A GENTZEN SYSTEM EQUIVALENT TO THE BCK-LOGIC

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1 Romà J. Adillon Ventura Verdú A GENTZEN SYSTEM EQUIVALENT TO THE BCK-LOGIC Abstract The sequent calculus L BCK is obtained by deleting the contraction rule and the introduction rules of the connectives of meet, join and negation from the sequent calculus for the Intuitionistic Propositional logic, LJ. In this paper we show that the Gentzen system G BCK, naturally associated with L BCK, is equivalent to the Hilbert-style logic BCK, in the sense that G BCK and BCK are interpretable one another, the interpretations being essentially inverse in each other. This result is a strengthening of the one obtained by H. Ono and Y. Komori [8], which concerned only the derivable sequents of L BCK and the theorems of BCK. We obtain, as a corollary of this result, that the quasivariety of BCKalgebras is the equivalent quasivariety semantics of G BCK in the sense of [10]. Let L be the propositional language with a single binary connective,. The logic BCK ( L, BCK, for short) is the 1-dimensional deductive system over L defined by the axioms: (B) (X Y ) ((Y Z) (X Z)) (C) (X (Y Z)) (Y (X Z)) (K) X (Y X) and the modus ponens rule: {X, X Y }, Y, where X, Y, Z F m L. Recall that it is well known that the following formulas are theorems of BCK: (I) (X Y ) ((Z X) (Z Y )) (II) (X 1 (..(X n Y )..)) ((Y Z) (X 1 (..(X n Z)..))), where X 1,..., X n, X, Y, Z F m L. Recall that the quasivariety BCK is the class of algebras of type {2} which satisfy the following equations and quasiequation: (1) (X Y ) ((Y Z) (X Z)) = X X (2) X ((X Y ) Y ) = X X 0 Work partially supported by grant PB from the Spanish DGICYT 73

2 (3) Y (X X) = X X (4) X Y = X X Y X = X X X = Y, where X, Y, Z F m L. It is known that the BCK is not a variety [11]. Next we introduce a Gentzen system G BCK, and we will show in Theorem 7 that it is equivalent to the deductive system BCK. Definition 1. Let L= { } and let Γ, Π be finite sequences of L-formulas and ϕ, ψ, ξ L-formulas. The sequent calculus L BCK is defined by the following axiom and rules: ϕ ϕ (Axiom) Γ ϕ ϕ, Π ξ Γ, Π ξ Γ, ϕ, ψ, Π ξ Γ, ψ, ϕ, Π ξ (cut) (e ) Γ ξ Γ, ϕ ξ (w ) Γ ϕ ψ, Π ξ ϕ ψ, Γ, Π ξ ( ) Γ, ϕ ψ Γ ϕ ψ ( ) This sequent calculus was studied by H. Ono and Y. Komori [8]. They proved the cut elimination theorem and that this sequent calculus is logically equivalent to the BCK-logic, that is, for any formula γ 1,.., γ n, ϕ, the sequent γ 1,.., γ n ϕ is derivable in L BCK if and only if γ 1 (.. (γ n ϕ)..) is a theorem of the BCK-logic. The sequent calculus L BCK allows us to define the consequence relation LBCK on the set of sequents in the following way: Let T be a set of sequents and Γ ϕ a sequent, then T LBCK Γ ϕ iff there is a finite sequence of sequents Γ 0 ϕ 0,..., Γ n 1 ϕ n 1 (which is called a proof of Γ ϕ from T ) such that Γ n 1 ϕ n 1 = Γ ϕ and for each i < n one of the following conditions holds: (i) Γ i ϕ i is an instance of the axiom; (ii) Γ i ϕ i T ; (iii) Γ i ϕ i is obtained from {Γ j ϕ j : j < i} by using a rule of L BCK. 74

3 We define the Gentzen system G BCK as the pair L, LBCK. Definition 2. Let Γ be a finite sequence of L-formulas and ϕ an L- formula. We define a (0, 1)-deductive system S GBCK = L, SGBCK associated to G BCK as the deductive system given by: Γ SGBCK ϕ { γ : γ Γ} LBCK ϕ. Theorem 3. Let τ and ρ be the translations from G BCK to S GBCK and from S GBCK to G BCK respectively, defined by: { p0 (p τ (m,1) (p 0,..., p m 1 q 0 ) = 1 (... (p m 1 q 0 )...)) if m 0 q 0 if m = 0 ρ(p 0 ) = { p 0 }. Then we have: (i) {Γ i γ i : i I} LBCK Γ γ {τ (Γ i γ i ) : i I} SGBCK τ (Γ γ). (ii) ϕ SGBCK τρ(ϕ). Proof. (a) It is easy to check property (i) by using that every sequent is equivalent, modulo G BCK, to a sequent with the empty sequence on the left of the turnstyle. (b) It is easy to check property (ii) by using the definitions of τ and ρ. According to the definition of equivalence between systems given by J. Rebagliato and V. Verdú [10], the theorem shows that this (0, 1)-deductive system S GBCK is equivalent to the Gentzen system G BCK. Now we will show that SGBCK = BCK. To prove this equality we will use the following lemmas. Lemma 4. Let τ be the translation of L-sequents in L-formulas defined in Theorem 3. Then for every rule of L BCK, { j δ j :j 2} δ, we have {τ ( j δ j ) : j 2} BCK τ ( δ) 75

