Logics preserving degrees of truth from varieties of residuated lattices


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1 Corrigendum Logics preserving degrees of truth from varieties of residuated attices FÉLIX BOU and FRANCESC ESTEVA, Artificia Inteigence Research Institute IIIA  CSIC), Beaterra, Spain. Emai: JOSEP MARIA FONT, Department of Probabiity, Logic and Statistics, Facuty of Mathematics, University of Barceona, Spain. Emai: ÀNGEL J. GIL, Departament d Economia i Empresa, Universitat Pompeu Fabra, Barceona, Spain. Emai: LLUÍS GODO, Artificia Inteigence Research Institute IIIA  CSIC), Beaterra, Spain. Emai: ANTONI TORRENS and VENTURA VERDÚ, Department of Probabiity, Logic and Statistics, Facuty of Mathematics, University of Barceona, Spain. Emai: J Logic Computation December 2009; 19: ; doi: /ogcom/exp030 Logics preserving degrees of truth from varieties of residuated attices FÉLIX BOU and JOSEP MARIA FONT, Department of Probabiity, Logic and Statistics, Facuty of Mathematics, University of Barceona. Emai: Abstract A wrong argument in the proof of one of the main resuts in the paper is corrected. The resut itsef remains true. The right proof incorporates the basic ideas in the originay aeged proof, but in a more restricted construction. eywords: substructura ogic, manyvaued ogic, degrees of truth, residuated attices, protoagebraic ogic. Vo. 22 No. 3, The Author, Pubished by Oxford University Press. A rights reserved. For Permissions, pease emai: Pubished onine March 1, 2011 doi: /ogcom/exr003
2 662 Corrigendum The proof of the ast impication in Theorem 4.4 of the referenced paper [1] is wrong. At a certain point, it performs a construction on an arbitrary agebra but the properties used in its deveopment impicity assume that the agebra is in fact a residuated attice, which it needs not be. Here, we present a correct proof done by working ony in the formua agebra, foowing the same ideas and performing essentiay the same construction, moduo a certain crucia emma that characterizes the supremum operation in the attice of theories of one of the ogics considered in [1]. Let us reca the necessary background. Let be an arbitrary variety of commutative, integra residuated attices. The paper [1] studies two finitary ogics associated with each such, which are denoted by their consequence reations: The first one, denoted by,isthetruthpreserving ogic determined by the agebras in when their maximum 1 which is aso the unit of the monoid structure of the fusion operation ) is taken as representing truth; the second one, denoted by =,istheogic preserving degrees of truth determined by the ordering reation of the agebras in which are aways attices). More precisey, for any n 1 and any formuas ψ,ϕ 0,...,ϕ n 1 : ψ A, v HomFm,A), vψ)=1. ϕ 0,...,ϕ n 1 ψ A, v HomFm,A), if vϕ i )=1 for a i <n, then vψ)=1. = ψ A, v HomFm,A), vψ)=1. ϕ 0,...,ϕ n 1 = ψ A, v HomFm,A), a A, if vϕ i ) a for a i <n, then vψ) a. The formua agebra is denoted by Fm, and its universe i.e. the set of a formuas) by Fm. The paper [1] focuses on the essknown ogic =. The ogic is the one customariy associated with, and has been extensivey studied, see [3]; in particuar, it is finitey and reguary agebraizabe, having as its argest equivaent agebraic semantics. The properties needed here that aready appear in [1] are summarized beow. 1) The two ogics have the same theorems, and the ogic is an extension of the ogic = by either of the foowing rues: Modus Ponens for α,α β β); Adjunction for α,β α β); or Squaring α α 2 ); the exponentia notation denotes iteration of the operation. Hence, every theory of is in particuar a theory of =. 2) For a α,β Fm, α β = α β α = β. The second expression means that A, v HomFm,A), vα) vβ). 3) The ogic satisfies the Loca DeductionDetachment Theorem: For a Γ {α,β} Fm,Γ,α β if and ony if there is some n ω such that Γ α n β. 4) The Leibniz congruence Ω T of a theory T of is defined from T with the hep of the equivaence connective as foows: for any α,β Fm, α,β Ω T α β T. 5) For any theory T of = there is a theory T + of such that T + T and Ω T + =Ω T. Moreover, we need the foowing properties, not expicity mentioned in [1]: 6) For a α Fm, α 1) α and 1 α) α. Reca that 1 is a constant term which is a theorem of both ogics. 7) For a α 1,...,α n Fm,α 1 α n α i for each i {1,...,n}.
