Maximality in finite-valued Lukasiewicz logics defined by order filters Marcelo E. Coniglio1, Francesc Esteva2, Joan Gispert3 and Lluis Godo2 1 Dept. of Philosophy IFCH and Centre for Logic, Epistemology and the History of Science, University of Campinas, Brazil coniglio@cle.unicamp.br 2 Artificial Intelligence Research Institute (IIIA) CSIC, Barcelona, Spain {esteva,godo}@iiia.csic.es 3 Dept. of Mathematics and Computer Science, University of Barcelona, Spain jgispertb@ub.edu Abstract In this paper we consider the logics Lin obtained from the (n + 1)valued Lukasiewicz logics Ln+1 by taking the order filter generated by i/n as the set of designated elements. In particular, the conditions of maximality and strong maximality among them are analyzed. We present a very general theorem which provides sufficient conditions for maximality between logics. As a consequence of this theorem it is shown that Lin is maximal w.r.t. CPL whenever n is prime. Concerning strong maximality between the logics Lin (that is, maximality w.r.t. rules instead of axioms), we provide algebraic arguments in order to show that the logics Lin are not strongly maximal w.r.t. CPL, even for n prime. Indeed, in such case, we show there is just one extension between Lin and CPL obtained by adding to Lin a kind of graded explosion rule. Finally, using these results, we show that the logics Lin with n prime and i/n < 1/2 are ideal paraconsistent logics. 1 Introduction In this paper we study the notion of maximality and strong maximality among finite-valued propositional logics. Recall the usual notion of maximality found in the literature: a propositional logic L1, that is a sublogic of another logic L2 (in the sense of inclusionship of their consequence relations over the same signature), is called maximal with respect to L2 if, roughly speaking, L1 extended with any theorem of L2 which is not a theorem of L1, coincides with L2. Similarly, recall the stronger notion of strong maximality following [2, 4, 35]: L1 is called strongly 1 ar X iv :1 80 3. 09 81 5v 2 [ m at h. L O ] 2 A pr 2 01 8 maximal with respect to L2 if, roughly speaking again, L1 extended with a rule of inference valid in L2 but not a valid in L1, coincides with L2. The problem of finding and characterizing maximal sublogics (in both senses) of a given logic has already been addressed in the literature, specially in the context of paraconsistent logics, where being maximal with respect to classical logic is felt as a desirable or ideal feature, c.f. [3, 10]. Indeed, being maximal means that, while still allowing non-trivial inconsistent theories, it retains as much as possible of classical logic. In the present paper we approach the general problem of characterizing maximality (not necessarily for paraconsistent logics) in two different scenarios. The first one considers a very general class of finite-valued logics, those defined by almost arbitrary finite logical matrices. In such a context, we provide a sufficient condition for a logic to be maximal w.r.t. another one with less truthvalues under very general conditions. This result, inspired on the notion of recovery operators from paraconsistent logics, turns out to be very powerful and encompasses many maximality results scattered in the literature. The second scenario considers a particular class of finite-valued logics, the class of n-valued Lukasiewicz logics Ln and their related logics defined by order filters. We show that these logics, for n being prime, are maximal but not strongly maximal with respect to classical logic. Actually, we show that each of these logics can always be uniquely extended with a sort of explosion inference rule such that the obtained logic is the unique one below classical logic, and hence strongly maximal. The paper is structured as follows. After this introduction, we provide in Section 2 a very general condition for a finite matrix logic to be maximal w.r.t. another one with less truth-values, and we analyze in particular the case of 3-valued logics. In the rest of the paper we focus our attention on the class of finite-valued Lukasiewicz logics Lin defined by order filters. In Section 3 we identify which of these logics are maximal with respect to classical logic, while in Section 4 we study their status regarding the property of strong maximality. It is in Section 5 where we fully characterize, by algebraic techniques, conditions of strong maximality. Finally, in Section 6 the question of ideal paraconsistent logics (as introduced in [3]) will be analized in the present framework. Specifically, it will be shown that the logics Lin with n prime and i/n < 1/2 are ideal paraconsistent logics. In addition, the case L13 will be discussed with more detail, and it will be argued that this logic constitutes the 4-valued version of the well-known 3-valued paraconsistent logic J3 (see [20]). We finish in Section 7 with some conclusions and prospects of future research. 2 Maximality and recovery operators Let us recall the usual notion of maximality of a (standard) logic with respect to another: Definition 1. Let L1 and L2 two standard propositional logics defined over the same signature Θ such that L1 is a proper sublogic of L2, i.e. such that 2 `L1 ( `L2 , where `Li denotes the consequence relation of Li (for i = 1, 2). Then, L1 is said to be maximal w.r.t. L2 if, for every formula φ over Θ, if `L2 φ but 0L1 φ, then the logic L + 1 obtained from L1 by adding φ as a theorem, coincides with L2. By L+1 above we mean the logic whose consequence relation is obtained from the one of L1 as follows: for every set of formulas Γ ∪ {ψ} over Θ, Γ `L+1 ψ if Γ, {σ(φ) : σ is a substitution over Θ} `L1 ψ. Remark 1. It should be noticed that, according to the above definition, if L1 is a proper sublogic of L2 such that they validate the same formulas (that is: `L1 φ iff `L2 φ, for every formula φ) then L1 is maximal w.r.t. L2. In this section, for the class of propositional logics induced by finite logical matrices, we will provide a very general sufficient condition for a logic to be maximal w.r.t. another one (see Theorem 1 below), its proof being inspired in the role played by the so-called recovery operators in paraconsistent and adaptive logics. Recall from [12] (see also [11, 10]) the definition of the class of paraconsistent logics called Logics of Formal Inconsistency (LFIs): a given logic, say L, is an LFI if it is paraconsistent w.r.t. some negation, say ¬ (that is, there exist formulas α and β such that β does not follows from {α,¬α} in L). In addition, there is a (primitive of definable) unary connective ◦ in L (called a consistency operator) such that every formula β follows in L from a set of the form{α,¬α, ◦α}.1 If L is an LFI which is sublogic of classical propositional logic (CPL), presented in the same signature of L,2 then the consistency operator ◦ allows to recover CPL inside L by adding additional hypothesis concerning the consistency (or 'classicality', or 'well-behavior') of some formulas. Namely, for every (finite) set Γ ∪ {ψ} of formulas, Γ `CPL ψ iff (∃Λ)[Γ, {◦α : α ∈ Λ} `L ψ], where Λ is a set of formulas. This is what is called a Derivability Adjustment Theorem (DAT). The idea of DATs was proposed by Battens in the context of Adaptive logics, but this technique (as well as the notion of consistency operator) was already used by da Costa for his well-known hierarchy of paraconsistent systems Cn (see [18]). A more interesting DAT (as, for instance, the ones obtained by da Costa) requires that the consistency (or well-behavior) operator ◦ can just be applied to the propositional variables occurring in Γ ∪ {ψ}. This suggests that, given two standard propositional logics L1 and L2 defined over the same signature Θ such that `L1 ⊆ `L2 , a DAT between both logics can be defined in terms of a recovery operator ◦ (generalizing the idea of LFIs): for every (finite) Γ ∪ {ψ}, Γ `L2 ψ iff Γ, {◦p1, . . . , ◦pm} `L1 ψ, 1This is a slightly simplified presentation of the original definition of LFIs. 2In this case, the formulas ◦α take the value 1 for every evaluation in CPL. 3 where {p1, . . . , pm} is the set of propositional variables occurring in Γ ∪ {ψ}. The idea then is that if one of such recovery operators ◦φ can be defined as a family of instances of a theorem φ of L2 which is not derivable in L1, and if this process can be reproduced for any of such formulas φ, then it will follow that L2 is maximal w.r.t. L1. To be more general, a finite recovery set ©(p) of formulas depending only on one variable p will be considered instead of a single formula ◦(p), following the original definition of LFIs. Actually, in Theorem 1 below some sufficient conditions are given in order to define such recovery sets, which will allow us to determine if one logic is maximal w.r.t. another. In what follows, L(Θ) will denote the term algebra generated by a propositional signature Θ from a fixed set P = {pn : n ≥ 1} of propositional variables. If A is an algebra over Θ then the set of homomorphisms from L(Θ) to A will be denoted by Hom(L(Θ),A). Given an algebra A over Θ and a non-empty subset F ⊆ A, the pair 〈A, F 〉 is called a logical matrix [39]. The logic L defined by the matrix 〈A, F 〉 over L(Θ) is given by the following consequence relation: for every set of formulas Γ ∪ {φ} ⊆ L(Θ), Γ `L φ if, for all e ∈ Hom(L(Θ),A), e(ψ) ∈ F for all ψ ∈ Γ implies e(φ) ∈ F. From now on, with no danger of confusion, given a logical matrix 〈A, F 〉 we will write L = 〈A, F 〉 to refer to the corresponding induced logic defined as above. We will also use the term matrix logic to refer a logic defined by a logical matrix. Lemma 1. Let L1 = 〈A1, F1〉 and L2 = 〈A2, F2〉 be two matrix logics defined over a signature Θ such that A2 is a subalgebra of A1 and F2 = F1 ∩A2. Then `L1 ⊆ `L2 , that is: for every Γ ∪ {ψ}, if Γ `L1 ψ then Γ `L2 ψ. Proof. Assume that Γ`L1ψ. Let e ∈ Hom(L(Θ),A2) be an evaluation for L2 such that e[Γ] ⊆ F2. Let ē : L(Θ) → A1 such that ē(φ) = e(φ) for every φ ∈ L(Θ). Then ē ∈ Hom(L(Θ),A1), so ē is an evaluation for L1 such that ē[Γ] ⊆ F1. By hypothesis, ē(ψ) ∈ F1 and so e(ψ) ∈ F1 ∩ A2 = F2. This shows that Γ `L2 ψ. After this previous lemma, we can state the main result on this section. Theorem 1. Let L1 = 〈A1, F1〉 and L2 = 〈A2, F2〉 be two distinct finite matrix logics over a same signature Θ such that A2 is a subalgebra of A1 and F2 = F1 ∩A2. Assume the following: 1. A1 = {0, 1, a1, . . . , ak, ak+1, . . . , an} and A2 = {0, 1, a1, . . . , ak} are finite such that 0 6∈ F1, 1 ∈ F2 and {0, 1} is a subalgebra of A2. 