Swap structures semantics for Ivlev-like modal logics Marcelo E. Coniglio1,2 and Ana Claudia Golzio2 1 Department of Philosophy, IFCH, University of Campinas, Brazil 2 Centre for Logic, Epistemology and the History of Science (CLE), University of Campinas, Brazil coniglio@cle.unicamp.br anaclaudiagolzio@yahoo.com.br Abstract In 1988 J. Ivlev propose some (non-normal) modal systems which are semantically characterized by four-valued non-deterministic matrices in the sense of A. Avron and I. Lev. Swap structures are multialgebras (a.k.a. hyperalgebras) of a special kind, which were introduced in 2016 by W. Carnielli and M. Coniglio in order to give a non-deterministic semantical account for several paraconsistent logics known as logics of formal inconsistency, which are not algebraizable by means of the standard techniques. Each swap structure induces naturally a non-deterministic matrix. The aim of this paper is to obtain a swap structures semantics for some Ivlev-like modal systems proposed in 2015 by M. Coniglio, L. Fariñas del Cerro and N. Peron. Completeness results will be stated by means of the notion of Lindenbaum-Tarski swap structures, which constitute a natural generalization to multialgebras of the concept of Lindenbaum-Tarski algebras. 1 Introduction In 1981 and 1988 respectively, J. Kearns (see [13]) and J. Ivlev (see [12]) independently propose a new semantics for some modal systems defined by means of four-valued multivalued truth-functions. Both contributions anticipated the non-deterministic matrices (a.k.a. Nmatrices) introduced in 2001 by A. Avron and I. Lev. Kearns proposes a semantics of four-valued Nmatrices with a restriction on the valuations (which are called level valuations) for the systems T, S4 and S5. On the other hand, Ivlev proposes a genuine semantics of four-valued Nmatrices (in the sense of Avron and his collaborators, see Definition 2.7 below) for several weak modal systems which do not have the necessitation rule. Afterwards, Kearns and Ivlev's approaches were revisited and extended in [7], [8] and [15]. Non-deterministic matrices were considered by Avron and his collaborators in order to semantically characterize logics. In particular, some paraconsistent logics known as Logics of Formal Inconsistency (LFIs for short, see for instance [6]) were characterized by this kind of semantics. In formal terms, Nmatrices are logical matrices in which the underlying algebra is replaced by a multialgebra, that is, an algebra in which the 1 operations interpreting the connectives are multiple-valued: instead of returning a single value, a non-empty set of outputs is obtained from a single input. This change requires to consider a new notion of valuation, which choose, for any complex formula, a possible value generated by the multioperators from the values already given to its subformulas (see Definition 2.5 below). Multialgebras (also known as hyperalgebras) where introduced in 1934 by F. Marty (see [14]), by means of the notion of hypergroup. Afterwards, several classes of multialgebras were proposed in the literature. This concept was studied from many points of view and applied to several areas of Mathematics and Computer Science. In the field of Logic, besides the already mentioned Nmatrix semantics obtained by Avron and his collaborators, Carnielli and Coniglio propose in [6, Chapter 6] a class of semantical structures based on multialgebras called swap structures, suitable for characterizing several LFIs. A formal study of swap structures by using tools from category theory and universal algebra was started in [10]. The present paper generalizes the finite Nmatrix semantics introduced in [7] and [8] to some Ivlev-like systems by considering swap structures semantics for them. After defining the class of swap structures for each of such modal logics, soundness and completeness theorems will be obtained for the Hilbert calculi associated to these logics with respect to the corresponding swap structures semantics. In order to obtain the completeness theorem, the notion of Lindenbaum-Tarski swap structures will be introduced, which generalizes in a quite natural way the classical Lindenbaum-Tarski technique to these logics which are not algebraizable in the usual sense. 2 Multialgebras and non-deterministic matrices In this section the notion of multialgebra and submultialgebra adopted in this paper will be recalled, taken from [10]. The related notion of non-deterministic matrices will also be recalled, for the reader's convenience. Notation 1 If f : A → B is a function and X ⊆ A then f [X] will stand for the set {f(x) : x ∈ X}. The powerset of any set A will be denoted by P(A). Definition 2.1 (Signature) Let Ξ be a denumerable set of propositional variables. A (propositional) signature is a family of sets Σ = {Σn}n∈N, such that Ξ∩Σn = ∅ for every n and, if n 6= m, then Σn ∩ Σm = ∅. The free algebra of terms generated by Ξ over a signature Σ will be denoted by For(Σ).1 Elements in Σn are called connectives of arity n. In particular, elements in Σ0 are called constants. Elements in For(Σ) are called formulas over Σ. From now on, and whenever |Σ| = ⋃ n≥0 Σn is finite, the signature Σ will be identified with |Σ|, assuming that the arity of each connective is clear from the context. Definition 2.2 (Multialgebras) Let Σ be a signature. A multialgebra (or hyperalgebra) over Σ is a pair A = 〈A, σA〉 such that A is a nonempty set (the universe or support of A) and σA is a mapping assigning to each c ∈ Σn a function (called multioperation or hyperoperation) cA : An → (P(A) \ {∅}). In particular, ∅ 6= cA ⊆ A if c ∈ Σ0. 