Undefinability of standard sequent calculi for Paraconsistent logics
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1 Undefinability of standard sequent calculi for Paraconsistent logics Michele Pra Baldi (with S.Bonzio) University of Padua Topology, Algebra and Categories in Logic. Prague, June 6, 07 M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 /
2 Outline Kleene Logics The starting point: PWK 3 Deductive calculi for paraconsistent Kleene logics 4 Our results M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 /
3 Kleene tables Strong Kleene tables: M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 3 /
4 Kleene tables Strong Kleene tables: Weak Kleene tables: M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 3 /
5 Kleene s family M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 4 /
6 Kleene s family Strong Kleene logic: SK, {} M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 4 /
7 Kleene s family Strong Kleene logic: SK, {} The Logic of Paradox, LP: SK, {, } M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 4 /
8 Kleene s family Strong Kleene logic: SK, {} The Logic of Paradox, LP: SK, {, } Bochvar s logic: WK, {} M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 4 /
9 Kleene s family Strong Kleene logic: SK, {} The Logic of Paradox, LP: SK, {, } Bochvar s logic: WK, {} Paraconsistent Weak Kleene logic, PWK: WK, {, } M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 4 /
10 Paraconsistent Week Kleene The language:,,, 0, M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 5 /
11 Paraconsistent Week Kleene The language:,,, 0, The algebra WK M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 5 /
12 Paraconsistent Week Kleene The language:,,, 0, The algebra WK The matrix: PWK = WK, {, /} 0 0 M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 5 /
13 Paraconsistent Week Kleene The language:,,, 0, The algebra WK The matrix: PWK = WK, {, /} 0 0 Γ PWK α for every v, v[γ] {, /} v(α) {, /} M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 5 /
14 A closer look to WK WK = {0,, },,,, 0, 0 0 M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 6 /
15 A closer look to WK WK = {0,, },,,, 0, a b a b = b 0 0 M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 6 /
16 A closer look to WK WK = {0,, },,,, 0, a b a b = b and a b a b = a 0 0 M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 6 /
17 A closer look to WK WK = {0,, },,,, 0, a b a b = b and a b a b = a a b b a 0 0 M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 6 /
18 A closer look to WK WK = {0,, },,,, 0, a b a b = b and a b a b = a 0 0 a b b a Counterexample to absorption: ( ) = M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 6 /
19 Involutive bisemilattices Definition An involutive bisemilattice is an algebra B = B,,,, 0, of type (,,,0,0), satisfying: M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 7 /
20 Involutive bisemilattices Definition An involutive bisemilattice is an algebra B = B,,,, 0, of type (,,,0,0), satisfying: I x x x; I x y y x; I3 x (y z) (x y) z; I7 0 x x; M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 7 /
21 Involutive bisemilattices Definition An involutive bisemilattice is an algebra B = B,,,, 0, of type (,,,0,0), satisfying: I x x x; I x y y x; I3 x (y z) (x y) z; I4 x x; I7 0 x x; I8 0. M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 7 /
22 Involutive bisemilattices Definition An involutive bisemilattice is an algebra B = B,,,, 0, of type (,,,0,0), satisfying: I x x x; I x y y x; I3 x (y z) (x y) z; I4 x x; I5 x y ( x y); I7 0 x x; I8 0. M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 7 /
23 Involutive bisemilattices Definition An involutive bisemilattice is an algebra B = B,,,, 0, of type (,,,0,0), satisfying: I x x x; I x y y x; I3 x (y z) (x y) z; I4 x x; I5 x y ( x y); I6 x ( x y) x y; I7 0 x x; I8 0. M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 7 /
24 Involutive bisemilattices Definition An involutive bisemilattice is an algebra B = B,,,, 0, of type (,,,0,0), satisfying: I x x x; I x y y x; I3 x (y z) (x y) z; I4 x x; I5 x y ( x y); I6 x ( x y) x y; I7 0 x x; I8 0. Theorem V(WK) = IBSL. M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 7 /
25 Algebraizability A formula-equation transformer is a map τ : Fm P(Fm ) (given by a set of equations in one variable). M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 8 /
26 Algebraizability A formula-equation transformer is a map τ : Fm P(Fm ) (given by a set of equations in one variable). An equation-formula transformer is a map ρ : Fm P(Fm) (given by a set of formulas in two variables). M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 8 /