2. There are formulas >(p) and ⊥(p) in L(Θ) depending at most on one variable p such that e(>(p)) = 1 and e(⊥(p)) = 0, for every evaluation e for L1. 3. For every k+ 1 ≤ i ≤ n and 1 ≤ j ≤ n (with i 6= j) there exists a formula αij(p) in L(Θ) depending at most on one variable p such that, for every evaluation e, e(αij(p)) = aj if e(p) = ai. 4 Then, L1 is maximal w.r.t. L2. Proof. Let us begin by observing that the family of evaluations for L1 which take values in A2 for every propositional variable can be identified with the family of evaluations for L2. 3 Notice that, by Lemma 1, `L1 ⊆ `L2 . Suppose that there is some formula φ(p1, . . . , pm) such that `L2 φ but 0L1 φ (otherwise the proof is done, by Remark 1). Then, e(φ) ∈ F2 for every evaluation e ∈ Hom(L(Θ),A2), but there is an homomorphism e0 ∈ Hom(L(Θ),A1) such that e0(φ) 6∈ F1. By the observation at the beginning of the proof (and by considering that F2 ⊆ F1), there exists a propositional variable pi (for 1 ≤ i ≤ m) such that e0(pi) 6∈ A2. Consider now a substitution σ0 such that σ0(p) =  >(p1) if e0(p) = 1,⊥(p1) if e0(p) = 0, pj if e0(p) = aj (for 1 ≤ j ≤ n) and let γ(p1, . . . , pn) = σ0(φ). Observe that some of the variables pj may not appear in γ, but at least one variable pj (with k + 1 ≤ j ≤ n) must occur in γ, by the hypothesis over e0. Now we can state two immediate facts: Fact 1: Given an evaluation e for L1, if e(pj) ∈ A2 for every 1 ≤ j ≤ n then e(γ) ∈ F2. Proof: follows from the observation at the beginning of the proof, and by noting that γ is an instance of a tautology of L2. Fact 2: Given an evaluation e for L1, if e(pj) = aj for 1 ≤ j ≤ n then e(γ) = e0(φ) 6∈ F1. Proof: Observe that, from the hypothesis, it follows that e(σ0(pi)) = e0(pi) for every 1 ≤ i ≤ m. Now, for any propositional variable p, let αjj(p) = p for every 1 ≤ j ≤ n, and let ©(p) be the finite set of formulas ©(p) = {γ(αi1(p), . . . , αin(p)) : k + 1 ≤ i ≤ n}. Let e be an evaluation in L1. Observe the following: (i) If e(p) ∈ A2 then e(αij(p)) ∈ A2 (since A2 is a subalgebra). For each k + 1 ≤ i ≤ n let ei be an evaluation for L1 such that ei(pj) = e(αij(p)), for every 1 ≤ j ≤ n. Then ei(γ) ∈ F2 , by Fact 1. But ei(γ) = e(γ(αi1(p), . . . , αin(p))) and so e(γ(αi1(p), . . . , α i n(p))) ∈ F2 for every k + 1 ≤ i ≤ n. This means that e[©(p)] ⊆ F1 if e(p) ∈ A2. (ii) If e(p) /∈ A2 then e(p) = ai for some k+1 ≤ i ≤ n. From this, e(αij(p)) = aj for all 1 ≤ j ≤ n. Let e′ be an evaluation for L1 such that e′(pj) = aj , 3This fact was used in the proof of Lemma 1. 5 for every 1 ≤ j ≤ n. Then e′(γ) = e(γ(αi1(p), . . . , αin(p))). But, by Fact 2, e′(γ) = e0(φ) /∈ F1 and so e(γ(αi1(p), . . . , αin(p))) /∈ F1. Thus, e[©(p)] 6⊆ F1 if e(p) /∈ A2. Equivalently, e(p) ∈ A2 if e[©(p)] ⊆ F1. From the observations (i) and (ii) it follows that (∗) e[©(p)] ⊆ F1 iff e(p) ∈ A2. Finally, let L+1 be the logic obtained from L1 by adding φ (and all of its instances) as a theorem. As observed above, Γ `L+1 ψ iff Γ, {σ(φ) : σ is a substitution in L(Θ)} `L1 ψ. Fact 3: Let Γ ∪ {ψ} be a finite a set of formulas in L(Θ) depending on the variables p1, . . . , pt. Then (∗∗) Γ `L2 ψ iff Γ,©(p1), . . . ,©(pt) `L1 ψ. Proof: Assume that Γ `L2 ψ and let e ∈ Hom(L(Θ),A1) such that e[Γ ∪⋃t i=1©(pi)] ⊆ F1. By (∗), e(pi) ∈ A2 for every 1 ≤ i ≤ t. Consider now an evaluation ē ∈ Hom(L(Θ),A2) such that ē(p) = e(p) if p ∈ {p1, . . . , pt}, and ē(p) = 0 otherwise. Then ē(β) = e(β) for every β in L(Θ) depending on the variables p1, . . . , pt. Thus, ē[Γ] ⊆ F1 ∩ A2 = F2 whence ē(ψ) ∈ F2, by hypothesis. That is, e(ψ) ∈ F1 and so Γ,©(p1), . . . ,©(pt) `L1 ψ. Conversely, assume that Γ,©(p1), . . . ,©(pt) `L1 ψ and consider an evaluation ē ∈ Hom(L(Θ),A2) such that ē[Γ] ⊆ F2. Define an evaluation e ∈ Hom(L(Θ),A1) such that e(p) = ē(p) for every variable p. Then e(β) = ē(β) for every β in L(Θ) and so e[Γ] ⊆ F1 and also e[©(pi)] ⊆ F1 for every 1 ≤ i ≤ t, by (∗). By hypothesis, e(ψ) ∈ F1 and then ē(ψ) ∈ F1 ∩ A2, that is, ē(ψ) ∈ F2. This shows that Γ `L2 ψ, proving Fact 3. Consider now a finite a set of formulas Γ ∪ {ψ} in L(Θ) depending on the variables p1, . . . , pt. Suppose that Γ `L2 ψ. Then Γ,©(p1), . . . ,©(pt) `L1 ψ, by Fact 3. But the latter implies that Γ, {σ(φ) : σ is a substitution in L(Θ)} `L1 ψ, because each ©(pi) is a set of instances of φ. From this, it follows that Γ `L+1 ψ, by definition of L + 1 . On the other hand, suppose that Γ `L+1 ψ. Given that `L1 ⊆ `L2 (by Lemma 1) and that `L2 φ (by hypothesis) then Γ `L2 ψ, by definition of L+1 . This shows that L+1 coincides with L2 and so L1 is maximal w.r.t. L2. In the next example we show an application of Theorem 1 in order to prove some maximality conditions for two logics related to the well-known 4-valued logic FOUR introduced by Belnap and Dunn [21, 5, 6]. Example 1. Consider Belnap-Dunn's matrix logic BD = 〈M4, {1, B}〉, where M4 = 〈M4,∧,∨,¬〉 is the algebra associated to the logical lattice M4 (see Fig. 1) expanded with the De Morgan negation ¬ defined as: 6 Figure 1: Lattice M4. ¬ 1 0 B B N N 0 1 Much later, De and Omori considered in [19] the expansion BD∼ of BD by adding the strong negation ∼, given by the following table: x ∼x 0 1 N B B N 1 0 On the other hand, before Belnap and Dunn's investigations, L. Monteiro already considered in 1963 (see [33]) the 4-valued algebra M4m obtained from M4 by adding a modal operator  defined as follows:  1 1 B 0 N 0 0 0 This led to A. Monteiro to consider the variety TMA of tetravalent modal algebras, which is the one generated by M4m (cf. [28]). As proven by Font and Rius in [23], the (degree-preserving) logic of TMA is characterized by the matrix logic MB = 〈M4m, {B, 1}〉. Previous to [19] and with a different motivation, Coniglio and Figallo define in [16] the logicM∼B = 〈M∼4m, {B, 1}〉, the expansion ofMB with the strong negation ∼ described above, characterizing the (degreepreserving) logic of the variety generated by M∼4m (which was independently introduced by A. Monteiro in [32] and by G. Moisil in [31].) 7 By using Theorem 1, it is easy to show that both M∼B and BD ∼ are maximal relative to CPL presented in the signature Θ = {∧,∨,¬ ∼,} and Θ′ = {∧,∨,¬ ∼} over the two-element Boolean algebra B2, respectively (where p is equivalent to p and ¬p is equivalent to ∼p). Indeed, observe that B2 (expanded by ∼ and ) is a subalgebra of M∼4m, and >(p) = p ∨ ∼p and ⊥(p) = p ∧ ∼p are as required. Notice that, since there are in M4 just two values besides the 'classical' ones, namely a1 = N and a2 = B, the formulas α 1 2(p) = α 2 1(p) = ∼p are such that e(α12(p)) = B if e(p) = N , e(α 2 1(p)) = N if e(p) = B. Therefore, it follows from Theorem 1 that M∼B is maximal reative to CPL presented over the signature Θ. Similarly, it also follows that BD∼ is maximal relative to CPL presented over the signature Θ′ (the latter corresponding to [19, Theorem 3]).  As an immediate consequence of Theorem 1, it follows that any 3-valued logic which extends CPL and it can express the top and the bottom formulas, is maximal w.r.t. CPL. Corollary 1. Let A1 be an algebra defined over a signature Θ with domain A1 = {0, 1/2, 1}, and consider the matrix logic L1 = 〈A1, F1〉 where 0 6∈ F1 and 1 ∈ F1. Further, let A2 be a subalgebra of A1, with A2 = {0, 1}, and assume that the matrix logic L2 = 〈A2, {1}〉 is a presentation of classical propositional logic CPL over signature Θ such that L2 is distinct from L1. Suppose additionally there are formulas >(p) and ⊥(p) in L(Θ) on one variable p such that e(>(p)) = 1 and e(⊥(p)) = 0, for every evaluation e for L1. Then, L1 is maximal w.r.t. CPL (presented as L2). Proof. Observe that L1 and L2 are matrix logics as in Lemma 1, since {1} = F1∩A2. Given that A1 contains just one element out of {0, 1}, namely a1 = 1/2, then Theorem 1 can be applied (since requirement (3) is satisfied by vacuity). As a consequence of Theorem 1, L1 is maximal w.r.t. CPL (presented as L2). In the next example some instances of Corollary 1 are analyzed, showing the strength of this result: indeed, several well-known 3-valued logics which are known to be maximal w.r.t. CPL fall inside the scope of Corollary 1. Example 2. (1) Let us begin with Lukasiewicz 3-valued logic L3 = 〈 LV3, {1}〉, where LV3 is the usual 3-valued algebra for L3 over Θ = {¬,→} with domain {0, 1/2, 1}. Let L11 = 〈B2, F 〉 be a presentation of CPL, where B2 is the twoelement Boolean algebra