1This notation makes sense, provided that the set Ξ of generators will keep fixed along this paper. 2 Definition 2.3 (Submultialgebras, [10]) Let B = 〈B, σB〉 and A = 〈A, σA〉 be two multialgebras over Σ. Then B is said to be a submultialgebra of A if the following conditions hold: (i) B ⊆ A, (ii) if c ∈ Σn and ~b ∈ Bn, then cB(~b) ⊆ cA(~b); in particular, cB ⊆ cA if c ∈ Σ0. The general notion of non-deterministic matrix (or Nmatrix) was formally introduced in [3] (see [5] for a survey on non-deterministic matrices). However, Nmatrices were already used in the literature for some specific cases: see, for instance, N. Rescher's nondeterministic implication proposed in [16] and the modal Nmatrices proposed by J. Kearns and J. Ivlev to be discussed in Section 3. In formal terms, Nmatrices are nothing else than multialgebras with a non-empty set of designated elements (generalizing the idea of logical matrices, which are algebras together with a set of designated values). Definition 2.4 (Non-deterministic matrices, [3]) Let Σ be a signature. A non-deterministic matrix (or Nmatrix) over Σ is a pair M = 〈A, D〉 such that A = 〈A, σA〉 is a multialgebra over Σ with support A, and D is a nonempty subset of A. The elements in D are called designated elements. Recall now the definition of valuations over non-deterministic matrices proposed in [3] (see also [4], [2]). Definition 2.5 (Valuations, [3]) Let M = 〈A, D〉 be a non-deterministic matrix over a signature Σ. A valuation2 over M is a function v : For(Σ)→ |A| such that, for every c ∈ Σn and every α1, . . . , αn ∈ For(Σ): v(c(α1, . . . , αn)) ∈ cA(v(α1), . . . , v(αn)). In particular, v(c) ∈ cA, for every c ∈ Σ0. Remark 2.6 It is worth noting that the same notion of valuation over Nmatrices was already considered (in a somewhat informal way) in the pioneering works [16], [13] and [12]. In the case of [13], a restricted subset of valuations was also considered, as it was mentioned in Section 1. Definition 2.7 (Consequence relation, [3]) Let M = 〈A, D〉 be a non-deterministic matrix over a signature Σ, and let Γ ∪ {α} ⊆ For(Σ). We say that α is a consequence of Γ in the non-deterministic matrixM, denoted by Γ |=M α, if the following holds: for every valuation v over M, if v[Γ] ⊆ D then v(α) ∈ D. In particular, α is valid in M, denoted by |=M α, if v(α) ∈ D for every valuation v over M. 2Avron and his collaborators use in [3], [1], [5] the term legal valuation to refer to valuations over an Nmatrix. 3 3 Nondeterministic Ivlev-like modal logics As mentioned in Section 1, J. Ivlev propose in [12] a family of (non-normal) modal logics (that is, modal logics in which the necessitation rule is no longer valid) which are characterized by four-valued Nmatrices. Some generalizations of Ivlev's logics were proposed in [7] (see also [8] and [15]). In order to define four-valued non-deterministic matrices, Ivlev considers the set {T, t, f, F} of truth-values with the following intended meaning: T : Necessarily true t: Possibly true f : Possibly false F : Necessarily false The set of designated truth-values is {T, t}, while {F, f} is the set of non-designated truth-values. Let Σ′ = {¬,→,} be a propositional signature for modal systems. The following (four-valued) non-deterministic matrices where independently proposed by Ivlev (system Sa+, see [12]) and Kearns (system T, see [13]): 3 → T t f F T T t f F t T {T, t} f f f T {T, t} {T, t} t F T T T T α ¬α α T F {T, t} t f {f, F} f t {f, F} F T {f, F} The connectives ∨ and ♦ can be defined, as in the classical case by α ∨ β = ¬α → β and ♦α = ¬¬α, respectively, thus producing the following multioperators: ∨ T t f F T T T T T t T {T, t} {T, t} t f T {T, t} f f F T t f F α ♦α T {T, t} t {T, t} f {T, t} F {f, F} A Hilbert-style deductive system for Sa+ was introduced in [7], under the name of Tm. A slightly different version will be introduced here, by following the modifications suggested in [15] and [8]. Definition 3.1 The system Tm over Σ′ = {¬,→,} is given by the following axiom schemes: α→ (β → α) (Ax1) 3As observed in Section 1, Kearns restricts the valuations over the Nmatrices in order to deal with the necessitation rule. 4 (α→ (β → σ))→ ((α→ β)→ (α→ σ)) (Ax2) (¬β → ¬α)→ ((¬β → α)→ β) (Ax3) (α→ β)→ (α→ β) (K) (α→ β)→ (¬β → ¬α) (K1) ¬¬(α→ β)→ (α→ ¬¬β) (K2) ¬α→ (α→ β) (M1) β → (α→ β) (M2) ¬(α→ β)→ ¬β (M3) ¬(α→ β)→ α (M4) α→ α (T) α→ ¬¬α (DN1) ¬¬α→ α (DN2) together with the following inference rule: α, α→ β β (MP) The systems T4m, T45m, TBm and Dm were introduced in [7] (see also [8]), obtained from Tm by considering the following axioms: α→ α (4) ¬¬α→ α (5) ¬¬α→ α (B) α→ ¬¬α (D) Let us consider firstly the systems T4m, T45m and TBm. A Hilbert calculus for each of them is given, respectively, by: T4m = Tm ∪ {(4)} T45m = T4m ∪ {(5)} TBm = Tm ∪ {(B)} The non-deterministic matrices for the systems T4m, T45m and TBm are obtained from the one for Tm by respectively changing the multioperation  to: α T4mα T T t {f, F} f {f, F} F {f, F} α T45mα T T t F f F F F α TBmα T {T, t} t {f, F} f F F F 5 Remark 3.2 The operation T45m coincides with the interpretation for  proposed by Kearns in [13] for the system S5 and by Ivlev for the system Sb+ in [12]. By taking (F ) = {F} instead of (F ) = {f, F} in T4m, the resulting multioperation coincides with the one proposed by Kearns [13] in order to interpret  in the system S4. As usual, ♦α will be an abbreviation of the modal formula ¬¬α. Recall that deontic logics are modal logics in which the operators  and ♦ are intuitively