Plain Fibring and Direct Union of Logics with Matrix Semantics
Marcelo E. Coniglio1 and V´ıctor L. Fern´andez2
1 Centre for Logic, Epistemology and the History of Science and Department of Philosophy, State University of Campinas, P.O. Box 6133, 13081-970, Campinas, SP,
Brazil,
WWW home page:http://www.cle.unicamp.br/prof/coniglio/
2 Basic Sciences Institute – Mathematical Area, National University of San Juan, Av. I. de la Roza 230 (O), 5400, San Juan, Argentina,
Abstract. In this paper a variation of the fibred semantics of D. Gabbay calledplain fibring is proposed, with the aim of combining logics given by matrix semantics. It is proved that the plain fibring of matrix logics is also a matrix logic. Moreover, it is proved that any logic obtained by plain fibring is a conservative extension of the original logics. It is also proposed a simpler version of plain fibring of matrix logics calleddirect union. This technique is applied to the study of the class of fuzzy logics defined byt-norms.
Introduction
Among the different techniques for combining logics, fibring has deserved the attention of a great number of logicians. This method 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 [15] and [16]). Essentially, the under- lying idea of fibring is 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 bothL1andL2. So, the formulas ofLi (i= 1,2) are also formulas ofL1~L2. But in the new logic there exist new formulas (that we call “hybrid formulas”), obtained by mixing the con- nectives of the two original logics. Thus, it is possible to evaluate a connectivec of the language ofL1applied to formulas of the language ofL2. This can be done by means of a function which associates, to each world of any model of Kr1, a world of a model of Kr2. Using this method, the hybrid formulas of the new language can always be evaluated. Moreover,L1~L2 is a weak extension ofL1
andL2, in the sense that all the tautologies of the original logics are tautologies of the new logic.
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 [19] and widely studied afterwards (see for instance [2], [23], [4], [9] and [3]).
However, D. Gabbay used the expression “fibring” for another techniques (see the above mentioned [16]), applied to fuzzy logics and temporal logics among others. In the present article the original notion of fibring is extended to another class of logics: the logics characterized by matrix semantics. Thus, in this paper the term “fibring” will mean “Gabbay’s fibring”. Moreover, one of the techniques to be introduced here will be called plain fibring because its inspiration on the original notion of fibring.
Briefly, the method proposed here is defined as follows: given two matrix log- icsL1andL2, the mixed language is obtained as usual from the given languages of both logics. Then, since each Li has a matrix semantics Mi (i = 1,2), we define functions f :M1→M2 and g : M2→M1 which allow us “to jump” from a valuation of M1 to a valuation ofM2 and vice-versa. Considering these func- tions we can obtain the truth-value of any hybrid formula of the mixed language.
The class of valuations obtained in this manner define a consequence relation
`M1~M2, corresponding to the logic that will be denoted byL1~L2 (the plain fibring of L1 andL2).
The following two sections are devoted to introduce the basic notions about propositional logics, matrix semantics and fibring of modal logics.
1 Preliminaries
Definition 1.
(a) The set of propositional variables V is any countable set (fixed from now on). Its elements will be denoted byp, q, r, p1, p2, . . .. A signatureis a familyC
= {Ck}k∈N, where eachCk is a set ofconnectives of arity k. We assume that Ck∩Cn=∅=Ck∩ V for everyk6=n. Thedomain of the signatureCis the set
|C|=S
k∈NCk. Given two signaturesC1 andC2, we say that C1is included in C2(indicated byC1⊆C2) if, for everyk∈N,C1k ⊆C2k. The signaturesC1∪C2
and C1]C2 (the union and the disjoint union of C1 and C2, respectively) are defined as expected.
(b)A propositional language with signatureC (denoted byL(C)) is the algebra of words, freely generated by C over V, considering each set Ck as the set of k-ary operations of that algebra.
It should be noted that it is possible to haveC6=C0such thatL(C) =L(C0).
By simplicity, sometimes we will identify a signatureC with its domain|C|.
Following the Tarskian perspective, the other component that characterizes a logic is the consequence relation.
Definition 2. Fixed a signature C, a consequence relation over the signature C (or inC) is a relation` ⊆℘(L(C))×L(C)verifying: 3
– Ifϕ∈Γ thenΓ `ϕ (Extensiveness).
3 As usual, (Γ, α)∈ `will be denoted byΓ `α.
– IfΓ `ϕandΣ`ψ, for every ψ∈Γ, then Σ`ϕ (Transitivity).
It is worth noting that any consequence relation defined as above satisfies the following useful property:
– IfΓ `ϕandΓ ⊆Σ thenΣ`ϕ (Monotonicity).
In fact, If Γ `ϕ andΓ ⊆Σ thenΣ `ψfor every ψ∈Γ, by Extensiveness.
ThusΣ`ϕ, byTransitivity.
A substitution in C is any function σ: V→L(C). Since L(C) is freely gen- erated from V, σcan be extended to a unique endomorphism bσ:L(C)→L(C).
Using this, we arrive to the following definition.
Definition 3. A (propositional) logic is a pair L = hCL,`Li, where CL is a signature and `L is a consequence relation in CL in the sense of Definition 2.
A logic L is said to be astructural logic if it satisfies:
– For every substitutionσinCL and every Γ ∪ {ϕ} ⊆L(CL):
IfΓ `Lϕthenσ(Γb )`Lbσ(ϕ) (Structurality).
On the other hand,L is called afinitary logicif it satisfies:
– For everyΓ ∪ {ϕ} ⊆L(CL):
IfΓ `LϕthenΓ0`Lϕfor some finite setΓ0⊆Γ (Finitariness).