over Θ with domain {0, 1} and F = {1}. It is easy to see that L3 satisfies the requirements of Corollary 1 by taking >(p) = (p → p) and ⊥(p) = ¬(p→ p). This produces a new proof of the maximality of L3 w.r.t. CPL. In order to illustrate this fact consider by instance φ(p1) = p1 ∨ ¬p1 := (p1 → ¬p1) → ¬p1, a formula which is valid in CPL but it is not valid in L3. Indeed, any evaluation e0 in L3 where e0(p1) = 1/2 is such that e0(φ) = 1/2, a non-designated truth-value. By following the construction described in the proof of Theorem 1 (where α11(p) = p), it follows that γ(p1) = φ(p1), and so ◦(p) = p ∨ ¬p is a recovery operator for L3 w.r.t. CPL defined in terms of φ. 8 Thus, L3 plus φ coincides with CPL. Notice that the truth-table of the recovery operator ◦ is as follows: ◦ 1 1 1/2 1/2 0 1 (2) Consider now the logic L12 = 〈 LV3, {1, 1/2}〉. As it is well known, the matrices of L3 are functionally equivalent to that of the 3-valued paraconsistent logic J3, introduced by da Costa and D'Ottaviano, see [20]. This means that L 1 2 coincides with J3 up to language. By item (1) and Corollary 1 it follows that L12 is maximal w.r.t. CPL. This constitutes a new proof of the maximality of J3 (and all of its alternative presentations, such as LFI1 or MPT, see [17]) w.r.t. CPL. A generalization of J3 to L4, called J4, will be proposed in Subsection 6.2. As an illustration of how the technique of the proof works, let φ(p1) = ¬((¬p1 → p1) ∧ (p1 → ¬p1)). It is easy to see that φ(p1) is valid in CPL but it is not valid in L12: any evaluation e0 in L 1 2 with e0(p1) = 1/2 is such that e0(φ) = 0. Then, by the proof of Theorem 1 (where α 1 1(p) = p), γ(p1) = φ(p1) and so ◦(p) = ¬((¬p→ p) ∧ (p→ ¬p)) is a recovery operator for L12 w.r.t. CPL defined in terms of instances of φ. This means that L12 plus φ coincides with CPL. The truth-table of ◦ is as follows: ◦ 1 1 1/2 0 0 1 (3) In an unpublished draft, J. Marcos [29] (see also [11, Section 5.3]) proposes a family of 8,192 logics which are 3-valued and paraconsistent, belonging to the hierarchy of LFIs. Among these logics, there is J3 (whose truth-tables can define the matrices of all the other logics in the family) and Sette's logic P1 (see [37]), whose truth-tables are definable by the matrices of any of the logics in the family. All these logics are maximal w.r.t. CPL presented in the signature {∧,∨,→,¬, ◦} such that ◦φ is valid for every φ (that is, algebraically, ◦(x) = 1 for all x ∈ {0, 1}). The proof of maximality of all these logics w.r.t. CPL follows easily from Corollary 1 by taking >(p) = p→ p and ⊥(p) = p ∧ ¬p ∧ ◦p. (4) Let I1 be the 3-valued paracomplete logic introduced by A.M. Sette and W.A. Carnielli in [38]. It is defined over Θ = {→,¬} with domain {0, 1/2, 1} and designated value 1, and whose operations are given by the tables below. → 1 1/2 0 1 1 0 0 1/2 1 1 1 0 1 1 1 ¬ 1 0 1/2 0 0 1 Once again, the maximality of I1 w.r.t. CPL follows from Corollary 1 by taking >(p) = p→ p and ⊥(p) = ¬(p→ p). 9 (5) Let Gn+1 = 〈Gn+1, {1}〉 be the (n+ 1)-valued Gödel logic defined over the algebra Gn+1 for Θ = {∧,∨,→,¬} with domain { 0, 1n , . . . , n−1 n , 1 } such that x ∧ y = min{x, y}; x ∨ y = max{x, y}; x → y = 1 if x ≤ y and x → y = y otherwise; and ¬x = 1 if x = 0 and ¬x = 0 otherwise. In particular, G3 is defined over {0, 1/2, 1} with the following tables for → and ¬: → 1 1/2 0 1 1 1/2 0 1/2 1 1 0 0 1 1 1 ¬ 1 0 1/2 0 0 1 Clearly G3 falls within the scope of Corollary 1 (where >(p) = p → p and ⊥(p) = p∧¬p) and so it is maximal w.r.t. CPL presented over Θ. Observe that for n ≥ 3 the algebra Gn+1 does not have enough expressive power to define all the formulas αij in order to apply Theorem 1. For instance, in G4 there are no formulas α12(p) and α 2 1(p) such that e(α 1 2(p)) = 2/3 if e(p) = 1/3 and e(α21(p)) = 1/3 if e(p) = 2/3.  Example 3. In [3] the authors introduced the notion of ideal paraconsistent logics. Together with this, they presented a family Mn+2 of (n + 2)-valued matrix logics (with n ≥ 2) which are ideal paraconsistent (and so, from the very definition, they are also maximal w.r.t. CPL, see Definition 4 in Section 6). The fact that all these logics are maximal w.r.t. CPL (as proved in [3]) can also be proved by applying Theorem 1, as it will be shown in what follows. Given n ≥ 2 consider the algebras An+2 over the signature Θ = {¬, ,⊃} with domain An+2 = {0, 1, a1, . . . , an} such that the operations are defined as follows: ¬0 = 1, ¬1 = 0 and ¬x = x otherwise; 0 = 1, 1 = 0, ai = ai+1 if i < n and an = a1; x ⊃ y = 1 if x /∈ D = {1, a1} and x ⊃ y = y otherwise. The logic Mn+2 is defined by the logical matrix 〈An+2, D〉 for every n ≥ 2. Let us see that the conditions of Theorem 1 are satisfied for every logic Mn+2 w.r.t. CPL. It is easy to see that {0, 1} is a subalgebra of An+2 and so, by Lemma 1, Mn+2 is a sublogic of CPL presented in the signature Θ in which  coincides with negation and 1 is the designated value. In addition, it is easy to see that, given a propositional variable p, the formulas >(p) = (p ⊃ p) ⊃ (p ⊃ p) and ⊥(p) = ¬>(p) are such that e(>(p)) = 1 and e(⊥(p)) = 0, for every evaluation e. Consider now the formulas αij(p) = j−ip if i < j and αij(p) = n−i+jp if i > j, where 0p = p and i+1p = i p, for every i. An easy computation shows that e(αij(p)) = aj if e(p) = ai, for every i 6= j. Therefore, the conditions of Theorem 1 are fullfilled and so each logic Mn+2 is maximal w.r.t. CPL. The question of ideal paraconsistent logics in the present framework will be treated again in Section 6.  The examples given above show the value of Theorem 1 in order to state maximality of logics under certain hypothesis concerning the expressive power of the given logics. Indeed, several proofs of maximality found in the literature can be easily obtained as a consequence of Theorem 1: for instance, the ones given for the 3-valued paraconsistent logic P1 in [37, Proposition 11], for the 3-valued 10 logic I1 in [38, Proposition 17] and for J3 (formulated as the equivalent logic LFI1) in [13, Theorem 4.6], respectively. It is worth noting that all the examples of maximality of a logic L1 w.r.t. another logic L2 given in this section, as well as the examples to be given in the rest of the paper, are non-vacuous in the sense of Remark 1. Indeed, in all the examples of maximality presented here the set of theorems of L1 is strictly contained in the set of theorems of L2, thus the notion of maximality holds in a non-trivial way. For instance, the formula p→ ◦p is a theorem of CPL which does not hold in any of the logics presented in Example 2(3), while the formula p→ ¬  p is a theorem of CPL which does not hold in any of the systems Mn+2 presented in Example 3. On the other hand, it should be observed that the set of designated values may not play a relevant role with respect to maximality, for instance, when analyzing maximality with respect to CPL (recall e.g. Corollary 1 or see Proposition 2 in next section). As observed above, Theorem 1 cannot be applied to logics which do not have enough expressive power, as seen in Examples 2(5) for Gödel's logics Gn (with n ≥ 4). This is not the case for finite-valued Lukasiewicz logics, as it will be shown in the next section. 3 Maximality between finite-valued Lukasiewicz logics induced by order filters In the rest of the paper we will deal with matrix logics based on the family of finite-valued Lukasiewicz logics Ln with n ≥ 2. The (n+ 1)-valued Lukasiewicz logic can be semantically defined as the matrix logic Ln+1 = 〈 LVn+1, {1}〉, where LVn+1 = ( LVn+1,¬,→) with LVn+1 = { 0, 1n , . . . , n−1 n , 1 } , and the operations are defined as follows: for every x, y ∈ LVn+1, ¬x = 1− x ( Lukasiewicz negation) x→ y = min{1, 1− x+ y} ( Lukasiewicz implication) The following operations can be defined in every algebra LVn+1: x⊗ y = ¬(x→ ¬y) = max{0, x+ y − 1} (strong conjunction) x⊕ y = ¬x→ y = min{1, x+ y} (strong disjunction) x ∨ y = (x→ y)→ y = max{x, y} (lattice disjunction) x ∧ y = ¬((¬x→ ¬y)→ ¬y) = min{x, y} (lattice conjunction) Observe that L2 is the usual presentation of classical propositional logic CPL as a matrix logic over the two-element Boolean algebra B2 with domain {0, 1} with signature {¬,→}. 