Fibring by Functions over Matrix Logics (Draft)

Fibring by Functions over Matrix Logics (Draft) Marcelo E. Coniglio1 and V´ıctor L. Fern´andez2 1

Center of Logic, Epistemology and the History of Science and Departament of Philosophy State University of Campinas P.O. Box 6133, 13081-970 Campinas, SP, Brazil E-mail: [email protected] 2

Basic Sciences Institute (Mathematical Area), Philosophy College National University of San Juan Av. Ignacio de la Roza 230 (O) San Juan, Argentina E-mail: [email protected]

Abstract In this article we introduce some operations for combining matrix logics, in the same spirit that Gabbay’s original formulation of fibring. The unrestricted fibring by functions became a simple generalization of the original definition of fibring, which produces a weak conservative extension of the given logics. In order to obtain a strong conservative extension of the given logics it is necessary to impose additional conditions to the fibring pairs. This originates the so-called plain fibring of logics. We can generalize these methods in a natural way, extending our first definition of fibring to sets of fibring pairs. The main results obtained with these new definitions encompasses the similar results for the simpler cases.

Introduction The method of combination of logics known as fibring was introduced by D. Gabbay in the 90’s, with the aim of combining logics having Kripke semantics, such as intuitionistic and modal logics (see [6] and [7]). The underlying idea of this technique is, roughly speaking, the following: given two logics L1 and L2 , with Kripke semantics Kr1 and Kr2 respectively), a new logic L1 ~L2 is defined over the language obtained from the union of the connectives of both L1 and L2 . So, the formulas of Li (i = 1, 2) are also formulas of L1 ~ L2 . But in the new logic there exist new formulas (that we call “hybrid formulas”), obtained by 1

mixing the connectives of the two original logics. Thus, it is possible to evaluate a connective c of the language of L1 applied to formulas of the language of L2 . This can be done by means of a pair of functions: the first one associates, to each world of any model of Kr1 , a world of a model of Kr2 , while the second one works analogously from Kr2 to Kr1 . Using this method, the hybrid formulas of the new language can always be evaluated. Moreover, L1 ~ L2 is a weak extension of L1 and L2 , in the sense that all the tautologies of the original logics are tautologies of the new logic. As much as modal logics (or intuitionistic logic) are considered this is enough, since these logics are characterized through theoremhood. Right after the introduction of fibring in the literature, the original notion was investigated and modified by several authors. In particular, an interesting adaptation of fibring using the framework of Category Theory was introduced in [11] and widely studied afterwards (see for instance [17] or [1]). However, in the present article, we return to the original definition of fibring (which, according to [14], we will call fibring by functions), with a small modification: instead of dealing with Kripke semantics, we will work with logics induced by matrix semantics. This kind of semantics is usually associated to the method of “truth-tables” for propositional logics, but is somewhat more general. The abstract theory of logical matrices goes back to the seminal paper [10], and was posteriorly developed by the Polish school of logic (see [9] [15], for example) 1 . In this paper we will work with both approaches to logical matrices: the “truth-table” approach and the abstract one. Briefly, the method proposed here (that is motivated by the preliminary article [3]) is defined as follows. Firstly, consider two logics Li , defined by the sets of connectives Ci (i = 1, 2), whose respective consequence relations |=Li are defined by matrices Mi = (Ai , Di ). To define a new “combined logic” (indicated here as L1 ~(λ,µ) L2 ), the new language is obtained from the given ones of both logics. Now, consider a fixed pair (λ, µ) of functions from A1 to A2 and vice versa. These functions will allow us “to jump” from a valuation of M1 to a valuation of M2 and vice-versa. Then, we can obtain the truth-value of any hybrid formula of the mixed language. In this way, we can define a consequence relation |=(λ,µ) in the new language. This new consequence relation is, precisely, the fibring of the original ones. Of course, we can generalize this basic idea in different ways: for example, by considering not just a pair of functions (λ, µ), but a set of pairs. Also, we can work with classes of matrices instead of just one matrix for each original logic. These generalizations are also be treated in this article. In addition, we study some technical questions, such as the identification of two truth-values in the combined logic, and the identification of two different connectives in L1 ~(λ,µ) . The following two sections are devoted to introduce the basic notions about propositional logics and matrix semantics. 1 Very complete surveys on matrix logics can be found in [16] and in [4], whose formalism is the basis of our work.

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1

Preliminaries

Basically, our notation and preliminary concepts are the usual for the abstract theory of matrix logics, and follow mainly the books [16] and[4]. The fundamental concepts for this paper are the following: Definition 1.1 (a) The set of propositional variables V is any countable set (fixed from now on). Its elements will be denoted by p, q, r, p1 , p2 , . . . . A signature is a family C = {C k }k∈N , where each C k is a set of connectives of arity k. We assume that n C k ∩ CS = ∅ = C k ∩ V for every k 6= n. The domain of the signature C is the set |C| = k∈N C k . Given two signatures C1 and C2 , we say that C1 is included in C2 (indicated by C1 ⊆ C2 ) if, for every k ∈ N, C1k ⊆ C2k . The signatures C1 ∪ C2 and C1 ] C2 (the union and the disjoint union of C1 and C2 , respectively) are defined as expected. (b) A propositional language with signature C (denoted by L(C)) is the algebra of words, freely generated by C over V, considering each set C k as the set of k-ary operations of that algebra. It should be noted that it is possible to have C 6= C 0 such that L(C) = L(C 0 ). By simplicity, sometimes we will identify a signature C with its domain |C|. Now, for the following definition, we consider algebras that are similar to the algebra L(C), that is, algebras having their operations interpreting the connectives in C. For simplicity, these operations are denoted by the same symbols that the corresponding connectives in C. Definition 1.2 Given a signature C, a C-matrix is a pair M = (A, D), where A = (A, C) is an algebra similar to L(C), and D ⊆ A If A = (A, C) is an algebra similar to L(C), we will denote A simply by its universe A (when there is no risk of confusion). From the notion of C-matrix we obtain consequence relations for the languages L(C). That is, we can define propositional logics from C-matrices, in the following way: Definition 1.3 Let C be a signature, and consider M = (A, D) a C-matrix. A M -valuation is any function v : V−→A. Since L(C) is a free algebra we can consider that a M -valuation is just a homomorphism v : L(C)−→A. The consequence relation |=M ⊆ ℘(L(C)) × L(C) induced by M is given by: (Γ, α) ∈ |=M iff, for every M -valuation v, if v(Γ) ⊆ D, then v(α) ∈ D. As usual, we indicate (Γ, α) ∈ |=M by Γ |=M α. This definition can be naturally generalized as follows. If K is a class of C-matrices, the consequence relation |=K ⊆ ℘(L(C)) × L(C) is defined as: Γ |=K α iff, for every C-matrix M in K, Γ |=M α. Note that, if K = {M }, then |=K = |=M .

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Definition 1.4 A matrix logic is a pair L = (C, |=K ), where C is a signature, and |=K is the consequence relation as in Definition 1.3, being K a class of C-matrices. A matrix logic L can also be characterized as a pair L = hC, Ki instead of a pair hC, |=K i, for a class K of C-matrices. In particular, if K = {M }, we will write hC, M i. Some well-know facts about matrix logics are the following: Proposition 1.5 Let L = hC, |=K i be a matrix logic. For every Γ∪{ϕ} ⊆ L(C) we have: - If ϕ ∈ Γ then Γ |=K ϕ

(Extensiveness).

- If Γ |=K ϕ and Σ |=K ψ, for every ψ ∈ Γ, then Σ |=K ϕ - If Γ |=K ϕ and Γ ⊆ Σ then Σ |=K ϕ

(Transitivity).

(Monotonicity)2 .

Another important property of matrix logics is structurality: Definition 1.6 A substitution in L(C) is any function σ : V−→L(C). Again, since L(C) is an absolutely free algebra, we can consider a substitution as an endomorphism σ : L(C)−→L(C). A logic L = hC, |=K i is structural iff, for every substitution σ, if Γ |=K α then σ(Γ) |=K α. Proposition 1.7 For every signature C, for every matrix consequence relation |=K , the logic L = hC, |=K i is structural.

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Direct union of matrix logics

In the sequel we introduce an adaptation of fibring to logics defined by means of matrix semantics. An important feature of this process of combining logics to be defined below is that the obtained logic will depend on the choice of the matrices characterizing the semantics of the given logics. The basic idea is the following: given two matrix logics L1 and L2 such that Li is characterized by a single matrix Mi with domain Ai and designated values Di (i = 1, 2), it is possible to extend the original operators of the algebra Mi to the disjoint union A1 ] A2 by means of mappings fi : Aj → Ai (i 6= j). In this section we briefly analyze the simpler case in which the domain and designated values of the matrices involved are the same. In such cases, the combined logic can be simply obtained by putting together both matrices. This technique is what we call direct union of matrix logics. Formally: 2 This shows that |= K (and, in particular, |=M ) is a consequence relation, in the sense of [9], [16] or [4]. This justifies the nomenclature of Definition 1.3.