Finally,L is astandard logicif it is structural and finitary.
The following notions will be used from now on:
Definition 4. Let L=hCL,`Libe a logic, and letC⊆CL. TheC-fragment of Lis the logicL|C:=hC,`L|Ciwhere`L|C =`L ∩(℘(L(C))×L(C)). Thus, for every Γ∪ {ϕ} ⊆L(C),Γ `L|C ϕif and only if Γ `Lϕ. Given two logics Land L0, we will say thatL0 is a strong extension ofLifCL ⊆CL0 and`L⊆ `L0. The logic L0 is a weak extensionof L if CL⊆CL0 and, for every ϕ∈L(CL),`L ϕ implies that`L0 ϕ. We say thatL0 is a conservative extension ofLifCL⊆CL0
and L=L0|CL. Finally, L0 is a weak conservative extension of L if CL⊆CL0
and, for everyϕ∈L(C),`Lϕiff`L0 ϕ.
The proof of the following result is straightforward:
Proposition 1. The following holds:
(a)EachC-fragment of any (structural, finitary, standard) logic is also a (struc- tural, finitary, standard) logic.
(b) Every logicLis a conservative extension of any of its C-fragments. ut We briefly recall the basic facts about matrix semantics.
Definition 5. Given a signature C, a C-matrix is a pair M =hA, Di, where A =hA, Ci is an algebra over C, 4 and D⊆A. The set D is usually referred
4 That is, we can identify the set ofk-ary operations ofAwithCk. ThereforeL(C), considered as an algebra, has the same type of similarity thanA.
as the set of designated values of M. The M-valuations of L(C) are the C- homomorphismsv:L(C)→A.
For simplicity, sometimes we will writeM =hA, Diinstead of M =hA, Di in concrete examples. Additionally, the interpretation of a connective c in M will be frequently written ascM, or will be simply written asc.
Definition 6. Let C be a signature and let K be a class of C-matrices. The matrix semantics induced by K (denoted by`K) is defined by: Γ `K ϕ iff, for every C-matrix M = hAM, DMi belonging to K and every M-valuation v of L(C),v(Γ)⊆DM implies that v(ϕ)∈DM.
A logicL is said to be matrix logicif there exists a classK ofCL-matrices such that`L=`K. WhenK={M}is a singleton then`M will stand for`{M}. Clearly, every matrix logic is a logic in the sense of Definition 3. Moreover, the following fundamental result due to J. Lo´s and R. Suszko (see [18]) shows that a matrix logic is, in fact, structural:
Proposition 2. LetC be a signature, and letK be a class ofC-matrices. Then
`K is a structural consequence relation inCand`K=inf{`M : M ∈ K}.5 ut
Note that`K do not need to be finitary and, therefore,L =hC,`Ki is not necessarily standard. A sufficient condition for a matrix logic be standard was obtained by R. W´ojcicki (see [22]):
Proposition 3. Every consequence relation induced by a finite set of finite ma- trices is finitary and then it defines a standard logic. ut
2 Fibring of modal logics
In this section we briefly recall the fibred semantics as defined by D. Gabbay, applied to Kripke semantics.
Definition 7. Amodal signatureis a signatureCsuch thatC1={¬,};C2= {⇒}; Ck =∅ in any other case. A Kripke model (for modal logics) is a triple m=hWm, Rm, hmisuch thatWmis a nonempty set (the set of possible-worlds of m);Rm⊆Wm×Wm (the accessibility relation ofm); and hm :V→℘(Wm) is a mapping (the m-valuation). A Kripke semantics is a class Kr of Kripke models.
Fixed a modal signatureCand a Kripke semanticsKr, a consequence relation
`Kr inC can be defined as usual.
5 The infimum is taken with respect to the inclusion ordering ⊆.
It is worth noting that the consequence relation of a modal logic can be obtained from several Kripke semantics: for instance, the well-known modal logic S5can be obtained by means ofKr1(the Kripke semantics such that everyRm is an equivalence relation) or by means of Kr2 (where there is no restrictions to Rm). This is a relevant fact because the fibring of modal logics depends not only on the consequence relations but also on the given Kripke semantics of each logic. Therefore, in the following definition, the modal logics will be denoted by pairsL=hCL, Kri.
Definition 8 (Gabbay). Let Li = hCi, Krii (i = 1,2) be two modal logics.
The fibred signatureC~ ofL1 andL2 is defined by:
C~1 ={¬,1,2}; C~2 ={⇒}; C~k =∅ in any other case.