11 The logics Ln can also be presented as Hilbert calculi that are axiomatic extensions of the infinite-valued Lukasiewicz logic L∞. Recall that L∞ is algebraizable and the class MV of all MV-algebras is its equivalent quasivariety semantics [36, 14]. Since algebraizability is preserved by finitary extensions then every finite valued Lukasiewicz logic Ln is also algebraizable, and we will denote by MVn its corresponding subvariety of algebras. In this section, finite-valued Lukasiewicz logics with a set of designated values possibly different to {1} will be studied from the point of view of maximality. First, some notation will be introduced. For every i ≥ 1 and for every x ∈ LVn+1, ix will stand for x ⊕ * * * ⊕ x (i-times), while xi will stand for x⊗ * * * ⊗ x (i-times). For 1 ≤ i ≤ n let Fi/n = {x ∈ LVn+1 : x ≥ i/n} = { i n , . . . , n− 1 n , 1 } be the order filter generated by i/n, and let Lin = 〈 LVn+1, Fi/n〉 be the corresponding matrix logic. From now on, the consequence relation of Lin is denoted by Lin . Observe that Ln+1 = L n n for every n. In particular, CPL is L11 (that is, L2). If 1 ≤ i,m ≤ n, we can also consider the following matrix logic: Li/nm = 〈 LVm+1, Fi/n ∩ LVm+1〉. Since Fi/n ∩ LVm+1 = Fj/m for some 1 ≤ j ≤ m, L i/n m = Ljm for that j. It is interesting to notice that some of these logics are paraconsistent, and some are not. Indeed, it is easy to prove the following characterization. Proposition 1. The logic Lin is paraconsistent w.r.t. ¬ iff i/n ≤ 1/2. Proof. Lin is paraconsistent w.r.t. ¬ iff there exists x ∈ LVn+1 such that x ≥ i/n and ¬x ≥ i/n, iff i/n ≤ x ≤ (n− i)/n for some x ∈ LVn+1, iff i/n ≤ (n− i)/n, iff 2i ≤ n. Thus, for instance, for n = 5 it follows that L15 and L 2 5 are paraconsistent, while L35, L 4 5 and L 5 5 = L6 are explosive. By its turn, if n = 3 then L 1 3 is paraconsistent, while L23 and L 3 3 = L4 are explosive. The paraconsistent logics of this form which are maximal w.r.t. CPL will be analyzed with more detail in Section 6. Theorem 1 can be used in order to analyze the maximality of the logic Lin w.r.t. L i/n m whenever m|n (taking into account that LVm+1 is a subalgebra of LVn+1 iff m|n). In particular, the maximality of certain instances of Lin w.r.t. CPL can be obtained by using Theorem 1. The following examples deal with the algebras LVn which, as observed above, can define a meet operator ∧ such that, for any order filter F , (a∧b) ∈ F iff a, b ∈ F . Because of this, a recovery operator ◦(p) will be considered instead of a recovery set ©(p), consisting of the conjunction of all of its members. 12 Example 4. Let us first consider the case of LV4. For 1 ≤ i ≤ 3 let Li3 = 〈 LV4, Fi/3〉. Then F1/3 = {1/3, 2/3, 1}, F2/3 = {2/3, 1} and F3/3 = F1 = {1}. As in the previous example, it can be proved that each Li3 satisfies the requirements of Theorem 1 w.r.t. CPL and so each Li3 is maximal w.r.t. CPL, presented as CPL = 〈B2, {1}〉. Indeed, B2 is a subalgebra of LV4 and >(p) = (p → p) and ⊥(p) = ¬(p → p) are as required. Finally, the formulas α12(p) = α21(p) = ¬p are such that e(α12(p)) = 2/3 if e(p) = 1/3, e(α21(p)) = 1/3 if e(p) = 2/3. (Observe that there are in LV4 just two 'non-classical' values: a1 = 1/3 and a2 = 2/3.) Fix 1 ≤ i ≤ 3. Thus, given a theorem φ(p1, . . . , pm) of CPL which is not valid in Li3, consider the formula γ(p1, p2) as in the proof of Theorem 1. Then, the formula ◦(p) = γ(p,¬p)∧γ(¬p, p) defines an operator (in terms of a conjunction of instances of φ) which allows to recover classical logic inside Li3.  From Komori's characterization of axiomatic extensions of (infinite-valued) Lukasiewicz logic L∞ [27], it directly follows that the logic Ln+1 is maximal w.r.t. CPL iff n is a prime number. By adapting our previous arguments, we can obtain the following extension of this classical result for logics matrix logics over LVn+1 with (almost) arbitrary filters. Proposition 2. Let n ≥ 2 and ∅ 6= F ⊆ LVn+1. Then, the logic L = 〈 LVn+1, F 〉 is maximal w.r.t. CPL provided that 0 /∈ F and n is a prime number. Observe that, as a direct consequence, all the logics Liq with q prime are maximal w.r.t. classical logic. Corollary 2. Let q be a prime number, and 1 ≤ i ≤ q. Then, the logic Liq is maximal w.r.t. CPL. Remark 2. Note that, for a given prime q, if i < j the set of theorems of Ljq is strictly included in the set of theorems of Liq. However this does not contradict the fact that both logics are maximal w.r.t. CPL, since their consequence relations are in fact incomparable. For example, the set of theorems of L23 is included in the set of theorems of L13, but the inclusion is strict: |=L13 (p ∨ ¬p)⊗ (p ∨ ¬p) while 6|=L23 (p ∨ ¬p)⊗ (p ∨ ¬p). It suffices to consider an evaluation e such that e(p) = 1/3; then e((p∨¬p)⊗(p∨¬p)) = 1/3 6≥ 2/3. On the other hand, L23 is not a sublogic of L13: p |=L23 (p⊗p)⊕(p⊗p) but p 6|=L13 (p⊗p)⊕(p⊗p). In order to see this, consider an evaluation e such that e(p) = 1/3; then e((p⊗p)⊕(p⊗p)) = 0. Next examples exploit the fact that Ln+1 is a sublogic of Lm+1 iff m divides n, considering additional filters as designated values, and obtaining maximality in some cases. Example 5. Now, the logics asociated to the algebra LV5 will be analyzed. For 1 ≤ i ≤ 4 let Li4 = 〈 LV5, Fi/4〉 such that F1/4 = {1/4, 1/2, 3/4, 1}, F2/4 = F1/2 = {1/2, 3/4, 1}, F3/4 = {3/4, 1}, and F4/4 = F1 = {1}. Since 2 divides 4 then LV3 is a subalgebra of LV5 and L5 is a sublogic of L3. We will prove that, indeed, any Li4 (for 1 ≤ i ≤ 4) is maximal w.r.t. L i/4 2 = 〈 LV3, Fi/4 ∩ LV3〉, by using Theorem 1. 13 By Lemma 1, each Li4 is a sublogic of L i/4 2 . LV3 is a subalgebra of LV5 and >(p) = (p → p) and ⊥(p) = ¬(p → p) are as required. Let a1 = 1/2, a2 = 1/4 and a3 = 3/4, and consider the formulas α 2 1(p) = p⊕p, α23(p) = α32(p) = ¬p, and α31(p) = p⊗ p. Finally, let αii(p) = p for i = 2, 3. Then, the formulas αij defined above are such that e(αij(p)) = aj if e(p) = ai, for i = 2, 3 and j = 1, 2, 3. Fix 1 ≤ i ≤ 4. Thus, given a theorem φi(p1, . . . , pmi) of L i/4 2 which is not valid in Li4, consider the formula γi(p1, p2, p3) as in the proof of Theorem 1. Then, the formula ◦i(p) = γi(p⊕ p, p,¬p) ∧ γi(p⊗ p,¬p, p) defines a recovery operator (in terms of a conjunction of instances of φi) which allows to recover L i/4 2 inside L i 4. This shows that the latter is maximal w.r.t. the former.  Example 6. Consider now the case of LV7. Since 2 and 3 divide 6, it follows that LV3 and LV4 are subalgebras of LV7 and so L = 〈 LV7, F 〉 is a sublogic of both 〈 LV3, F ∩ LV3〉 and 〈 LV4, F ∩ LV4〉 for any non-trivial filter F of LV7. However, it is not possible to prove the maximality of L by applying Theorem 1 since, for every formula α(p) and every evaluation e in LV7, e(α(p)) 6= 1/2 if e(p) ∈ {1/3, 2/3} (since LV4 is a subalgebra), while e(α(p)) /∈ {1/3, 2/3} if e(p) = 1/2 (since LV3 is a subalgebra).  As another example of application of Theorem 1, we can obtain the following maximality condition of a logic Lin with respect to a logic L i/n m . Proposition 3. Let 1 ≤ i,m ≤ n. Then Lin = 〈 LVn+1, Fi/n〉 is maximal w.r.t. L i/n m = 〈 LVm+1, Fi/n ∩ LVm+1〉 if the following condition holds: there is some prime number q and k ≥ 1 such that n = qk, and m = qk−1. Proof. We recall that LVn+1 is singly generated by any element 0 < l n < 1 such that l and n are mutually prime [24, Lemma 1.2]. Then, since q is prime, LVqk+1r LVqk−1+1 = {0 < rqk < 1 : r and q are mutually prime} and therefore all conditions of Theorem 1 are satisfied. 4 On strong maximality and explosion rules in the logics Liq Along this section q will denote a prime number. In the previous section we have seen that all the logics of the form Liq = 〈 LVq+1, Fi/q〉 are maximal w.r.t. CPL. However, there are maximal logics that are not maximal w.r.t. CPL in an stronger sense, as firstly considered in [4, 3] in the context of paraconsistent logics, or in [35] in a more general context of belief revision techniques for change of logics. 14 Definition 2. Let L1 and L2 two standard propositional logics defined over the same signature Θ such that L1 is a proper sublogic of L2, i.e. such that `L1 ( `L2 . Then, L1 is said to be strongly maximal w.r.t. L2 if, for every finitary rule φ1, . . . , φn/ψ over Θ, if φ1, . . . , φn `L2 ψ but φ1, . . . , φn 0L1 ψ, then the logic L∗1 obtained from L1 by adding φ1, . . . , φn/ψ as structural rule, coincides with L2. By L∗1 above we mean the logic whose consequence relation `L∗1 is the minimal extension of `L1 such that σ(φ1), . . . , σ(φn) `L∗1 σ(ψ) for any substitution σ over Θ (see e.g. [39, 3]). For instance, as observed in [19, Remark 14], the logic BD∼ introduced in Section 2, that is maximal w.r.t. CPL, it is not strongly maximal relative to CPL. Thus, a natural question is whether a given logic is strongly maximal w.r.t. another logic. In particular, in this section, we are interested in studying the status of the logics Liq = 〈 LVq+1, Fi/q〉 with q prime in relation to the notion of strong maximality w.r.t. CPL. We will show that the answer is negative, as each of them admits a proper extension by a finitary rule related to the explosion law w.r.t. Lukasiewicz negation. In fact, in Section 5.2 it will be shown that such proper extensions are strongly maximal w.r.t. CPL. Remark 3. By using the techniques presented in [15], a sound and complete Hilbert calculus for each Liq (where i < q) can be defined from the one for L ≤ q+1 (the degree-preserving counterpart of Lq+1) by adding additional inference rules. The negative feature of such approach is that these Hilbert calculi have "global" inference rules, that is, inference rules such that one of its permises need to be a theorem of Lq+1. By a general result by Blok and Pigozzi (see Theorem 4.3 in [8]) and from Theorem 2 in Section 5 below, a standard Hilbert calculus for Liq (for i < q) can be obtained from the usual one for L q q = Lq+1 by means of translations. That is, such calculi have no "global" inference rules. The negative side of this approach is that the resulting axiomatization is obtained by translating connectives from the other logic, and so the resulting calculus can appear as very artificial. As an alternative, it seems that a direct method for defining a sound and complete "more natural" Hilbert calculus for each Liq over a suitable signature can also be obtained by means of a 'separation' technique for the truth-values, similar to the one used in Subection 6.2 to define an alternative axiomatization for L13. To verify this conjecture is left as an open problem. Anyway, from now on it will be assumed the existence of a standard Hilbert calculus Hiq which is sound and complete for the logic L i q, where i < q. Of course Hqq will stand for the usual axiomatization