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Definition 2.1 Let Li = hCi , Mi i (i = 1, 2) be two matrix logics, where each Mi = hAi , Di i is a Ci -matrix. Assume that A1 = A = A2 and D1 = D = D2 .3 The direct union of L1 and L2 is the logic L1 + L2 = hC1 ] C2 , |=M1 +M2 i where |=M1 +M2 is defined by the C1 ]C2 -matrix M1 +M2 = hA, Di such that, if c ∈ Cik and a1 , . . . , ak ∈ A, then cM1 +M2 (a1 , . . . , ak ) = cMi (a1 , . . . , ak ) (i = 1, 2). Proposition 2.2 Let L = hC, |=i be a logic such that |= is characterized by a C-matrix M . Let L1 and L2 be two fragments of L defined over signature C1 and C2 , respectively, such that C1 ] C2 = C. Then L1 + L2 = L. In particular, the same result holds if C1 ∪ C2 = C. Proof: Straightforward, from the definitions.4



Example 2.3 Let Li = hCi , Mi i (with i = 1, 2) such that |C1 | = {¬, ∨} (negation and disjunction, respectively) and |C2 | = {∧, →} (conjunction and implication, respectively). Suppose that M1 is the matrix for classical negation and disjunction, and that M2 is the matrix for classical conjunction and implication, where both matrices are defined over A = {1, 0} with D = {1}. Then L1 + L2 turns out to be the matrix presentation L of classical propositional logic over A and D and signature {¬, ∨, ∧, →}. The logics L1 and L2 are two (simpler) factors of L. By its turn, L1 and L2 can of course split into two elementary logics: on one hand, L11 (the logic of classical negation) and L21 (the logic of classical disjunction); and, on the other hand, L12 (the logic of classical conjunction) and L22 (the logic of classical implication). That is, L1 = L11 + L21 and L2 = L12 + L22 . Therefore, L splits into L11 , L21 , L12 and L22 , and so L = L11 + L21 + L12 + L22 . From the previous result, the reader could think that the process of combining logics is motivated by the idea of decomposition of the involved logics. However, the idea behind plain fibring and direct union is not just to decompose a logic into fragments, but also to combine given logics in order to obtain a bigger logic such that the given logics are fragments of it. A related discussion was addressed in [2], where a stronger notion of logical translations was proposed in order to recover a logic from its fragments through the process of (categorial) fibring.

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Fibring by Functions over Matrix Logics: the simplest case

A more interesting situation is to combine two matrix logics L1 = hC1 , M1 i and L2 = hC2 , M2 i, where M1 and M2 are defined over different algebras whose domains can be consider disjoint. Moreover, we will suppose that the signatures 3 Note

that this does not mean that the operations defined in M1 and M2 coincide. is a minor technical detail to be considered in the case that C1 ∪ C2 = C and C1k ∩ C2k 6= ∅ for some k ∈ N. In this case, there will be duplicate connectives in L1 + L2 and so this logic is not identical to L. However, L1 + L2 can be identified with L up to logical translations. 4 There

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C1 and C2 are also disjoint. Now suppose that we wish to define a new logic L, from L1 and L2 , with the following properties: - As in disjoint union, the signature C of L is C1 ] C2 , the disjoint union of C1 and C2 . - L is defined by a matrix logic M , where M is determined (in some reasonable way) by M1 and M2 . The fundamental problem here (that we could call “the problem of the combinations of the languages”) arises from the conjunction of the requirements above indicated: if we have a formula ϕ ∈ LhC1 ] C2 i, and we wish to evaluate it, a first approximation suggests to consider A = A1 ] A2 as the domain of M . The M -valuations would be either the M1 or the M2 valuations, according to the form of ϕ. Obviously, a new problem appears in this case: ϕ could be a “hybrid formula”, generated by connectives of both signatures C1 and C2 . So, the set of M -valuations to be defined must consider the sets of M1 -valuations and of M2 -valuations at the same time. In which form would interact both sets? A possible answer (of course, not the only one) is based in Gabbay’s fibring approach to the problem of combinations of modal languages, which will be called here the (unrestricted) plain fibring by functions. Roughly speaking, this process is determined by a pair of functions: λ : A1 −→A2 and µ : A2 −→A1 . These functions “translate”, when necessary, the “truth-values” of M1 and M2 . The formal definition follows in the sequel. Definition 3.1 Let Li = hCi , Mi i (with i = 1, 2) be two matrix logics, where each Mi = hAi , Di i is a Ci -matrix with domain Ai . The fibred signature (induced by C1 and C2 ) is C1 ] C2 , and the fibred language is L(C1 ] C2 ). Every pair (λ, µ) ∈ A2 A1 × A1 A2 is denominated a pair of fibring functions. Let (λ, µ) be a pair of fibring functions, fixed from now on: an unrestricted fibred valuation (u.f.v) is any function v : V−→A1 ]A2 . This function v is extended to a function v : L(C1 ] C2 )−→D1 ] D2 (which is determined not only by v but also by the pair (λ, µ)), according to this recursive definition: • If ϕ ∈ V then v(ϕ) = v(ϕ); • If ϕ = c(β1 , . . . , βk ) then v(ϕ) = c((β1 )(v,c) , . . . , (βk )(v,c) )5 where, for every formula βj (j = 1, . . . , k): – If c ∈ Cik and v(βj ) ∈ Ai then (βj )(v,c) = v(βj ) (for i = 1, 2); – If c ∈ C1k and v(βj ) ∈ A2 then (βj )(v,c) = µ(v(βj )); – If c ∈ C2k and v(βj ) ∈ A1 then (βj )(v,c) = λ(v(βj )). We say that a u.f.v v satisfies ϕ if v(ϕ) ∈ D1 ] D2 . The unrestricted plain fibred consequence relation |=(λ,µ) ⊆ ℘(L(C1 ] C2 )) × L(C1 ] C2 ) is defined as follows: Γ |=(λ,µ) ϕ if, for every u.f.v v satisfying simultaneously all the formulas 5 Again,

we use here the same symbol for a connective and for its matrix interpretation.

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of Γ, we have that v satisfies ϕ. Finally, the pair L1 ~(λ,µ) L2 = hC1 ]C2 , |=(λ,µ) i is the unrestricted fibring of L1 and L2 , determined by (λ, µ)6 . Example 3.2 Consider the paraconsistent logic P 1 , introduced in [12]. Its signature is such that |CP 1 | = {¬P 1 , →P 1 }. The logic P 1 is determined by the matrix MP 1 = hAP 1 , {T, T1 }i such that AP 1 = {T, T1 , F }, and the operations corresponding to ¬P 1 and →P 1 are defined through the tables below.

¬P 1

T F

T1 T

→P 1 T T1 F

F T

T T T T

T1 T T T

F F F T

Now, consider the classical propositional logic CP L defined over signature CCP L such that |CCP L | = {∧CP L , ¬CP L }, and with the usual matrix semantics over {0, 1}. Consider, in addition, a pair (λ, µ) such that: • λ(T ) = 1; λ(T1 ) = 1; λ(F ) = 0; • µ(1) = T ; µ(0) = F . Let P 1 ~(λ,µ) CP L be the unrestricted fibring of P 1 and CP L by the pair (λ, µ), let ϕ be the formula →P 1 (p, ∧CP L (¬CP L r, ∧CP L (q, ¬P 1 r))) in the fibred language, where p, q, r ∈ V, and let v be an u.f.v such that: • v(p) = T ; v(q) = 0; v(r) = T1 ; Then, by Definition 5.17 , v(ϕ)

= = = = = =

v(→P 1 (p, ∧CP L (¬CP L r, ∧CP L (q, ¬P 1 r)))) →P 1 (T, (∧CP L (¬CP L (λ(T1 )), ∧CP L (0, λ(¬P 1 (T1 )))))) →P 1 (T, µ(∧CP L (¬CP L (1), ∧CP L (0, λ(T ))))) →P 1 (T, µ(∧CP L (0, ∧CP L (0, 1)))) →P 1 (T, µ(∧CP L (0, 0))) →P 1 (T, µ(0)) = →P 1 (T, F ) = F.

Remark 3.3 It is worth noting that the functions v : L(C1 ] C2 )−→A1 ] A2 cannot be considered as homomorphisms, according to Definition 5.1. This is because the operations associated to the connectives of C1 ]C2 are just partially defined in A1 ] A2 , as the following example shows. Example 3.4 Consider the same logics of Example 3.2, and the same pair (λ, µ). Let v an u.f.v. such that v(p) = 1 and v(q) = F . Then (using infix 6 Briefly, we say that L ~ 1 (λ,µ) L2 is the fibring by the functions λ and µ, which justifies the title of this article. 7 In this example (and in some other ones along this paper) we use prefix notation, for a better illustration of Definition 5.1.