The fibred language ofC1andC2 isL(C~). Afibred model ofKr1 andKr2 is a triple (f, g, h)such that
f : ]
m∈Kr1
Wm→ ]
m∈Kr2
Wm; g: ]
m∈Kr2
Wm→ ]
m∈Kr1
Wm; h:V→℘(W),
whereW := (U
m∈Kr1Wm)](U
m∈Kr2Wm).6 The fibred structure ofKr1 and Kr2 (symbolized byKr1~Kr2) is the class of all the fibred models of Kr1 and Kr2. The consequence relation`Kr1~Kr2 ⊆℘(L(C~))×L(C~) is defined from the notion of(f, g, h)wϕ(for(f, g, h)∈Kr1~Kr2,w∈W andϕ∈L(C~)) as expected, with the following relevant cases: if w ∈ Wm and m ∈ Kr2 then (f, g, h) w 1ψ iff (f, g, h) w0 ψ for every w0 ∈ Wm0 such that g(w)Rm0w0 (the dual case is treated analogously, using f instead of g). The logic L~ = hC~,`Kr1~Kr2iis calledthe fibring ofL1and L2.7
Example 1. LetL1be the modal logicT. Then,Kr1can be taken as the class of all the modelsmsuch thatRmis reflexive. LetL2be the von Wright’s temporal logic VW (withVW meaning “in all future time”). In this case Kr2 can be taken as the class of all the models m such that Wm = N and Rm is a total ordering. The fibred signatureC~ is given by:C~1 ={¬,T,VW};C~2 ={⇒}
and C~k =∅ in any other case. Let (f, g, h) be a fibred model, w ∈Wm, with m∈Kr1, and letϕbe the formulaVW(Tq⇒p). Then:
(f, g, h) w VW(Tq ⇒ p) iff (f, g, h) w0 Tq ⇒ p for every w0 ∈ Wm0 such that f(w)Rm0w0. On the other hand, givenm0 ∈Kr2 andw0 ∈Wm0 then (f, g, h)w0 Tq⇒piff eitherw0 ∈h(p) orw006∈h(q) for somew00∈Wm00such that g(w0)Rm00w00.
The logic L~ is a weak extension of L1 and L2 (i.e., it just preserves the tautologies, cf. Definition 4). On the other hand, by the very definition, L~ distinguishes the different modal connectives, but identifies the implication and the negation of both logics as being the same. In the plain fibring of matrix semantics we propose here there are no identifications of connectives.
6 Here, and in the rest of the paper,U
denote the disjoint union of sets.
7 Note thatL~is a logic in the sense of Definition 3.
3 Direct union of matrix logics
In the sequel we introduce an adaptation of fibring to logics defined by means of matrix semantics. The basic motivation is to consider the set of worlds as being an analogous to the algebra of truth-values. Similarly to the modal case, the operations to be defined below will depend on the choice of the matrices characterizing the semantics of the given logics. Thus, from now on a matrix logic will be represented as a pairhC,Kiinstead of a pairhC,`Ki, for a classK ofC-matrices. In particular, ifK={M}we will write hC, Mi.
The basic idea is that, given two matrix logics L1 and L2 such that Li is characterized by a single matrixMi with domain Ai and designated values Di
(i= 1,2), it is possible to extend the original operators of the algebraMito the disjoint unionA1]A2by means of mappingsfi :Aj →Ai (i6=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 com- bined logic can be simply obtained by putting together both matrices. This technique is what we calldirect unionof matrix logics. Formally:
Definition 9. LetLi =hCi, Mii(withi= 1,2) be two matrix logics, where each Mi=hAi, Dii is aCi-matrix. Assume thatA1=A=A2 andD1=D=D2.8 The direct union ofL1 andL2 is the logic L1+L2=hC1]C2,`M1+M2iwhere
`M1+M2 is the consequence relation 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 4. Let L=hC,`ibe a logic such that ` is characterized by a C- matrixM. Let L1 andL2 be two fragments ofL defined over signatureC1 and C2, respectively, such thatC1]C2=C. Then L1+L2 =L. In particular, the same result holds if C1∪C2=C.
Proof. Straightforward, from the definitions.9 ut
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 [8], where a stronger notion of logical translations was proposed in order to recover a logic from its fragments through the process of (categorial) fibring. On the other hand, in [6] the generic operations of decomposition and composition of logics were called splitting and splicing logics, respectively. In the next section we will give an example of the direct union of two fuzzy logics defined throught-norms, in order to obtain a bigger logic which contains both
8 Note that thisdoes not meanthat the operations defined inM1andM2 coincide.
9 There is a minor technical detail to be considered in the case thatC1∪C2=C and C1k∩Ck2 6= ∅for some k ∈N. In this case, there will be duplicate connectives in L1+L2 and so this logic is not identical toL. However,L1+L2 can be identified withLup to logical translations.
logics as its fragments. This example is useful in the realm of abstract algebraic logic, as studied in [11].
Example 2. Let Li =hCi, Mii (withi= 1,2) such that C1 ={¬,∨} (negation and disjunction, respectively) and C2 ={∧,⇒} (conjunction and implication, respectively). Suppose thatM1 is the matrix for classical negation and disjunc- tion, and thatM2is the matrix for classical conjunction and implication, where both matrices are defined over A={1,0} withD ={1}. Then L1+L2 turns out to be the matrix presentation L of classical propositional logic overA and D and signature{¬,∨,∧,⇒}. The logicsL1andL2are two (simpler) factors of L. By its turn,L1and L2 can of course be split into two elementary logics:L11 (the logic of classical negation) andL21(the logic of classical disjunction), on one hand; and L12 (the logic of classical conjunction) and L22 (the logic of classical implication), on the other hand. That is, L1 = L11+L21 and L2 = L12+L22. Therefore,Lsplits intoL11,L21,L12 andL22, and soL=L11+L21+L12+L22.
4 Direct union of logics induced by t-norms
By the very definition, the direct union of logics deals with matrix logics, sayL1
andL2, whereLiis characterized by a single matrixMi(i= 1,2), such thatM1
andM2share the same domain and set of designated truth-values. An interesting example comes from fuzzy logics, which are usually defined by means of a matrix over the interval [0,1] with 1 as the unique designated value. The logics induced byt-norms constitute a particular case of this (cf. [17]; see also [11] for a study within the context of abstract algebraic logic). In what follows we briefly recall the main definitions and properties of logics induced byt-norms:
Definition 10. A t-norm is a mapping ∗ : [0,1]×[0,1]→[0,1] satisfying, for any x, y, z∈[0,1]:
(i) x∗y=y∗x (commutativity);
(ii) x∗(y∗z) = (x∗y)∗z (associativity);
(iii) If x≤y then x∗z≤y∗z (monotonicity);
(iv) 1∗x=x (identity).