of Lq+1. According to the notation introduced at the beginning of Section 3, iα is an abbreviation for the formula α⊕ * * *⊕α (i-times), and the consequence relation of Lin is denoted by Lin . Recall the following basic property of LVn+1. Lemma 2. For every 1 ≤ i ≤ n and x ∈ LVn+1: ix < i/n iff x = 0. Thus, e(iα) < i/n iff e(α) = 0 for every evaluation e in LVn+1 and every formula α. From now on, ⊥ will denote any formula of the form ¬(p→ p), for a propositional variable p. Observe that e(⊥) = 0 for every evaluation in LVn+1, 15 every n ≥ 1 and every propositional variable p. This is why the choice of p is inessential for a concrete construction of ⊥. Consider, for 1 ≤ i ≤ q, the i-explosion law (expi) i(φ ∧ ¬φ) ⊥ . It is not hard to prove that this rule is not derivable in any Hiq, the sound and complete Hilbert calculus given for the logic Liq (see Remark 3). Corollary 3. For every 1 ≤ i ≤ q, the rule (expi) is not derivable in Hiq. Proof. We first observe that if p, p′ are two different propositional variables, then i(p ∧ ¬p) 6Lin p ′ for 1 ≤ i ≤ n. Indeed, let e be an evaluation in LVn+1 such that e(p) /∈ {0, 1} and e(p′) = 0. Since e(p ∧ ¬p) 6= 0 then e(i(p ∧ ¬p)) ≥ i/n, by Lemma 2. Hence, i(p ∧ ¬p) 6Lin p ′. Finally, the corollary then follows from soundness and completeness of Hiq w.r.t. L i q. However, the i-explosion rule is clearly admissible in Hiq since it is a passive rule, that is: for no instance of the (expi) rule, the premisse can be a theorem of Hiq. Indeed, for any classical evaluation over {0, 1} it is the case that e(φ) ∈ {0, 1} for every formula φ and so e(i(φ ∧ ¬φ)) < i/n, by Lemma 2. This leads us to consider the following definition. Definition 3. H i q is the Hilbert calculus obtained from H i q by adding the iexplosion rule (expi). We will denote by `Hiq and `Hiq the notions of proof associated to the Hilbert calculi Hiq and H i q, respectively. The following is a characterization of ` H i q in terms of `Hiq . Proposition 4. Let Γ ∪ {φ} be a set of formulas. Then Γ ` H i q φ iff either Γ `Hiq φ, or Γ `Hiq i(ψ ∧ ¬ψ) for some formula ψ. Proof. 'Only if' part: Suppose that Γ ` H i q φ such that Γ 0Hiq φ. Then, any derivation in H i q of φ from Γ must use the rule (expi). Let φ1 . . . φn be a derivation in H i q of φ from Γ. Thus, there exists 1 ≤ m < n such that φm = i(ψ ∧ ¬ψ) for some formula ψ, allowing so the first application of (expi) in the given derivation. This means that Γ `Hiq i(ψ ∧ ¬ψ), since it was assumed that (expi) was not applied before φm in the given derivation. 'If' part: Suppose that Γ `Hiq φ. Then, clearly Γ `Hiq φ. Now, suppose that Γ `Hiq i(ψ ∧ ¬ψ) for some formula ψ. Then Γ `Hiq ⊥, by using (expi). But ⊥ Liq φ and so ⊥ `Hiq φ, by completeness of H i q w.r.t. L i q. This means that Γ ` H i q φ. The following question is how to characterize semantically the logic H i q with respect to Liq, the original sematics for H i q. The answer will be obtained in the 16 next section by algebraic arguments (Theorem 7 and Remark 4). Indeed, it will be shown there that H i q is sound and complete w.r.t. L i q where, for every i and n with 1 ≤ i ≤ n, Lin = 〈 LVn+1 × LV2, Fi/n × {1}〉 such that LV2 is the two-element Boolean algebra B2 with domain {0, 1}. 5 Translations, equivalent logics and strong maximality 5.1 Preliminaries Blok and Pigozzi introduce the notion of equivalent deductive systems in [8] (see also [9]). Two propositional deductive systems S1 and S2 in the same language L are equivalent iff there are two translations τ1, τ2 (finite subsets of L-propositional formulas in one variable) such that: • Γ `S1 φ iff τ1(Γ) `S2 τ1(φ), • ∆ `S2 ψ iff τ2(∆) `S1 τ2(ψ), • φ àS1 τ2(τ1(φ)), • ψ àS2 τ1(τ2(ψ)). From very general results stated in [8] it follows that two equivalent logic systems are indistinguishable from the point of view of algebra, provided that one of them is algebraizable. Indeed, in such case if one of the systems is algebraizable then the other will be also algebraizable w.r.t. the same quasivariety. By applying this fact to the systems of the form Lin studied in the previous sections, several results on relative maximality between these systems and classical logic will be obtained in the next Subsection 5.2. Actually, these results will be generalized in Subsection 5.3 to obtain relative maximality results among the systems Lin. However, for the sake of self containment, we prefer to leave the results of Subsection 5.2 with their simpler proofs as well. In the rest of this subsection, we provide the necessary preliminaries that will be needed in the subsequent subsections. We recall that L∞ is algebraizable and the class MV of all MV-algebras is its equivalent quasivariety semantics [36, 14]. Since algebraizability is preserved by finitary extensions then every finite valued Lukasiewicz logic is also algebraizable. Now we can prove that the deductive systems Lin and L j n are in fact equivalent in the above sense. First, observe that, by the McNaughton functional representation theorem [30], for every n ≥ 2 and every 1 ≤ m ≤ n there is an MV-term λm,n(p) such that for every a ∈ [0, 1], λm,n(a) =  0, if a ≤ m−1 n ; na− (m− 1), if m−1n < a < m n ; 1, if mn ≤ a. 17 Lemma 3. The restrictions of the λi,m and λn,n functions on LVn+1 are the characteristic functions of the order filters Fi/n and F1 respectively, i.e. for each a ∈ LVn+1, λi,n(a) = { 0, if a < in 1, if a ≥ in λn,n(a) = a n = { 0, if a < 1 1, if a = 1 Theorem 2. For every n ≥ 2 and every 1 ≤ i, j ≤ n, Lin and Ljn are equivalent deductive systems. Proof. It is enough to prove that for every 1 ≤ i ≤ n − 1, Lin is equivalent to Lnn = Ln+1. Let the translations τ and σ be given by τ = {λi,n(p)} and σ = {λn,n(p)}. It is easy to check that for every set of formulas Γ ∪ {φ}, Γ Lin φ iff {τ(ψ) : ψ ∈ Γ} Lnn τ(φ) Γ Lnn φ iff {σ(ψ) : ψ ∈ Γ} Lin σ(φ) φ Lin σ(τ(φ)) and φ Lnn τ(σ(φ)). Thus, Lin and L n n = Ln+1 are equivalent deductive systems. From the equivalence among Lin and Ln+1, we can obtain, by translating the axiomatization of the finite valued Lukasiewicz logic Ln+1, a calculus sound and complete with respect Lin that we denote by H i n (see [8, Theorem 4.3]). Since L∞ is algebraizable and the class MV of all MV-algebras is its equivalent quasivariety semantics, finitary extensions of L∞ are in 1 to 1 correspondence with quasivarieties of MV-algebras . Actually, there is a dual isomorphism from the lattice of all finitary extensions of L∞ and the lattice of all quasivarieties of MV . Moreover, if we restrict this correspondence to varieties of MV we get the dual isomorphism from the lattice of all varieties of MV and the lattice of all axiomatic extensions of L∞. Since Ln+1 = L n n is an axiomatic extension of L∞, Ln+1 is an algebraizable logic with the class MVn = Q( LVn+1), the quasivariety generated by LVn+1, as its equivalent variety semantics. It follows from the previous theorem and from [8] that Lin, for every 1 ≤ i ≤ n, is also algebraizable with the same class of MVn-algebras as its equivalent variety semantics. Thus, the lattices of all finitary extensions of Lin are isomorphic, and in fact, dually isomorphic to the lattice of all subquasivarieties of MVn, for all 0 < i < n. Therefore maximality conditions in the lattice of finitary (axiomatic) extensions correspond to minimality conditions in the lattice of subquasivarieties (subvarieties). Thus, given two finitary extensions L1 and L2 of a given logic Lin, where KL1 and KL2 are its associated MVn-quasivarieties, L1 is strongly maximal with respect L2 iff KL1 is a minimal subquasivariety of MVn among those MVn-quasivarieties properly containing KL2 . Moreover, if L1 and L2 are axiomatic extensions of Lin, then KL1 and KL2 are indeed MVn-varieties. In that case, L1 is maximal with respect L2 iff KL1 is a minimal subvariety of MVn among those MVn-varieties properly containing KL2 . 18 All the axiomatic extensions of L∞ are characterized by Komori in [27], where it is shown that every axiomatic extension is finitely axiomatizable and depends only on two finite sets of natural numbers I, J not both empty. Moreover, Panti proved in [34] that every axiomatic extension can be axiomatized relative to L∞ by a single axiom γI,J with a single propositional variable. For the case of finite valued Lukasiewicz logics, Komori's characterization depends on just a finite set of natural numbers in the following sense: given n > 1, every axiomatic extension of Ln+1 is of the form⋂ 1≤j≤k Lmj+1 for some natural number k where mj |n for every 1 ≤ j ≤ k. Moreover, from the equivalence of Theorem 2, it follows that every axiomatic extension of Lin is of the form ⋂ 1≤j≤k Li/nmj for some natural number k where mj |n for every 1 ≤ j ≤ k, and it is axiomatized by a single axiom γ i/n m1,...,mk which depends on one variable. We denote by H i/n m1,...,mk the calculus obtained from H i n by adding the axiom γ i/n m1,...,mk . Note that for every m ≥ 1 such that m|n, the calculus Hi/nm is the same logic as Hjm, where j is the natural number such that Fj/m = Fi/n ∩ LVm. The lattice of all axiomatic extensions L∞ is fully described also by Komori in [27], thus from the equivalence of Theorem 2, we can obtain the following maximality conditions for all axiomatic extensions of Lin. Theorem 3. Let 0 < i,m ≤ n be natural numbers such that m|n. If L is an axiomatic extension of Lin, then • L is maximal with respect to Li/nm iff L = Li/nm ∩ Li/nqk+1 for some prime number q with q|n and