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notation now), v(p →P 1 q) = p(v,→P 1 ) →P 1 q(v,→P 1 ) = µ(1) →P 1 F = T →P 1 F = F ∈ A1 ] A2 . However, v(p) →P1 v(q) is not defined, since the domain of →P 1 is A1 , and v(p) = v(p) = 1 ∈ A2 . Despite the indicated in Remark 3.3, we can consider L1 ] L2 a matrix logic, in a somewhat hidden way. This result will be developed in the sequel. Definition 3.5 Let L = hC, M i be a matrix logic, where M is a C-matrix with domain A and set of designated values D ⊆ A. Let A0 and D0 be two sets such that D0 ⊆ A0 . Suppose, without loss of generality, that A ∩ A0 = ∅. Finally, let µ : A0 → A be a mapping. The C-matrix Mµ is defined as follows: its domain is A ] A0 ; the set of designated values is D ] D0 and, for c ∈ C k and a1 , . . . , ak ∈ A ] A0 , cMµ (a1 , . . . , ak ) = cM (a1µ , . . . , akµ ) where, for every aj (j = 1, . . . , k): - If aj ∈ A, then aj µ = aj . - If aj ∈ A0 , then aj µ = µ(aj ). Proposition 3.6 Let Li = hCi , Mi i (with i = 1, 2) be two matrix logics, where each Mi is a Ci -matrix, and let be (λ, µ) a pair of fibring functions. Then L1 ~(λ,µ) L2 is a matrix logic. Proof: We will prove the following: |=(λ,µ) can be characterized by a certain C1 ] C2 matrix M(λ,µ) of the form M(λ,µ) = hA1 ] A2 , D1 ] D2 ) such that, if a1 , . . . , ak ∈ Ai and c ∈ Cik then cM(λ,µ) (a1 , . . . , ak ) = cMi (a1 , . . . , ak ) (with i = 1, 2). Thus, M(λ,µ) extend the original matrices. Moreover: (i) For each u.f.v. v there is a M(λ,µ) -valuation v 0 in M such that v(ϕ) = v 0 (ϕ) for every formula ϕ ∈ L(C1 ] C2 ). (ii) Conversely, for each M(λ,µ) -valuation v there exists an unrestricted fibred valuation v 0 such that v 0 (ϕ) = v(ϕ) for every formula ϕ ∈ L(C1 ] C2 ). It is clear that, from (i) and (ii), the desired result follows easily. We proceed now to define M(λ,µ) , from the pair of fibring functions (λ, µ). We define our desired matrix as M(λ,µ) = (M1 )λ + (M2 )µ (recall Definitions 2.1 and 3.5). Thus, M(λ,µ) = hA, D1 ] D2 i where A = A1 ] A2 and the operations are defined as follows: - If c ∈ C1k and a1 , . . . , ak ∈ A, then cM(λ,µ) (a1 , . . . , ak ) = cM1 (a1(µ,c) , . . . , ak(µ,c) ) where, for every aj (j = 1, . . . , k): - If aj ∈ A1 , then aj (µ,c) = aj . - If aj ∈ A2 , then aj (µ,c) = µ(aj ). - If c ∈ C2k and a1 , . . . , ak ∈ A, then cM(λ,µ) (a1 , . . . , ak ) = cM2 (a1(λ,c) , . . . , ak(λ,c) ) where, for every aj (j = 1, . . . , k): - If aj ∈ A2 , then aj (λ,c) = aj . - If aj ∈ A1 , then aj (λ,c) = λ(aj ). Now we will prove that M(λ,µ) satisfies property (i). Given an u.f.v. v, consider the valuation v 0 defined in M(λ,µ) such that v 0 (p) = v(p) for every p ∈ V. It is 8

straightforward to see that, for every ϕ ∈ L(C1 ] C2 ), v 0 (ϕ) = v(ϕ). In order to prove that M(λ,µ) satisfies (ii), let v : L(C1 ] C2 )−→A1 ] A2 be a M(λ,µ) -valuation. Then we define v 0 (p) = v(p) for every p ∈ V and so v 0 (ϕ) = v(ϕ). From the considerations above we have that |=(λ,µ) = |=M(λ,µ) , which is defined by a matrix. So, L1 ~(λ,µ) L2 is a matrix logic.  Corollary 3.7 From Proposition 3.6 we have: (a) L1 ~ L2 is a structural logic. (b) If the matrix Mi is finite (i = 1, 2) then |=M(λ,µ) is finitary and so L1 ~(λ,µ) L2 is a standard logic. Proof: (a) As it was said, every matrix logic is structural. (b) It follows from item (a) and a result due to R. W´ojcicki (see [15]), which establishes that every consequence relation defined by a finite class of finite matrices is finitary: since the domains of M1 and M2 are finite, the domain of M(λ,µ) is also finite.  The result above shows that the unrestricted plain fibring is obtained as follows: (1) the original logics are extended to the disjoint union of the domains of the matrices, using the pair of fibring functions (λ, µ) (as in Definition 3.5); and (2) the direct union of the extended matrices is computed, obtaining the matrix M(λ,µ) for the fibred signature. Now we will analyze the relationship between the logic obtained by the fibring determined by (λ, µ) and the original given logics. Firstly, we obtain two useful lemmas. Lemma 3.8 Let v be an unrestricted fibred valuation, and consider the M1 valuation v 0 : L(C1 )−→A1 such that

v 0 (p) =

  µ(v(p)) 

v(p)

if v(p) ∈ A2 otherwise

for every p ∈ V. Then v 0 (ϕ) = v(ϕ) for every ϕ ∈ L(C1 ) \ V. An analogous result holds for M2 (using λ instead of µ). Proof: Use induction on the complexity of ϕ and recall Definition 5.1.



Lemma 3.9 Let v : L(C1 )−→A1 be a M1 -valuation, and let v 0 : V−→A an u.f.v such that v 0 (p) = v(p) for every p ∈ V. Then v 0 (ϕ) = v(ϕ) for every ϕ ∈ L(C1 ). An analogous result holds for M2 . Proof: Again, it follows straightforwardly by induction on the complexity of ϕ and Definition 5.1.  9

Proposition 3.10 Let Li be a nontrivial logic8 induced by Mi (i = 1, 2). Then L1 ~(λ,µ) L2 is a weak conservative extension of both logics L1 and L2 , that is: |=Li ϕ iff |=(λ,µ) ϕ, for every ϕ ∈ L(Ci ) (i = 1, 2). Proof: Since Li are structural nontrivial logics (by hypothesis and by propositions 1.5 and 1.7), no propositional variable is a tautology of Li (i = 1, 2). In fact, if |=Li p for some propositional variable then, by structurality, |=Li ϕ for every formula ϕ and so, by monotonicity, Γ |=Li ϕ for every Γ ∪ {ϕ}. Thus, suppose that ϕ ∈ L(C1 ) is such that |=L1 ϕ, and let v be a u.f.v. Since ϕ 6∈ V there exists a M1 -valuation v 0 such that v(ϕ) = v 0 (ϕ), by Lemma 3.8. But ϕ is a M1 -tautology and so v 0 (ϕ) ∈ D1 . That is, v(ϕ) ∈ D and then |=(λ,µ) ϕ. Analogously, it can be proved that |=L2 ϕ implies that |=(λ,µ) ϕ, for every ϕ ∈ L(C2 ). Conversely, let ϕ ∈ L(C1 ) such that |=(λ,µ) ϕ, and let v be a M1 -valuation. Let v 0 : V−→A the mapping defined as in Lemma 3.9. Since v 0 (ϕ) = v(ϕ) then v(ϕ) ∈ D1 . Therefore, |=L1 ϕ. The proof for L2 is analogous.  Note that, if exactly one of the logics (say, L1 ) is trivial, then the last result is no longer true for any pair (λ, µ). In fact: assuming that A1 = D1 6= ∅ and L2 is not trivial then the propositional variable p is a L1 -tautology but not a L2 -tautology. Now, for any pair (λ, µ), it is easy to define a unrestricted fibred valuation v such that v(p) = v(p) ∈ A2 \ D2 and then v(p) 6∈ D1 ] D2 . The Proposition 3.10 cannot be improved in general terms. That is, L1 ~(λ,µ) L2 is not in general a strong extension of the given logics, as the following example shows. Example 3.11 Let L1 = h{∨}, |=1 i be the disjunction-fragment of the classical propositional logic (induced, therefore, by the matrix M1 = h{0, 1}, {1}i with its usual truth-table). On the other hand, let L2 = hC2 , |=2 i be any logic such that |=2 is defined by a matrix M2 = h{T, T1 , F }, {T, T1 }i (the signature C2 and the operations of M2 are irrelevant here). Obviously, p1 |=1 p1 ∨ p2 . Now, let v : V−→{0, 1, T, T1 , F } be a mapping such that v(p1 ) = T , v(p2 ) = 0, and consider a pair (λ, µ) such that µ : {T, T1 , F }−→{0, 1}, verifying µ(T ) = 0, (where λ is any mapping λ : {0, 1}−→{T, T1 , F }). Consider now the u.f.v. v. Then v(p1 ) = v(p1 ) = T ∈ D1 ] D2 . On the other hand, v(p1 ∨ p2 ) = p1 (c,v) ∨ p2 (c,v) = µ(T ) ∨ 0 = 0 ∨ 0 = 0 ∈ / D1 ] D2 . Hence, p1 6|=(λ,µ) p1 ∨ p2 .

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Additional Conditions for Fibring Functions

A situation as the one described in Example 3.11 is not desirable, in general, in the context of combination of logics: any logic obtained by a combination process should be a strong extension of the given logics. Sometimes (see, for instance, [7]) it is required that the obtained logic should be a conservative 8A

logic is trivial if Γ |= ϕ for every Γ ∪ {ϕ}.

