an associated fully adequate Gentzen calculus that is algebraizable w.r.t. the variety DMLat of De Morgan lattices Alg B ˆ AlgB DMLat. Remark DMLat is not the equivalent algebraic semantics of any algebraizable logic; the same holds for any sub-quasi-variety of DMLat (except Boolean algebras). U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

6 The Belnap-Dunn logic Algebraic models of B (Font 1997) Any matrix xa, Dy such that A P DMLat and D A is a lattice filter (or is empty) is a model of B. xa, Cy is a reduced full model of B iff A P DMLat and C is the set of all lattice filters of A plus the empty set. xa, Dy is a reduced matrix for B iff A P DMLat and D is a lattice filter satisfying that for all a, b P A with a b, at least one of the following conditions holds: (i) there is c P A such that a _ c R D and b _ c P D (ii) there is c P A such that a _ c P D and b _ c P D. U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

7 Extensions of B Well-known ones Priest s (1979) logic of paradox LP Kleene s logic of order K (a.k.a. the implicationless fragment of the relevant logic RM) the strong three-valued (assertional) Kleene logic K classical logic CPC. U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

8 Extensions of B Inclusions CPC LP K K B U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

9 Extensions of B Axiomatizations (Hilbert-style) LP B plus $ p _ p (Pynko 1995) K B plus pp ^ pq _ r $ q _ q _ r K K plus p ^ p p _ qq $ q CPC K plus $ p _ p. Algebraic models AlgLP AlgK AlgK KLat xa, Dy P Matr L iff xa, Dy P Matr B and D is closed under the corresponding additional rule(s) of L listed above. U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

10 Extensions of B Semantical presentations LP is defined by xk 3, tt, Kuy K by txk 3, tt, Kuy, xk 3, ttuu K by xk 3, ttuy CPC by xb 2, ttuy. U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

11 There are more... Some more extensions of B Any (set of) matrices belonging to Matr B defines an extension of B. For instance: xm 4, ttuy xk 6, tt, auy xm 8, tt, auy txm 4, ttuy, xk 3, tt, Kuyu... U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

12 There are more... Some more extensions of B We are going to consider the following new basic logics : B 4 defined by xm 4, ttuy B 6 defined by xk 6, tt, auy B 8 defined by xm 8, tt, auy together with the known ones: K defined by xk 3, ttuy LP defined by xk 3, tt, Kuy. U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

13 There are more... Some more extensions of B The above-mentioned basic logics generate the following: B 36 B 6 X K B 46 B 36 X B 4 K B 36 X LP B 38 K X B 8 B 48 B 8 X B 46 B 34 K X B 46 B 348 B 34 X B 38. U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

14 A hierarchy LP CPC B 6 K B 36 B 4 K B 8 B 46 B 38 B 34 B 48 B 348 U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

15 What we do (not) know Miscellaneous facts The above logics are obtained from B by adding some form of excluded middle and/or ex contradictione quodlibet. B 4 has no axiomatic extensions of except CPC. K has no finitary extensions of except CPC. The only logics with theorems are LP and B 6. All the logics (except K ) are non-selfextensional. All the logics are non-protoalgebraic. U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

16 What we do (not) know Relations with the order For any ϕ, ψ P Fm, it holds that ϕ ( L ψ if and only if: DMLat ( ϕ ψ for L B KLat ( ϕ ψ for L K DMLat ( ϕ _ ψ ϕ _ ψ for L B 34 DMLat ( ϕ ϕ _ ψ for L B 4 KLat ( ϕ ϕ _ ψ for L K KLat ( ϕ _ ψ ϕ _ ψ for L LP QpK 6 q ( ϕ ϕ ñ ψ ψ for L B 6. U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

17 What we do (not) know Axiomatizations (Hilbert-style) The previous results allow to prove that B 4 B 6 B plus p ^ p p _ qq $ q LP plus p ^ p $ q K B plus pp ^ pq _ r $ q _ q _ r K K plus p ^ p p _ qq $ q U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