a natural k ≥ 0 such that qk|m and qk+1 6 |m. Proof. Using the equivalence of Theorem 2 we obtain that the lattice of axiomatic extensions of Lin is isomorphic to the lattice of axiomatic extensions of Ln+1. As mentioned above, every axiomatic extension of Ln+1 is characterized by a finite set {m1, . . . ,mk} where all of its elements are divisors of n. Given two such sets {m1, . . . ,mk} and {n1, . . . , ns}, we define the following relation among finite subsets of divisors of n: {m1, . . . ,mk}  {n1, . . . , ns} iff for every 1 ≤ i ≤ k there is 1 ≤ j ≤ s such that mi|nj . This relation  is the dual order of the lattice of axiomatic extensions of Ln+1 in the following sense: {m1, . . . ,mk}  {n1, . . . , ns} iff ⋂ 1≤j≤s Lnj+1 ≤ ⋂ 1≤i≤k Lmi+1. Clearly, {m}  {m, q} and {m, q} 6 {m} if q is a prime number such that q|n and q 6 |m; Similarly, {m}  {m, qk+1} and {m, qk+1} 6 {m} if q is a prime number such that q|n, qk|m and qk+1 6 |m. Moreover if {m}  {m1, . . . ,mk} and {m1, . . . ,mk} 6 {m}, then there is mi such that m|mi. If m 6= mi then there 19 is a prime number q such that mq|mi|n. Thus {m, q}  {m1, . . . ,mk} if q 6 |m and {m, qk+1}  {m1, . . . ,mk} if qk|m and qk+1 6 |m. If m = mi, then there is an mj with 1 < j < k such that mj 6 |m. If there is a prime number q such that q|mj |n such that q 6 |m, then {m, q}  {m1, . . . ,mk}. Otherwise, there are a prime number q and a natural k > 0 such that qk+1|mj |n, qk|m and qk+1 6 |m, then {m, qk+1}  {m1, . . . ,mk}. Duality and Theorem 2 close the proof. As a corollary we obtain that the suficient condition of Proposition 3 is also necessary. Corollary 4. Let 1 ≤ i,m ≤ n. Then Lin = 〈 LVn+1, Fi/n〉 is maximal w.r.t. L i/n m = 〈 LVm+1, Fi/n∩ LVm+1〉 if and only if there is some prime number q and k ≥ 1 such that n = qk, and m = qk−1. The task of fully describing the lattice of all all finitary extensions of L∞, isomorphic to the lattice of all subquasivarieties of MV , turns to be an heroic task since the class of all MV-algebras is Q-universal (see [1]). For the finite valued case it is much simpler, since MVn is a locally finite discriminator variety (cf. [7, 25]). Any locally finite quasivariety is generated by its critical algebras (see [22]), where an algebra A is said to be critical iff it is a finite algebra not belonging to the quasivariety generated by all its proper subalgebras. A description of all critical MV-algebras can be found in [25]. Theorem 4. [25, Theorem 2.5] An MV-algebra A is critical if and only if A is isomorphic to a finite MV-algebra LVn0+1 × * * * × LVnl−1+1 satisfying the following conditions: 1. For every i, j < l, i 6= j implies ni 6= nj. 2. If there exists nj, j < l such that ni|nj for some i 6= j, then nj is unique. Moreover the following result characterizes the inclusion among locally finite quasivarieties. Lemma 4. [25, Lemma 2.9] Let F = {LVni1+1 × * * * × LVnil(i)+1 : i ∈ I} and G = {LVmj1+1×* * *×LVmjl(j)+1 : j ∈ J} be two finite families of critical MV-algebras. Then it holds that Q(F) ⊆ Q(G) if, and only if, for every i ∈ I there exists a non-empty H ⊆ J such that: 1. For any 1 ≤ k ≤ l(i) there are j ∈ H and 1 ≤ r ≤ l(j) such that nik|mjr. 2. For any j ∈ H and 1 ≤ r ≤ l(j) there exists 1 ≤ k ≤ l(i) such that nik|mjr. 20 5.2 Strong maximality among logics Liq, L i q, and classical logic As a direct application of Lemma 4, we have the following particular case that will be used later. Corollary 5. Consider the following two sets of one critical MV-algebra each: {LVq+1 × LV2} and {LVk+1}, where q is a prime number such that q > 1. Then Q({LVq+1 × LV2}) ⊆ Q(LVk+1) if and only if q|k. Proof. The two families of critical algebras above correspond in Lemma 4 to take I = {1} and J = {1}, with n11 = q, n12 = 1, m11 = k. Then one can check that these values satisfy the two conditions of the lemma only in the case that q|k. Now, for any k > 1, we are able to provide a full description of the minimal subquasivarieties of MVk = Q(LVk+1) strictly containing the variety of Boolean algebras. Theorem 5. Let k > 1. The set of all minimal subquasivarieties of MVk = Q(LVk+1) among those strictly containing the class of all the Boolean algebras B = Q(LV2) is Mk = {Q(LVq+1 × LV2) : q > 1 prime, q|k}. Proof. By Lemma 4 and the previous Corollary 5, every K ∈ Mk is a subquasivariety of Q(LVk+1) strictly containing B. Moreover, for every K1,K2 ∈Mk, if K1 6= K2 then K1 6⊆ K2 and K2 6⊆ K1. On the other hand, let K be a minimal subquasivariety of Q(LVk+1) strictly containing B. Since K 6= B, it must contain a critical algebra C that, by Theorem 4, it must be such that C ∼= LVm1+1 × * * * × LVms+1, where mi|k for every 1 ≤ i ≤ s, and mj > 1 for some 1 ≤ j ≤ s. Hence, for every prime number q such that q|mj , and hence q|k, we have LVq+1 ×LV2 ∈ Q(C) ⊆ K, and thus Q(LVq+1 ×LV2) ⊆ K. Since we are assuming the minimality of K, it must be Q(LVq+1 × LV2) = K. Theorem 6. If q > 0 is a prime number, then Q(LVq+1×LV2) is axiomatized by the MV quasi-identities plus: • γq(x) ≈ 1 (the identity axiomatizing V(LVq+1)) • q(x ∧ ¬x) ≈ 1⇒ y ∨ ¬y ≈ 1 Proof. It is easy to check that LVq+1×LV2 satisfies these two quasi-identities. Since the MV-identities and γq(x) ≈ 1 axiomatize V(LVq+1), and V(LVq+1) is a locally finite quasivariety, it is enough to prove that every critical MV-algebra 21 C ∈ V(LVq+1) where the quasi-equation q(x ∧ ¬x) ≈ 1 ⇒ y ∨ ¬y ≈ 1 holds, belongs to Q(LVq+1 × LV2). Let C be a critical MV-algebra satisfying the axiomatization, then C is such that C ∼= LVm1+1 × * * * × LVmk+1 satisfying conditions of Theorem 4. Moreover, every 1 ≤ i ≤ k, is such that either mi = 1 or mi = q because LVmi+1 belongs to V(LVq+1). If there is c ∈ C such that q(c ∧ ¬c) = 1 then, by the second quasi-equation of the above axiomatization, b ∨ ¬b ≈ 1 for any b ∈ C. Thus we have C ∈ B ⊆ Q(LVq+1 × LV2). Otherwise, recalling that either mi = 1 or mi = q for every i, if for every c ∈ C one has q(c ∧ ¬c) 6= 1 then mi = 1 for some 1 ≤ i ≤ k. In that case, by the characterization of critical algebras (Theorem 4), we have C ∼= LV2 or C ∼= LVq+1 × LV2. If C ∼= LV2, then trivially C ∈ Q(LVq+1 × LV2). If C ∼= LVq+1 × LV2, then clearly C ∈ Q(LVq+1 × LV2). Above, note that the identity y ∨ ¬y ≈ 1 corresponds to the previously mentioned Panti's axiom γI,J(y), with I = {1} and J = ∅, axiomatizing CPL as an axiomatic extension of Ln+1 for any n > 1. Finally, we obtain the following characterization result about strong maximality of logics Ljq with respect to classical logic. Theorem 7. Let q > 1 be a prime number. Then, for every j such that 0 < j ≤ q: • Ljq is strongly maximal with respect to CPL and it is axiomatized by Hjq plus the rule j(φ ∧ ¬φ)/(ψ ∨ ¬ψ)q. • Ljq is strongly maximal w.r.t. Ljq. Proof. By using the equivalence of Theorem 2 and the algebraizability of Lq+1, the lattice of subquasivarieties of V(LVq+1) is dually order isomorphic to the lattice of all finitary extensions of Lq+1. Clearly CPL = L2 is the finitary extension of Ljq corresponding to the subvariety Q(LV2) of Q(LVq+1×LV2), and Ljq is the finitary extension of Ljq corresponding to the subquasivariety Q(LVq+1 × LV2) of V(LVq+1). By Theorem 4, the only critical algebras of V(LVq+1) are LVq+1, LV2 and LV2×LVq+1 and, by Lemma 4, all its subquasivarieties are Q(LV2) ( Q(LV2× LVq+1) ( Q(LVq+1). Therefore, by Theorem 5, Ljq is strongly maximal with respect to CPL, while Ljq is strongly maximal with respect to L j q. Finally, the axiomatization of Ljq follows from Theorem 6 and the facts that j φ Ljq q φ holds for every formula φ and that the equation (qx) q = qx is valid in the class MVq. From the above proof, it readily follows the next corollary. Corollary 6. Ljq is the unique strongly maximal logic w.r.t. CPL above L j q. In fact, Ljq is the only logic between L j q and CPL. 22 Remark 4. It is worth noting that the rule j(φ ∧ ¬φ)/(ψ ∨ ¬ψ)q exactly corresponds to the explosion rule (expj) introduced in Section 4. Indeed, the rule j(φ ∧ ¬φ)/(ψ ∨ ¬ψ)q is clearly derivable from (expj). On the other hand, assuming j(φ ∧ ¬φ), by this rule it follows that (ψ ∨ ¬ψ)q for every ψ. Hence the logic becomes CPL because the translation of the classical axiom ψ ∨ ¬ψ is precisely (ψ ∨¬ψ)q, and thus ⊥ follows from j(φ∧¬φ). This does not come as a surprise, since as we have proved above, Ljq is strongly maximal w.r.t. L j q and so the latter is the only proper extension of Ljq (with a finitary rule) properly contained in CPL. As a corollary of the previous remark, it follows the completeness of H j q. Corollary 7. H j q is sound and complete w.r.t. L j q. 5.3 Strong maximality with respect to systems Lin Next theorems are generalizations of Theorems 5, 6 and 7 respectively. Theorem 8. Let n > 0 and k > 1. The set of all minimal subquasivarieties of MVnk = Q(LVnk+1) among those strictly containing Q(LVn+1) is Mnkn = {Q({LVn+1,LVq+1 × LV2}) : q prime, q|k and q 6 |n} ⋃ {Q({LVn+1,LVqr+1+1 × LV2}) : q prime, q|k, qr|n and qr+1 6 |n}. Proof. By Lemma 4, every K ∈Mnkn is a subquasivariety of Q(LVnk+1) strictly containing Q(LVn+1). Moreover, for every K1,K2 ∈ Mnkn , if K1 6= K2 then K1 6⊆ K2 and K2 6⊆ K1. LetK be a minimal subquasivariety ofQ(LVnk+1) strictly containingQ(LVn+1). Trivially, LVn+1 ∈ K. Since K 6= Q(LVn+1), then it must contain a critical algebra C ∼= LVm1+1 × * * * × LVms+1 such that mi|nk for every 1 ≤ i ≤ s and mj 6 |n for some 1 ≤ j ≤ s. If there is a prime number q|mj such that q 6 |n, then LVq+1 × LV2 ∈ Q(C) ⊆ K. Otherwise, there is a