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extension of the given logics.9 The reason of the failure in Example 3.11 is that we are dealing with a “inconvenient” pair of fibring functions, and then a designated value could be mapped into a nondesignated one, and vice-versa (note that, in Example 3.11, we define µ(T ) = 0). In order to obtain a combined semantics inducing a logic L which is a conservative extension of both L1 and L2 , me must to impose additional conditions to the pair (λ, µ). This is the key idea of the Conservative Fibring by Functions, to be defined below. Definition 4.1 Let Li = hCi , Mi i (with i = 1, 2) be two matrix logics as in A2 1 Definition 5.1. A pair (λ, µ) ∈ AA 2 × A1 is admissible if it satisfies: λ(x) ∈ D2 iff x ∈ D1 , for every x ∈ A1 ; and µ(y) ∈ D1 iff y ∈ D2 , for every y ∈ A2 . If (λ, µ) is an admissible pair, and v : L(C1 ] C2 )−→A1 ] A2 is a u.f.v. we say that v (as its extension v) is a fibred valuation. In addition, the resulting fibring L1 ~(λ,µ) L2 will be indicated as conservative fibring determined by the pair of functions (λ, µ). To indicate that the fibring is determined by an admissible pair, we change slightly the notation: L1 ~(λ,µ) L2 will be denoted as L1 (λ,µ) L2 . The condition required for a pair (λ, µ) to be admissible is related to the definition of strict homomorphisms between matrices (cf. [4]). The adjective “conservative” of this restriction of fibring will be justified in Proposition 5.7. Definition 4.2 Two logics L1 and L2 as in Definition 4.1 are said to be comA2 1 patible if there is at least one admissible pair (λ, µ) in AA 2 × A1 . Note that L1 and L2 are compatible iff: (i) D1 6= ∅ iff D2 6= ∅; and (ii) (A1 \ D1 ) 6= ∅ iff (A2 \ D2 ) 6= ∅. Observe that, if L1 and L2 are not compatible, then one of the logics is trivial; therefore any pair of nontrivial logics is compatible. Now, we will prove that Proposition 3.10 can be improved by considering conservative fibrings. Proposition 4.3 Let Li = hCi , Mi i (with i = 1, 2) be two matrix logics as in Definition 5.1, and let (λ, µ) an admissible pair. Then L1 (λ,µ) L2 is a conservative extension of both L1 and L2 . Proof: Let v be a fibred valuation, and consider v 0 the M1 -valuation defined as in Lemma 3.8. Then, for every ϕ ∈ L(C1 ), v(ϕ) ∈ D iff v 0 (ϕ) ∈ D1 .

(∗)

In fact, if ϕ ∈ V then suppose that v(ϕ) ∈ A1 . Then v 0 (ϕ) = v(ϕ) = v(ϕ) and the result holds. On the other hand, if v(ϕ) 6∈ A1 then v 0 (ϕ) = µ(v(ϕ)) ∈ D1 iff v(ϕ) = v(ϕ) ∈ D2 and the result holds. On the other hand, if ϕ 6∈ V then the result follows from Lemma 3.8. Now, let Γ ∪ {ϕ} ⊆ L(C1 ) such that Γ |=L1 ϕ 9 In [2], however, it is argued against the desideratum of obtaining conservative extensions of the given logics through a combination process.

11

and let v be a fibred valuation such that v(Γ) ⊆ D. Let v 0 be the M1 -valuation obtained from v as in Lemma 3.8. Using (∗) it follows that v 0 (Γ) ⊆ D1 and then v 0 (ϕ) ∈ D1 . From (∗) again we get v(ϕ) ∈ D. This shows that Γ |=(λ,µ) ϕ. Conversely, suppose that Γ ∪ {ϕ} ⊆ L(C1 ) such that Γ |=(λ,µ) ϕ, and let v be a M1 -valuation such that v(Γ) ⊆ D1 . Now, consider v 0 be the fibred valuation defined as in Lemma 3.9. Then v 0 (Γ) ⊆ D and so v 0 (ϕ) ∈ D, that is, v(ϕ) ∈ D1 . This proves that Γ |=L1 ϕ.  Corollary 4.4 If L1 and L2 are compatible, then exists at least one conservative extension of both, L1 and L2 . It should be stressed that each matrix logic hC1 , (M1 )µ i coincides with L1 , and each matrix logic hC2 , (M2 )λ i coincides with L2 , provided that (λ, µ) is admissible. In fact: Proposition 4.5 For every admissible fibring pair (λ, µ) it holds: (a) |=M1 = |=(M1 )µ ; (b) |=M2 = |=(M2 )λ . Proof: (a): let Γ ∪ {ϕ} ⊆ L(C1 ). Suppose that Γ |=(M1 )µ ϕ, and let v be a valuation over M1 such that v(Γ) ⊆ D1 . Then v is a valuation over (M1 )µ such that v(Γ) ⊆ D1 ] D2 and then v(ϕ) ∈ D1 ] D2 . Thus v(ϕ) ∈ D1 and so Γ |=M1 ϕ. Conversely, suppose that Γ |=M1 ϕ and let v 0 be a valuation over (M1 )µ such that v 0 (Γ) ⊆ D1 ] D2 . The valuation v over M1 such that v(p) = v 0 (p) if v 0 (p) ∈ A1 , and v(p) = µ(v 0 (p)) if v 0 (p) ∈ A2 is such that, for every ψ ∈ L(C1 ): v 0 (ψ) ∈ D1 ] D2 iff v(ψ) ∈ D1 . Therefore v(Γ) ⊆ D1 and then v(ϕ) ∈ D1 . From this we get v 0 (ϕ) ∈ D1 ] D2 .  This shows that Γ |=(M1 )µ ϕ. The proof of item (b) is entirely analogous. Example 4.6 Recall the paraconsistent matrix logic P 1 considered in Example 3.2. Let L1 be the fragment of P 1 defined over signature {¬P 1 } given by the matrix M1 with domain A1 = {T, T1 , F } defined below, where D1 = {T, T1 } is the set of designated values.

¬P 1

T F

T1 T

F T

On the other hand, let L2 be the fragment of classical propositional logic defined over signature {→} given by the usual matrix M2 , with domain A2 = {0, 1} and set D2 = {1} of designated values, displayed below. → 1 1 1 0 1 12

0 0 1

A2 1 Now, let A = {T, T1 , F, 1, 0} and D = {T, T1 , 1}, and let (λ, µ) ∈ AA 2 × A1 such that λ(T ) = λ(T1 ) = 1, λ(F ) = 0, µ(1) = T and µ(0) = F . Then (λ, µ) is admissible and (M1 )µ and (M2 )λ are given by the tables below, respectively.

T ¬ F

T1 T

1 F

F T

→ T T1 1 F 0

0 T

T 1 1 1 1 1

T1 1 1 1 1 1

1 1 1 1 1 1

F 0 0 0 1 1

0 0 0 0 1 1

Let L be the logic over {¬, →} characterized by M(λ,µ) = (M1 )µ + (M2 )λ given by the two tables above, with {T, T1 , 1} as the set of designated values. It is easy to see that the reduced matrix for L produces the 3-valued logic P 1 , because T and 1 are congruent 10 , as well as F and 0. A2 0 1 It should be noted that the pair (λ, µ0 ) ∈ AA 2 × A1 such that µ (1) = T1 0 0 and µ (0) = F is also admissible (moreover, (λ, µ) and (λ, µ ) are the only admissible pairs). In the resulting matrix M(λ,µ0 ) = (M1 )µ0 +(M2 )λ the formula ϕ = (p1 → p2 ) → ¬¬(p1 → p2 ) is not valid, and so this logic does not coincide with P 1 .

Now we will analyze the converse property of admissibility, that is: if (λ, µ) is not admissible then M(λ,µ) is not a strong conservative extension of M . Proposition 4.7 If ck is a k-ary connective of Ci and β1 , . . . , βk ∈ L(Ci ) \ V, then for every u.f.v. v, we have that v(ck (β1 , . . . , βk )) = ck (v(β1 ), . . . , v(βk )). Proof: Since β1 , . . . , βk ∈ L(Ci )\V, then v(βt ) ∈ A1 , for every 1 ≤ t ≤ k. Now, from definition of v, we have that v(ck (β1 , . . . , βk )) = ck ((β1 )(v,c) , . . . , (βk )(v,c) ) = ck (v(β1 ), . . . , v(βk )). 2 The last result shows that, if φ is a pure (non-hybrid) non-atomic formula then the u.f.v.’s act as homomorphisms in the algebra A1 ] A2 . The next proposition states that, if φ is a pure formula, the truth-value v(φ) depends exclusively of the value of its atomic subformulas (and of the transformations applied to them). Proposition 4.8 Let Li = (Ci , Mi ) be matrix logics (i = 1, 2), and consider the fibring pair (λ, µ). For every formula φ(p1 , . . . , pk ) ∈ L(Ci ) \ V, for every u.f.v. v we have that v(φ(p1 , . . . , pk ) = φ((p1 )(v,c) , . . . , (pk )(v,c) ) 10 That is, the matrix M (λ,µ) is not reduced. We will study the relationship between Fibring by Functions and Reduced Matrices in the last sections. See [4] for an updated survey about Reduced Matrices.