27 Algebraizability A formula-equation transformer is a map τ : Fm P(Fm ) (given by a set of equations in one variable). An equation-formula transformer is a map ρ : Fm P(Fm) (given by a set of formulas in two variables). A class of algebras K is an algebraic semantics of a logic L if: Γ L α τ[γ] K τ(α). M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 8 /
28 Algebraizability A formula-equation transformer is a map τ : Fm P(Fm ) (given by a set of equations in one variable). An equation-formula transformer is a map ρ : Fm P(Fm) (given by a set of formulas in two variables). A class of algebras K is an algebraic semantics of a logic L if: Γ L α τ[γ] K τ(α). K is an equivalent algebraic semantics of L if moreover: α β K τ[ρ(α, β)]. M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 8 /
29 Algebraizability A formula-equation transformer is a map τ : Fm P(Fm ) (given by a set of equations in one variable). An equation-formula transformer is a map ρ : Fm P(Fm) (given by a set of formulas in two variables). A class of algebras K is an algebraic semantics of a logic L if: Γ L α τ[γ] K τ(α). K is an equivalent algebraic semantics of L if moreover: α β K τ[ρ(α, β)]. L is algebraizable if it has an equivalent algebraic semantics. M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 8 /
30 PWK, IBSL, and Algebraic Logic Theorem IBSL is not the equivalent algebraic semantics of any algebraizable logic L. M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 9 /
31 PWK, IBSL, and Algebraic Logic Theorem IBSL is not the equivalent algebraic semantics of any algebraizable logic L. Theorem PWK is not protoalgebraic. Thus not algebraizable. M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 9 /
32 PWK, IBSL, and Algebraic Logic Theorem IBSL is not the equivalent algebraic semantics of any algebraizable logic L. Theorem PWK is not protoalgebraic. Thus not algebraizable. Theorem PWK is not selfextensional, i.e. the interderivability relation L is not a congruence on Fm. M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 9 /
33 Deductive calculi M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 0 /
34 Deductive calculi PWK: Hilbert style (Bonzio et al.), sequent calculi (Coniglio, Corbalan) M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 0 /
35 Deductive calculi PWK: Hilbert style (Bonzio et al.), sequent calculi (Coniglio, Corbalan) LP: Hilbert style (Font), sequent calculi (Avron, Beall) M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 0 /
36 Sequent calculus for PWK Axioms α α M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 /
37 Sequent calculus for PWK Axioms α α Structural rules Γ Γ, α LW Γ Γ α, Γ, α Γ, α Cut Γ RW M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 /
38 Sequent calculus for PWK Axioms Structural rules Γ Γ, α Operational rules LW α α Γ Γ α, Γ, α Γ, α Cut Γ Γ α, L Γ, α Γ, α Γ, β L Γ, α β Proviso for L : var(α) var( ) RW Γ, α R Γ α, Γ α, β, Γ α β, M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 / R
39 Sequent calculi for LP Avron (04). Axioms and structural rules as for PWK, logical rules including: Γ, α Γ, β L Γ, (α β) Γ, α L Γ, α M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 /
40 Our starting question Both LP and PWK have sequent calculi with some limitations on operational rules. Is it possible to give standard sequent calculi for LP and PWK? M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 3 /
41 Standard Sequent Calculi A standard Gentzen calculus for a logic L has the foll. 5 properties: Axioms: α α (α atomic). M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 4 /
42 Standard Sequent Calculi A standard Gentzen calculus for a logic L has the foll. 5 properties: Axioms: α α (α atomic). Premises (active) of logical rules with only subformulas of the conclusion; exactly one connective at time M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 4 /
43 Standard Sequent Calculi A standard Gentzen calculus for a logic L has the foll. 5 properties: Axioms: α α (α atomic). Premises (active) of logical rules with only subformulas of the conclusion; exactly one connective at time 3 No linguistic restrictions on rules M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 4 /
44 Standard Sequent Calculi A standard Gentzen calculus for a logic L has the foll. 5 properties: Axioms: α α (α atomic). Premises (active) of logical rules with only subformulas of the conclusion; exactly one connective at time 3 No linguistic restrictions on rules 4 Sequents are interpreted in the object language M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 4 /