interpreted as it is obligatory that and it is permitted that, respectively. Under this interpretation, it is natural to consider the principle α → ♦α instead of the stronger principles α → α and α→ ♦α. Now, swap structures will be applied to a deontic-like modal system. The goal is to provide a slightly more general example. Let us consider the modal system Dm introduced in [7] and [8] which constitute a weaker version of the deontic logic KD. It is obtained from Tm by changing axiom (T) by (D). From the semantical point of view, it produces a six-valued characteristic non-deterministic matrix semantics. Given a proposition p, some usual deontic-like operators can be defined in the system Dm: • p is fulfilled whenever it is obligatory and it is the case. In symbols: (p ∧ p). • p is infringed whenever it is obligatory and it is not the case. In symbols: (p∧¬p) • p is optional if it is neither obligatory nor forbidden. In symbols: ¬(p ∨ ¬p). Note that p is optional iff ¬p is optional. • p is preferred whenever it is optional and it is the case. In symbols: (¬(p∨¬p)∧p). From this, a plausible interpretation for the six truth-values in the system Dm can be proposed: T+: p is fulfilled; C+: p is preferred; F+: ¬p is infringed; T−: p is infringed; C−: ¬p is preferred; and F−: ¬p is fulfilled. As mentioned above, a Hilbert calculus for Dm is given by: Dm = ( Tm \ {(T)} ) ∪ {(D)} The Nmatrix semantics for the system Dm proposed in [7] is given by the multioperations →Dm and Dm and by the operation ¬Dm displayed below: 6 →Dm T+ T− C+ C− F+ F− T+ T+ T− C+ C− F+ F− T− T+ T+ C+ C+ F+ F+ C+ T+ T− {T+, C+} {T−, C−} C+ C− C− T+ T+ {T+, C+} {T+, C+} C+ C+ F+ T+ T− T+ T− T+ T− F− T+ T+ T+ T+ T+ T+ α ¬Dmα T+ F− T− F+ C+ C− C− C+ F+ T− F− T+ α Dmα T+ {T+, C+, F+} T− {T+, C+, F+} C+ {T−, C−, F−} C− {T−, C−, F−} F+ {T−, C−, F−} F− {T−, C−, F−} in which {T+, C+, F+} is the set of designated values. Remark 3.3 It is worth noting that the six-valued Nmatrix for Dm extends the fourvalued Nmatrix for Tm in the following sense: the truth-values T , t, f and F corresponds to T+, C+, C− and F−, respectively. Indeed, the multialgebra underlying the Nmatrix for Tm is a submultialgebra of the multialgebra underlying the Nmatrix for Dm (recall the notion of submultialgebra given in Definition 2.3 taken from [10]). Moreover, the respective sets DTm and DDm of designated values are related as follows: DTm = DDm ∩{ T, t, f, F } . Notation 2 From now on, L will denote the set{ Tm,T4m,T45m,TBm,Dm } of modal logics. In [7] and [8] it was obtained the following characterization result: Theorem 3.4 Let L ∈ L and let ML be the Nmatrix for L as defined above. Then, for every α ∈ For(Σ′), `L α⇔ ML α. In the next section, swap structures semantics for the systems in L will be introduced, which generalizes the Nmatrix semantics analyzed above. 7 4 From Nmatrices to swap structures semantics As it was discussed in Section 1, the notion of swap structures was introduced in [6, Chapter 6] in order to give a semantical account (with an "algebraic flavour") to several paraconsistent logics in the class of logics of formal inconsistency (LFIs) which cannot be algebraizable by the standard methods. These structures are multialgebras formed by triples z = (z1, z2, z3) of elements over a given Boolean algebra A, which are called snapshots.4 The intended meaning of a snapshot z is that z1 represents a given (algebraic) truth-value for a formula φ in A, while z2 and z3 represent a (possible) truth-value for the formulas ¬φ and ◦φ, respectively.5 Each swap structure for a given LFI induces a nondeterministic matrix in a natural way, where the class of all such Nmatrices characterizes the given logic. In [10] a formal study of the category of swap structures for some LFIs was developed, by adapting notions and tools from universal algebra. In this section, the original approach to swap structures semantics for LFIs will be adapted to the family of non-normal modal systems presented in Section 3. As in the case of LFIs, the swap structures proposed here are multialgebras whose elements are triples (snapshots) in a given Boolean algebra. In the present case, the intended meaning of a snapshot z = (z1, z2, z3) is that z1 represents a given truth-value for a formula φ in A, while z2 and z3 represent a (possible) truth-value for the formulas φ and ¬φ, respectively. Definition 4.1 (Swap structures for Tm) Let A = 〈A,⊃,t,u,∼, 0, 1〉 be a Boolean algebra and let BTmA = {(a1, a2, a3) ∈ A3 : a2 ≤ a1 and a1 u a3 = 0}. A swap structure for Tm over A is any multialgebra B = 〈B, →, ¬, 〉 over Σ′ = {→,¬,} such that B ⊆ BTmA and the multioperations satisfy the following, for every (a1, a2, a3) and (b1, b2, b3) in B: (i) (a1, a2, a3) → (b1, b2, b3) = {(c1, c2, c3) ∈ B : c1 = a1 ⊃ b1, c3 = a2 u b3 and a3 t b2 ≤ c2 ≤ (a1 ⊃ b1) u (a2 ⊃ b2) u (b3 ⊃ a3)} (ii) ¬(a1, a2, a3) = {(∼a1, a3, a2)} (iii) (a1, a2, a3) = {(c1, c2, c3) ∈ B : c1 = a2} The unique swap structure for Tm with domain BTmA will be denoted by BTmA . It will be called the full swap structure for Tm over A. 4This terminology is inspired by its use in computer science to refer to states. 