13

Proof: By induction on the number of connectives in φ. Suppose first that φ ∈ L(C1 ) \ V (for L(C2 ) \ V the proof is similar). If n = 1 then φ = ck (p1 , . . . , pk ). Let v be an u.f.v. Then, it is clear that v(φ) = ck ((p1 )(v,c) , . . . , (pk )(v,c) ). Suppose now that our claim is valid for every formula with m connectives such that m < n, and suppose that φ has n connectives. We can consider that φ = cj (β1 , . . . , βj ), where βt = βt (p1 , . . . , pk ) (1 ≤ t ≤ j). Then, v(φ) = v(cj (β1 , . . . , βj )) = cj ((β1 )(v,c) , . . . , (βj )(v,c) ). Now, for every βt we have two possibilities; either βt ∈ V or βt ∈ L(C1 ) \ V. In the first case βt = pr (1 ≤ r ≤ k), and therefore (βt )(v,c) = (pr )(v,c) . In the second case (βt )(v,c) = v(βt ) = βt ((p1 )(v,c) , . . . , (pk )(v,c) ) (by Induction Hypothesis). So, in both cases, (βt )(v,c) = βt ((p1 )(v,c) , . . . , (pk )(v,c) ) and therefore v(φ) = φ((β1 )(v,c) , . . . , (βj )(v,c) ). 2 As in the previous cases, the next result is shown for L1 but it is also valid for L2 : Proposition 4.9 Consider the fibring pair (λ, µ) for L1 and L2 such that: (i) There exist a ∈ D2 and b ∈ A1 \ D1 such that µ(a) = b. (ii) There exists a formula φ(p1 , . . . , pk ) ∈ L(C1 ) \ V such that: (ii.a) p1 |=1 φ. (ii.b) There is a M1 -valuation w such that w(p1 ) = b and w(φ) ∈ / D1 . Under the previous conditions, |=(λ,µ) is not a strong extension of L1 . Proof: We will define an u.f.v. v such that v(p1 ) ∈ D1 ]D2 , but v(φ) ∈ / D1 ]D2 . So, our result follows from (ii.a). For every pi ∈ V we define v as:

v(pi ) =

  a 

if i = 1

w(pi )

otherwise

Now, v(p1 ) = v(p1 ) = a ∈ D2 ⊆ D (by (i)). On the other hand, v(φ) = v(φ(p1 , . . . , pk )) = φ((p1 )(v,c) , . . . , (pk )(v,c) ), since φ ∈ L(C1 )\V and Proposition 4.8. Considering that v(p1 ) ∈ / A1 , and v(pi ) ∈ A1 (when i 6= 1), we have that v(φ) = φ(µ(a), v(p2 ), . . . , v(pk )) = φ(b, w(p2 ), . . . , w(pk )) = φ(w(p1 ), w(p2 ), . . . , w(pk )) = w(φ(p1 , . . . , pk )) ∈ / D1 (use (ii.b) here and the fact that w is M1 -valuation). From this it follows that p1 6|=(λ,µ) φ, and this concludes the proof. 2

5

General Fibring by Functions

At this point, note the following: to combine the logics Li = hCi , Mi i (i = 1, 2), we fixed a pair (λ, µ) ∈ A2 A1 × A1 A2 , obtaining in this way the matrix M(λ,µ) = M1λ + M2µ . So, the logic L1 ~(λ,µ) L2 is a matrix logic, because is defined

14

by M(λ,µ) . Now, what happens if, instead of consider just one pair (λ, µ) we proceed to consider a set Σ ⊆ A2 A1 × A1 A2 ? We will prove in the sequel that, basically, no important property is lost with this more general form of fibring by functions. The basic idea motivating the following results it that a matrix logic can be characterized by means of classes of matrices, and not merely by single ones. Firstly, we include some definitions that generalize the previous ones. Definition 5.1 Let Li = hCi , Mi i (with i = 1, 2) be two matrix logics, where each Mi = hAi , Di i is a Ci -matrix with domain Ai . Let Σ be a set of unrestricted A1

fibred valuations. That is, Σ ⊆ A1 A2 × A2 . We define the unrestricted fibring ~ induced by Σ as the logic L = hC1 ] C2 , |=~ Σ i, where |=Σ is defined as follows: ~ Γ |=Σ α iff Γ |=(λ,µ) α for every u.f.v (λ, µ) ∈ Σ. When Σ = A1 A2 × A2 A1 , we ~ denote |=~ Σ simply by |= , and L is denominated the general unrestricted fibring of L1 and L2 . T Remark 5.2 We can consider that |=~ |=~ Σ= (λ,µ) . This fact will be (λ,µ)∈Σ

applied in the sequel. Now we can characterize the unrestricted fibring induced by Σ as being a matrix logic: Proposition 5.3 Given the logics Li = hCi , Mi i (i = 1, 2), the logic LΣ (~) is a matrix logic, and therefore is structural. Moreover, If the domains of M1 and M2 are finite matrices, then LΣ (~) is finitary, and therefore is a standard logic. Proof: From the previous remark and the characterization of |=~ (λ,µ) given in ~ Proposition 3.6, we can consider that |=Σ = |=KΣ , where KΣ = {M(λ,µ) }(λ,µ)∈Σ . That is, |=~ Σ is a matrix consequence relation (and, so, is structural). Now, if A1 and A2 are finite, A2 A1 × A1 A2 is also finite. Then, the class KΣ is finite. And considering that, for every pair (λ, µ), the domain of M(λ,µ) is A1 ] A2 (which is finite), we have that every matrix of KΣ is finite. By the same considerations of Corollary 3.7, |=~  Σ is a finitary (and standard) consequence relation. The Proposition 3.10 can also be generalized. Proposition 5.4 If Li are nontrivial logics induced by the matrix Mi (i = 1, 2), Then L1 ~Σ L2 is a weak conservative extension of both logics L1 and L2 . Proof: Consider Li , Mi and Ai as indicated, andΣ ⊆ A2 A1 × A1 A2 . If |=Li ϕ then, for every fibring pair (λ, µ) ∈ Σ, |=~ (λ,µ) ϕ, since Proposition 3.10 is valid for every (λ, µ) ∈ A2 A1 × A1 A2 . So, |=~ Σ ϕ (by the definition of |=Σ ). If 6|=Li ϕ, then 6|=~ ϕ (again, by Proposition3.10), and therefore 6|=~  Σ ϕ. (λ,µ) Remark 5.5 Please note that we have different consequence relations over the same signature (that is, C1 ] C2 ), according the set Σ chosen. It is obvious that, 15

~ if Σ1 ⊆ Σ2 , then |=~ Σ2 ϕ implies |=Σ2 ϕ. Therefore, the least fibred logic by L1 and L2 is, obviously, L1 ~ L2 := L1 ~Σ L1 , when Σ = A2 A1 . This logic will be called the optimal unconstrained fibring of L1 and L2 .

Definition 5.6 Let Li = hCi , Mi i (with i = 1, 2) be two matrix logics, and let Σ be a set of admissible pair of fibring functions, cf. Definition 4.1. Then, the fibring L1 ~Σ L2 (according to Definition 5.1) will be called the conservative fibring induced by Σ, and denoted by L1 Σ L2 . If Σ is the set of all the admissible fibring pairs, we denote this fibring simply as L1 L2 , and will be indicated as the general conservative fibring of L1 and L2 . Proposition 5.7 L1 L2 is a conservative extension of both L1 and L2 . Proof: We just prove that L1 L2 is a conservative extension of L1 , because the proof for L2 is analogous. Suppose firstly that D1 = A1 . Then D2 = A2 (because L1 and L2 are compatible) and then both L1 and L1 L2 are trivial, thus the result follows. Suppose now that D1 = ∅. Then D2 = ∅ and, again, both L1 and L1 L2 are trivial. Suppose now that both A1 \ D1 and D1 are nonempty. For every fibred valuation (f, g, v) let v 0 the M1 -valuation defined as in Lemma 3.8. Then, for every ϕ ∈ L(C1 ), (f, g, v)(ϕ) ∈ D iff v 0 (ϕ) ∈ D1 .

(∗)

In fact, if ϕ ∈ V then suppose that v(ϕ) ∈ A1 . Then v 0 (ϕ) = v(ϕ) = (f, g, v)(ϕ) and the result holds. On the other hand, if v(ϕ) 6∈ A1 then v 0 (ϕ) = g(v(ϕ)) ∈ D1 iff v(ϕ) = (f, g, v)(ϕ) ∈ D2 and the result holds. On the other hand, if ϕ 6∈ V then the result follows from Lemma 3.8. Now, let Γ ∪ {ϕ} ⊆ L(C1 ) such that Γ `L1 ϕ and let (f, g, v) be a fibred valuation such that (f, g, v)(Γ) ⊆ D. Let v 0 be the M1 -valuation obtained from (f, g, v) as in Lemma 3.8. Using (∗) it follows that v 0 (Γ) ⊆ D1 and then v 0 (ϕ) ∈ D1 . From (∗) again we get (f, g, v)(ϕ) ∈ D. This shows that Γ `M1 M1 ϕ. Conversely, suppose that Γ ∪ {ϕ} ⊆ L(C1 ) such that Γ `M1 M1 ϕ, and let A2 1 v be a M1 -valuation such that v(Γ) ⊆ D1 . Let (f, g) ∈ AA 2 × A1 admissible 0 (recall that L1 and L2 are compatible) and let (f, g, v ) be the fibred valuation defined as in Lemma 3.9. Then (f, g, v 0 )(Γ) ⊆ D and so (f, g, v 0 )(ϕ) ∈ D, that is, v(ϕ) ∈ D1 . This proves that Γ `L1 ϕ.  Example 5.8 In [5] a hierarchy of paraconsistent logics generalizing P 1 was introduced, called {P n }n∈N . Each logic P n is defined over CP n = {¬P n , →P n }, with semantics given by the matrix MP n = hAP n , {T0 , T1 , . . . , Tn }i such that AP n = {T0 , T1 , . . . , Tn , f }. The corresponding operations are displayed in the tables below.