prime q such that q|mj and qr|n, and for some r ≥ 1, qr+1 6 |n and qr+1|mj , whence LVqr+1+1 × LV2 ∈ Q(C) ⊆ K. Thus, in both cases K contains some Ki ∈ Mnkn , from which it follows that K ∈Mnkn since we are assuming minimality of K. Theorem 9. For every n > 0. • If q is a prime number such that q 6 |n, then Q({LVn+1,LVq+1×LV2}) is axiomatized by the MV identities plus – γ{n,q},∅(x) ≈ 1 (the identity axiomatizing V({LVn+1,LVq+1})) – nq(x ∧ ¬x) ≈ 1⇒ γ{n},∅(y) ≈ 1. • If q is a prime number such that qr|n and qr+1 6 |n, Q({LVn+1,LVqr+1+1× LV2}) is axiomatized by the MV identities plus – γ{n,qr+1},∅(x) ≈ 1 (the identity axiomatizing V({LVn+1,LVqr+1+1})) 23 – nq(x ∧ ¬x) ≈ 1⇒ γ{n},∅(y) ≈ 1. Proof. We prove the first item, the other is proved in a analogous way. It is easy to check that LVn+1 and LVq+1 × LV2 satisfy all the quasi-identities. Since the MV-identities with γ{n,q},∅(x) ≈ 1 axiomatize V({LVn+1,LVq+1}) and V({LVn+1,LVq+1}) is a locally finite quasivariety, it is enough to prove that every critical MV-algebra C ∈ V({LVn+1,LVq+1}) where the quasiequation nq(x ∧ ¬x) ≈ 1 ⇒ γ{n},∅(y) ≈ 1 holds, belongs to Q({LVn+1,LVq+1 × LV2}). Therefore, let C be a critical MV-algebra satisfying the axiomatization. Then, C is such that C ∼= LVm1+1 × * * * × LVmr+1 satisfying conditions of Theorem 4, and moreover for every 1 ≤ i ≤ k, either mi|n or mi = q because C ∈ V({LVn+1,LVq+1}). If there is c ∈ C such that nq(c ∧ ¬c) = 1 then, by the second quasi-equation of the axiomatization above, γ{n},∅(b) ≈ 1 for any b ∈ C, thus C ∈ V({LVn+1}) = Q({LVn+1}) ⊆ Q({LVn+1,LVq+1 × LV2}). If for every c ∈ C, nq(c ∧ ¬c) 6= 1 then mi = 1 for some 1 ≤ i ≤ k. In that case, by the characterization of critical algebras (Theorem 4), either C ∼= LV2 or C ∼= LVm+1×LV2. If C ∼= LV2, then trivially C ∈ Q({LVn+1,LVq+1×LV2}). Otherwise, if C ∼= LVm+1×LV2, since C ∈ V({LVn+1,LVq+1}), either m|n or m = q. If m|n then C ∈ V({LVn+1}) = Q({LVn+1}) ⊆ Q({LVn+1,LVq+1 × LV2}). If m = q then C ∼= LVq+1 × LV2 ∈ Q({LVn+1,LVq+1 × LV2}). If 1 ≤ i,m ≤ n, by analogy with Li/nm , we define the matrix logic Li/nm = 〈 LVm+1 × LV2, (Fi/n ∩ LVm+1)× {1}〉. Then we have the following generalization of Theorem 7. Theorem 10. Let 0 < i ≤ n be natural numbers and let q be a prime number. Then we have: • If q 6 |n then, for every j such that (i− 1)q < j ≤ iq, Lin ∩ L j/nq q is strongly maximal with respect to Lin, and it is axiomatized by H j/nq n,q plus the rule j(φ ∧ ¬φ)/γj/nqn (ψ). • If qr|n and qr+1 6 |n then, for every j such that (i−1)q < j ≤ iq, Lin∩ L j/nq qr+1 is strongly maximal with respect to Lin, and it is axiomatized by H j/nq n,qr+1 plus the rule j(φ ∧ ¬φ)/γj/nqn (ψ). Recall that in the above rules γ j/nq n (ψ) refers to the axiom in one variable that axiomatizes L j/nq n relative to Ljnq. Moreover, every finitary extension of some L j k is strongly maximal with respect Lin iff it is of one of the two preceeding types. Proof. Notice that Lin = L j/nq n for every j such that (i − 1)q < j ≤ iq. Thus, Lin is an extension of L j nq. Now, by using the equivalence of Theorem 2 and the algebraizability of Lnq + 1, the lattice of subquasivarieties of V(LVnq+1) is dually order isomorphic to the lattice of all the finitary extensions of Ljnq. Moreover, Lin∩ L j/nq q and Lin∩ L j/nq qr+1 are the finitary extensions of L j nq associated 24 to Q({LVn+1,LVq+1×LV2}) and Q({LVn+1,LVqr+1+1×LV2}), respectively. Hence, they are strongly maximal with respect to Lin, by Theorem 8. The axiomatization follows from Theorem 9 and the facts that j φ Ljnq nq φ holds for every formula φ and that the equation (nqx)nq = nqx is valid in the class MVnq. Finally, the last statement of this theorem follows from Theorem 8 and Theorem 2. 6 An application to ideal paraconsistent logics As mentioned in Example 3, Arieli et al. have introduced in [3] the concept of ideal paraconsistent logics. We recall here this notion. Definition 4 (c.f. [3]). Let L be a propositional logic defined over a signature Θ (with consecuence relation `L) containing at least a unary connective ¬ and a binary connective → such that: (i) L is paraconsistent w.r.t. ¬ (or simply ¬-paraconsistent), that is, there are formulas φ,ψ ∈ L(Θ) such that φ,¬φ 0L ψ; (ii) → is an implication for which the deduction-detachment theorem holds in L, that is, Γ ∪ {φ} `L ψ iff Γ `L φ → ψ, for every set for formulas Γ ∪ {φ,ψ} ⊆ L(Θ). (ii) There is a presentation of CPL as a matrix logic L′ = 〈A, {1}〉 over the signature Θ such that the domain of A is {0, 1}, and ¬ and→ are interpreted as the usual 2-valued negation and implication of CPL, respectively. (iv) L is a sublogic of CPL in the sense that `L⊆ `L′ , that is, Γ `L φ implies Γ `L′ φ, for every set for formulas Γ ∪ {φ} ⊆ L(Θ). Then, L is said to be an ideal paraconsistent logic if it is maximal w.r.t. L′, and every proper extension of L over Θ is not ¬-paraconsistent. An implication connective satisfying the above condition (ii) will be called deductive implication in the rest of the paper.4 Thus, a ¬-paraconsistent logic L with a deductive implication is ideal if it is maximal w.r.t. CPL (presented over the signature Θ of L) and, if L′′ is another logic over Θ properly containing L, with Γ ∪ {φ} ⊆ L(Θ) such that Γ `L′′ φ but Γ 0L φ, then the logic obtained from L by adding Γ/φ as an inference rule is not ¬-paraconsistent. As already noticed, the logics Lni with i/n ≤ 1/2 are paraconsistent. In this section, using the results of the previous sections, we study the status of the logics Lin in relation to ideal paraconsistency. Namely, in the following subsection, we will show that the logics of the form Liq, where q is prime and i/q ≤ 1/2 are ideal paraconsistent, while in subsection 6.2 the special case of L13, renamed as J4, is analyzed in more detail. 4Such an implication is called deductive in [11, 16] and proper in [3]. 25 6.1 The ideal paraconsistent logics Liq By combining Proposition 1 with Corollary 2 we know a logic Liq is ¬-paraconsistent and maximal w.r.t. CPL, provided that q is prime and i/q ≤ 1/2. From now on we will assume this is the case when referring to a logic Liq. Recall that H i q is the Hilbert calculus obtained from the calculus H i q for L i q by adding the i-explosion rule (expi). Since φ ∧ ¬φ `Hiq i(φ ∧ ¬φ), the logic H i q is explosive. Then, taking into account Corollary 6, it follows that every proper extension of Liq defined over its signature is either L i q or CPL, and hence not ¬-paraconsistent. In addition, by Lemma 3, we know there is a definable unary connective ∼iq such that, for every evaluation e, e(∼iq p) = 0 if e(p) ≥ i/q, and e(∼iq p) = 1 otherwise, for every propositional variable p.5 This is a kind of "classical" negation defined on Liq. Using this negation, one can define in turn a new implication ⇒iq by stipulating φ ⇒iq ψ = ∼iqφ ∨ ψ. In fact, one can easily check that ⇒iq is a deductive implication on Liq in the sense of Definition 4 and that over {0, 1} it coincides with the classical implication. All the above considerations lead to the following result. Proposition 5. Let q is a prime number, and let 1 ≤ i < q such that i/q ≤ 1/2. Then, Liq is a (q + 1)-valued ideal paraconsistent logic. 6 Therefore we have a large family of examples of ideal paraconsistent logics. In particular, for each prime q, all the logics in the set PCq+1 = {Liq : i < q/2} are (q + 1)-valued ideal paraconsistent logics. Moreover, if we consider "the more theorems a paraconsistent logic has, the more well-behaved is the logic" as a valid further criterion, then we can still refine the set PCq+1. Indeed, if we denote by Th(L) the set of theorems of a logic L then, as noticed in Remark 2, we have the strict inclusions Th(Liq) ( Th(Ljq) ( Th(CPL) whenever i > j. Therefore the logic Jq+1 = L 1 q appears to be the "best" ideal logic in the set PCq+1, 7 since it is the logic in that set having the biggest set of theorems from classical logic. Finally, it is worth mentioning that all the paraconsistent logics of the form Lin are, indeed, LFIs (recall Section 2): Proposition 6. Suppose that i/n ≤ 1/2. Then, the logic Lin is an LFI w.r.t. ¬ and where the consistency operator is defined as ◦α = ∼in(α ∧ ¬α). Proof. Straightforward. 5Namely, ∼iq p = ¬λi,q(p). 6Strictly speaking, in this claim we implicitly assume that the signature of Liq has been changed by adding the definable implication ⇒iq as a primitive connective. 7We have chosen the name Jq+1 to denote the logic L1q inspired in the 3-valued case, where the ideal paraconsistent logic J3 coincides with L12. 