A t-norm iscontinuous if it is a continuous mapping with respect to the usual topologies of[0,1]and[0,1]×[0,1].
The following are trivial consequences of the definition above:
(iii)0 Ifx≤y thenz∗x≤z∗y;
(iv)0 0∗x= 0.
Proposition 5. Let ∗ be a continuoust-norm. Then x⇒y:=sup{z∈[0,1] : x∗z≤y}
is the unique operation⇒: [0,1]×[0,1]→[0,1]satisfying, for everyx, y, z∈[0,1]:
x∗z≤y iffz≤x⇒y. ut
Given a continuous t-norm ∗, the operation⇒ defined above is called the residuum of∗.
Definition 11. Let ∗ be a continuous t-norm and let ⇒ be its residuum. The logic generated by∗ is the logicL(∗) =hCL(∗),`L(∗)idefined as follows:
(i) CL(∗)1 ={∗},CL(∗)2 ={⇒},CL(∗)k =∅ in any other case;
(ii) `L(∗) is the consequence relation defined by the matrix h[0,1],{1}i, where the matrix operations are the mappings∗ and⇒.10
Previous to study the direct union of logics induced byt-norms it is necessary to state some technical results, whose proof is left to the reader.
Lemma 1. For everyt-norm∗ it holds, for everyx, y∈[0,1]:
a) x⇒x= 1.
b) x∗y= 1iffx=y= 1.
c) x⇒y= 1 iffx≤y. ut
We will state below a fundamental result relating different t-norms: their respective residua are interderivable in a certain sense (see Proposition 7).
Definition 12. Let ∗ be a continuous t-norm, and let ⇒ be its correspondent residuum. The equivalence operatoris given by x∆y:= (x⇒y)∗(y⇒x).
Note that the equivalence operator ∆ induces a (derived) connective ∆ in the logic L(∗), called theequivalence connective: ϕ∆ψ := (ϕ⇒ψ)∗(ψ ⇒ϕ).
Using this connective, it is possible to obtain a very interesting result (cf. [11]
and [7]):
Proposition 6. Any logic of the form L(∗)for a continuous t-norm ∗ is alge- braizable (in the sense of Blok-Pigozzi, cf. [1]). ut
The direct union of logics can therefore be applied to obtain another inter- esting result in abstract algebraic logic (see again [11] and [7]):
Proposition 7. Let L(∗i)be the logic induced by a continuous t-norm ∗i (i= 1,2). Let∆i be the equivalence connective of L(∗i)(i= 1,2). If L is the direct union L(∗1) +L(∗2) thenp∆1q`Lp∆2q andp∆2q`Lp∆1q for everyp, q∈ V, and soL(∗1) +L(∗2)is algebraizable in the sense of Blok-Pigozzi. ut
A detailed discussion about combination of algebraizable logics can be found in [7] (see also [11], [12] and [14]).
10Here, as usual, we use the same symbol for a connective and its matrix interpretation.
5 Unrestricted plain fibring of matrix logics
A more interesting situation is to combine two matrix logics characterized by single matrices defined over different domains. In this case, each matrix logic is extended to the disjoint union of the domains of the matrices by means of a pair of mappings, and then the direct union of the extensions is computed.
The set of matrices obtained in this way is the matrix semantics of the so-called (unrestricted) plain fibring.
Definition 13. Let Li = hCi, Mii (with i = 1,2) be two matrix logics, where each Mi = hAi, Dii is a Ci-matrix with domain Ai. The fibred signature is given by C1]C2 and the fibred language is L(C1]C2). A unrestricted fibred valuation is a triple (f, g, v), where (f, g) ∈ AA21 ×AA12 and v ∈ (A1]A2)V. Given ϕ ∈ L(C1 ]C2) and a unrestricted fibred valuation (f, g, v), we define (f, g, v)(ϕ)∈A1]A2 by recursion on the complexity of ϕ:
– Ifϕ∈ V then(f, g, v)(ϕ)= v(ϕ);
– If ϕ = c(β1, . . . , βk) then (f, g, v)(ϕ) = c((f, g, v)(β1), . . . ,(f, g, v)(βk)) 11 where, for every formulaβj (j= 1, . . . , k):
• If c ∈ Cik and (f, g, v)(βj) ∈ Ai then (f, g, v)(βj) = (f, g, v)(βj) (for i= 1,2);
• If c∈C1k and(f, g, v)(βj)∈A2 then(f, g, v)(βj) =g((f, g, v)(βj));
• If c∈C2k and(f, g, v)(βj)∈A1 then(f, g, v)(βj) =f((f, g, v)(βj)).
We say that a unrestricted fibred valuation (f, g, v) satisfies ϕ if (f, g, v)(ϕ) ∈ D1]D2. Theunrestricted plain fibred consequence relation`M1~M2 ⊆℘(L(C1] C2))×L(C1]C2)is defined as follows: Γ `M1~M2 ϕ if, for every unrestricted fibred valuation(f, g, v)satisfying simultaneously all the formulas ofΓ, we have that(f, g, v)satisfiesϕ. The pairL1~L2=hC1]C2,`M1~M2iis theunrestricted plain fibringof L1 andL2.
Example 3. Consider the paraconsistent matrix logicP1, introduced in [20]. The signature of P1 is such that |CP1| = {¬P1,⇒P1}. The P1-matrix is MP1 = hAP1,{T, T1}isuch thatAP1 ={T, T1, F}, and the operations corresponding to
¬P1 and⇒P1 are defined through the tables below.
