Fibring Non-Truth-Functional Logics: Completeness Preservation

Fibring Non-Truth-Functional Logics:

Completeness Preservation

C. Caleiro1, W.A. Carnielli2, M.E. Coniglio2, A. Sernadas1 and C. Sernadas11CLC, Department of Mathematics, IST, UTL, Portugal2CLE, Department of Philosophy, IFCH, UNICAMP, Brazil

Abstract. Fibring has been shown to be useful for combining logics endowed with truth-functional semantics. However, the techniques used so far are unable to cope with fibringof logics endowed with non-truth-functional semantics as, for example, paraconsistent logics.The first main contribution of the paper is the development of a suitable abstract notionof logic, that may also encompass systems with non-truth-functional connectives, and wherefibring can still be dealt with. Furthermore, it is shown that this extended notion of fibringpreserves completeness under certain reasonable conditions. This completeness transfer result,the second main contribution of the paper, generalizes the one established in (Zanardo et al.,2001) but is obtained using new techniques that explore the properties of a suitable meta-logic (conditional equational logic) where the (possibly) non-truth-functional valuations arespecified. The modal paraconsistent logic of (da Costa and Carnielli, 1988) is studied in thecontext of this novel notion of fibring and its completeness is so established.

Keywords: non-truth-functional logics, fibring, completeness.

1. Introduction

In recent years, the problem of combining logics has gained the attention ofmany researchers in mathematical logic. Besides leading to very interestingapplications whenever it is necessary to work with different logics at the sametime, combinations of logics are also of great interest on purely theoreticalgrounds (Blackburn and de Rijke, 1997).

The practical impact of the problem is clear, at least from the point of viewof those working in knowledge representation (within artificial intelligence) andin formal specification and verification (within software engineering). Namely,in a knowledge representation problem it may be necessary to work with bothtemporal and deontic aspects. And in a software specification problem it maybe necessary to work with both equational and temporal specifications. Indeed,in these fields, the need for working with several formalisms at the same timeis the rule rather than the exception. We refer the reader, for instance, to(Finger and Gabbay, 1992; Goguen and Burstall, 1992; Sannella and Tarlecki,1993; Astesiano and Cerioli, 1994) for a discussion of this and related problems,including some early attempts at their solution.

Obviously, an approach to the combination of logics will be of significanceonly if general preservation results are available. For example, if it had beenestablished that completeness is preserved by the combination mechanism ◦and it is known that logic L is given by L′ ◦ L′′, then the completeness of thecombination L would follow from the completeness of L′ and L′′. No wonderthat so much theoretical effort has been dedicated to establishing preservation

c© 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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results and/or finding preservation counterexamples within the community oflogicians working in the problem of combining logics.

Among the different techniques for combining logics, fibring (Gabbay, 1996;Gabbay, 1998; Sernadas et al., 1999; Sernadas et al., 2000; Zanardo et al.,2001) deserves close attention. But what is fibring? The answer can be givenin a few paragraphs for the special case of logics with a propositional base,that is, with propositional variables and connectives of arbitrary arity. Fibringis a mechanism that produces a new logic by mixing up two given logics. Asmentioned above, ideally, the fibred logic would inherit the properties, namelycompleteness (soundness and adequacy), of its two component logics. Unfor-tunately, it is well known that it is not always the case. Still, it is sometimespossible to recover some lost property by further manipulation of the fibredlogic. Let us first explain the mechanism of fibring by itself and delay the issueof preservation of properties for a few paragraphs.

The language of the fibring is obtained by the free use of the languageconstructors (atomic symbols and connectives) from the given logics. For ex-ample, when fibring a temporal logic and a deontic logic, mixed formulae like((Gα) ⊃ (O(Fβ))) appear in the resulting logic. Naturally, in many cases, onewants to share some of the symbols. The previous example would involve theconstrained form of fibring imposed by sharing a common propositional part.

At the deductive system level, provided that the two given logics are endowedwith deductive systems of the same type, the deductive system of the fibringwill be obtained by the free use of the inference rules from both. This approachwill be of interest only if the two given deductive systems are schematic inthe sense that their inference rules are open for application to formulae withforeign symbols. For instance, when one represents Modus Ponens by the ruleMP, {(ξ1 ⊃ ξ2), ξ1} ` ξ2, in some Hilbert system, one may implicitly assumethat the instantiation of the schema variables ξ1, ξ2 by any formulae, possiblywith symbols from both logics, is allowed when applying MP in the fibring.

Although the most basic form of fibring is quite simple at the syntactic levelas described above, the semantics of fibring is much more complex and it isadvisable to consider only the special case where both logics have semanticswith similar models. Following (Sernadas et al., 1999; Zanardo et al., 2001), aconvenient, but quite general, model for a wide class of logics with propositionalbase is provided by a triple 〈U,B, ν〉 where U is a set (of points, worlds, states,whatever), B ⊆ ℘U , and ν(c) : Bn → B for each language constructor c of arityn ≥ 0. We look at the pair 〈B, ν〉 as an algebra of truth-values. It is preciselyin this sense that such models are said to be truth-functional . Given two logicsL′,L′′ with models of this type, what is the semantics of their fibring? As firstshown in (Sernadas et al., 1999), it is a class of models of the same type, suchthat at each point u ∈ U it is possible to extract a model from L′ and one fromL′′. Clearly, if symbols are shared, the two extracted models should agree onthem. In order to visualize the semantics of fibring, consider the fibring of apropositional linear temporal logic with a propositional linear space logic. Eachmodel of the fibring will be a cloud U of points such that at each point oneknows the time line and the space line crossing there. For instance, at the point〈Berlin, 10h15m 25 March 2000〉 one knows the time line (of past, present and

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Fibring Non-Truth-Functional Logics: Completeness Preservation 3

future) of Berlin and the space line (the universe taken as a line for the sake ofthe example) at that time.

It is well known that, contrarily to soundness, adequacy is in general notpreserved by fibring. Still, adequacy can sometimes be recovered by addingfurther interaction axioms and/or inference rules to the fibred logic (see forinstance the modal fibring rule in (Gabbay, 1998), Chapter 3). Another approachto adequacy preservation consists in imposing reasonable extra conditions to thegiven logics that may be sufficient to guarantee that the fibred logic turns outto be adequate (Zanardo et al., 2001).

In this paper we aim at broadening fibred semantics in order to cope withnon-truth-functional logics like paraconsistent logics. Paraconsistent logics wereintroduced in (da Costa, 1963) and since then have been the object of continuedattention, because of their theoretical and practical significance. In particular,the paraconsistent systems Cn of (da Costa, 1963) are subsystems of proposi-tional classical logic in which the principle of Pseudo Scotus γ,¬ γ ` δ does nothold. It is well known that, in all the Cn systems, negation cannot be given atruth-functional semantics (Mortensen, 1980).