5Here ¬ represents the paraconsistent negation of a given LFI, while ◦ represents a consistency operator w.r.t. ¬. See for instance [6] for more information about LFIs. 8 Remark 4.2 The intuition behind the universe BTmA and the definition of the multioperations for swap structures for Tm is the following: suppose that z = (a1, a2, a3) is a snapshot in a given swap structure B for Tm. Then, as explained above, a1, a2 and a3 represent a possible truth-value (over the Boolean algebra A) for φ, φ and ¬φ, respectively, for a given formula φ. By axiom (T) it follows that φ→ φ is valid, which means that a2 ⊃ a1 = 1 under such interpretation. But the latter is equivalent to say that a2 ≤ a1. By an analogous argument, a3 ≤ ∼a1, where ∼a1 represents the truth-value of ¬φ under such interpretation (recall by Definition 4.1 that the Boolean complement in the Boolean algebra A is denoted by ∼). Hence, a1 u a3 = 0. With respect to the multioperations, the definition of the (standard) operation ¬ in a given swap structure B for Tm is natural under such interpretation: indeed, since z = (a1, a2, a3) represents the truth-values of φ, φ and ¬φ, then (∼a1, a3, a2) represents the truth-values of ¬φ, ¬φ and ¬¬φ, given that ¬¬φ is equivalent to φ in Tm. Concerning the multioperation associated to , it is clear that any snapshot w = (c1, c2, c3) in the set z must satisfy that c1 = a2, since a2 represents φ. The truth-values c2 and c3 for φ and ¬φ cannot be expressed in Tm in terms of a Boolean combination of the given truth-valued for φ, φ and ¬φ, and this is why the second and third coordinates of w cannot be additionally restricted. As we shall see below, the axiomatic extensions of Tm anayzed here will impose some retrictions to the multioperation . The definition of the multioperation associated to → is not so intuitive, and it is obtained by generalizing the definition of Kearns and Ivlev's four-valued Nmatrix for Tm to arbitrary Boolean algebras. Indeed, as it will be seen in Remark 4.6, such Nmatrix corresponds to the swap structure BTmA2 defined over the twoelement Boolean algebra A2. Finally, it is worth noting that, by the very definition, the class of swap structures for Tm is closed under submultialgebras (recall Definition 2.3). Moreover, any swap structure for Tm over A is a submultialgebra of BTmA . As it will be shown in Corollary 4.5, the implication in any swap structure for Tm is a well-defined multioperation. Namely: (a1, a2, a3) → (b1, b2, b3) 6= ∅ for any (a1, a2, a3) and (b1, b2, b3) in BTmA . Proposition 4.3 : If (a1, a2, a3), (b1, b2, b3) ∈ BTmA then a3 t b2 ≤ (a1 ⊃ b1) u (a2 ⊃ b2) u (b3 ⊃ a3). Proof: Let be (a1, a2, a3), (b1, b2, b3) ∈ BTmA . As a1 u a3 = 0, then a3 ≤ ∼a1. But, as a2 ≤ a1, then ∼a1 ≤ ∼a2 and, therefore, a3 ≤ ∼a2. So, a3 t b2 ≤ ∼a2 t b2 = a2 ⊃ b2 (1) It also holds that (a3 t b2) u a1 = (a3 u a1) t (b2 u a1) = 0 t (b2 u a1) = b2 u a1 ≤ b2 ≤ b1. So, a3 t b2 ≤ a1 ⊃ b1 (2) On the other hand, (a3tb2)ub3 = (a3ub3)t(b2ub3) ≤ (a3ub3)t(b1ub3) = (a3ub3)t0 = a3 u b3 ≤ a3. So, a3 t b2 ≤ b3 ⊃ a3 (3) From (1), (2) and (3), we have a3 t b2 ≤ (a1 ⊃ b1) u (a2 ⊃ b2) u (b3 ⊃ a3).  Proposition 4.4 : If (a1, a2, a3), (b1, b2, b3) ∈ BTmA , then (a1 ⊃ b1) u (a2 u b3) = 0. 9 Proof: Straightforward.  Corollary 4.5 If (a1, a2, a3), (b1, b2, b3) ∈ BTmA then ∅ 6= (a1, a2, a3) → (b1, b2, b3) ⊆ BTmA . Remark 4.6 Observe that, in the case of the two-element Boolean algebra A2 with domain A2 = {0, 1}, eight triples are possible: (1, 1, 1), (1, 0, 1), (0, 1, 0), (0, 1, 1), (1, 1, 0), (1, 0, 0), (0, 0, 0) and (0, 0, 1). However, in order to satisfy the requirements of Definition 4.1, the snapshots (0, 1, 0), (0, 1, 1), (1, 1, 1) and (1, 0, 1) must be discarded. So, BTmA2 = {(1, 1, 0), (1, 0, 0), (0, 0, 0), (0, 0, 1)}. By identifying the values (1, 1, 0), (1, 0, 0), (0, 0, 0) and (0, 0, 1) with T , t, f and F , respectively, it is immediate to see that the multioperations of the Nmatrix MTm for Tm coincide with the ones for BTmA2 . Definition 4.7 (Swap structures for T4m) A swap structure for T4m over A is a swap structure for Tm where the multioperation  is given by (a1, a2, a3) = {(c1, c2, c3) ∈ B : c1 = a2 and a2 ≤ c2} for every (a1, a2, a3) in B. Definition 4.8 (Swap structures for T45m) A swap structure for T45m over A is a swap structure for Tm such that the multioperation  is given by (a1, a2, a3) = {(c1, c2, c3) ∈ B : c1 = a2, a2 ≤ c2 and c3 t c1 = 1} for every (a1, a2, a3) in B. Definition 4.9 (Swap structures for TBm) A swap structure for TBm over A is a swap structure for Tm such that the multioperation  is given by (a1, a2, a3) = {(c1, c2, c3) ∈ B : c1 = a2 and a1 t c3 = 1} for every (a1, a2, a3) in B. Remark 4.10 As in the case of Tm, assume that a snapshot z = (a1, a2, a3) represents the truth-values of φ, φ and ¬φ. Then, the multioperation associated to  in the previous definitions is natural under such interpretation: In the case for T4m, the additional axiom (4) imposes the condition a2 ≤ c2 in Definition 4.7. In the case for T45m, the additional axioms (4) and (5) impose the conditions a2 ≤ c2 and c3 t c1 = 1 in Definition 4.8. Finally, in the case for TBm, the axiom (B) imposes the condition a1 t c3 = 1 in Definition 4.9. Remark 4.11 Let L ∈ { T4m,T45m,TBm } and consider again the case of the twoelement Boolean algebra A2. Thus, BLA2 = B Tm A2 = {(1, 1, 0), (1, 0, 0), (0, 0, 0), (0, 0, 1)}. The unique swap structure for L with domain BTmA will be denoted by BLA. It will be called the full swap structure for L over A. As in the case of Tm, its multioperations coincide with the ones for the Nmatrix ML for L. Definition 4.12 (Swap structures for Dm) Let A = 〈A,⊃,t,u,∼, 0, 1〉 be a Boolean algebra and let BDmA = {(a1, a2, a3) ∈ A3 : a2 u a3 = 0}. A swap structure for Dm over A is any multialgebra B = 〈B, →, ¬, 〉 over Σ′ = {→,¬,} such that B ⊆ BDmA and the multioperations satisfy the following, for every (a1, a2, a3) and (b1, b2, b3) in B: 10 (i) (a1, a2, a3) → (b1, b2, b3) = {(c1, c2, c3) ∈ B : c1 = a1 ⊃ b1, c3 = a2 u b3 and a3 t b2 ≤ c2 ≤ (a2 ⊃ b2) u (b3 ⊃ a3)} (ii) ¬(a1, a2, a3) = {(∼a1, a3, a2)} (iii) (a1, a2, a3) = {(c1, c2, c3) ∈ B : c1 = a2} Remark 4.13 As in the case of Tm, the intuition behind the universe BDmA is the following: if z = (a1, a2, a3) is a snapshot in a given swap structure B for Dm then a1, a2 and a3 represent a possible truth-value (over the Boolean algebra A) for φ, φ and ¬φ, respectively, for a given formula φ. By axiom (D) it follows that φ → ¬¬φ is valid, which means that ¬(φ∧¬φ) is valid under such interpretation. But the latter is equivalent to say that a2 u a3 = 0. With respect to the multioperations in a swap structure for Dm: ¬ and  are defined as in Definition 4.1, and → is also defined as in Definition 4.1, but now without requiring that c2 ≤ (a1 ⊃ b1). As in the case of Tm, it will be shown that the implication (a1, a2, a3) → (b1, b2, b3) in Dm returns a non-empty set, for (a1, a2, a3), (b1, b2, b3) ∈ BDmA : Proposition 4.14 : If (a1, a2, a3), (b1, b2, b3) ∈ BDmA , then a3 t b2 ≤ (a2 ⊃ b2) u (b3 ⊃ a3). Proof: Let be (a1, a2, a3), (b1, b2, b3) ∈ BDmA . We have a3 u ∼a2 = 0 t (a3 u ∼a2) = (a3 u a2) t (a3 u ∼a2) = a3 u (a2 t ∼a2) = a3 u 1 = a3. But, a3 u ∼a2 = a3 iff a3 ≤ ∼a2. Then a3 t b2 ≤ ∼a2 t b2 = a2 ⊃ b2 (4) On the other hand, b2 u∼b3 = 0t (b2 u∼b3) = (b2 u b3)t (b2 u∼b3) = b2 u (b3 t∼b3) = b2 u 1 = b2. But, b2 u ∼b3 = b2 iff b2 ≤ ∼b3. Then a3 t b2 ≤ a3 t ∼b3 = b3 ⊃ a3 (5) From (4) and (5), a3 t b2 ≤ (a2 ⊃ b2) u (b3 ⊃ a3).  Proposition 4.15 If (a1, a2, a3), (b1, b2, b3) ∈ BDmA , then (a3 t b2) u (a2 u b3) = 0. Proof: (a3tb2)u(a2ub3) = (a2ub3ua3)t(a2ub3ub2) = (0ub3)t(a2u0) = 0t0 = 0.  Corollary 4.16 If (a1, a2, a3), (b1, b2, b3) ∈ BDmA then ∅ 6= (a1, a2, a3) → (b1, b2, b3) ⊆ BDmA . The unique swap structure for Dm with domain BDmA will be denoted by BDmA . It will be called the full swap structure for Dm over A. Remark 4.17 Recall that L denotes the set { Tm,T4m,T45m,TBm,Dm } (see Notation 2). In the case of the two-element Boolean algebra A2, it is clear that BDmA2 = BTmA2 ∪ {(1, 0, 1), (0, 1, 0)} such that (1, 0, 1) and (0, 1, 0) correspond to the "new" truthvalues F+ and T−, respectively. As in the four-valued cases, the multioperations of the Nmatrix MDm for Dm coincide with the ones for BDmA2 . It is possible to give an intuitive reading for the definition of the multioperations in the class of swap structures for any L ∈ L \ {Tm}, analogous to the analysis for Tm found in Remark 4.2. 11 Definition 4.18 If L ∈ L, we denote by KL the class of swap structures for L. Definition 4.19 Let L ∈ L and let DB = {(z1, z2, z3) ∈ |B| : z1 = 1} for each B ∈ KL. The non-deterministic matrix associated to B is M(B) = 〈B, DB〉. Definition 4.20 If L ∈ L, then we denote by Mat(KL) = {M(B) : B ∈ KL} the class of non-deterministic matrices associated to the class of swap structures for the system L. Recall the semantics associated to non-deterministic matrices where the relation ∆ |=M α is the consequence relation defined over a non-deterministic matrix M (see Definition 2.7): Definition 4.21 For L ∈ L, let ∆ ∪ {α} ⊆ For(Σ′) be a set of formulas of L. We say that α is a consequence of ∆ in the class Mat(KL) of non-deterministic matrices, and we denote it by ∆ |=Mat(KL) α, if ∆ |=M α for every M∈Mat(KL). In particular, α is valid in Mat(KL), denoted by |=Mat(KL) α, if it is valid in every M∈Mat(KL). Theorem 4.22 (Deduction-detachment theorem) Let L ∈ L. Then, for every set of formulas ∆ ∪ {α, β}: ∆ ∪ {α} `L β iff ∆ `L α→ β. Proof: Since every system in L is an axiomatic extension of classical propositional logic in which Modus Ponens is the only inference rule, the result follows easily.  Corollary 4.23 Let L ∈ L and letML be the Nmatrix for L. Then, for every ∆∪{α} ⊆ For(Σ′), ∆ `L α⇔ ∆ ML α. Proof: It follows from theorems 3.4 and 4.22, and the fact that the syntactical consequence relation `L for each L is finitary (since it is given by a standard Hilbert calculus).  Lemma 4.24 Let πi : A 3 → A be the canonical projection on the ith coordinate, namely πi((a1, a2, a3)) = ai for i = 1, 2, 3. Let α, β ∈ For(Σ′) and let v be any valuation for a swap structure for L ∈ L (in the sense of Definition 2.5). Then, i) π1(v(α→ β)) = π1(v(α)) ⊃ π1(v(β)); ii) π1(v(¬α)) = ∼π1(v(α)). Proof: Immediate from the definitions.  Theorem 4.25 (Soundness) Let L ∈ L. For every ∆ ∪ {φ} ⊆ For(Σ′): if ∆ `L α then ∆ Mat(KL) α. 12 Proof: Let v be a valuation for a swap structure B for L such that v(γ) ∈ DB for every γ ∈ ∆. We will prove, by induction on the length of a given deduction of φ from ∆ in L, that v(φ) ∈ DB. In order to do this, it is enough to verify that: 1) v(γ) ∈ DB for every instance of an axiom of L; and 2) (MP) preserves trueness, that is: if v(γ → δ) ∈ DB and v(γ) ∈ DB then v(δ) ∈ DB. To simplify the proof, the following notation will be adopted: if γ is a formula and v is a valuation, we will write |γ|i instead of πi(v(γ)), for i = 1, 2, 3. Item 2) is obviously true, by the very definitions. Indeed, suppose that v(γ → δ) ∈ DB and v(γ) ∈ DB. By definition of DB and by Lemma 4.24, |γ → δ|1 = |γ|1 ⊃ |δ|1 = 1, and |γ|1 = 1. Then 1 = |γ|1 ≤ |δ|1 whence |δ|1 = 1 as well.6 In order to conclude the proof, item 1) must be verified. The task of checking the axioms will be divided into three parts: Part 1: If α is an axiom in {(Ax1), (Ax2), (Ax3), (K2), (M1), (M2), (M3), (M4), (DN1), (DN2)}, the proof is the same for any system L. The proof for (Ax1), (Ax2) and (Ax3) is immediate, by definition of DB and by Lemma 4.24. For the other axioms: • If α is (K2) ¬¬(δ → β)→ (δ → ¬¬β): note that |¬¬(δ → β)|1 = ∼|¬(δ → β)|1 = ∼|¬(δ → β)|2 = ∼|(δ → β)|3 = ∼(|δ|2u|β|3) = ∼|δ|2t∼|β|3 = |δ|2 ⊃ ∼|β|3. And |δ → ¬¬β|1 = |δ|1 ⊃ |¬¬β|1 = |δ|2 ⊃ ∼|¬β|1 = |δ|2 ⊃ ∼|¬β|2 = |δ|2 ⊃ ∼|β|3. In particular, |¬¬(δ → β)|1 ≤ |δ → ¬¬β|1. • If α is (M1) ¬δ → (δ → β): then |¬δ|1 = ∼|δ|2 = |δ|3 and |(δ → β)|1 = |δ → β|2. But |δ|3 ≤ |δ|3 t |β|2 ≤ |δ → β|2. Therefore, |¬δ|1 ≤ |(δ → β)|1. • If α is (M2) β → (δ → β): then |β|1 = |β|2 and |(δ → β)|1 = |δ → β|2. On the other hand, |β|2 ≤ |δ|3 t |β|2 ≤ |δ → β|2. From this, |β|1 ≤ |(δ → β)|1. • If α is (M3) ¬(δ → β)→ ¬β: then |¬β|1 = |¬β|2 = |β|3 and |¬(δ → β)|1 = |¬(δ → β)|2 = |δ → β|3 = |δ|2u|β|3. So, |¬(δ → β)|1 = |δ|2u|β|3 ≤ |β|3 = |¬β|1. • If α is (M4) ¬(δ → β)→ δ: note that |δ|1 = |δ|2 and |¬(δ → β)|1 = |¬(δ → β)|2 = |δ → β|3 = |δ|2 u |β|3. Then, |¬(δ → β)|1 = |δ|2 u |β|3 ≤ |δ|2 = |δ|1. • If α is (DN1) δ → ¬¬δ or (DN2) ¬¬δ → δ: we have |δ|1 = |δ|2 and |¬¬δ|1 = |¬¬δ|2 = |¬δ|3 = |δ|2. In particular, |δ|1 ≤ |¬¬δ|1 and |¬¬δ|1 ≤ |δ|1. Part 2: If α is an axiom in {(K), (K1)} then the proof is the same for any system L ∈ L \ {Dm}. So, the proof of each one will be divided into two parts. • If α is (K) (δ → β)→ (δ → β) and if L ∈ L \ {Dm}: we have |(δ → β)|1 = |δ → β|2. So, |δ|3 t |β|2 ≤ |δ → β|2 ≤ (|δ|1 ⊃ |β|1) u (|δ|2 ⊃ |β|2) u (|β|3 ⊃ |δ|3). On the other hand, |δ → β|1 = |δ|1 ⊃ |β|1 = |δ|2 ⊃ |β|2. So, |(δ → β)|1 ≤ (|δ|1 ⊃ |β|1) u |δ → β|1 u (|β|3 ⊃ |δ|3). Therefore, |(δ → β)|1 ≤ |δ → β|1. If L = Dm, from |(δ → β)|1 = |δ → β|2 we have |δ|3 t |β|2 ≤ |δ → β|2 ≤ (|δ|2 ⊃ |β|2)u (|β|3 ⊃ |δ|3). On the other hand, |δ → β|1 = |δ|1 ⊃ |β|1 = |δ|2 ⊃ |β|2. So, |(δ → β)|1 ≤ |δ → β|1 u (|β|3 ⊃ |δ|3). Therefore, |(δ → β)|1 ≤ |δ → β|1. 6Recall that, in any Boolean algebra, a ⊃ b = 1 iff a ≤ b. 13 • If α is (K1) (δ → β)→ (¬β → ¬δ): the proof is quite similar to the previous case. Part 3: In this part, we will check the specific axioms of each system. • If α is (T) δ → δ and L ∈ L \ {Dm}: as |δ|1 = |δ|2 then, by definition of BTmA , we have |δ|2 ≤ |δ|1. So, |δ|1 ≤ |δ|1. • If α is (D) δ → ¬¬δ and L = Dm: note that |δ|1 = |δ|2 and |¬¬δ|1 = ∼|¬δ|1 = ∼|¬δ|2 = ∼|δ|3 and by definition of BDmA , we have |δ|2 u |δ|3 = 0. From this |δ|2 ≤ ∼|δ|3. Therefore, |δ|1 ≤ |¬¬δ|1. • If α is (B) ¬¬δ → δ and L = TBm: since |¬¬δ|1 = ∼|¬δ|1 = ∼|¬δ|2 = ∼|δ|3, such that |δ|1 t |δ|3 = 1 (by Definition 4.9). But, then ∼|δ|3 ⊃ |δ|1 = 1. And so, ∼|δ|3 ≤ |δ|1. Therefore, |¬¬δ|1 ≤ |δ|1. • If α is (5) ¬¬δ → δ and L = T45m: since |¬¬δ|1 = ∼|¬δ|1 = ∼|¬δ|2 = ∼|δ|3, such that |δ|1 t |δ|3 = 1 (by Definition 4.8). But, then ∼|δ|3 ⊃ |δ|1 = 1. And so, ∼|δ|3 ≤ |δ|1. Therefore, |¬¬δ|1 ≤ |δ|1. • If α is (4) δ → δ and L ∈ {T4m, T45m}: as |δ|1 = |δ|2 such that (by Definitions 4.7 and 4.8) |δ|2 ≤ |δ|2 and |δ|1 = |δ|2. So, |δ|1 ≤ |δ|1.  Definition 4.26 Let L ∈ L and let ∆ be a theory in L. The relation ≡L∆ between formulas of L is defined as follows: α ≡L∆ β iff ∆ `L α→ β and ∆ `L β → α. Proposition 4.27 Let L ∈ L. Then, the relation ≡L∆ is a congruence w.r.t. connectives in Σ′′ = {→,¬}. Proof: It follows from the fact that L contains classical logic over the signature Σ′′.  Notation 5 If L ∈ L and ≡L∆ is the congruence defined above, then [α]∆ = α/≡L∆ = {β ∈ For(Σ′) : α ≡L∆ β} is the equivalence class of α ∈ For(Σ′), while For(Σ′)/≡L∆ = {α/≡L∆ : α ∈ For(Σ′)} denotes the set of all the equivalence classes. By Proposition 4.27, ≡L∆ is a congruence w.r.t. the operations in Σ′′ = {→,¬}. However, the connective  is not congruential: Proposition 4.28 Let L ∈ L. Then, the connective  is not congruential, that is: there are formulas α and β such that α ≡L∆ β but α 6≡L∆ β. Proof: The proof will be divided into two cases, the first for L ∈ L \ {Dm} and the second for L = Dm. So, if L ∈ L \ {Dm}, then we have p ≡∆ ¬p → p but p 6≡∆ (¬p → p), because there exist a valuation v : For(Σ′)→ {T, t, f, F} for L, such that: 14 • If L ∈ {Tm, TBm}, then v(p) = t and for v(¬p→ p) = T , we have v(p) ∈ {F, f}, but v((¬p→ p)) ∈ {T, t}; • If L = T4m, then v(p) = t and for v(¬p → p) = T , we have v(p) ∈ {F, f}, but v((¬p→ p)) ∈ {T}; • If L = T45m, then v(p) = t and for v(¬p → p) = T , we have v(p) ∈ {F}, but v((¬p→ p)) ∈ {T}. If L = Dm, we have once again that p ≡∆ ¬p → p but, p 6≡∆ (¬p → p). Indeed, there exist a valuation v : For(Σ′) → {T+, T−, C+, C−, F+, F−} for Dm such that if v(p) = C+ and for v(¬p → p) = T+, we have v(p) ∈ {T−, F−, C−}, but v((¬p→ p)) ∈ {T+, F+, C+}.  By means of this example it follows that the connective  does not induce a welldefined operator on For(Σ′)/≡L∆ . As Carnielli and Coniglio did in [6, Chapter 6], in order to circumvent a similar problem with the operators ◦ and ¬ in the context of LFIs, the connective  will be interpreted as a multioperation instead as an algebraic operator, by defining a swap structure over the Boolean algebra generated by the Lindenbaum-Tarski equivalence relation ≡L∆ (see Definition 4.31). Proposition 4.29 Let L ∈ L and let ∆ be a theory in L. Let AL∆ = 〈For(Σ′)/≡L∆ ,⊃,t,u,∼, 0∆, 1∆〉 be the structure given by [α]∆ ⊃ [β]∆ = [α→ β]∆ [α]∆ t [β]∆ = [α ∨ β]∆ [α]∆ u [β]∆ = [α ∧ β]∆ ∼[α]∆ = [¬α]∆ 0∆ = [¬(α→ α)]∆ 1∆ = [α→ α]∆. Then, AL∆ is a Boolean algebra. Proof: The operations and the constants are well defined, by Proposition 4.27. Since L contains classical logic, the rest of the proof is straightforward.  