T T1F
¬P1 F T T
⇒P1T T1F T T T F T1 T T F F T T T
Now, consider the classical propositional logicCP Ldefined over signatureCCP L such that |CCP L| ={∧CP L,¬CP L}, and with the usual matrix semantics over {0,1}. LetP1~CP Lbe the unrestricted plain fibring ofP1andCP L, letϕbe the formula⇒P1(p,∧CP L(¬CP Lr,∧CP L(q,¬P1r))) in the fibred language, where p, q, r∈ V, and let (f, g, v) be a unrestricted fibred valuation such that:
11Again, we use here the same symbol for a connective and its matrix interpretation.
– v(p) =T;v(q) = 0;v(r) =T1; – f(T) = 1;f(T1) = 1;f(F) = 0;
– g(1) =T;g(0) =F. Then, by Definition 13,
(f, g, v)(ϕ) = (f, g, v)(⇒P1(p,∧CP L(¬CP Lr,∧CP L(q,¬P1r))))
= ⇒P1(T, g(∧CP L(¬CP L(f(T1)),∧CP L(0, f(¬P1(T1))))))
= ⇒P1(T, g(∧CP L(¬CP L(1),∧CP L(0, f(T)))))
= ⇒P1(T, g(∧CP L(0,∧CP L(0,1))))
= ⇒P1(T, g(∧CP L(0,0)))
= ⇒P1(T, g(0)) = ⇒P1(T, F) =F.
6 Some results about unrestricted plain fibring
In this section we present some basic results about the logics obtained by means of unrestricted plain fibring.
Firstly, consider the following questions: isL1~L2 a logic in the sense of Definition 3? Is it structural? Is it finitary? Is it standard? Proposition 8 below gives us some answers.
Definition 14. Let L=hC, Mibe a matrix logic, whereM is aC-matrix with domain A and set of designated values D⊆A. LetA0 and D0 be two sets such that D0 ⊆A0. Suppose, without loss of generality, thatA∩A0 =∅. Finally, let f :A0→A be a mapping. TheC-matrix Mf is defined as follows: its domain is A]A0; the set of designated values isD]D0 and, for c∈Ck anda1, . . . , ak ∈ A]A0,cMf(a1, . . . , ak) =cM(a1, . . . , ak)where, for everyaj (j= 1, . . . , k):
- Ifaj∈A, thenaj = aj. - Ifaj∈A0, thenaj =f(aj).
Proposition 8. Let Li = hCi, Mii (with i= 1,2) be two matrix logics, where each Mi is aCi-matrix. Then:
(a)L1~L2 is a matrix logic and so it is a structural logic.
(b) If the matrixMi is finite (i= 1,2) then `M1~M2 is finitary and soL1~L2
is a standard logic.
Proof. (a) The basic idea is to prove the following:`M1~M2 can be characterized by a certain set K~ of C1]C2-matrices of the form M = hA, D1]D2) such that A = A1]A2 and, if a1, . . . , ak ∈ Ai and c ∈ Cik then cM(a1, . . . , ak) = cMi(a1, . . . , ck) (i= 1,2). Thus, the matrices of the setK~ extend the original matrices. Moreover:
(i) For each unrestricted fibred valuation (f, g, v) there exists aC1]C2-matrix M inK~and a valuationv0 inM such that (f, g, v)(ϕ) =v0(ϕ) for every formula ϕ∈L(C1]C2).
(ii) Conversely, for each C1]C2-matrix M ∈ K~ and each valuation v in M there is a unrestricted fibred valuation (f, g, v0) such that (f, g, v0)(ϕ) =v(ϕ) for
every formulaϕ∈L(C1]C2).
It is clear that, from (i) and (ii), the desired result follows easily. We proceed now to define the setK~ of matrices. Given (f, g)∈AA21×AA12 (assuming, with- out loss of generality, thatA1∩A2=∅) consider theC1]C2-matrixM(f,g)= (M1)g+ (M2)f (recall Definitions 9 and 14). Thus,M(f,g)=hA, D1]D2iwhere A=A1]A2 and the operations are defined as follows:
- Ifc∈C1kanda1, . . . , ak∈A, thencM(f,g)(a1, . . . , ak) =cM1(a1, . . . , ak) where, for everyaj (j = 1, . . . , k):
- Ifaj ∈A1, thenaj =aj. - Ifaj ∈A2, thenaj =g(aj).
- Ifc∈C2kanda1, . . . , ak∈A, thencM(f,g)(a1, . . . , ak) =cM2(a1, . . . , ak) where, for everyaj (j = 1, . . . , k):
- Ifaj ∈A2, thenaj =aj. - Ifaj ∈A1, thenaj =f(aj).
LetK~={M(f,g) : (f, g)∈AA21×AA12}. Now we will prove that K~ satisfies property (i). Given a unrestricted fibred valuation (f, g, v), consider the matrix M(f,g) and the valuation v0 defined inM(f,g) such that v0(p) = v(p) for every p ∈ V. It is straightforward to see that, for every ϕ ∈ L(C1]C2), v0(ϕ) = (f, g, v)(ϕ).
In order to prove thatK~ satisfies (ii), letM(f,g)∈ K~and letv be a valuation in M(f,g). Then we definev0(p) = v(p) for every p∈ V and so the unrestricted fibred valuation (f, g, v0) satisfies the requirements.
From the considerations above we have that`M1~M2 is defined by a class of matrices. Using Proposition 2 it follows that`M1~M2 is structural.