The first main contribution of this paper is the definition of a general notionof logic that also encompasses non-truth-functional logics. In previous workon the semantics of fibring this kind of logics has never been considered. Infact, non-truth-functional logics could not even be represented using those ap-proaches. In order to overcome this limitation we consider a broader notion oflogic system that accommodates this novelty. The main ingredient is the useof a suitable auxiliary logic, that we call the meta-logic, where the (possibly)non-truth-functional valuations are defined. Since it is enough for the presentpurposes, we choose conditional equational logic (CEQ, (Goguen and Meseguer,1985; Meseguer, 1998)) as the meta-logic. Furthermore, we manage to recoverfibring in this wider context and also to prove that this extended notion offibring preserves completeness under reasonable conditions. This completenesstransfer result, the second main contribution of this paper, generalizes the oneestablished in (Zanardo et al., 2001) and is obtained using a new adequacypreservation technique exploiting the properties of the meta-logic, in this caseCEQ. We should stress that the present approach is not just an adaptationof previous work but it involves the conceptual breakthrough of dropping thewidely accepted principle of truth-functionality.

As an example of application we analyze the system CD1 of paraconsistent

modal logic of (da Costa and Carnielli, 1988). One might wonder if we couldrecover such a mixed logic by fibring the underlying modal and paraconsistentlogics. In fact, it turns out that by simply fibring the two, using the method wepropose, the fibred logic obtained is a little weaker than the original paracon-sistent modal logic CD

1 . This problem has to do with the simple fact that CD1

contains an essential interaction axiom that cannot even be expressed in eitherof the logics being fibred. As a consequence, the target paraconsistent modallogic can be easily recovered by adding that axiom to the obtained fibred logic(similarly to the above mentioned technique for completeness preservation),together with a corresponding semantic restriction. This process is part of theessential idea of fibring as proposed in (Gabbay, 1998), Chapter 1.

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4 C. Caleiro et al.

Following the same methodology used in previous work, namely (Sernadaset al., 1999), we advocate that the basic form of fibring must be characterized, inprecise terms, by means of a categorial construction with a universal property.Still, in this paper, we shall keep the categorial apparatus to the minimumin order to keep the focus of attention on the issue of non-truth-functionality,rather than on the category theoretic details. Therefore, we believe that the pa-per can still be fully assessed by the reader not conversant with the elementarylanguage of categories (MacLane, 1971; Barr and Wells, 1990).

The paper is organized as follows. In Section 2, the notion of interpretationsystem presentation as a specification of the intended valuations within themeta-logic is introduced. The interpretation structures appear as models of thespecification. Section 3 defines the notions of unconstrained and constrainedfibring of interpretation system presentations. The main example, fibring theparaconsistent system C1 and the modal system KD, is also discussed in Sec-tion 3 at the semantic level. Section 4 contains a brief account of the appropriateproof-theoretic notions and returns to the main example at the deductive level.Section 5 establishes the completeness preservation theorem and applies it forproving the completeness of the modal paraconsistent logic CD

1 of (da Costaand Carnielli, 1988). Section 6 discusses applications of self-fibring, namely inthe context of the Cn hierarchy of paraconsistent systems. Section 7 concludeswith an assessment of what was achieved and what lays ahead.

2. Specifying valuation semantics

Observe that, when setting-up an algebraic semantics for a truth-functionallogic, we endow it with models that are algebras (of truth-values) over thesignature of the logic and evaluate formulae homomorphically. This approachdoes not work when the logic is not truth-functional. But still within thespirit of “algebraic semantics”, there is a solution: work instead with two-sorted algebras of formulae and truth-values and include the valuation mapas an operation between the two sorts! This new approach, first sketched in(Coniglio et al., 2000), captures, as a special case, truth-functional logics byimposing the homomorphism conditions on the valuation map which can bedone with equations. Looking at examples of non-truth-functional logics we findthat the envisaged requirements on the valuation map could also be imposed by,albeit conditional, equations. Therefore, we are led to the following algebraicnotion of possibly non-truth-functional semantics: each model is a two-sortedalgebra (of formulae and truth-values) including a valuation operation thatsatisfies some requirements written in a suitable conditional equational meta-logic. As mentioned in the introduction, we adopt CEQ (Goguen and Meseguer,1985; Meseguer, 1998) as the meta-logic.

Let us start by setting up the syntax that we need to use. Since the ob-ject logics under investigation are propositional-based, the following notion ofsignature suffices for our purposes:

DEFINITION 1. An object signature is a family C = {Ck}k∈N where each Ck

is a set (of connectives of arity k).

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In particular, the set of propositional symbols is included in C0.We assume given once and for all the set Ξ = {ξ1, ξ2, . . .} of propositional

schema variables, to be used in inference rules, such that Ξ ∩ C0 = ∅.We denote by L(C, Ξ) the set of schema formulae inductively built from C

and Ξ. For example, ξ1 ⊃ (p∨ ¬ ξ2) is a schema formula, if p ∈ C0, ¬ ∈ C1 and⊃,∨ ∈ C2.

Our next step will be to define an equational signature induced by a givenobject signature C as a meta-linguistic device which permits to talk about thesemantics of logics based on C. For this purpose it is convenient to consider twosorts, sort φ (for formulae) and sort τ (for truth-values). As usual, given a setof sorts S, we write the Kleene closure S∗ to denote the set of all strings overS and ε to denote the empty string. In the following definition, if w ∈ S∗ ands ∈ S then Ow s denotes the set of operations with domain w and codomain s.

DEFINITION 2. Given an object signature C, the induced meta-signature isthe 2-sorted equational signature Σ(C, Ξ) = 〈S, O〉 where S = {φ, τ} and:

− Oε φ = C0 ∪ Ξ;

− Oφk φ = Ck for k > 0;

− Oφ τ = {v};− Oε τ = {>,⊥};− Oτ τ = {−};− Oττ τ = {u,t,⇒};− Oω s = ∅ in the other cases.

We shall use Σ(C) to denote the subsignature Σ(C, ∅), that is, where Oε φ = C0.

The symbols >, ⊥, −, u, t and ⇒ are used as generators of truth-values.The symbol v will be interpreted as a valuation map.

We consider the following sets of variables for Σ(C) and Σ(C, Ξ): Xφ ={y1, y2, . . .} and Xτ = {x1, x2, . . .}. For ease of notation we simply use X todenote the two-sorted family {Xφ, Xτ}. Recall that a term t is called a groundterm if it does not contain variables, and that a substitution θ is said to beground if it replaces every variable by a ground term.

We want to write valuation specifications (within the adopted meta-logicCEQ) over Σ(C) and X. Recall that a CEQ-specification is composed of con-ditional equations of the general form:

(equation1 & . . . & equationn → equation)

with n ≥ 0. Each equation is of the form t = t′ where t, t′ are terms of thesame sort built over Σ(C) and X. The sort of each equation is defined tobe the sort of its terms. A conditional equation that only involves equationsof a given sort is said to be a conditional equation of that sort. Conditional

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6 C. Caleiro et al.

equations are universally quantified, although, for the sake of simplicity, weomit the quantifier, contrarily to the notation used in (Meseguer, 1998). Forexample, ( → v(y1 ∧ y2) = u(v(y1), v(y2))) is a conditional equation of sortτ , supposing that ∧ ∈ C2. It is clear, from this and also the forthcomingexamples, that we only need to consider specifications containing exclusivelyconditional equations (or meta-axioms) of sort τ . Such specifications are calledτ -specifications in the sequel. The deductive system of CEQ (Meseguer, 1998)is a system for deriving equations from a given specification of conditionalequations. It consists of the usual rules for reflexivity, symmetry, transitivityand congruence of equality, plus a form of Modus Ponens that allows us toobtain an equation eq θ from already obtained equations eq1θ, . . . , eqnθ, givena conditional equation (eq1 & . . . & eqn → eq) in the specification anda substitution θ. In the sequel, we use `CEQ

Σ(C,Ξ) to denote the correspondingconsequence relation.