4 Proof. We distinguish cases: Cut rule: we have to show that {τ (Γ ϕ), τ (ϕ, Π ξ)} BCK τ (Γ, Π ξ) (cuth). If Γ = we have to show that {ϕ, ϕ τ (Π ξ)} BCK τ (Π ξ). By using modus ponens, the result follows. If Γ = {γ 1,..., γ n } then τ (Γ, Π ξ) = γ 1 (... (γ n τ (Π ξ))...), τ (ϕ, Π ξ) = ϕ τ (Π ξ), and τ (Γ ϕ) = γ 1 (... (γ n ϕ)...). By applying the property (II) of BCK, we obtain BCK τ (Γ ϕ) (τ (ϕ, Π ξ) τ (Γ, Π ξ)). Now by applying modus ponens {τ (Γ ϕ), τ (ϕ, Π ξ)} BCK τ (Γ, Π ξ). Exchange rule: we have to show that τ (Γ, ϕ, ψ, Π ξ) BCK τ (Γ, ψ, ϕ, Π ξ) (eh). If Γ =, then by using (C) we have: BCK τ(ϕ, ψ, τ(π ξ)) τ(ψ, ϕ, τ(π ξ)) and applying modus ponens, the result follows. If Γ, then from the last expression, the property (I) and modus ponens we obtain τ (Γ, ϕ, ψ, Π ξ) BCK τ (Γ, ψ, ϕ, Π ξ). Weakening rule: we have to show that {τ (Γ ξ)} BCK τ (Γ, ϕ ξ) (wh). If Γ =, then by using (K) we have: BCK ξ (ϕ ξ) and by applying modus ponens, we obtain {τ ( ξ)} BCK τ (ϕ ξ). If Γ, then from the last expression and property (I) we obtain {τ (Γ ξ)} BCK τ (Γ, ϕ ξ). Left introduction rule for : we have to show that {τ (Γ ϕ), τ (ψ, Π ξ)} BCK τ (ϕ ψ, Γ, Π ξ) ( H). From property (I) we have: BCK (ψ τ(π ξ)) ((ϕ ψ) (ϕ τ(π ξ))) and by applying modus ponens, we obtain τ (ψ, Π ξ) BCK τ (ϕ ψ, ϕ, Π ξ). Now by using (eh) we obtain τ (ψ, Π ξ) BCK τ (ϕ, ϕ ψ, Π ξ) and by applying (cuth): {τ (Γ ϕ), τ (ϕ, ϕ ψ, Π ξ)} BCK τ (Γ, ϕ ψ, Π ξ), we obtain {τ (Γ ϕ), τ (ψ, Π ξ)} BCK τ (Γ, ϕ ψ, Π ξ). By using (eh), the result ( H) follows. Right introduction rule for : we have to show that {τ (Γ, ϕ ψ)} BCK τ (Γ ϕ ψ) (H ). And this is true from the definition of τ. 76

5 Lemma 5. Let τ be the translation of L-sequents in L-formulas defined in Theorem 3 and let Σ i, Σ be finite sequences of L-formulas and σ i, σ L-formulas, then {Σ i σ i : i I} LBCK Σ σ {τ (Σ i σ i ) : i I} BCK τ (Σ σ) Proof. We use induction on the length of the proof. n = 1 i. If Σ σ is any of Σ i σ i then it is evident. ii. If Σ σ is the axiom, then we have to prove that BCK τ (ϕ ϕ), that is, BCK ϕ ϕ, and this is true in BCK. n > 1 If Σ σ is obtained by applying a rule { j δ j : j 2} Σ σ of L BCK, then using induction hypothesis we have {τ (Σ i σ i ) : i I} BCK τ ( j δ j ) for all j 2 and applying Lemma 4 we obtain {τ (Σ i σ i ) : i I} BCK τ (Σ σ). Now we are ready to prove the equality mentioned above. Theorem 6. SGBCK = BCK. Proof. ): Suppose Γ BCK ξ. We will prove that Γ SGBCK ξ by induction on the length of the proof. n = 1 i. If ξ Γ then it is trivial. ii. If ξ is an instance of three axioms called B, C and K, then by [7, corollary 2.8.] we obtain that LBCK ξ, and applying the definition of S GBCK we obtain SGBCK ξ, so Γ SGBCK ξ. n > 1 If ξ is obtained by modus ponens from the previous formulas ψ and ψ ξ, we have by the induction hypothesis Γ SGBCK ψ and Γ SGBCK ψ ξ. Finally, from {ψ, ψ ξ} SGBCK ξ, we obtain Γ SGBCK ξ. ): We suppose that Γ SGBCK ξ, that is, { γ : γ Γ} LBCK ξ, then using Lemma 5 we have {τ ( γ) : γ Γ} BCK τ ( ξ), so Γ BCK ξ. Now we are ready to prove the main result of this paper. 77