(b) Follows from item (a) and Proposition 3. ut
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 a pair of mappings (f, g) (as in Definition 14); and
(2) the direct union of the extended matrices is computed, obtaining a matrix M(f,g) for the fibred signature.
(3) The unrestricted plain fibring is characterized by the set of all the matrices obtained in this way (by varying the pair (f, g)).
Now we will analyze the relationship between the logic obtained by unre- stricted plain fibring and the given logics. Firstly, we obtain two useful lemmas.
Lemma 2. Let(f, g, v)be a unrestricted fibred valuation, and consider theM1- valuationv0:L(C1)→A1 such that
v0(p) =
g(v(p)) ifv(p)∈A2 v(p) otherwise
for everyp∈ V. Thenv0(ϕ) = (f, g, v)(ϕ)for everyϕ∈L(C1)\ V. An analogous result holds for M2 (using f instead of g).
Proof. Straightforward, by induction on the complexity ofϕand Definition 13.
u t Lemma 3. Let v : L(C1)→A1 be a M1-valuation, and let v0 :V→A such that v0(p) = v(p) for every p ∈ V. Then (f, g, v0)(ϕ) = v(ϕ) for every (f, g) ∈ AA21×AA12 and every ϕ∈L(C1). An analogous result holds for M2.
Proof. Again, it follows straightforwardly by induction on the complexity ofϕ
and Definition 13. ut
Proposition 9. Let Li be a nontrivial logic12 induced by the matrixMi (i= 1,2). Then L1~L2 is a weak conservative extension of both logics L1 andL2, that is:`Liϕiff `M1~M2ϕ, for every ϕ∈L(Ci)(i= 1,2).
Proof. By hypothesis, 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 formulaϕand so, by monotonicity,Γ `Li ϕfor everyΓ∪ {ϕ}. Thus, suppose thatϕ∈L(C1) is such that`L1 ϕ, and let (f, g, v) be a fibred valuation.
Thenϕ6∈ V and so there exists aM1-valuationv0such that (f, g, v)(ϕ) =v0(ϕ), by Lemma 2. Butϕis aM1-tautology and sov0(ϕ)∈D1. That is, (f, g, v)(ϕ)∈D and then `M1~M2 ϕ. Analogously, it can be proved that `L2 ϕ implies that
`M1~M2 ϕ, for everyϕ∈L(C2).
Conversely, letϕ∈L(C1) such that`M1~M2ϕ, and letv be aM1-valuation.
Letv0 :V→A the mapping defined as in Lemma 3, and let (f, g)∈AA21×AA12 (observe that, by hypothesis, bothA1andA2are nonempty, soAA21×AA12 6=∅).
Since (f, g, v0)(ϕ) =v(ϕ) thenv(ϕ)∈D1. Therefore,`L1 ϕ. The proof forL2 is
analogous. ut
Note that, if exactly one of the logics (say,L1) is trivial, then the last result is no longer true. In fact: assuming that A1 = D1 6= ∅ and L2 is not trivial then the propositional variablepis aL1-tautology but not aL2-tautology. Since AA21×AA12 6=∅ it is easy to define a unrestricted fibred valuation (f, g, v) such that (f, g, v)(p) =v(p)∈A2\D2and then (f, g, v)(p)6∈D1]D2.
The Proposition 9 cannot be improved. In fact, L1~L2 is not in general a strong extension of the given logics, as the following example shows.
Example 4. Let L1 = h{∨},`1i 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,`2i 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
12A logic istrivialifΓ `ϕfor everyΓ ∪ {ϕ}.
let g : {T, T1, F}→{0,1} such that g(T) = 0. Consider now the unrestricted fibred valuation (f, g, v) (wheref is any mapping f :{0,1}→{T, T1, F}). Then (f, g, v)(p1) =T ∈D1]D2. On the other hand, (f, g, v)(p1∨p2) = (f, g, v)(p1)∨ (f, g, v)(p2) =g(T)∨0 = 0∨0 = 0∈/ D1]D2. Hence,p16`M1~M2p1∨p2.
7 Plain fibring
A situation as the one described in Example 4 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, [16]) it is required that the obtained logic should be a conservative extension of the given logics.13 The reason of the failure in Example 4 is that we are using unrestricted fibred valuations, and then a designated value could be mapped into a nondesignated one, and vice-versa (note that, in Example 4, we define g(T) = 0). In order to obtain a fibred semantics inducing a logic L which is a conservative extension of bothL1andL2, the set of unrestricted fibred valuations must be refined. This is the key idea of theplain fibring, to be defined below.
Definition 15. Let Li = hCi, Mii (with i = 1,2) be two matrix logics as in Definition 13. A pair(f, g)∈AA21×AA12 isadmissible if it satisfies:f(x)∈D2
iffx∈D1, for everyx∈A1; andg(y)∈D1iffy∈D2, for everyy∈A2. Afibred valuationis a unrestricted fibred valuation(f, g, v)such that(f, g)is admissible.
The plain fibring of L1 andL2 is the pair L1 L2 =hC1]C2,`M1M2i such that`M1M2is the consequence relation obtained by using fibred valuations. That is, for every Γ ∪ {ϕ} ⊆L(C1]C2) it holds: Γ `M1M2 ϕ iff, for every fibred valuation(f, g, v),(f, g, v)(Γ)⊆D implies that (f, g, v)(ϕ)∈D.
The condition required for a pair (f, g) to be admissible is related to the definition of strict homomorphisms between matrices (cf. [10]). In Proposition 10 it will be proved that, in normal cases, the plain fibring is a conservative extension of its factors.