An important remark is that, in the context of the meta-signature Σ(C, Ξ),it might seem that we have two different ways to represent arbitrary formulae:by means of propositional schema variables (i.e., ξ1, ξ2, etc.) and by means ofvariables of sort φ (i.e., y1, y2, etc.). The former shall indeed represent arbitraryformulae but only in the context of Hilbert calculi (to be defined in Section 4).The latter represent arbitrary formulae in the meta-language of CEQ. In thismeta-language, propositional schema variables appear as constants.

DEFINITION 3. An interpretation system presentation (isp) is a pair S =〈C, S〉 where C is an object signature and S is a τ -specification over Σ(C).

DEFINITION 4. Given an isp S, the class Int(S) of interpretation structurespresented by S is the class of all Heyting algebras over Σ(C, Ξ) satisfying thespecification S.

We denote by S• the specification composed of the meta-axioms in S plusτ -equations over Σ(C) specifying the class of all Heyting algebras. Note thatInt(S), that is, the class of all algebras over Σ(C, Ξ) satisfying S•, is alwaysnon-empty. Indeed, the trivial algebra with singleton carrier sets for all sortssatisfies any set of conditional equations.

In the sequel, we need to refer to the denotation [[t]]ρA of a meta-term t givenan assignment ρ over an algebra A. As expected, an assignment maps eachvariable to an element in the carrier set of the sort of the variable. In the caseof a ground term t, as usual, we just write [[t]]A for its denotation in A.

For the sake of economy of presentation, we introduce the following abbrevi-ations: x1 ≤ x2 for u(x1, x2) = x1, and ⇔(x1, x2) for u(⇒(x1, x2),⇒(x2, x1)).The relation symbol ≤ denotes a partial order on truth-values. Furthermore, thepartial order is a bounded lattice with meet u, join t, top > and bottom ⊥ (cf.(Birkhoff, 1967)). As expected, given an algebraA, a1 ≤A a2 and⇔A(a1, a2) areabbreviations of uA(a1, a2) = a1 and uA(⇒A(a1, a2),⇒A(a2, a1)), respectively.It is also well known that the Heyting algebra axioms further entail the followingresult:

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PROPOSITION 1. Let S be an isp, t1 and t2 terms of sort τ and A ∈ Int(S).Then, for every assignment ρ over A:

[[t1]]ρA ≤A [[t2]]

ρA iff ⇒A ([[t1]]

ρA, [[t2]]

ρA) = >A

and[[t1]]

ρA = [[t2]]

ρA iff ⇔A ([[t1]]

ρA, [[t2]]

ρA) = >A.

As explained, our framework is intended to study properties of fibring ofnon-truth-functional logics in general. We now illustrate the notion of isp withtwo examples that will be used throughout the rest of the paper.

EXAMPLE 1. Paraconsistent system C1 (da Costa, 1963):

− Object signature - C:

• C0 = {pn : n ∈ N} ∪ {t, f};• C1 = {¬};• C2 = {∧,∨,⊃}.

− Meta-axioms - S:

• Truth-values axioms – further axioms in order to obtain a specificationof the class of all Boolean algebras, e.g., adding the equation:

∗ ( → −(−(x1)) = x1).

• Valuation axioms:

∗ ( → v(t) = >);∗ ( → v(f) = ⊥);∗ ( → v(y1 ∧ y2) = u(v(y1), v(y2)));∗ ( → v(y1 ∨ y2) = t(v(y1), v(y2)));∗ ( → v(y1 ⊃ y2) = ⇒(v(y1), v(y2)));∗ ( → −(v(y1)) ≤ v(¬ y1));∗ ( → v(¬¬ y1) ≤ v(y1));∗ ( → u(v(y◦1),u(v(y1), v(¬ y1))) = ⊥);∗ ( → u(v(y◦1), v(y◦2)) ≤ v((y1 ∧ y2)◦));∗ ( → u(v(y◦1), v(y◦2)) ≤ v((y1 ∨ y2)◦));∗ ( → u(v(y◦1), v(y◦2)) ≤ v((y1 ⊃ y2)◦)).

As usual in the Cn systems, γ◦ is an abbreviation of ¬(γ ∧ ¬ γ).

The reader should be warned that we are using Boolean algebras here as ametamathematical environment sufficient to carry out the computations oftruth-values for the formulae in C1. Specifically we are not introducing anyunary operator in the Boolean algebras corresponding to paraconsistentnegation, but we are computing the values of formulae of the form ¬ γ bymeans of conditional equations in the algebras. In other words, ¬ does not

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8 C. Caleiro et al.

correspond to the Boolean algebra complement −. Therefore we are notattempting to algebraize C1 in any usual way.1

It is straightforward to verify that every paraconsistent bivaluation introducedin (da Costa and Alves, 1977) has a counterpart in Int(S). Furthermore, theadditional interpretation structures do not change the semantic entailment (asdefined below). Note that it is easy to extend this example in order to set up theisp’s for the whole hierarchy Cn by specifying the paraconsistent n-valuationsintroduced in (Loparic and Alves, 1980). 4

After this example, we can now clarify the meaning of non-truth-functionalsemantics. To be as general as possible we shall not only consider primitiveconnectives (as given by the object signature) but also derived ones. As usual,a derived connective of arity k is a λ-term λy1 . . . yk . δ, where the variablesoccurring in the schema formula δ are taken from y1, . . . , yk. Of course, if c ∈ Ck

is a primitive connective it can also be considered as the derived connectiveλy1 . . . yk . c(y1, . . . , yk).

DEFINITION 5. A derived connective λy1 . . . yk . δ is said to be truth-functionalin a given isp S if

S• `CEQΣ(C,Ξ) v(δ) = t θv(y)

x

for some τ -term t written only on the variables x1, ..., xk, where θv(y)x is the

substitution such that θv(y)x (xn) = v(yn) for every n ≥ 1.

If it is not possible to fulfill the above requirement, the connective is said tobe non-truth-functional in S.

For obvious reasons, showing that a certain connective is non-truth-functionalcan be a very hard task. In C1, classical negation ∼ := λy1 . ¬ y1 ∧ y◦1 (take t as−(x1)) and equivalence ≡ := λy1y2 . (y1 ⊃ y2) ∧ (y2 ⊃ y1) (take t as ⇔(x1, x2))are both truth-functional. And, of course, so are the primitive conjunctionλy1y2 . y1 ∧ y2, disjunction λy1y2 . y1 ∨ y2, and implication λy1y2 . y1 ⊃ y2.On the other hand, paraconsistent negation λy1 . ¬ y1 is known to be non-truth-functional. We refer the reader to (Mortensen, 1980) for a proof of thisfact.