Remark 4.30 It is worth noting that, if ∆ is the trivial theory For(Σ′), then AL∆ is the trivial Boolean algebra with a single element. Definition 4.31 Let L ∈ L and let ∆ be a theory in L. Given the Boolean algebra AL∆, the unique swap structure over AL∆ with domain BLAL∆ is called the Lindenbaum-Tarski swap structure for ∆ in L, and it will denoted by BL∆. Definition 4.32 (Canonical Nmatrix) Let L ∈ L and let ∆ be a theory in L. Let BL∆ be the Lindenbaum-Tarski swap structure for ∆ in L. The canonical Nmatrix associated to ∆ in L is M(BL∆), which will be denoted by ML∆ = 〈BL∆, DL∆〉. 15 Lemma 4.33 Let L ∈ L and let AL∆ be as in Proposition 4.29. Then [α]∆ = 1∆ iff ∆ `L α. Proof: It is immediate.  Lemma 4.34 Let L ∈ L and let AL∆ be as in Proposition 4.29. Then, the following conditions are equivalent: (i) [α]∆ ≤ [β]∆ (ii) [α]∆ ⊃ [β]∆ = 1∆ (iii) [α→ β]∆ = 1∆ (iv) ∆ `L α→ β Proof: It is immediate from the definitions and from the properties of Boolean algebras.  Lemma 4.35 Let L ∈ L and let AL∆ be as in Proposition 4.29. The following holds in the indicated Boolean algebras: (i) [¬¬α]∆ = [α]∆ in AL∆, for L ∈ L; (ii) [α]∆ ≤ [α]∆ in AL∆, for L ∈ {T4m, T45m}; (iii) [¬α]∆ t [α]∆ = 1∆ in AT45m∆ ; (iv) [α]∆ t [¬α]∆ = 1∆in ATBm∆ ; (v) [¬(α→ β)]∆ = [α]∆ u [¬β]∆ in AL∆, for L ∈ L; (vi) [¬α]∆ t [β]∆ ≤ [(α→ β)]∆ in AL∆, for L ∈ L; (vii) [(α→ β)]∆ ≤ [α]∆ ⊃ [β]∆ in AL∆, for L ∈ L \ {Dm}; (viii) [(α→ β)]∆ ≤ [α]∆ ⊃ [β]∆ in AL∆, for L ∈ L; (ix) [(α→ β)]∆ ≤ [¬β]∆ ⊃ [¬α]∆ in AL∆, for L ∈ L. Proof: (i) [¬¬α]∆ = [α]∆ iff ∆ `L ¬¬α → α and ∆ `L α → ¬¬α. But the latter holds by axioms (DN1) and (DN2). (ii) [α]∆ ≤ [α]∆ iff ∆ `L α→ α. But this holds by Axiom (4). (iii) [¬α]∆ t [α]∆ = 1∆ iff [¬α ∨ α]∆ = 1∆ iff [¬¬α → α]∆ = 1∆ iff ∆ `T45m ¬¬α→ α. But this holds by Axiom (5). (iv) [α]∆ t [¬α]∆ = 1∆ iff [α ∨ ¬α]∆ = 1∆ iff [¬¬α → α]∆ = 1∆ iff ∆ `TBm ¬¬α→ α. But this holds by Axiom (B). 16 (v) [¬(α → β)]∆ = [α]∆ u [¬β]∆ iff [¬(α → β)]∆ = [α ∧ ¬β]∆ iff ∆ `L ¬(α→ β)→ (α ∧¬β) and ∆ `L (α ∧¬β)→ ¬(α→ β). See the derivation below: 1. ∆ `L ¬¬(α→ β)→ (α→ ¬¬β) (K2) 2. ∆ `L ¬(α→ ¬¬β)→ ¬¬¬(α→ β) 1, CPC 3. ∆ `L ¬(¬α ∨ ¬¬β)→ ¬(α→ β) 2, CPC 4. ∆ `L (¬¬α ∧ ¬¬¬β)→ ¬(α→ β) 3, CPC 5. ∆ `L (α ∧¬β)→ ¬(α→ β) 4, CPC Now, consider the following derivation: 1. ∆ `L ¬(α→ β)→ ¬β (M3) 2. ∆ `L ¬(α→ β)→ α (M4) 3. ∆ `L ¬(α→ β)→ (α ∧¬β) 1,2, CPC (vi) [¬α]∆ t [β]∆ ≤ [(α → β)]∆ iff [¬α ∨β]∆ ≤ [(α → β)]∆ iff ∆ `L (¬α ∨ β)→ (α→ β). See the derivation below: 1. ∆ `L ¬α→ (α→ β) (M1) 2. ∆ `L β → (α→ β) (M2) 3. ∆ `L (¬α ∨β)→ (α→ β) 1,2, CPC (vii) [(α → β)]∆ ≤ [α]∆ ⊃ [β]∆ iff [(α → β)]∆ ≤ [α → β]∆ iff ∆ `L (α → β) → (α→ β). But this is a consequence of Axiom (T). (viii) [(α → β)]∆ ≤ [α]∆ ⊃ [β]∆ iff [(α → β)]∆ ≤ [α → β]∆ iff ∆ `L (α → β)→ (α→ β). But this follows from Axiom (K). (ix) [(α → β)]∆ ≤ [¬β]∆ ⊃ [¬α]∆ iff [(α → β)]∆ ≤ [¬β → ¬α]∆ iff ∆ `L (α→ β)→ (¬β → ¬α). But this is a consequence of Axiom (K1).  Proposition 4.36 Let L ∈ L. Then, the function vL∆ : For(Σ′)→ BLAL∆ given by v L ∆(α) = ([α]∆, [α]∆, [¬α]∆) is a valuation in ML∆. That is, for every α, β ∈ For(Σ′): (i) vL∆(¬α) ∈ ¬vL∆(α); (ii) vL∆(α) ∈ vL∆(α); (iii) vL∆(α→ β) ∈ vL∆(α) → vL∆(β). Proof: (i) vL∆(¬α) = ([¬α]∆, [¬α]∆, [¬¬α]∆). If L ∈ L then, by Proposition 4.27, ¬vL∆(α) = ¬([α]∆, [α]∆, [¬α]∆) = {([¬α]∆, [¬α]∆, [α]∆)} and by Lemma 4.35 (i), [¬¬α]∆ = [α]∆. This implies that vL∆(¬α) ∈ ¬vL∆(α); 17 (ii) vL∆(α) = ([α]∆, [α]∆, [¬α]∆). If L belongs to {Tm, Dm}, vL∆(α) = ([α]∆, [α]∆, [¬α]∆) = {([β1]∆, [β2]∆, [β3]∆) ∈ BL∆ : [β1]∆ = [α]∆}. If L = T4m, vL∆(α) = ([α]∆, [α]∆, [¬α]∆) = {([β1]∆, [β2]∆, [β3]∆) ∈ BL∆ : [β1]∆ = [α]∆ and [α]∆ ≤ [β2]∆} and by Lemma 4.35 (ii) [α]∆ ≤ [α]∆. If L = T45m, vL∆(α) = ([α]∆, [α]∆, [¬α]∆) = { ([β1]∆, [β2]∆, [β3]∆) ∈ BL∆ : [β1]∆ = [α]∆, [α]∆ ≤ [β2]∆ and [β3]∆ t [β1]∆ = 1∆ } and by Lemma 4.35 (ii) and (iii) [α]∆ ≤ [α]∆ and [¬α]∆ t [α]∆ = 1∆. If L = TBm, vL∆(α) = ([α]∆, [α]∆, [¬α]∆) = {([β1]∆, [β2]∆, [β3]∆) ∈ BL∆ : [β1]∆ = [α]∆ and [α]∆t[β3]∆ = 1∆} and by Lemma 4.35 (iv) [α]∆t[¬α]∆ = 1∆. (iii) vL∆(α→ β) = ([α→ β]∆, [(α→ β)]∆, [¬(α→ β)]∆). If L ∈ L \ {Dm}, vL∆(α) → vL∆(β) = ([α]∆, [α]∆, [¬α]∆) → ([β]∆, [β]∆, [¬β]∆) ={ ([γ1]∆, [γ2]∆, [γ3]∆) ∈ BLA∆ : [γ1]∆ = [α]∆ ⊃ [β]∆, [γ3]∆ = [α]∆ u [¬β]∆ and [¬α]∆ t [β]∆ ≤ [γ2]∆ ≤ ([α]∆ ⊃ [β]∆) u ([α]∆ ⊃ [β]∆)u ([¬β]∆ ⊃ [¬α]∆) } . By Lemma 4.35 (v) and (vi), we have [¬(α → β)]∆ = [α]∆ u [¬β]∆ and [¬α]∆ t [β]∆ ≤ [(α→ β)]∆. And by Lemma 4.35 (vii), (viii) and (ix), we have [(α→ β)]∆ ≤ ([α]∆ ⊃ [β]∆) u ([α]∆ ⊃ [β]∆) u ([¬β]∆ ⊃ [¬α]∆). If L = Dm, the proof is quite similar to the previous cases.  Proposition 4.37 Let L ∈ L. Then, the function vL∆ : For(Σ′) → BLAL∆ given by vL∆(α) = ([α]∆, [α]∆, [¬α]∆) is a canonical valuation in ML∆, that is: vL∆(α) ∈ DL∆ iff ∆ `L α. Proof: By Proposition 4.36, vL∆(α) is a valuation inML∆. On the other hand, by the very definitions, vL∆(α) ∈ DL∆ iff [α]∆ = 1∆ iff ∆ `L α, by Lemma 4.33.  Definition 4.38 (Canonical model) Let L ∈ L and let ∆ be a theory in L. The function vL∆ is called the canonical valuation associated to ∆ in L. The pair (ML∆, vL∆) is called the canonical model associated to ∆ in L. The canonical model allows to prove the completeness of L ∈ L w.r.t. swap structures in a straightforward way: Theorem 4.39 (Completeness) For every L ∈ L and for every ∆ ∪ {α} ⊆ For(Σ′), if ∆ |=Mat(KL) α then ∆ `L α. 18 Proof: Suppose that ∆ 6`L α. Then, by Proposition 4.37, there exist a valuation vL∆ over the canonical NmatrixML∆ for L such that vL∆(β) ∈ DL∆ for every β ∈ ∆, but vL∆(α) 6∈ DL∆. From this, ∆ 6|=Mat(KL) α.  Corollary 4.40 Let L ∈ L and let ∆ ∪ {α} ⊆ For(Σ′). Then, are equivalent: 1. ∆ `L α; 2. ∆ Mat(KL) α; 3. ∆ ML∆ α; 4. ∆ M(BLA2 ) α. Proof: (1) ⇔ (2) By Theorems 4.25 and 4.39; (1) ⇔ (3) By Theorem 4.25 and the proof of Theorem 4.39; (1) ⇔ (4) By Corollary 4.23 and by remarks 4.6, 4.11 and 4.17.  