Two logicsL1 and L2 as in Definition 15 are said to be compatibleif there exist admissible pairs inAA21×AA12. Note thatL1andL2are compatible iff:
(i)D16=∅ iffD26=∅; and
(ii) (A1\D1)6=∅ iff (A2\D2)6=∅.
Observe that, ifL1and 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 9 can be improved by considering plain fibrings.
Proposition 10. Let Li =hCi, Mii (with i = 1,2) be two matrix logics as in Definition 13 such thatL1andL2are compatible. ThenL1L2is a conservative extension of both L1 andL2.
13In [8], however, it is argued against thedesideratumof obtaining conservative exten- sions of the given logics through a combination process.
Proof. We just prove thatL1 L2is a conservative extension ofL1, because the proof forL2is analogous. Suppose firstly thatD1=A1. ThenD2=A2(because L1 and L2 are compatible) and then bothL1 and L1 L2 are trivial, thus the result follows. Suppose now thatD1=∅. ThenD2=∅and, again, bothL1 and L1 L2 are trivial. Suppose now that bothA1\D1 andD1 are nonempty. For every fibred valuation (f, g, v) let v0 the M1-valuation defined as in Lemma 2.
Then, for everyϕ∈L(C1),
(f, g, v)(ϕ)∈D iff v0(ϕ)∈D1. (∗)
In fact, ifϕ∈ V then suppose thatv(ϕ)∈A1. Thenv0(ϕ) =v(ϕ) = (f, g, v)(ϕ) and the result holds. On the other hand, ifv(ϕ)6∈A1thenv0(ϕ) =g(v(ϕ))∈D1 iff v(ϕ) = (f, g, v)(ϕ)∈D2 and the result holds. On the other hand, if ϕ6∈ V then the result follows from Lemma 2. Now, let Γ ∪ {ϕ} ⊆L(C1) such that Γ `L1 ϕand let (f, g, v) be a fibred valuation such that (f, g, v)(Γ)⊆D. Letv0 be theM1-valuation obtained from (f, g, v) as in Lemma 2. Using (∗) it follows that v0(Γ)⊆D1 and thenv0(ϕ)∈D1. From (∗) again we get (f, g, v)(ϕ)∈D.
This shows that Γ `M1M1 ϕ.
Conversely, suppose thatΓ ∪ {ϕ} ⊆L(C1) such that Γ `M1M1 ϕ, and let v be a M1-valuation such that v(Γ)⊆D1. Let (f, g) ∈AA21×AA12 admissible (recall that L1 andL2 are compatible) and let (f, g, v0) be the fibred valuation defined as in Lemma 3. Then (f, g, v0)(Γ)⊆D and so (f, g, v0)(ϕ)∈D, that is,
v(ϕ)∈D1. This proves thatΓ `L1 ϕ. ut
By the observation above, Proposition 10 still holds if, in particular, both logics are nontrivial.
It is worth noting that Proposition 8 can be adapted to plain fibring. Thus, given matrix logicsLi=hCi, Mii(i= 1,2) then the plain fibringL1 L2is the matrix logic characterized by the set ofC1]C2-matrices
M1M2={(M1)g+ (M2)f : (f, g)∈AA21×AA12 is admissible}
(recall Definitions 9, 14 and 15). That is,L1 L2=hC1]C2, M1M2i.
It should be stressed that each matrix logichC1,(M1)gi coincides with L1, and each matrix logic hC2,(M2)fi coincides with L2, provided that (f, g) is admissible. In fact:
Proposition 11. For every admissible pair(f, g)it holds:
(a)`M1 =`(M1)g; (b) `M2 =`(M2)f.
Proof. (a) Let Γ ∪ {ϕ} ⊆L(C1). Suppose that Γ `(M1)g ϕ, and let v be a valuation overM1 such thatv(Γ)⊆D1. Thenv is a valuation over (M1)g such thatv(Γ)⊆D1]D2and thenv(ϕ)∈D1]D2. Thusv(ϕ)∈D1and soΓ `M1 ϕ.
Conversely, suppose thatΓ `M1 ϕand letv0be a valuation over (M1)gsuch that
v0(Γ)⊆D1]D2. The valuationv overM1 such thatv(p) =v0(p) ifv0(p)∈A1, andv(p) =g(v0(p)) ifv0(p)∈A2is such that, for everyψ∈L(C1):
v0(ψ)∈D1]D2 iff v(ψ)∈D1.
Therefore v(Γ)⊆D1 and then v(ϕ)∈D1. From this we get v0(ϕ)∈D1]D2. This shows that Γ `(M1)g ϕ. The proof of item (b) is entirely analogous. ut
Example 5. Recall the paraconsistent matrix logicP1considered in Example 3.
LetL1 be the fragment ofP1defined over signature{¬P1}given by the matrix M1 with domainA1={T, T1, F} defined below, where D1={T, T1} is the set of designated values.
T T1F
¬P1 F T T
On the other hand, letL2be the fragment of classical propositional logic defined over signature{⇒}given by the usual matrixM2, with domainA2={0,1}and setD2={1} of designated values, displayed below.
⇒1 0 1 1 0 0 1 1
Now, let A = {T, T1, F,1,0} and D = {T, T1,1}, and let (f, g) ∈ AA21×AA12 such thatf(T) =f(T1) = 1, f(F) = 0, g(1) =T andg(0) =F. Then (f, g) is admissible and (M1)g and (M2)f are given by the tables below, respectively.