EXAMPLE 2. Modal system KD (Hughes and Cresswell, 1996; Lemmon andScott, 1977):

− Object signature - C:

• C0 = {pn : n ∈ N} ∪ {t, f};• C1 = {¬, L};• C2 = {∧,∨,⊃}.

− Meta-axioms - S:1 The question of algebraizing paraconsistent logic is a separate issue and we refer the

interested reader to (Mortensen, 1980; Lewin et al., 1991).

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• Truth-values axioms:∗ Further axioms in order to obtain a specification of the class of

all Boolean algebras as in the previous example.• Valuation axioms:

∗ ( → v(t) = >);∗ ( → v(f) = ⊥);∗ ( → v(¬ y1) = −(v(y1)));∗ ( → v(y1 ∧ y2) = u(v(y1), v(y2)));∗ ( → v(y1 ∨ y2) = t(v(y1), v(y2)));∗ ( → v(y1 ⊃ y2) = ⇒(v(y1), v(y2)));∗ ( → v(L t) = >);∗ ( → v(L(y1 ∧ y2)) = u(v(L y1), v(Ly2)));∗ ( → u(v(L y1), v(¬L¬ y1)) = v(Ly1);∗ (v(y1) = v(y2) → v(Ly1) = v(Ly2)).

It is straightforward to verify that every Kripke model has a counterpart inInt(S): consider the algebra of truth-values given by the power set of the set ofworlds. Furthermore, every general model in (Zanardo et al., 2001) also has acounterpart in Int(S): take 〈B, ν〉 as the algebra of the truth-values. Again, theextra interpretation structures do not change the semantic entailment. 4

In the isp above, all derived connectives are truth-functional, but the modal-ity λy1 . L y1 would require in Σ(C) the extra generator ¤ in Oτ τ satisfying:

− ( → ¤(>) = >);

− ( → ¤(u(x1, x2)) = u(¤(x1), ¤(x2)));

− ( → u(¤(x1),−(¤(−(x1))) = ¤(x1));

− ( → v(L y1) = ¤(v(y1))).

Note that these axioms on ¤ are very closely related to the last four valuationaxioms used in Example 2, which allowed us to specify the intended modalalgebras and still avoid the use of ¤. Although such an operation ¤ can beeasily defined over the set of truth-values according to the axioms above, ourdefinition does not comply with its inclusion in the signature Σ(C).

We are now ready to define the (global and local) semantic entailments.

DEFINITION 6. Given an isp S, a set Γ of schema formulae and a schemaformula δ, we say that:

− Γ ²Sg δ (Γ globally entails δ) if, for every A ∈ Int(S), vA([[γ]]A) = >A foreach γ ∈ Γ implies vA([[δ]]A) = >A;

− Γ ²Sl δ (Γ locally entails δ) if, for every A ∈ Int(S) and every b ∈ Aφ,vA(b) ≤A vA([[γ]]A) for each γ ∈ Γ implies vA(b) ≤A vA([[δ]]A).

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10 C. Caleiro et al.

Observe that Γ ²Sl δ implies Γ ²Sg δ provided that for every A ∈ Int(S) thereexists b ∈ Aφ such that vA(b) = >A. On the other hand, if Γ = ∅ then Γ ²Sg δ

implies Γ ²Sl δ.

Before turning our attention to the problem of fibring isp’s, we first define thecategories Sig and Isp. The objects of the category Sig are object signatures.A morphism h : C → C ′ in Sig is of the form h = {hk : Ck → C ′

k}k∈N, each hk

being a map. The objects of the category Isp are isp’s. The appropriate notionof morphism in the category Isp is as follows: each h : 〈C, S〉 → 〈C ′, S′〉 is amorphism h : C → C ′ in Sig such that for each s ∈ S, h(s) is in S′•. Here, his the free extension of h to a map from the meta-language over Σ(C) to themeta-language over Σ(C ′). Note that such a morphism imposes the conditionthat for every A′ ∈ Int(S ′) its reduct to Σ(C) via h is in Int(S). As usual,A′|hΣ(C) denotes the corresponding reduct algebra, that is characterized up toisomorphism by the following property:

[[t]]A′|hΣ(C)

= [[h(t)]]A′ , for every term t over Σ(C).

In what follows, we shall also use the forgetful functor N from Isp to thecategory Sig of object signatures. This functor maps each S to the underlyingobject signature C and each morphism h to the underlying object signaturemorphism.

3. Fibring non-truth-functional logics

Fibring, as originally proposed by (Gabbay, 1996; Gabbay, 1998), may be arather complex form of combining given logics. Here, we consider only themost basic forms of fibring seen as “operations” between logics as in (Sernadaset al., 1999; Zanardo et al., 2001): unconstrained fibring where two logics arecombined by putting together their signatures and rules, and by picking up asmodels all structures over the new signature whose reducts are models in thetwo given logics; constrained fibring where two logics are combined as for theunconstrained fibring but requiring that some symbols are to be shared. Thesebasic forms of fibring lead to new logics that sometimes need fine tuning for theapplication at hand, namely by adding further interaction rules (axioms). Thisidea of agreement on the reducts when fibring logics endowed with homomorphicalgebraic semantics is already present in (Gabbay, 1998), Chapter 20.

Here we face the novel problem of defining these two basic forms of fibringas operations on logics endowed with non-truth-functional semantics as definedin the previous section. Recall that models are now two-sorted algebras (offormulae and truth-values).

In the fibring, like in the truth-functional case, we still expect to find two-sorted algebras over the new signature whose reducts are models of the logicsbeing fibred. Therefore, when fibring two isp’s, we expect to put together thesignatures and the requirements on the valuation map.

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Assume that we are given two isp’s S ′ and S ′′. We start by considering thenotion of unconstrained fibring that corresponds to combining the two isp’swithout sharing any of the symbols of the object signatures C ′ and C ′′. Thatis, if we so combine C1 and KD we shall obtain in the result of the fibring twodifferent symbols for conjunction, disjunction, etc. This construction appearsas the coproduct of S ′ and S ′′ in the category Isp. Therefore,

S ′ ⊕ S ′′ = 〈C, S〉where:

− C = C ′ ⊕ C ′′ is a coproduct within Sig with the injections i′ and i′′;

− S = i′(S′) ∪ i′′(S′′).

The following result confirms the intuitions that guided the definition:

PROPOSITION 2. Given S ′ and S ′′ as above, a Σ(C ′ ⊕ C ′′, Ξ)-algebra Abelongs to Int(S ′ ⊕ S ′′) if and only if:

− A|i′Σ(C′,Ξ) ∈ Int(S ′);

− A|i′′Σ(C′′,Ξ) ∈ Int(S ′′);

where A|i′Σ(C′,Ξ) and A|i′′Σ(C′′,Ξ) are the reducts of A to the signatures Σ(C ′, Ξ)and Σ(C ′′,Ξ), respectively, via the indicated inclusions.