¬P n

T0 f

Th Th−1

f T0

→P n T0 Th f 16

T0 T0 T0 T0

Th T0 T0 T0

f f f T0

(1 ≤ h ≤ n)

Also in [5], it was introduced a hierarchy of weakly-intuitionistic logics called {I n }n∈N , generalizing the weakly-intuitionistic logic I 1 introduced in [13] as a dual of P 1 . Each I n is defined over the signature CI n = {¬I n , →I n }, with semantics given by the matrix MI n = hAI n , {t}i, with AI n = {t, F0 , F1 , . . . , Fn }. The operations of the matrix MI n are given by the tables below.

¬I n

t F0

F0 t

→I n t F0 Fl

Fl Fl−1

t t t t

F0 F0 t t

Fl F0 t t

(1 ≤ l ≤ n)

Note that both P 0 and I 0 coincide with the classical propositional logic over {¬, →} with two-valued matrix semantics. Now we will analyze the conservative fibring of I n with P k , where Σ is the set of all the admissible fibring pairs. First note that, given I n and P k , the admissible pairs are of the form (λj , µi ) (for 0 ≤ j ≤ k and 0 ≤ i ≤ n) such that µi (f ) = Fi ; λj (t) = Tj ; µi (Th ) = t and λj (Fl ) = f for 0 ≤ h ≤ k and 0 ≤ l ≤ n. Let M(λj ,µi ) = (MI n )µi + (MP k )λj . Then the matrix M(λj ,µi ) is defined over the signature Cnk = {¬In , →In , ¬P k , →P k }, with domain {t, T0 , T1 , . . . , Tk , F0 , F1 , . . . , Fn , f } and designated values {t, T0 , T1 , . . . , Tk }. The operations are given below (the truth-tables of the negations consider the cases: i = 0 and i > 0; j = 0 and j > 0).

→iIn t T0 Th F0 Fl f

¬0In ¬iIn ¬0P k ¬jP k

t t t t t t t

T0 t t t t t t

Th t t t t t t

F0 F0 F0 F0 t t t

t F0 F0 f Tj−1

T0 F0 F0 f f

Th F0 F0 Th−1 Th−1

Fl F0 F0 F0 t t t

F0 t t T0 T0

→jP k t T0 Th F0 Fl f

f F0 F0 F0 t t t

Fl Fl−1 Fl−1 T0 T0

t T0 T0 T0 T0 T0 T0

T0 T0 T0 T0 T0 T0 T0

Th T0 T0 T0 T0 T0 T0

F0 f f f T0 T0 T0

Fl f f f T0 T0 T0

f f f f T0 T0 T0

f t Fi−1 T0 T0

(1 ≤ h ≤ k; 1 ≤ l ≤ n)

Each M(λj ,µi ) defines a matrix logic which is simultaneously paraconsistent (w.r.t. ¬P k ) and weakly-intuitionistic (w.r.t. ¬In ). The conservative fibring 17

I n P k of I n and P k is the matrix logic characterized by MI n MP k , being this set of matrices, the following: MI n MP k = {M(λj ,µi ) : 0 ≤ j ≤ k and 0 ≤ i ≤ n}. The relationships between the logic defined by each matrix M(λj ,µi ) , the logic I n P k and the logic I n P k (having a single negation which is simultaneously paraconsistent in the sense of P k and weakly-intuitionistic in the sense of I n ), introduced in [5], deserve further research.

We conclude this section by observing that both plain fibring and unrestricted plain fibring are operations defined over matrix logics characterized by a single matrix, but the result is a matrix logic characterized, in general, by a set of matrices (not necessarily being a singleton). We can easily correct this asymmetry by generalizing the operation of plain fibring to matrix logics in general. Definition 5.9 Let Li = hCi , Ki i (with i = 1, 2) be two matrix logics. For every matrix M of K1 and every matrix N of K2 , fix a set ΣM,N ⊆ AN AM × AM AN . Let S be the class of all the sets ΣM,N . The unrestricted fibring L1 ~S L2 = ~ ~ hC1 ] C2 , |=~ S i, where Γ |=S α iff Γ |=ΣM,N α, for every ΣM,N ∈ S. If every ΣM,N in S is a set of admissible pairs, we say that L1 ~S L2 is the conservative fibring (induced by S) of L1 and L2 , which can be denoted by L1 S L2 . The definitions of general unrestricted fibring and general conservative fibring are similar to Definitions 5.1 and 5.6. From the previous definition, the next result follows easily: Proposition 5.10 Suppose that, for every M1 ∈ K1 and every M2 ∈ K2 , the logics hC1 , M1 i and hC2 , M2 i are compatible (cf. Definition 4.2). Then L1 L2 is a conservative extension of both L1 and L2 . Proof: Let Γ ∪ {ϕ} ⊆ L(C1 ). Then Γ `L1 ϕ iff, for every M1 ∈ K1 and every M2 ∈ K2 , Γ `M1 M2 ϕ, by adapting the proof of Proposition 5.7, iff Γ `K1 K2 ϕ. The proof for L2 is analogous. The details are left to the reader.  A matrix M = hA, Di is said to be trivial if either D = ∅ or D = A. The following result is a direct consequence of the proposition above. Corollary 5.11 Let Li = hCi , Ki i (with i = 1, 2) be two matrix logics. Suppose that K1 and K2 do not contain trivial matrices. Then L1 L2 is a conservative extension of both L1 and L2 . 

18

6

Identification of truth-values: Fibring by Functions over Reduced Matrices

Definition 6.1 Let M = (A, D) be a C-matrix. A congruence Θ ⊆ A2 is compatible with D if, for every a, b ∈ A it holds: if (a, b) ∈ Θ and a ∈ D, then b ∈ D. That is, if D is an union of equivalence classes of Θ. Notation 6.2 From now on, we can denote (a, b) ∈ Θ either by aΘb or by a ≡ b(Θ) or even by a ≡ b(modΘ). Definition 6.3 Consider a C-matrix M = (A, D), and a congruence Θ in A2 , we define the quotient matrix M/Θ := (A/Θ, D/Θ). The function kΘ : A −→ A/Θ defined by kΘ (a) = a/Θ is an epimorphism, called the canonical application from M to M/Θ. Definition 6.4 Let A be a C-algebra11 . For every D ⊆ A, we define the Leibniz congruence in A over D (denoted by ΩA D), in this way: (a, b) ∈ ΩA (D) iff, for every formula φ(p, q1 , . . . , qk ) ∈ L(C), for every k-uple (c1 , . . . , ck ) of elements of A, the following fact is valid: φ(a, c1 , . . . , ck ) ∈ D if and only if φ(b, c1 , . . . , ck ) ∈ D. Theorem 6.5 For every C-algebra A, for every D ⊆ A, ΩA D it the greatest congruence (w.r.t the inclusion order) compatible with D. Definition 6.6 Let A be an algebra. The Leibniz operator in A is the function ΩA : ℘(A) −→ A2 defined in the obvious way: for every D ⊆ A, the image of D, according ΩA , is ΩA (D) (cf. Definition 6.4). Note that we can apply the Leibniz operator to L(C) itself. So, we get the following result: Corollary 6.7 Consider a signature C and the algebra L(C); for every T ∈ L(C) we have the following: (α, β) ∈ ΩL(C) (T ) iff, for every formula φ ∈ L(C), for every p ∈ V, φ(p/α) ∈ T si y solo si φ(p/β) ∈ T . Definition 6.8 A matrix M = (A, D) is reduced if ΩA (D) is the identity relation in A. A concept slightly stronger than the reduced matrix one is the following 12 , very useful for applications, because deals is a more concrete way with the truth - values of a matrix. Definition 6.9 A matrix M = (A, D) is strongly reduced iff, for {a, a0 } ⊆ A (a 6= a0 ) exists a n-ary connective d, and exist c1 , . . . , ck ∈ A such that it is valid some of these two conditions: (a) d(a, c1 , . . . , ck ) ∈ D but d(a0 , c1 , . . . , ck ) ∈ /D (b) d(a, c1 , . . . , ck ) ∈ / D but d(a0 , c1 , . . . , ck ) ∈ D. 11 That 12 To

is, an algebra similar to L(C). the best of our known, this notion is an original contribution of this paper.

19

Proposition 6.10 Let M = (A, D) be a matrix. If M is strongly reduced, then is reduced. Proof: From Definitions 6.4 and 6.8, we can infer this fact: M = (A, D) is not reduced if and only if there are a, a0 ∈ A, a 6= a0 such that is valid: (∗) For every formula φ(p, q1 , . . . , qk ) ∈ S, for every c1 , . . . , ck ∈ A, φ(a, c1 , . . . , ck ) ∈ D if and only if φ(a0 , c1 , . . . , ck ) ∈ D. (In fact, there are a, a0 (a 6= a0 ) validating (∗) iff exist a 6= a0 with (a, a0 ) ∈ ΩA D, iff ΩA D 6= ∆). On the other hand, suppose that M is not strongly reduced: this is equivalent to say that exist a, a0 ∈ A, a 6= a0 such that: (∗∗) For every n-ary connective d, for every c1 , . . . , ck ∈ A, d(a, c1 , . . . , ck ) ∈ D if and only if d(a0 , c1 , . . . , ck ) ∈ D. It is obvious now that if M is not reduced, then we have (∗) and, as a particular case, (∗∗). So, M is not strongly reduced.  Rougly speaking, the proposition above estabilished this result; if in A there are not “superflous” truth-values (w.r.t D), then M is reduced. An interesting question is if the converse of the following result is valid. Our opinion is negative about it. Formally: Conjecture 6.11 The sufficent condition above is not neccesary. That is, there is a matrix where (∗∗) is valid, but wherein (∗), is not valid and, so, M is reduced. Returning to Fibring by function, we can study the preservation of the definitions above after applications of fibring. More precisely, the problems that we will study here are the following: If L1 and L2 are reduced (strongly reduced); the matrix that defines L1 ~(λ,µ) L2 is, in general, reduced (strongly reduced)? If not: which conditions would have the pair (λ, µ) for obtain a matrix reduced (strongly reduced) by means of fibring? Some results above the questions above posed are the following: Proposition 6.12 It is not valid in general (that is, for every fibring pair (λ, µ)), that the fibring of reduced matrix is also a reduced matrix. Which additional conditions must to have the fibring pairs for the preservation of reduced matrices? Some sufficient and necessary conditions are given here. Proposition 6.13 Let Li (i = 1, 2) be two matrix logics, defined by the reduced matrices Mi = hAi , Di i, and consider L1 ~(λ,µ) L2 . If (λ, µ) is .... then M(λ,µ) is reduced.