6 Theorem 7. G BCK and BCK are equivalent, that is, the translations τ from G BCK to BCK and ρ from BCK to G BCK defined in Theorem 3 satisfy the following properties: (i) {Γ i γ i : i I} LBCK Γ γ {τ (Γ i γ i ) : i I} BCK τ (Γ γ). (ii) ϕ BCK τρ(ϕ). Proof. It is straightforward to check these properties from Theorems 3 and 6. Corollary 8. G BCK is algebraizable with equivalent quasivariety semantics BCK, with the translations τ from G BCK in BCK and ρ from BCK in G BCK defined by: { p0 (.. (p τ (m,1) (p 0,.., p m 1 q 0 ) = m 1 q 0 )..) q 0 q 0 if m 0 q 0 q 0 q 0 if m = 0 ρ(p 0 p 1 ) = {p 0 p 1, p 1 p 0 }. Proof. It follows from the theorem and the fact that the quasivariety BCK is an equivalent algebraic semantics of BCK, with defining equation X = X X and equivalence formulas {X Y, Y X} (see [3, theorem 5.11]). Corollary 9. G BCK does not have the deduction-detachment theorem (DDT). Proof. Suppose that G BCK has the DDT. Then, as G BCK is equivalent to BCK, it is easy to see that BCK has the DDT, contradicting the already known fact [4, example 7.3] that BCK does not have the DDT. G BCK has the following local deduction-detachment the- Corollary 10. orem (LDDT) T ; (Γ ϕ) LBCK Π ξ T LBCK E i (τ (Γ ϕ), τ (Π ξ)) for some i < ω, where E i (τ (Γ ϕ), τ (Π ξ)) = τ (Γ ϕ),..., τ (Γ ϕ) τ (Π ξ). }{{} i Proof. It is easy to see from the theorem and the fact that BCK has the LDDT [6, example 2.1], that G BCK has this LDDT. 78

7 References [1] R. J. Adillon and V. Verdú, Sistema de Gentzen algebrizables para las lógicas de Lambek, Girard, Relevancia y BCK, [in:] E. Bustos, J. Echeverría, E. Pérez Sedeño, M. I. Sánchez Balmaseda,(eds) Actas del I Congreso de la Sociedad de Lógica, Metodología y Filosofia de la Ciencia (Madrid 1993), pp [2] A. Avron, Gentzen-type system, resolution and tableaux, Journal of Automated Reasoning 10, (1993), pp [3] W. J. Blok and D. Pigozzi, Algebraizable logics, Memoirs of the American Mathematical Society 396 (1989). [4] W. J. Blok and D. Pigozzi, The Deduction Theorem in Algebraic Logics. Preprint, [5] W. J. Blok and D. Pigozzi, Abstract Algebraic Logic, A course given at the Summer School on Algebraic Logic and the Methodology of Applying it, Budapest, July Preprint, October 5, [6] W. J. Blok and D. Pigozzi, Local Deduction Theorems in Algebraic Logics, [in:] H. Andréka, J. D. Monk and I. Németi (eds) Algebraic Logic, Colloq. Math. Soc. János Bolyai, vol. 54, North-Holland (1991), pp [7] K. Iséki and S. Tanaka, An introduction to the theory of BCKalgebras, Math. Japonica 23 (1978), pp [8] H. Ono and Y. Komori, Logics without the contraction rule, The Journal of Symbolic Logic 50 (1985), pp [9] J. Rebagliato and V. Verdú, On the algebraization of some Gentzen systems, Fundamenta Informaticae, Special Issue: Algebraic Logic and its Applications 18 (1993), pp [10] J. Rebagliato and V. Verdú, Algebraizable Gentzen Systems and the Deduction Theorem for Gentzen Systems, Mathematics Preprint Series No. 175, University of Barcelona, [11] A. Wroński, BCK-algebras do not form a variety, Math. Japonica 28 (1983), pp

8 E.U.Estudis Empresarials Facultat de Matemàtiques Universitat de Barcelona Universitat de Barcelona Diagonal 696 Gran Via Barcelona, Spain Barcelona, Spain riscd2.eco.ub.es cerber.mat.ub.es 80