It is now easy to introduce the notion of constrained fibring by sharingconnectives and/or propositional symbols that corresponds to combining thetwo isp’s while sharing some of the symbols of the object signatures C ′ and C ′′.The construction appears as a co-Cartesian lifting by the functor N : Isp → Sigalong the signature coequalizer for the envisaged pushout of the signatures. Werefrain from dwelling further on the details of this construction since it doesnot bring any insight to the main issue of this paper (that is, the fibring oflogics possibly with non-truth-functional semantics). For illustration, considerthe following example of constrained fibring.

EXAMPLE 3. Modal paraconsistent logic:In (da Costa and Carnielli, 1988), a paraconsistent deontic logic called CD

1 isintroduced including the paraconsistent system C1 and the modal system KD(interpreting the modal operator L as “obligatory”). Let us see if we can recoverCD

1 as a fibring.The idea is to combine C1 and KD by fibring them while sharing the propo-

sitional symbols, conjunction, disjunction, implication, true and false. Let S ′ =〈C ′, S′〉 be the isp for C1 as described in Example 1 and S ′′ = 〈C ′′, S′′〉 the ispfor KD as described in Example 2.

We work first in the category Sig in order to set up the desired sharingof symbols. Consider the following propositional-based signature B of sharedsymbols:

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− B0 = {pn : n ∈ N} ∪ {t, f};− B2 = {∧,∨,⊃};− Bk = ∅ for the other values of k.

The matching signature inclusions f ′ : B → C ′ and f ′′ : B → C ′′ both mapthe symbols in B to the corresponding symbols in C ′ and C ′′.

The propositional-based signature C ′ f ′Bf ′′⊕ C ′′ of the envisaged constrained

fibring is now obtained by the pushout of C ′ f ′← Bf ′′→ C ′′. To compute it, we

first have to obtain the coproduct C ′ ⊕ C ′′, corresponding to the signature ofthe unconstrained fibring of C1 and KD, as follows:

− (C ′ ⊕ C ′′)0 = {p′n : n ∈ N} ∪ {p′′n : n ∈ N} ∪ {t′, t′′, f ′, f ′′};− (C ′ ⊕ C ′′)1 = {¬′,¬′′, L′′};− (C ′ ⊕ C ′′)2 = {∧′,∧′′,∨′,∨′′,⊃′,⊃′′}.

The corresponding injections i′ : C ′ → C ′⊕C ′′ and i′′ : C ′′ → C ′⊕C ′′ are suchthat i′ maps each constructor ] of C ′ to ]′ and i′′ maps each constructor ] ofC ′′ to ]′′.

Finally, the envisaged signature C ′ f ′Bf ′′⊕ C ′′ is obtained by identifying inC ′ ⊕C ′′ all the constructors obtained via f ′ and f ′′ from the same constructorof the shared signature B:

− (C ′ f ′Bf ′′⊕ C ′′)0 = {pn : n ∈ N} ∪ {t, f};

− (C ′ f ′Bf ′′⊕ C ′′)1 = {¬′,¬′′, L};

− (C ′ f ′Bf ′′⊕ C ′′)2 = {∧,∨,⊃}.

The unique compatible morphism z from C ′ ⊕ C ′′ to C ′ f ′Bf ′′⊕ C ′′ is definedby z(p′n) = z(p′′n) = pn for each n ∈ N, z(t′) = z(t′′) = t, z(f ′) = z(f ′′) = f ,z(¬′) = ¬′, z(¬′′) = ¬′′, z(L′′) = L, z(∧′) = z(∧′′) = ∧, z(∨′) = z(∨′′) = ∨ andz(⊃′) = z(⊃′′) = ⊃. To be precise, we should have written z0(p′n), z1(¬′) andso on, but we have omitted the arity subscripts to improve the readability.

This signature morphism z : C ′⊕C ′′ → C ′ f ′Bf ′′⊕ C ′′ is finally used to obtainthe envisaged fibred isp

S ′ f ′Bf ′′⊕ S ′′ = 〈C ′ f ′Bf ′′⊕ C ′′, z(i′(S′) ∪ i′′(S′′)〉corresponding to the respective co-Cartesian lifting briefly described at theend of the previous section. Expectedly, since z is a morphism, we have thatA|zΣ(C′⊕C′′,Ξ) ∈ Int(i′(S′) ∪ i′′(S′′)) for every A ∈ Int(z(i′(S′) ∪ i′′(S′′)), andtherefore the interpretation structures presented by this isp are precisely thosealgebras in the unconstrained fibring that agree on the shared symbols.

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Note that we end up having two negations: ¬′ coming from C ′ and ¬′′ comingfrom C ′′. The former is a paraconsistent negation and the latter is the classicalnegation inherited from KD. Clearly, the derived (classical) strong negationλy1 . ¬′ y1 ∧ y◦1 inherited from C1 collapses into ¬′′. Note that now γ◦ is anabbreviation of ¬′(γ ∧ ¬′ γ).

In order to recover CD1 , we have to add one additional meta-axiom on valu-

ations to the previously obtained fibred isp:

− ( → v(y◦1) ≤ v((Ly1)◦)).

Using the terminology introduced in (Carnielli and Coniglio, 1999), thisprocedure can be seen as a splitting of CD

1 in the components KD and C1.This idea is also in the spirit of Gabbay’s proposal on the broad meaning offibring, as described in (Gabbay, 1998), Chapter 1. 4

There are other interesting examples of combination of modal and paracon-sistent reasoning that would deserve to be analyzed from this point of view,namely those in (Deutsch, 1979; Deutsch, 1984; Puga et al., 1988) that, usingparaconsistent techniques, deal with problems of deontic logic having to do withdeontic paradoxes and moral dilemmas.

4. Logic systems

This section is devoted to extending fibring to the proof-theoretical counterpartof isp’s. For their simplicity and ubiquity we use a suitable notion of Hilbert cal-culus. As we have hinted before, we shall use the propositional schema variablesin Ξ = {ξ1, ξ2, . . .} to write inference rules.

DEFINITION 7. A Hilbert calculus over Ξ is a triple 〈C, P, D〉 in which (1) Cis an object signature, (2) P is a subset of ℘finL(C, Ξ) × L(C, Ξ), (3) D is asubset of (℘finL(C, Ξ) \ ∅)× L(C, Ξ), and (4) D ⊆ P .

Given any r = 〈Γ, γ〉 in P , the (finite) set Γ is the set of premises of r and γis the conclusion; we will often write r = 〈Prem(r),Conc(r)〉. If Prem(r) = ∅,then r is said to be an axiom schema; otherwise, it is said to be a proof ruleschema. Each r in D is said to be a derivation rule schema.