20

Proposition 6.14 Let Li (i = 1, 2) be two matrix logics, defined by the reduced matrices Mi = hAi , Di i, and consider L1 ~(λ,µ) L2 . If M(λ,µ) is reduced, then (λ, µ) is .... . Remark 6.15 At this point, we must note the following fact: the fibring method used here is based on the combinations of the matrices itself: from this matricial combination we can define a consequence relation. So, strictly speaking, we are not fibring logics but matrices, in fact. It is common to say, in the specific literature about fibring, that we are working with the presentations of the logics, but the logics itself. Actually, in a certain sense, the more abstract categorical approach mentioned in the Introduction of this paper, has as one of its motivations, the fibring of the pure consequence relations, avoiding to deal with the presentations.

7

Identification of connectives: Constrained Fibring by Functions

The key concept in the previous section was the identification of certain truthvalues in such a way that it is possible to obtain the Reduced matrix for the fibring of two given logics. Now, we proceed in a similar way, but identifying connectives in this case. Actually, this technique is strongly studied in the literature of Combinations of Logics (see [11], [17] or [1], for example), but in a more abstract approach. In our chosen approach for this paper we will identify connectives in a very concrete way, by a simple adaptation of the ideas of the Fibring over Reduced Matrices. Actually, from the mentioned adaptation, we can easily get the following definitions and results: Definition 7.1 Let C be a signature, and A a C-algebra. We say that C is strongly reduced (with respect to A) iff, for every pair c1 ,c2 of k-ary connectives, exists a k-uple (a1 , . . . , ak ), such that c1 (a1 , . . . , ak ) 6= c2 (a1 , . . . , ak ). If M = (A, D) is a C-matrix, we say that C is strongly reduced (with respect to M ) iff, for every pair c1 , c2 of k-ary connectives, exists a k-uple such that c1 (a1 , . . . , ak ) ∈ D iff c2 (a1 , . . . , ak ) ∈ / D. Remark 7.2 From the previous definition, C is not strongly reduced with respect to A iff there are two k-ary connectives c1 , c2 such that c1 (a1 , . . . , ak ) = c2 (a1 , . . . , ak ), for every k-uple (a1 , . . . , ak ) of elements of A. In a similar way, C is not strongly reduced with respect to M iff there are two k-ary connectives c1 , c2 such that c1 (a1 , . . . , ak ) ∈ D iff c2 (a1 , . . . , ak ) ∈ D. Now, we can consider the same kind of problems of the previous section. That is: what happens if we combine two logics where its respective signatures are reduced. Is C1 ] C2 reduced for L1 ~(λ,µ) L2 ? Some answers can be found in the sequel.

21

Proposition 7.3 Let Li (i = 1, 2) be two matrix logics, defined by the reduced matrices Mi = hAi , Di i, and consider L1 ~(λ,µ) L2 . If (λ, µ) is .... then M(λ,µ) is reduced. Proposition 7.4 Let Li (i = 1, 2) be two matrix logics, defined by the reduced matrices Mi = hAi , Di i, and consider L1 ~(λ,µ) L2 . If M(λ,µ) is reduced, then (λ, µ) is .... .

To elucidate the previous notion, let us see some examples, considering now the logics L3 and G3 (of Lusasiewicz and G¨odel, resp.). To simplify notation, the logic L3 will be denoted as La 13 . On the other hand, G3 will be indicated as Lb . The next definitions, are based on [8]. The signatures involved are Ca = {→a , &a , 0a } and Cb = {→b , &b , 0b }. Every one of these signatures is interpreted for the following three-valued matrices, Ma = (Aa , Da ) and Mb = (Ab , Db ), defined in this way: For the case of La : →a 0a

0a 1a

1 2a 1a

1 2a 0a

1 2a

1a 1a 1 2a

1a 1a 1a 1a

&a 0a

1b 1b 1b 1b

&b 0b

1 2a 1a

0a 0a 0a 0a

1 2a

0a 0a 1 2a

1a 0a 1 2a 1a

With respect to Lb : →b 0b 1 2b 1b

0b 1b 0b 0b

1 2b

1b 1b 1 2b

1 2b 1b

0b 0b 0b 0b

1 2b

0b

1b 0b

1 2b 1 2b

1 2b 1b

Let us see different forms of combinations of these matrices: A) The direct union La ] Lb : if we suppose that Aa = Ab = A = {0, 12 , 1}, then C = {→a , →b , &a , &b , 0a , 0b }. In this case, we have a logic defined by an algebra A with 6 functions, whose truth-tables are simply the collection of the original truth-tables. Remark 7.5 Please note the following fact: the interpretation of 0-ary operations in A of the connectives 0a and 0b is the same, so the signature C is not strongly reduced w.r.t M1 ] M2 . Then, we have here an example that, if we combine logics with irredundant sets of connectives is not preserved by direct union, and therefore by fibring. 13 This subindex will be applied to all the components of L , such as its signature, its matrix, 3 the operations and consequence relation

22

B) Fibring by functions (by means of admissible pairs) La Lb : Since this technique depends on the chosen fibring pair (λ, µ) admissible, in the case of admissible fibring pairs we have four cases. In fact: b.1) There are two admissible functions λ1 and λ2 from La to Lb . Actually (with x varying in {0a , 21 a , 1a }):   1   0b if x = 0a if x = 0a       2b        1  1   1 if x = 0b si x = λ (x) = λ1 (x) = 2b 2a 2 2a             1 if x = 1a   1b si x = 1a  b    b.2) In a similar way, there are two admissible functions µ1 and µ2 from Lb to La 14 :   1   0 if x = 0 if x = 0b a b     2a           1  1  1 if x = 0a if x = µ1 (x) = µ (x) = 2a 2b 2 2b            1a  if x = 1b   1a if x = 1b     So, the four admissible fibring pairs are (λ1 , µ1 ), (λ1 , µ2 ), (λ2 , µ1 ) and (λ2 , µ2 ). Every one of these pairs defines a different matrix such that, as in direct union, interpret 4 connectives. Every matrix have six elements in its domain:

14 In

this case, x is varying in {0b ,

1 ,1 } 2b b

23

A) The matrix Mλ1 ,µ1 : This matrix have two important properties: first, λ1 are µ2 mutually inverse. On the other hand, every function associate a given truth-value to the same truth-value of the another matrix. Hence, λ1 and µ1 are isotone functions 15 Here we can note that the resulting matrix is, precisely, reduced (see the truth-tables below): &a 0a

1 2a

1 2b 1b

0a 0a 0a 0a 0a 0a 0a

→a 0a

0a 1a

1 2a 1a

1 2a 0a

0b

1a

1 2b 1b

1 2a 0a

1a

&b 0a

0a 0b 0b 0b 0b 0b 0b

1 2a

1 2a 1a

0b

1 2a 1a

0b 1 2b 1b

→b 0a 1 2a 1a

0b 1 2b 1b

0a 1b 0b 0b 1b 0b 0b

0a 0a 1 2a 0a

0a 1 2a 1 2a

1a 1a 1 2a 1a 1 2a

0b 1 2b 1 2b 0b 1 2b 1 2b 1 2a

1b 1b 1 2b 1b

1b 1 2b

1a 0a 1 2a 1a

0a 1 2a 1a

1a 1a 1a 1a 1a 1a 1a 1a 0b

0b 0a 0a 0a 0a 0a 0a 0b 1a 1 2a 0a

1a 1 2a 0a

1 2b 1b

0b 0b 0b 0b 0b 0b 0b

1a 1b 1b 1b 1b 1b 1b

0b 1b 0b 0b 1b 0b 0b

1 2b 1b

0b

1 2b

0a 0a 1 2a 0a

0a 1 2a 1 2b

1a 1a 1 2a 1a

1a 1 2a 1 2b

0b

1b 0a 1 2a 1a

0a 1 2a 1a

1b 1a 1a 1a 1a 1a 1a 1b 0b

1 2b 1 2b 0b 1 2b 1 2b

1 2b 1b

1 2b

1b 1b 1b 1b 1b 1b 1b

1b 1b 1 2b 1b

1b 1 2b

0b 1 2b 1b

15 It is important to remark that usually, when the elements of the domain of a matrix take “numerical values”, they are not just indicating names but they are being used to define the truth-functions by means of mathematical formulas. Moreover, in these cases, the truth-values are organized by a certain “order”. So, for the matrix logics considered here, the isotonicity of λ1 and µ1 deserves a detailed study.