18 What we do (not) know: algebraic models Algebraic reducts of educed g-models AlgB 34 AlgB 4 AlgB 46 AlgB 48 AlgB 348 DMLat AlgB 8, AlgB 38 DMLat (?) AlgB 6, AlgB 36 KLat (?) U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

19 What we do (not) know: algebraic models Reduced models For A P DMLat, let A : ta P A : a au. Let xa, Dy be a reduced matrix: if xa, Dy is a model of B, then D A B 34 or B 48 and D X A H, then D A LP, then D A B 4 or K, then A is bounded and D t1u. U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

20 What we do (not) know: algebraic models Reduced models So, a matrix xa, Dy P Matr B is a reduced model of B 34 iff A P DMLat and D is a lattice filter such that b P D whenever a P D and a _ b a _ b B 4 iff A is bounded De Morgan lattice and D t1u K iff A P KLat and D is a lattice filter K iff A is a bounded Kleene lattice and D t1u LP iff A P KLat and D ta P A : a au. U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

21 What there is to know Open problems Complete the picture of all (?) the extensions of B (for instance: is there some logic between B and B 348?). Axiomatize B 34, B 48, B 8 etc. and characterize their reduced models. Characterize the extensions of B in terms of their metalogical properties. Introduce and study Gentzen calculi associated with these logics (study their algebraizability etc.). U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

22 References J. M. Font (1997): Belnap s four-valued logic and De Morgan lattices. Logic Journal of the I.G.P.L., 5(3): A. Pynko (1995a): On Priest s logic of paradox. Journal of Applied Non-Classical Logics, 5(2): A. P. Pynko (1995b): Characterizing Belnap s logic via De Morgan s laws. Mathematical Logic Quarterly, 41(4): A. P. Pynko (1999): Implicational classes of De Morgan lattices. Discrete Math., 205(1-3): U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

23 DMLat and KLat De Morgan lattices A De Morgan lattice is an algebra A xa, ^, _, y such that xa, ^, _y is a distributive lattice and the following equations are satisfied: (i) px ^ yq x _ y (ii) px _ yq x ^ y (iii) x x A Kleene lattice is a De Morgan lattice satisfying: (iv) x ^ x y _ y. U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

24 Some De Morgan lattices f K J t f K t f a a t M 4 K 3 K 4 U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

25 Some De Morgan lattices f b a t b a f a c t c a b b K 6 M 8 U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

26 Sub-quasi-varieties of DMLat Some facts The class DMLat is a variety, generated by M 4. Besides M 4, the the only subdirectly irreducible De Morgan lattices are K 3 and B 2. VpK 3 q KLat and VpB 2 q are the only subvarieties of DMLat. (Pynko 1999): DMLat has four sub-quasi-varieties (that are not varieties), i.e. 1 regular Kleene lattices QpK 4 q 2 non-idempotent Kleene lattices QpK 6 q 3 non-idempotent De Morgan lattices QpM 8 q 4 KLat Y non-idempotent De Morgan lattices QpM 8, K 3 q. U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28 Back

27 De Morgan matrices b a f t a xk 6, tt, auy b a t b a b c f xm 8, tt, auy c Back U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

28 A Hilbert calculus for B (Font, 1997) p ^ q (R1) p p (R4) p _ q p _ pq _ r q (R7) pp _ qq _ r (R10) p _ r p _ r p ^ q (R2) q p _ q (R5) q _ p p _ pq ^ r q (R8) pp _ qq ^ pp _ r q (R11) p _ r p _ r (R9) (R3) p q p ^ q (R6) p _ p p pp _ qq ^ pp _ r q p _ pq ^ r q pp _ qq _ r (R12) p p ^ qq _ r (R13) p p ^ qq _ r pp _ qq _ r pp ^ qq _ r p p _ qq _ r (R14) (R15) p p _ qq _ r pp ^ qq _ r Back U. Rivieccio (Università di Genova) Some extensions of the Belnap-Dunn logic November 11th, / 28

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