EXAMPLE 4. Paraconsistent system C1 revisited:Adapting the well known axiomatics presented in (da Costa, 1963; da Costaand Alves, 1977), a Hilbert calculus for C1 is easily defined:

− P = {〈∅, ξ1 ⊃ (ξ2 ⊃ ξ1)〉,〈∅, (ξ1 ⊃ (ξ2 ⊃ ξ3)) ⊃ ((ξ1 ⊃ ξ2) ⊃ (ξ1 ⊃ ξ3))〉,〈∅, (ξ1 ∧ ξ2) ⊃ ξ1〉,〈∅, (ξ1 ∧ ξ2) ⊃ ξ2〉,〈∅, ξ1 ⊃ (ξ2 ⊃ (ξ1 ∧ ξ2))〉,

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〈∅, ξ1 ⊃ (ξ1 ∨ ξ2)〉,〈∅, ξ2 ⊃ (ξ1 ∨ ξ2)〉,〈∅, (ξ1 ⊃ ξ3) ⊃ ((ξ2 ⊃ ξ3) ⊃ ((ξ1 ∨ ξ2) ⊃ ξ3))〉,〈∅,¬¬ ξ1 ⊃ ξ1〉,〈∅, ξ1 ∨ ¬ ξ1〉,〈∅, ξ◦1 ⊃ (ξ1 ⊃ (¬ ξ1 ⊃ ξ2))〉,〈∅, (ξ◦1 ∧ ξ◦2) ⊃ (ξ1 ∧ ξ2)◦〉,〈∅, (ξ◦1 ∧ ξ◦2) ⊃ (ξ1 ∨ ξ2)◦〉,〈∅, (ξ◦1 ∧ ξ◦2) ⊃ (ξ1 ⊃ ξ2)◦〉,〈∅, t ≡ (ξ1 ⊃ ξ1)〉,〈∅, f ≡ (ξ◦1 ∧ (ξ1 ∧ ¬ ξ1))〉,〈{ξ1, ξ1 ⊃ ξ2}, ξ2〉};

− D = {〈{ξ1, ξ1 ⊃ ξ2}, ξ2〉}. 4

EXAMPLE 5. Modal system KD revisited:Adapting from (Hughes and Cresswell, 1996; Lemmon and Scott, 1977), in themodal Hilbert calculus for KD we have:

− P = {〈∅, ξ1 ⊃ (ξ2 ⊃ ξ1)〉,〈∅, (ξ1 ⊃ (ξ2 ⊃ ξ3)) ⊃ ((ξ1 ⊃ ξ2) ⊃ (ξ1 ⊃ ξ3))〉,〈∅, (¬ ξ1 ⊃ ¬ ξ2) ⊃ (ξ2 ⊃ ξ1)〉,〈∅, L(ξ1 ⊃ ξ2) ⊃ (L ξ1 ⊃ Lξ2)〉,〈∅, L ξ1 ⊃ ¬L¬ ξ1〉,〈∅, (ξ1 ∨ ξ2) ≡ (¬ ξ1 ⊃ ξ2)〉,〈∅, (ξ1 ∧ ξ2) ≡ ¬(¬ ξ1 ∨ ¬ ξ2)〉,〈∅, t ≡ (ξ1 ⊃ ξ1)〉,〈∅, f ≡ (ξ1 ∧ ¬ ξ1)〉,〈{ξ1, ξ1 ⊃ ξ2}, ξ2〉,〈{ξ1}, L ξ1〉};

− D = {〈{ξ1, ξ1 ⊃ ξ2}, ξ2〉}. 4

DEFINITION 8. A schema formula δ ∈ L(C, Ξ) is provable from the set ofschema formulae Γ ⊆ L(C, Ξ) in the Hilbert calculus 〈C, P,D〉, denoted byΓ `PD

p δ, if there is a sequence γ1, . . . , γm ∈ L(C, Ξ)+ such that γm = δ and,for i = 1 to m, either(1) γi ∈ Γ, or(2) there exist a rule r ∈ P and a schema variable substitution σ : Ξ → L(C, Ξ)such that Conc(r)σ = γi and Prem(r)σ ⊆ {γ1, . . . , γi−1}.DEFINITION 9. A schema formula δ ∈ L(C, Ξ) is derivable from the set ofschema formulae Γ ⊆ L(C, Ξ) in the Hilbert calculus 〈C, P,D〉, denoted byΓ `PD

d δ, if there is a sequence γ1, . . . , γm ∈ L(C, Ξ)+ such that γm = δ and,for i = 1 to m, either(1) γi ∈ Γ, or(2) γi is provable from the empty set of formulae, or

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(3) there exist a rule r ∈ D and a schema variable substitution σ such thatConc(r)σ = γi and Prem(r)σ ⊆ {γ1, . . . , γi−1}.

Clearly, if Γ `PDd δ then also Γ `PD

p δ. Furthermore, if ∅ `PDp δ then ∅ `PD

d δ.As usual, we say that δ is a theorem schema whenever ∅ `PD

p δ (iff ∅ `PDd δ),

and simply write `PDp δ and `PD

d δ. The following structurality propertiesare also immediate: for every schema variable substitution σ, if Γ `PD

p δ thenΓσ `PD

p δσ, and if Γ `PDd δ then Γσ `PD

d δσ.

DEFINITION 10. The unconstrained fibring of the Hilbert calculi 〈C ′, P ′, D′〉and 〈C ′′, P ′′, D′′〉 is the Hilbert calculus

〈C ′, P ′, D′〉 ⊕ 〈C ′′, P ′′, D′′〉 = 〈C ′ ⊕ C ′′, i′(P ′) ∪ i′′(P ′′), i′(D′) ∪ i′′(D′′)〉.

DEFINITION 11. The constrained fibring of the Hilbert calculi 〈C ′, P ′, D′〉and 〈C ′′, P ′′, D′′〉 sharing C according to the injective morphisms f ′ : C → C ′

and f ′′ : C → C ′′ is the Hilbert calculus 〈C ′, P ′, D′〉 f ′Cf ′′⊕ 〈C ′′, P ′′, D′′〉 definedas follows:

〈C ′ f ′Cf ′′⊕ C ′′, z(i′(P ′)) ∪ z(i′′(P ′′)), z(i′(D′)) ∪ z(i′′(D′′))〉.

As a matter of fact, by adopting the notion of Hilbert calculus morphism pro-posed in (Sernadas et al., 1999), both forms of fibring appear again as universalcategorial constructions (coproduct and co-Cartesian lifting, respectively). Itfollows that there is a morphism from each given Hilbert calculus to the fibring,e.g., h′ from 〈C ′, P ′, D′〉 to 〈C, P, D〉 and therefore:

− if Γ `P ′D′p δ then h′(Γ) `PD

p h′(δ);

− if Γ `P ′D′d δ then h′(Γ) `PD

d h′(δ).

EXAMPLE 6. Modal paraconsistent logic revisited:The fibring of Hilbert calculi for C1 and KD, sharing the propositional symbols,conjunction, disjunction, implication, true and false, is the Hilbert calculuswhere we have all the proof and derivation rules for both C1 and KD. In orderto get the deontic paraconsistent system CD

1 of (da Costa and Carnielli, 1988),at the proof-theoretic level, we need to introduce the following proof rule:

− 〈∅, ξ◦1 ⊃ (Lξ1)◦〉.

This interaction axiom is already present in CD1 and could never be obtained

using the basic fibring operation since it makes full use of the mixed language.Note that the semantic counterpart of this axiom was also added to the corre-sponding fibred isp in Example 3. 4

DEFINITION 12. A logic system is a tuple L = 〈C, S, P,D〉 where the pair〈C, S〉 constitutes an isp and the triple 〈C,P, D〉 constitutes a Hilbert calculus.

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As expected, the (unconstrained and constrained) fibring of logic systemsis obtained by the corresponding fibring of the underlying isp’s and Hilbertcalculi.