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B) The matrix Mλ1 ,µ2 : &a 0a

1 2b 1b

0a 0a 0a 0a 0a 0a 0a

→a 0a

0a 1a

1 2a 1a

1 2a 0a 1 2a 1a

1 2a 1a

0b

0b 1 2b 1b

&b 0a 1 2a 1a

0b 1 2b 1b

→b 0a 1 2a 1a

0b 1 2b 1b

1 2a

0a 0a

0b 0a 0a

0a

1 2a 1a 1 2a 0a

1 2a

1a

1 2a

1a 1a 1a 1a 1a 1a 1a

0b 1a 1a

1a 0b

1 2b 1b

0b 0b 0b 0b 0b 0b 0b

1a 1b 1b 1b 1b 1b 1b

0b 1b 0b 0b 1b 0b 0b

1 2a 0a

1 2a

1a 1a 1 2a 1a

1a

0a

1 2a

0a 0b 0b 0b 0b 0b 0b

1 2a

0a 1b 0b 0b 1b 0b 0b

1a 0a

0b 1 2b 1 2b 0b 1 2b 1 2b 1 2a

1b 1b 1 2b 1b

1b 1 2b

1 2b 1b

0b

25

1 2a 0a

0a

1 2b

0a 0a 0a 0a 0a 0a 1 2b

1a

1a

1 2a 0a 1 2a 1a

1 2a

0a

1 2a 1a

1 2b

0b

1b 0a 1 2a 1a 1 2a 0a

1a 1b 1a 1a 1a 1a 1a 1a 1b 0b

1 2b 1 2b 0b 1 2b 1 2b

1 2b 1b

1 2b

1b 1b 1b 1b 1b 1b 1b

1b 1b 1 2b 1b

1b 1 2b

0b 1 2b 1b

C) The matrix Mλ2 ,µ1 : &a 0a

1 2a

1 2b 1b

0a 0a 0a 0a 0a 0a 0a

→a 0a

0a 1a

1 2a 1a

1 2a 0a

0b

1a

1 2b 1b

1 2a 0a

1a

&b 0a

0a

1 2a

1 2a 1a

0b

1 2a 1a

0b 1 2b 1b

→b 0a 1 2a 1a

0b 1 2b 1b

1 2b 0b 1 2b 0b 1 2b 1 2b

0a 1b 1b 1 2b 1b

1b 1 2b

0a 0a 1 2a 0a

0a 1 2a 1 2a

1a 1a 1 2a 1a 1 2a

0b 0b 0b 0b 0b 0b 1 2a

0b 1b 0b 1b 0b 0b

1a 0a 1 2a 1a

0a 1 2a 1a

1a 1a 1a 1a 1a 1a 1a

0b 0a 0a 0a 0a 0a 0a 0b 1a 1 2a 0a

1a 1 2a 0a

1 2b

0a 0a 1 2a 0a

0a 1 2a 1 2b

1a 1a 1 2a 1a

1a 1 2a

1b 0a 1 2a 1a

0a 1 2a 1a

1b 1a 1a 1a 1a 1a 1a

1 2b 1 2b 0b 1 2b 0b 1 2b 1 2b

1b

1 2b 1b

0b 0b 0b 0b 0b 0b 0b

1a 1b 1b 1b 1b 1b 1b

0b 0b 1b 0b 1b 0b 0b

1 2b

1b 1b 1b 1b 1b 1b 1b

1a 1b 0b 1b 0b

26

1b 1b 1 2b 1b

1b 1 2b

1 2b 0b

1b 0b 1 2b 1b

D) The matrix Mλ2 ,µ2 : &a 0a

1 2b 1b

0a 0a 0a 0a 0a 0a 0a

→a 0a

0a 1a

1 2a 1a

1 2a 0a 1 2a 1a

1 2a 1a

0b

0b 1 2b 1b

&b 0a 1 2a 1a

0b 1 2b 1b

→b 0a 1 2a 1a

0b 1 2b 1b

8

1 2a

0a 0a

1a 0a

1a

1 2a

0b 1a 1a

1 2b 1 2b 0b 1 2b 0b 1 2b 1 2b

1b

1 2b 1b

0b 0b 0b 0b 0b 0b 0b

1a 1b 1b 1b 1b 1b 1b

0b 0b 1b 0b 1b 0b 0b

1 2b

1b 1b 1b 1b 1b 1b 1b

1 2a

0a

1 2a

1a

0b 0b 0b 0b 0b 0b

1b 0b

1b 1 2b

1b 0a

1 2a

1 2a

1 2b 1b

0a 0a 0a 0a 0a 0a

0a

0a

0a 1b 1b

1 2b

1 2a 1a 1 2a 0a

1 2a 0a

1a 1a 1a 1a 1a 1a 1a

1 2b 0b 1 2b 0b 1 2b 1 2b

0b 0a 0a

1a 1a 1 2a 1a

1a

1 2a

0b 1b 0b 1b 0b 0b

1 2b 0b

1 2a 0a

0a

1 2b

1a

1a

1 2a 0a 1 2a 1a

1 2a

0a

1 2a 1a

1b 1b 1 2b 1b

1b 1 2b

1 2a 1a 1 2a 0a

1a 1b 1a 1a 1a 1a 1a 1a 1 2b 0b

1b 0b 1 2b 1b

Concluding Remarks

In this article we introduce some operations for combining matrix logics, in the same spirit that the original formulation of fibring. In this sense, this paper should be seen as a first step in the direction of obtaining mechanisms for combining matrix logics. The unrestricted fibring by functions became a simple generalization of the original definition of fibring, which produces a weak conservative extension of the given logics. In order to obtain a conservative extension of the given logics (as usually is expected for a “good” combination technique) it is necessary to impose additional conditions to the fibring pairs. This originates the so-called plain fibring of logics. We can generalize these methods in

27

a natural way, extending our first definition of fibring to sets of fibring pairs. The main results obtained with these new definitions encompasses the similar results for the simpler cases. The study of the general properties of the plain fibring and the direct union of logics deserves future research. In particular, the relationship between these operations and the categorical fibring (in the sense of [11] or [17]) should be analyzed. On the other hand, if the given logics have also a proof-theoretic presentation (such as Hilbert calculi) it would be interesting to give a prooftheoretic presentation of the logics obtained by plain fibring and direct union. Another topic to be studied is the possibility of sharing connectives through the combination process, in analogy to the constrained fibring (see [11]). Acknowledgements: This research was financed by FAPESP (Brazil), Thematic Project ConsRel 2004/1407-2, and by CICITCA, National University of San Juan, Argentina. The first author was also supported by an individual research grant from The National Council for Scientific and Technological Development (CNPq), Brazil.

References [1] C. Caleiro, W. A. Carnielli, M. E. Coniglio, A. Sernadas, and C. Sernadas. Fibring Non-Truth-Functional Logics: Completeness Preservation. Journal of Logic, Language and Information, 12(2):183–211, 2003. [2] M.E. Coniglio. Recovering a logic from its fragments by meta-fibring. Logica Universalis, 1(2):377–416, 2007. [3] M.E. Coniglio and V.L. Fern´andez. Plain fibring and direct union of logics with matrix semantics. In B. Prasad, editor, Proceedings of IICAI’05 Second Indian International Conference on Artificial Intelligence, pages 1590–1608, Pune, India, 2005. [4] J. Czelakowski. Protoalgebraic Logics. Kluwer Academic Publishers, Dordrecht, 2001. [5] V.L. Fern´ andez. Semˆ antica de sociedades para l´ogicas n-valentes (Society semantics for n-valued logics, in Portuguese). Master’s thesis, IFCH – State University of Campinas, Brazil, 2001. [6] D. Gabbay. Fibred semantics and the weaving of logics: Part 1. The Journal of Symbolic Logic, 61(4):1057–1120, 1996. [7] D. Gabbay. Fibring Logics. Clarendon Press - Oxford, 1999. [8] P. H´ ajek. Metamathematics of Fuzzy Logic. Kluwer, Amsterdam, 1998. [9] J. Lo´s and R. Suszko. Remarks on sentential logics. Indagationes Mathematicae, 20:177–183, 1958. 28

[10] J. Lukasiewicz and A. Tarski. Untersuchungen u ¨ber den aussagenkalk¨ ul. Comptes Rendus des S´eances de la Societ´e des Sciences et des Lettres de Varsovie, 23:30–50, 1930. [11] A. Sernadas, C. Sernadas, and C. Caleiro. Fibring of logics as a categorical construction. Journal of Logic and Computation, 9 (2):149–179, 1999. [12] A. M. Sette. On the propositional calculus P1 . Mathematica Japonicae, 18:173–180, 1973. [13] A. M. Sette and W. A. Carnielli. Maximal weakly-intuitionistic logics. Studia Logica, 55:181–203, 1995. [14] W.A W.A. Carnielli, M.E Coniglio, D. Gabbay, P. Gouveia, and C. Sernadas. Analysis and Synthesis of Logics. How to Cut and Paste Reasoning Systems. Applied Logic Series. Springer, New York, 2008. [15] R. W´ ojcicki. Matrix approach in methodology of sentential calculi. Studia Logica, 32:7 – 37, 1973. [16] R. W´ ojcicki. Theory of Logical Calculi. Synthese Library. Kluwer Academic Publishers, 1988. [17] A. Zanardo, A. Sernadas, and C. Sernadas. Fibring: Completeness preservation. The Journal of Symbolic Logic, 66(1):414–439, 2001.

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