Corollary 4.41 For L ∈ L and for every α ∈ For(Σ′), are equivalent: 1. `L α; 2. Mat(KL) α; 3. ML∆ α; 4. M(BLA2 ) α. The results stated in Corollary 4.40 show in a precise way that the class of Nmatrices associated to the swap structures for each modal logic L ∈ L generalizes the original Nmatrix semantics presented in [7] and [8]. Moreover, it also provides an interesting family of examples, besides the ones obtained for LFIs in [10], that the usual Linbenbaum-Tarski construction can be generalized to non-algebraizable logics, by considering multialgebras instead of ordinary algebras. 5 Concluding remarks and future research Non-deterministic semantic structures such as Nmatrices, swap structures and Fidel structures (see [6, Chapter 6]) can be considered as complementary to the algebraic structures associated to logics by means of the standard tools of abstract algebraic logic. Indeed, non-deterministic semantics can be applied to logics without non-trivial congruences, obtaining so algebraic-like interpretation for logics which lie outside the scope of the traditional approach to algebraic logic. It is worth observing that, as it was shown in [10], swap structures semantics for a given logic coincides with twist structures semantics, when the 19 logic is Blok-Pigozzi algebraizable. This justifies the study of this kind of structures from the standpoint of universal algebra and category theory, thus broadening the horizons of abstract algebraic logic. In this paper, new classes of swap structures were introduced as a suitable semantics for a family of non-normal modal systems. These classes of non-deterministic structures generalize the original semantical characterization of each of such logics by means of a single finite-valued non-deterministic matrix. Indeed, each finite characteristic Nmatrix is recovered from the respective class of models by considering the full swap structure over the two-element Boolean algebra. An analogous situation was observed in [6, Chapter 6] with respect to LFIs. Thus being so, the results obtained in the present paper suggest that a Birkhoff-like theorem for KL could be obtained for each L ∈ L, similar to the ones given in [10] for some LFIs. That is, to determine whether the finite-valued characteristic matrixML generates the whole classM(KL) is an interesting question to be analyzed in future research. In the present paper, just the deontic system Dm was analyzed under the perspective of swap structures. The extension of such structures to its axiomatic extensions D4m, D45m and DBm proposed in [7] and [8] should be obtained in a straightforward way. The 8-valued modal systems Km, K4m and K45m proposed in [9] can also be studied from the viewpoint of modal swap structures. It should be observed that, in principle, the nature of the snapshots associated to a given logic does not seem to be intrinsic to that logic. In the cases studied in the present paper, the snapshots are conceived as triples (a1, a2, a3) representing the truth values (over a given Boolean algebra A) of given formulas α, α and ¬α, respectively. But this choice seems to be rather arbitrary: it would be possible to change the interpretation of a3 by ¬¬α, that is, ♦α. It would produce a finite-valued characteristic Nmatrix M′L different to ML. It is an interesting topic of future research to relate the different classes of swap structures for a given logic which arise by modifying the nature of the corresponding snapshots. The structures presented in this paper constitute the first examples of swap structures defined for logics outside the scope of the LFIs, the class of paraconsistent logics for which swap structures were originally proposed. Based on the ideas of the completeness proofs of LFIs with respect to Fidel structures semantics found in [6, Chapter 6], a quotient swap structure that we called Lindenbaum-Tarski swap structure, as well as a canonical valuation over it, were proposed. A similar technique was already used in [10] in the context of swap structures for LFIs. It is interesting to observe that these results show a way in which the standard Lindenbaum-Tarski construction can be generalized to non-congruential logics. The key is considering, for the non-congruential connectives, multioperations (or relations, in the case of Fidel structures) instead of operations. We conjecture that this generalized Lindenbaum-Tarski technique could be applied to a wide class of non-algebraizable logics, offering an interesting new paradigm for algebraizing logics by means of multialgebras and Fidel structures. Acknowledgements Marcelo E. Coniglio was financially supported by a research grant from CNPq, Brazil (308524/2014-4). Ana Claudia Golzio was financially supported by scholarship grant from FAPESP, Brazil (2013/04568-1) and by a post-doctoral grant from CNPq, Brazil (150064/2018-7). 20 References [1] A. Avron. Non-deterministic matrices and modular semantics of rules. In J.-Y. Beziau, editor, Logica Universalis: Towards a General Theory of Logic, pages 149–167, Basel, 2005. Birkhäuser Basel. [2] A. Avron and B. Konikowska. Multi-valued calculi for logics based on nondeterminism. Logic Journal of the IGPL, 13(4):365–387, 2005. [3] A. Avron and I. Lev. Canonical propositional Gentzen-type systems. In Proceedings of the First International Joint Conference on Automated Reasoning, IJCAR '01, pages 529–544, London, UK, 2001. Springer-Verlag. [4] A. Avron and I. Lev. Non-deterministic multiple-valued structures. J. Log. and Comput., 15(3):241–261, June 2005. [5] A. Avron and A. Zamansky. 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