EXAMPLE 7. The logic systems for C1 and KD will be denoted by LC1 andLKD, respectively, and their fibring while sharing B will be denoted by LC1⊕KD.

DEFINITION 13. Given a logic system L = 〈C, S, P, D〉, we say that thedeductive system 〈C, P, D〉 is sound w.r.t. the isp 〈C, S〉, or simply that L issound , if for every set Γ of schema formulae and every schema formula δ:

− Γ `PDp δ implies Γ ²Sg δ;

− Γ `PDd δ implies Γ ²Sl δ.

We say that 〈C,P, D〉 is adequate w.r.t. 〈C,S〉, or simply that L is adequate, iffor every set Γ of schema formulae and every schema formula δ:

− Γ ²Sg δ implies Γ `PDp δ;

− Γ ²Sl δ implies Γ `PDd δ.

Furthermore, we say that 〈C, P, D〉 is complete w.r.t. 〈C, S〉, or simply that Lis complete, if it is both sound and adequate.

EXAMPLE 8. The logic systems LC1 and LKD are complete.

5. Preservation results

The main goal of this section is to establish sufficient conditions for the preser-vation of completeness by fibring. To this end, it is convenient to take advantageof the completeness of the meta-logic CEQ, as proved for instance in (Goguenand Meseguer, 1985; Meseguer, 1998), by encoding the relevant part of thedeductive system of CEQ in the object Hilbert calculus.

In order to deal with local reasoning at the meta-level, we shall take advan-tage of the following two schema variable substitutions:

− σ+1 such that σ+1(ξi) = ξi+1 for every i ≥ 1;

− σ−1 such that σ−1(ξ1) = ξ1 and σ−1(ξi) = ξi−1 for every i ≥ 2.

Note that if γ is a schema formula then γσ+1 is a variant of γ where ξ1 doesnot occur. Furthermore, easily, γσ+1σ−1 = γ.

5.1. Encoding

First we analyze what can be obtained proof-theoretically within CEQ. Given anisp S = 〈C, S〉, we adopt the following abbreviations, where Γ∪ {δ} ⊆ L(C, Ξ):

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− Γ `Sg δ for S• ∪ {( → v(γ) = >) : γ ∈ Γ} `CEQΣ(C,Ξ) v(δ) = >;

− Γ `Sl δ for S• ∪ {( → v(ξ1) ≤ v(γσ+1)) : γ ∈ Γ} `CEQΣ(C,Ξ) v(ξ1) ≤ v(δσ+1).

THEOREM 1. Given an isp S = 〈C,S〉 and Γ ∪ {δ} ⊆ L(C, Ξ), we have:

− Γ ²Sg δ iff Γ `Sg δ;

− Γ ²Sl δ iff Γ `Sl δ.

Proof: This is an immediate consequence of the completeness of CEQ. In thelocal case it is essential to note that, since schema variables cannot occur in S•,we can freely change the denotation of schema variables given by an algebraA ∈ Int(S) (namely according to σ+1 or σ−1) and still obtain an algebra inInt(S). The fact that ξ1 cannot occur in schema formulae instantiated by σ+1

does the rest. QED

For the envisaged encoding, it is necessary to assume that the logic systemat hand is sufficiently expressive:

DEFINITION 14. A logic system L = 〈C,S, P,D〉 is said to be rich if:

1. t, f ∈ C0 and ∧,∨,⊃∈ C2;

2. S• `CEQΣ(C,Ξ) v(t) = >;

3. S• `CEQΣ(C,Ξ) v(f) = ⊥;

4. S• `CEQΣ(C,Ξ) v(y1 ∧ y2) = u(v(y1), v(y2));

5. S• `CEQΣ(C,Ξ) v(y1 ∨ y2) = t(v(y1), v(y2));

6. S• `CEQΣ(C,Ξ) v(y1 ⊃ y2) = ⇒(v(y1), v(y2));

7. 〈{ξ1, ξ1 ⊃ ξ2}, ξ2〉 ∈ D.

EXAMPLE 9. Both logic systems LC1 and LKD, as well as many other commonlogics, are rich.

Within a rich logic system it is possible to translate from the meta-logic levelto the object logic level. A ground term of sort τ over Σ(C, Ξ) is mapped to aformula in L(C, Ξ) according to the following rules:

v(γ)∗ is γ;

>∗ is t;

⊥∗ is f ;

−(t)∗ is t∗ ⊃ f ;

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u(t1, t2)∗ is t∗1 ∧ t∗2;

t(t1, t2)∗ is t∗1 ∨ t∗2;

⇒(t1, t2)∗ is t∗1 ⊃ t∗2.

Moreover, a ground τ -equation (t1 = t2) is translated to (t1 = t2)∗ given byt∗1 ≡ t∗2. Finally, if E is a set of ground τ -equations, then E∗ will denote the set{eq∗ : eq ∈ E}.

LEMMA 1. Let L be a rich logic system and t a ground τ -term over Σ(C, Ξ).Then:

S• `CEQΣ(C,Ξ) v(t∗) = t.

Proof: Immediate by definition, taking into account the requirements of rich-ness and the completeness of CEQ. QED

LEMMA 2. Let L be a rich logic system, t1 and t2 ground τ -terms over Σ(C, Ξ)and A ∈ Int(S). Then:

[[t1]]A ≤A [[t2]]A iff vA([[t∗1 ⊃ t∗2]]A) = >Aand

[[t1]]A = [[t2]]A iff vA([[t∗1 ≡ t∗2]]A) = >A.

Proof: Direct corollary of Proposition 1 using the previous lemma and takinginto account the completeness of CEQ. QED

In a rich logic system, under certain conditions (cf. Definition 15 below), onecan encode the relevant part of the meta-reasoning into the object calculus.

DEFINITION 15. A rich logic system L is said to be equationally appropriateif for every conditional equation (eq1 & . . . & eqn → eq) in S• and everyground substitution θ:

{(eq1θ)∗, . . . , (eqnθ)∗} `PD

p (eq θ)∗.

Finally, we obtain the main results of this section relating adequacy to equa-tional appropriateness. Such results are important because it is much easierto analyze the preservation by fibring of equational appropriateness than ofadequacy directly.

THEOREM 2. Every rich and adequate logic system is equationally appropri-ate.

Proof: Assume that L is a rich and adequate logic system and A ∈ Int(S), andlet (t1 = s1 & . . . & tn = sn → t = s) be a conditional equation in S• and θa ground substitution.

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If it is the case that vA([[(tiθ)∗ ≡ (siθ)∗]]A) = >A for i = 1, . . . , n, then,according to the previous lemma, this means precisely that [[tiθ]]A = [[siθ]]A fori = 1, . . . , n. Consider the assignment ρ = [[ ]]A◦θ. It is straightforward to verifythat [[rθ]]A = [[r]]ρA, for every τ -term r over Σ(C, Ξ). So, since by definition ofInt(S) we know that A is a model of the conditional equation, it immediatelyfollows that also [[tθ]]A = [[sθ]]A, or equivalently, vA([[(tθ)∗ ≡ (sθ)∗]]A) = >A.