45 Standard Sequent Calculi A standard Gentzen calculus for a logic L has the foll. 5 properties: Axioms: α α (α atomic). Premises (active) of logical rules with only subformulas of the conclusion; exactly one connective at time 3 No linguistic restrictions on rules 4 Sequents are interpreted in the object language 5 Only classical structural rules, i.e. contraction, weakening and cut are (possibly) allowed. M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 4 /
46 IFL Logics We can generalise our question to a wider class of logics. M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 5 /
47 IFL Logics We can generalise our question to a wider class of logics. Definition IFL is the family of logics in the language {,, } s.t.: M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 5 /
48 IFL Logics We can generalise our question to a wider class of logics. Definition IFL is the family of logics in the language {,, } s.t.: Have the same theorems of classical logic. The connective has a fixed point on a designated value and k k. M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 5 /
49 IFL: basic facts Remark If L IFL then L is paraconsistent, i.e. α, α L β M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 6 /
50 IFL: basic facts Remark If L IFL then L is paraconsistent, i.e. α, α L β Lemma Let L IFL. Then there exists at least a designated truth value k s.t. k is not designated. M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 6 /
51 IFL: basic facts Remark If L IFL then L is paraconsistent, i.e. α, α L β Lemma Let L IFL. Then there exists at least a designated truth value k s.t. k is not designated. Lemma A logic L IFL can not have a non designated value k such that k = k. M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 6 /
52 Standard Gentzen Calculi In a standard calculus (L ) and (R ) must be of the form: Γ, α, β L Γ, α β Γ α, β, Γ α β, R M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 7 /
53 IFL: rules and soundess What does a standard R -rule for an IFL-logic look like? M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 8 /
54 IFL: rules and soundess What does a standard R -rule for an IFL-logic look like? Any possible sound R rule, whose conclusion is Γ, α, possess at least one premise of the following form: Γ, α. M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 8 /
55 IFL: rules and soundess What does a standard R -rule for an IFL-logic look like? Any possible sound R rule, whose conclusion is Γ, α, possess at least one premise of the following form: That is Γ, α. P,..., Γ, α..., P n Γ α, M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 8 /
56 A limitative result Theorem Let L IFL and S be a standard Gentzen calculus for L. Then, if S is sound, it is incomplete. M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 9 /
57 A limitative result Theorem Let L IFL and S be a standard Gentzen calculus for L. Then, if S is sound, it is incomplete. Sketch of the Proof L (α α) β and α, α L β (L IFL). Therefore S shall derive the sequent (α α) β. M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 9 /
58 A limitative result Theorem Let L IFL and S be a standard Gentzen calculus for L. Then, if S is sound, it is incomplete. Sketch of the Proof L (α α) β and α, α L β (L IFL). Therefore S shall derive the sequent (α α) β. The derivation shall necessarily contain a tree with a branch of the form: α, α β R α α β (α α), β (α α) β R (Lemma) R M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 9 /
59 A limitative result Theorem Let L IFL and S be a standard Gentzen calculus for L. Then, if S is sound, it is incomplete. Sketch of the Proof L (α α) β and α, α L β (L IFL). Therefore S shall derive the sequent (α α) β. The derivation shall necessarily contain a tree with a branch of the form: α, α β R α α β (α α), β (α α) β R (Lemma) Since S is sound, the above derivation tree cannot terminate with axioms. M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 9 / R
60 Work in Progress the algebraizability of the Gentzen system associated with PWK. M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 0 /
61 Work in Progress the algebraizability of the Gentzen system associated with PWK. Can the limitative result be extended? M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 0 /
62 Work in Progress the algebraizability of the Gentzen system associated with PWK. Can the limitative result be extended? 3 Which is the strongest paraconsistent logic admitting a standard calculus? M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 0 /
63 Work in Progress the algebraizability of the Gentzen system associated with PWK. Can the limitative result be extended? 3 Which is the strongest paraconsistent logic admitting a standard calculus? 4 The regularization of a logic. M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 0 /
64 Thank you! M.Pra Baldi (with S.Bonzio) TACL 07 June 6, 07 /
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