26 6.2 The four-valued ideal paraconsistent logic J4 As mentioned in Remark 3, we know from Theorem 4.3 in [8] that it is possible to obtain a standard (that is, without "global" inference rules) Hilbert calculus for a logic Lin for i < n from the usual one for Ln+1 by using translations. However, the calculi obtained in this manner can lack an intuitive meaning since they are defined in terms of the implication connective → of Ln+1, that is naturally associated to the filter F1 = {1} but not to the filter Fi/n = {i/n, . . . , 1}, which is the one at work in Lin. Actually, the implication naturally associated to the filter Fi/n is ⇒in, which was considered above, for which modus ponens (MP) and the deduction-detachament theorem hold. In this section we focus on the particular case of the (ideal paraconsistent) logic J4 = L 1 3. J4 can be considered as a generalization to four values of the paraconsistent 3-valued logic J3 introduced by da Costa and D'Ottaviano in [20] and briefly mentioned in Example 2. For this logic a more natural signature Σ will be considered for describing it axiomatically in terms of a deductive implication connective (in the sense of Definition 4 item (ii)) and a unary connective ∗ representing the square operation x ⊗ x, which can be seen as a kind of 'truth stresser' (see e.g. [26]). A soundness and completeness result for this calculus proved by using a 'separation' technique for truth-values will be presented. Note that dealing with logics Jq = L 1 q for a prime q > 3 appears to be much more complicated, and certainly it lies outside the scope of this paper. The signature Σ that will be used in the rest of the section is given by two unary connectives ∗ (square) and ¬ (negation), plus a binary connective ∨ for disjunction. Abusing the notation, we formally define next J4 over this signature, and we will show later that it is an equivalent presentation of L13. Definition 5. J4 is the matrix logic 〈A4, F1/3〉 over Σ, where the algebra is A4 = ( LV4,∨,¬, ∗), with operations defined by the tables below: ∨ 1 2/3 1/3 0 1 1 1 1 1 2/3 1 2/3 2/3 2/3 1/3 1 2/3 1/3 1/3 0 1 2/3 1/3 0 ¬ ∗ 1 0 1 2/3 1/3 1/3 1/3 2/3 0 0 1 0 Observe that ¬ is Lukasiewicz negation in LV4, while ∗x = x ⊗ x (with ⊗ being Lukasiewicz strong conjunction) and ∨ is the lattice join in LV4. In this signature Σ the following derived connectives can be defined (as usual, the corresponding operators will be denoted using the same symbol): 27 - ∆(p) = ∗∗p ; - ∼p = ∆(¬p) ; p⇒ r = ∼p ∨ r ; p⇔ r = (p⇒ r) ∧ (r ⇒ p) ; p ∧ r = ¬(¬p ∨ ¬r) ; - ∇(p) = ¬∼p; α1/3(p) = ∇(p) ∧ ∼∗p; β1/3(p) = α1/3(p) ∧ ∗¬p. It is easy to see that ∆ is Monteiro-Baaz Delta-operator) and ∼ is Gödel negation (∼x = 1 if x = 0, and 0 otherwise). Note that ∼ actually coincides with ∼13, and thus ⇒ is nothing but ⇒13. Furthermore, ∇(x) = 0 if x = 0, and 1 otherwise; α1/3(x) = 1 and β1/3(x) = 1/3 if x = 1/3, and 0 otherwise. It is worth to remark that Lukasiewicz implication is definable from these operators in the following way: p→ r = ((∇(¬p)∨r)∧(¬p∨∇(r))∧¬β1/3(r))∨((∼p∧α1/3(r))∨(α1/3(p)∧α1/3(r))). Then, the following result follows easily: Proposition 7. The algebras LV4 and A4 are functionally equivalent. This means that the proposed operators over Σ constitute an alternative presentation of the algebra LV4 underlying L4. Next we define an axiomatic system for J4. Definition 6. The Hilbert calculus H4 for the logic J4, defined over the signature Σ, is given as follows: Axiom schemas: those of CPL over the signature {∨,⇒,∼} plus (Ax1) ¬∼α⇒ α (Ax2) α ∨ ¬α (Ax3) ¬¬α⇔ α (Ax4) ¬(α ∨ β)⇒ ¬α (Ax5) ¬(α ∨ β)⇒ ¬β (Ax6) ¬α⇒ (¬β ⇒ ¬(α ∨ β)) (Ax7) ∗α⇒ α (Ax8) ∗(α ∨ ¬α) (Ax9) ∗α⇒ ∼∗¬α (Ax10) ∗∗α⇔ ∼¬α (Ax11) ¬∗α⇔ ¬α (Ax12) ∗(α ∨ β)⇔ (∗α ∨ ∗β) Inference rule: (MP) α α⇒ β β 28 Observe that, since (MP) is the only inference rule, H4 satisfies the deductiondetachment theorem w.r.t. the implication ⇒: Γ ∪ {α} `H4 β iff Γ `H4 α ⇒ β, for every set of formulas Γ ∪ {α, β}. On the other hand, it can be proved that ∗(α ⇒ β) ⇒ (∗α ⇒ ∗β) is derivable in H4, which gives additional support to consider ∗ as a truth stresser. Soundness of H4 can be proved straightforwardly. Proposition 8 (Soundness of H4). The calculus H4 is sound w.r.t. J4, that is: Γ `H4 φ implies that Γ J4 φ, for every finite set of formulas Γ ∪ {φ}. In order to prove completeness, since H4 is a finitary Tarskian logic, one can use the technique of maximal consistent sets of formulas. Indeed, for any set of formulas Γ ∪ {φ}, if Γ 0H4 φ then, by LindenbaumLos theorem, Γ can be extended to a maximal set Λ such that Λ 0H4 φ. We will call the set Λ maximal non-trivial with respect to φ in H4. Maximal sets w.r.t. a formula enjoy remarkable properties which directly follow from the axioms and rules of H4. Proposition 9. Let Λ be a maximal set non-trivial with respect to φ in H4. Then, Λ is closed, i.e. for every formula ψ, Λ ` ψ iff ψ ∈ Λ. Moreover, for any formulas α and β the following conditions hold: (1) α ∨ β ∈ Λ iff α ∈ Λ or β ∈ Λ; (2) α 6∈ Λ iff ∼α ∈ Λ; (3) α⇒ β ∈ Λ iff α 6∈ Λ or β ∈ Λ; (4) α 6∈ Λ implies ¬α ∈ Λ; (5) α ∈ Λ iff ¬¬α ∈ Λ; (6) ¬∼α ∈ Λ implies α ∈ Λ; (7) ¬(α ∨ β) ∈ Λ iff ¬α ∈ Λ and ¬β ∈ Λ; (8) ∗α ∈ Λ implies α ∈ Λ; (9) ∗(α ∨ β) ∈ Λ iff ∗α ∈ Λ or ∗β ∈ Λ; (10) ∗∗α ∈ Λ iff ¬α 6∈ Λ; (11) ¬∗α ∈ Λ iff ¬α ∈ Λ; (12) ∗α 6∈ Λ iff ∗¬α ∈ Λ. Next we prove a Truth Lemma for H4. Lemma 5 (Truth Lemma for H4). Let Λ be a maximal set of formulas nontrivial with respect to φ in H4. Consider the following evaluation eΛ of propositional variables for J4: (T ) eΛ(γ) =  1 iff γ ∈ Λ, and ¬γ 6∈ Λ 2/3 iff γ ∈ Λ, ¬γ ∈ Λ, and ∗γ ∈ Λ 1/3 iff γ ∈ Λ, ¬γ ∈ Λ, and ∗γ 6∈ Λ 0 iff γ 6∈ Λ. Then, (T) holds for every complex formula γ. Proof. The proof is done by induction on the complexity of the formula γ. If γ is atomic then (T) holds by hypothesis. Now, suppose (T) holds for every 29 formula with complexity ≤ n (induction hypothesis – IH) and let γ be a formula with complexity n. In order to prove (T) from (IH) by analyzing all the possible cases (namely, γ = ¬α or γ = ∗α or γ = α ∨ β), each item of Proposition 9 should be used.8 The details are left to the reader. Theorem 11 (Completeness of H4). The calculus H4 is complete w.r.t. J4, that is: Γ J4 φ implies that Γ `H4 φ, for every finite set of formulas Γ ∪ {φ}. Proof. Let Γ∪{φ} be a set of formulas in the language of J4 such that Γ 0H4 φ. By LindenbaumLos, there exists a set Λ maximal non-trivial with respect to φ in H4 such that Γ ⊆ Λ. Let eΛ be the evaluation defined as in the Truth Lemma 5. Then, it follows that eΛ(γ) ∈ F1/3 iff γ ∈ Λ, for every formula γ. Therefore eΛ is an evaluation such that eΛ[Γ] ⊆ F1/3 but eΛ(φ) = 0 since φ 6∈ Λ, hence Γ 6J4 φ. Recall that, from Theorem 7 and Remark 4, the Hilbert calculus H4 obtained from H4 by adding the explosion rule (exp1) φ ∧ ¬φ ⊥ (see Definition 3) is the axiomatization of the (only) proper extension of H4 which is strongly maximal w.r.t. CPL, and that is semantically characterized by the matrix logic J4 = 〈A4 ×A2, F1/3 × {1}〉, where A2 is the Boolean algebra over {0, 1} in the signature Σ, where the operator ∗ is defined as ∗x = x. 7 Conclusions In this paper we have been concerned with the study of maximality and strong maximality conditions among finite-valued Lukasiewicz logics Lin with order filters as designated values. In particular, we have characterized the conditions under which a logic Lin is maximal w.r.t. CPL and its unique extension L i n by an inference rule is strongly maximal w.r.t. classical logic. This allows us to show that, although they are not strongly maximal w.r.t. CPL, the logics Lin with n prime and i/n ≤ 1/2 are in fact ideal paraconsistent logics. Thus, they provide interesting and well-motivated examples of ideal paraconsistent logical systems which are (n+ 1)-valued, in contrast with the (n+ 2)-valued logics Mn+2 presented in [3] and reproduced here in Example 3, whose definition is somewhat ad hoc. 8Observe that it is enough to prove the 'only if' part of (T), since the four conditions on the right-hand side are pairwise incompatible, and e(γ) can only take one of the values 0, 1, 1/3, 2/3. Thus, if, for instance, the first condition on the right-hand side of (T) holds for a given formula γ then the other 3 conditions are false and so eΛ(γ) 6∈ {1/3, 1, 0}, by the 'only if' part of (T). Hence, eΛ(γ) must be 2/3. This shows that the 'if' part of (T) follows from the 'only if' part. 30 As for future work, there are several interesting problems that we leave open in this paper. Concerning maximality, a natural question is how to obtain a stronger version of Theorem 1 which give us sufficient conditions to guarantee that a given matrix logic L1 is strongly maximal w.r.t. another matrix logic L2. On the other hand, notice that the study of strong maximality developed in Section 5 was heavily based on results on the algebraic semantics associated to these systems by means of the Blok and Pigozzi's techniques. Thus, another interesting issue to be explored in future work is to obtain more examples of strong maximality for different families of algebraizable logics Another question raised here is the axiomatization of the ideal paraconsistent logics Jq+1 for q > 3 in a "natural" signature containing a deductive implication. As it was shown in Subsection 6.2, the signature Σ = {∨,¬, ∗} is suitable for the case q = 3. Moreover, besides being apt for axiomatizing J4 = L 1 3, it can be proved that the (non-paraconsistent) logic L23 can also be axiomatized over Σ in a relatively simple way. Note that α ⇒ β = ¬α ∨ β defines a deductive implication in L23. The fact that Lukasiewicz implication is definable in Σ justifies the convenience of using that signature for dealing with the case q = 3. However, this property does not hold for any prime q > 3. Indeed, there are primes q in which Lukasiewicz implication of Lq+1 cannot be defined over Σ, e.g. q = 17. The study of the fragments of Liq in the signature Σ is thus a different but closely related problem, which deserves future research. Acknowledgements The authors acknowledge partial support by the H2020 MSCA-RISE-2015 project SYSMICS. Coniglio was also financially supported by an individual research grant from CNPq, Brazil (308524/2014-4). Esteva and Godo also acknowledge partial support by the Spanish MINECO/FEDER project RASO (TIN201571799-C2-1-P). 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