3 8) For a α 1,...,α n Fm and a k, α k 1 α k n) α1 α n ) k. 9) For a α,β Fm,if α β then α k β k for a k. 10) For a α,α,β,β Fm,if α β and α β, then α α ) β β ). 11) For a α,β Fm, α β) β α). 12) For a α,β Fm, α β β α) ). Corrigendum 663 A these properties are easiy shown using the first equivaence in 2) and the corresponding properties of the ordering reation and the operations and in commutative, integra residuated attices; for instance, 10) foows from the monotonicity of with respect to order, 11) corresponds to its commutative character, and so on. Lemma. Let S and T be two theories of. Then, the smaest theory of which contains S T is the set { ϕ Fm: γ S, δ T such that γ δ) ϕ }. Proof. Let us denote the dispayed set by L. That this set contains both S and T is a consequence of 6). In order to show that L is cosed under the rues of, et us assume that ϕ 1,...,ϕ n ϕ and that ϕ 1,...,ϕ n L; we have to prove that ϕ L. Using 7) one can see that the assumption impies that ϕ 1 ϕ n ϕ. Now, by 3), there is some ω such that ϕ 1... ϕ n ) ϕ, and by 8) this impies that ϕ 1 ϕ n) ϕ. Since for every i {1,...,n},ϕ i L, there are formuas γ i S and δ i T such that γ i δ i ) ϕ i. Using 9), 8), 10) and 11) in succession we obtain γ 1... γ n δ 1... δ n) ϕ 1... ϕ n). We have aready seen that ϕ1... ϕ n ) ϕ, therefore γ 1... γ n δ 1... δ n) ϕ. By one of the properties in 1), the theories of are cosed under fusion, hence γ1... γ n S and δ1... δ n T. Thus, by definition, ϕ L. This shows that L is a theory of. Finay, L is the smaest such theory containing S T:IfT is a theory of such that S T T, and ϕ L, by definition there are γ S and δ T such that γ δ) ϕ. These facts impy that γ,δ T and aso that γ δ) ϕ T. Since by 1) the theories of are cosed under fusion and Modus Ponens, we obtain first that γ δ T, and then that ϕ T. This coses the proof. Observe that, by 2), the property that γ δ) ϕ can be equivaenty stated as γ δ = ϕ,so that, even if it is written using, it tes us something about the theories of =. It is in this form that it wi be used in a crucia step in the next proof. We can now give the right proof that corrects the origina one in [1]. This proof uses some standard facts on agebraizabiity and protoagebraicity that can be found in [2]. Theorem 4.4 of [1]. Let be a variety of residuated attices. Then the foowing conditions are equivaent: 1. The ogic = is protoagebraic.
4 664 Corrigendum 2. For a A and a F Fi = A), F + =max{s Fi A):G F}. 3. A matrix A,F is a reduced mode of = if and ony if A and F is a attice fiter of A such that {1} is the ony impicative fiter of A contained in F. 4. For every A and every F,G Fi = A), if G F and Ω A F =Id then Ω A G=Id. Proof of 4 1. We are going to show that the Leibniz operator Ω is monotonic over the theories of = ; Theorem of [2] guarantees that this amounts to = being protoagebraic. Let Fm be the formua agebra and et S,T be two theories of = with S T; we have to prove that Ω S Ω T. It is easy to see that, using 5), we can repace S with S + to the same effect; therefore, it wi be simper to assume, without oss of generaity, that S is a theory of. Now, using 5) again, we can consider the theory T + of associated with T, and denote by L the smaest theory of that contains S T +. Caim: T + L T. Proof of the caim. It is cear that T + L, so we have to prove that L T. Using the previous Lemma, it is enough to prove that if γ S, δ T + and γ δ) ϕ, then ϕ T. Since T + is a theory of, by 12) and Modus Ponens it foows that γ γ δ) T +. Now by 4) this tes us that γ,γ δ Ω T + =Ω T. But γ S T, so by the compatibiity of Ω T with T we get that γ δ T. Finay, by 2), the assumption that γ δ) ϕ can be equivaenty stated as γ δ = ϕ, and this aows us to concude that ϕ T. This competes the proof of the caim. By agebraizabiity of, the Leibniz operator Ω is monotonic over its theories. Since both T + and L are theories of, from the caim it foows that Ω T =Ω T + Ω L; hence, Ω T + is compatibe with L, and it is aso compatibe with T by definition; so, it makes sense to factor out L and T by Ω T + and we wi obtain fiters of the respective ogics on the quotient agebra. Put A:=Fm/Ω T +, L :=L/Ω T + and T :=T/Ω T +. Then L is a fiter of and hence by 1) a fiter of =, whie T is a fiter of = ; thus, L, T Fi = A), and by the caim, L T. Since is the argest equivaent agebraic semantics of, we know that A. And since Ω T =Ω T + we know that Ω A T =Ω T/Ω T + =Id. Thus, a the conditions required in item 4 of the present theorem are satisfied, and the assumption impies that Ω A L =Id. But since Ω A L =Ω L/Ω T +, it foows that Ω T + =Ω L. Agebraizabiity of impies that Ω is onetoone over its theories, therefore T + =L, which is the same as S T +. Since both are theories of, we can use monotonicity again and obtain that Ω S Ω T + =Ω T. This competes the proof that Ω is monotonic over the theories of =. Funding Spanish Ministerio de Ciencia e Innovación TIN C0401 to F.B.); EU Eurocores FP006/FFI E/FILO to F.B.); Spanish Ministerio de Ciencia e Innovación MTM to J.M.F.); Cataan Departament d Innovació, Universitats i Empresa 2009SGR1433 to F.B. and J.M.F.). References [1] F. Bou, F. Esteva, J. M. Font, A. Gi, L. Godo, A. Torrens, and V. Verdú. Logics preserving degrees of truth from varieties of residuated attices. Journa of Logic and Computation, 19, , 2009.
5 Corrigendum 665 [2] J. Czeakowski. Protoagebraic ogics, Vo. 10 of Trends in Logic  Studia Logica Library. uwer Academic Pubishers, [3] N. Gaatos, P. Jipsen, T. owaski, and H. Ono. Residuated attices: an agebraic gimpse at substructura ogics, Vo. 151 of Studies in Logic and the Foundations of Mathematics. Esevier, 2007.
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