T T1 1 F 0
¬F T F T T
⇒T T11F 0 T 1 1 1 0 0 T1 1 1 1 0 0 1 1 1 1 0 0 F 1 1 1 1 1 0 1 1 1 1 1
Let L be the logic over{¬,⇒} characterized by the matrix M(f,g) = (M1)g+ (M2)f 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 forLproduces the 3-valued logic P1, becauseT and 1 are congruent, as well as F and 0. On the other hand, it is clear that the pair (f, g0)∈AA21×AA12 such thatg0(1) =T1 andg0(0) =F is also admissible; moreover, (f, g) and (f, g0) are the unique admissible pairs, and soL1 L2 is characterized by the set of matrices
M1M2={(M1)g+ (M2)f, (M1)g0+ (M2)f}.
It is easy to see that, for instance, the formulaϕ= (p1 ⇒p2)⇒ ¬¬(p1 ⇒p2) is not valid in (M1)g0+ (M2)f and soϕis not valid inL1 L2.
Example 6. In [13] a hierarchy of paraconsistent logics generalizing P1 was in- troduced, called {Pn}n∈N. Each logic Pn is defined over the signature CPn = {¬Pn,⇒Pn}, with semantics given by the matrixMPn=hAPn,{T0, T1, . . . , Tn}i such thatAPn={T0, T1, . . . , Tn,f}. The corresponding operations are displayed in the tables below.
T0 Th f
¬Pn f Th−1T0
⇒PnT0Th f T0 T0T0 f Th T0T0 f f T0T0T0
(1≤h≤n)
Also in [13], it was introduced a hierarchy of weakly-intuitionistic logics called {In}n∈N, generalizing the weakly-intuitionistic logicI1 introduced in [21] as a dual ofP1. EachInis defined over the signatureCIn={¬In,⇒In}, with seman- tics given by the matrix MIn =hAIn,{t}isuch thatAIn={t, F0, F1, . . . , Fn}.
The operations of the matrixMIn are given by the tables below.
t F0 Fl
¬InF0 t Fl−1
⇒In tF0 Fl t tF0F0
F0 t t t Fl t t t
(1≤l≤n)
Note that both P0 and I0 coincide with the classical propositional logic over {¬,⇒}with two-valued matrix semantics. Now we will analyze the plain fibring ofIn withPk. First note that, givenIn andPk, the admissible pairs are of the form (fj, gi) (for 0≤j ≤k and 0 ≤i ≤n) such that gi(f) =Fi; fj(t) = Tj; gi(Th) = t and fj(Fl) = f for 0 ≤ h ≤ k and 0 ≤ l ≤ n. Let M(fj,gi) = (MIn)gi+ (MPk)fj. Then the matrixM(fj,gi)is defined over the signatureCnk= {¬In,⇒In,¬P k,⇒P k}, having 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 andi >0;j= 0 andj >0).
⇒iIntT0ThF0 Fl f t t t t F0F0F0
T0 t t t F0F0F0
Th t t t F0F0F0
F0 t t t t t t Fl t t t t t t f t t t t t t
⇒jP k t T0ThF0 Fl f t T0T0 T0 f f f T0 T0T0 T0 f f f Th T0T0 T0 f f f F0 T0T0 T0 T0T0T0 Fl T0T0 T0 T0T0T0
f T0T0 T0 T0T0T0
t T0 Th F0 Fl f
¬0In F0 F0 F0 t Fl−1 t
¬iIn F0 F0 F0 t Fl−1Fi−1
¬0P k f f Th−1T0 T0 T0
¬jP kTj−1 f Th−1T0 T0 T0
(1≤h≤k; 1≤l≤n)
EachM(fj,gi)defines a matrix logic which is simultaneously paraconsistent (w.r.t.
¬P k) and weakly-intuitionistic (w.r.t.¬In). The plain fibringInPk ofIn and Pk is the matrix logic characterized by the set of matrices
MInMPk ={M(fj,gi) : 0≤j≤kand 0≤i≤n}.
The relationships between the logic defined by each matrix M(fj,gi), the logic InPk and the logicInPk (having a single negation which is simultaneously paraconsistent in the sense ofPk and weakly-intuitionistic in the sense ofIn), introduced in [13], 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 asetof ma- trices (not necessarily being a singleton). We can easily correct this asymmetry by generalizing the operation of plain fibring to matrix logics in general.
Definition 16. LetLi=hCi,Kii(withi= 1,2) be two matrix logics. Theplain fibring ofL1andL2 is the matrix logic L1 L2=hC1]C2,K1 K2isuch that K1 K2 is the class ofC1]C2-matrices
{(M1)g+ (M2)f : M1∈ K1, M2∈ K2 and (f, g)∈AA21×AA12 is admissible}.
Proposition 12. Suppose that, for every M1 ∈ K1 and every M2 ∈ K2, the logics hC1, M1i and hC2, M2i are compatible. Then L1 L2 is a conservative extension of both L1 andL2.
Proof. Let Γ ∪ {ϕ} ⊆L(C1). Then Γ `L1 ϕiff, for every M1 ∈ K1 and every M2∈ K2,Γ `M1M2 ϕ, by adapting the proof of Proposition 10, iffΓ `K1K2 ϕ.
The proof forL2 is analogous. The details are left to the reader. ut
A matrixM =hA, Diis said to betrivialifD=∅or D=A. The following result is a direct consequence of the proposition above.
Corollary 1. Let Li = hCi,Kii (with i = 1,2) be two matrix logics. Suppose that K1 andK2 do not contain trivial matrices. Then L1 L2 is a conservative
extension of both L1 andL2. ut