Therefore, we have {(t1θ)∗ ≡ (s1θ)∗, . . . , (tnθ)∗ ≡ (snθ)∗} ²Sg (tθ)∗ ≡ (sθ)∗.Now, equational appropriateness follows easily since, from adequacy, we musthave {(t1θ)∗ ≡ (s1θ)∗, . . . , (tnθ)∗ ≡ (snθ)∗} `PD

p (tθ)∗ ≡ (sθ)∗. QED

Before proving the converse of this theorem, we need to establish sometechnical lemmas.

LEMMA 3. Let L be an equationally appropriate logic system and Γ∪{δ} be aset of schema formulae where ξ1 does not occur. If {ξ1 ⊃ γ : γ ∈ Γ} `PD

p ξ1 ⊃ δ

then Γ `PDd δ.

Proof: First of all we note that, since S• must contain a specification of theclass of all Heyting algebras, equational appropriateness implies that everyintuitionistic theorem written with t, f ,∧,∨,⊃ must be provable in the Hilbertcalculus. In the sequel, we shall use this fact without further notice.

Let us assume that {ξ1 ⊃ γ : γ ∈ Γ} `PDp ξ1 ⊃ δ. Immediately, by the finite

character of derivations, {ξ1 ⊃ γ1, . . . , ξ1 ⊃ γn} `PDp ξ1 ⊃ δ, where each γi ∈ Γ.

Let now γ be the schema formula γ1 ∧ . . . ∧ γn and take the schema variablesubstitution σ such that σ(ξ1) = γ and σ(ξi) = ξi for every i ≥ 2. Since ξ1

does not occur in Γ or δ, the structurality of proofs easily implies that we mustalso have {γ ⊃ γ1, . . . , γ ⊃ γn} `PD

p γ ⊃ δ. But it is clear by easy intuitionisticreasoning that `PD

p γ ⊃ γi for i = 1, . . . , n. So, it follows that `PDp γ ⊃ δ, and

since by further intuitionistic reasoning we have `PDp γ1 ⊃ (. . . ⊃ (γn ⊃ γ) . . .),

the derivation rule of Modus Ponens immediately implies that Γ `PDd δ. QED

LEMMA 4. Let L be an equationally appropriate logic system, E a set ofground τ -equations over Σ(C, Ξ) and θ a ground substitution. If

S• ∪ {( → eq ) : eq ∈ E} `CEQΣ(C,Ξ) t1 = t2

then either t1, t2 are the same term of sort φ, or t1, t2 are of sort τ and

E∗ `PDp (t1θ)∗ ≡ (t2θ)∗.

Proof: Recall that the deduction rules of CEQ are Reflexivity, Symmetry,Transitivity, Congruence and Modus Ponens.

Given a ground substitution θ, we proceed by induction on the length n ofa proof of S• ∪ {( → eq ) : eq ∈ E} `CEQ

Σ(C,Ξ) t1 = t2.Base: n = 1.(i) t1 is s1θ

′, t2 is s2θ′ and t1 = t2 is obtained by CEQ Modus Ponens from

( → s1 = s2) ∈ S•.

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Obviously t1 and t2 are τ -terms. Immediately, by equational appropriateness,we have `PD

p (s1θ′θ)∗ ≡ (s2θ

′θ)∗ and therefore E∗ `PDp (t1θ)∗ ≡ (t2θ)∗ by the

monotonicity of provability.(ii) t1 is s1θ

′, t2 is s2θ′ and t1 = t2 is obtained by CEQ Modus Ponens from

( → s1 = s2) with s1 = s2 ∈ E.Obviously t1 and t2 are closed τ -terms, s1θ

′ is s1 and s2θ′ is s2. Thus, (t1θ)∗ ≡

(t2θ)∗ ∈ E∗, i.e., t∗1 ≡ t∗2 ∈ E∗ and E∗ `PDp t∗1 ≡ t∗2 by the extensiveness of

provability.(iii) t1 and t2 are the same term, of either sort φ or τ , and t1 = t2 is obtainedby Reflexivity.If the sort is φ we are done. Otherwise, obviously, (t1θ)∗ and (t2θ)∗ are thesame formula and trivial intuitionistic reasoning allows us to conclude that`PD

p ξ1 ≡ ξ1. Therefore, E∗ `PDp (t1θ)∗ ≡ (t2θ)∗ by the structurality and

monotonicity of provability.Step: n > 1.(i) t1 = t2 is obtained from S• ∪ {( → eq ) : eq ∈ E} `CEQ

Σ(C,Ξ) t2 = t1 bySymmetry.If the terms have sort φ, by induction hypothesis, they coincide. Otherwise, alsoby induction hypothesis, we know that E∗ `PD

p (t2θ)∗ ≡ (t1θ)∗. Elementaryintuitionistic reasoning allows us to conclude that `PD

p (ξ1 ≡ ξ2) ⊃ (ξ2 ≡ ξ1),and therefore also E∗ `PD

p (t1θ)∗ ≡ (t2θ)∗.(ii) t1 = t2 is obtained from S• ∪ {( → eq ) : eq ∈ E} `CEQ

Σ(C,Ξ) t1 = t3, t3 = t2 byTransitivity.If the terms have sort φ by induction hypothesis they coincide. Otherwise, alsoby induction hypothesis, we know that E∗ `PD

p (t1θ)∗ ≡ (t3θ)∗, (t3θ)∗ ≡ (t2θ)∗.Simple intuitionistic reasoning allows us to conclude that {ξ1 ≡ ξ2, ξ2 ≡ ξ3} `PD

p

ξ1 ≡ ξ3, and therefore also E∗ `PDp (t1θ)∗ ≡ (t2θ)∗.

(iii) t1 is f(t11, . . . , t1k), t2 is f(t21, . . . , t2k) and t1 = t2 is obtained fromS• ∪ {( → eq ) : eq ∈ E} `CEQ

Σ(C,Ξ) t11 = t21, . . . , t1k = t2k by Congruence.It t1 and t2 have sort φ then f ∈ Ck and therefore all the terms tij are alsoof sort φ. By induction hypothesis, then, t1j and t2j must be identical and t1coincides with t2. Otherwise, f can either be v or a generator among −,u,t,⇒.In the first case k = 1 and by induction hypothesis t11 coincides with t21

since they must have sort φ. Therefore, t1 and t2 also coincide and we repeatstep (iii) of the Base to obtain E∗ `PD

p (t1θ)∗ ≡ (t2θ)∗. Finally, if f is agenerator then all the terms tij are of sort τ . Thus, by induction hypothesis,E∗ `PD

p (t11θ)∗ ≡ (t21θ)∗, . . . , (t1kθ)∗ ≡ (t2kθ)∗. Using again intuitionistic rea-soning we have that {ξ1 ≡ ξ2, ξ3 ≡ ξ4} `PD

p (ξ1 ∧ ξ3) ≡ (ξ2 ∧ ξ4), (ξ1 ∨ ξ3) ≡(ξ2 ∨ ξ4), (ξ1 ⊃ ξ3) ≡ (ξ2 ⊃ ξ4). Given that − is translated using f and ⊃ thisis enough to guarantee that E∗ `PD

p (t1θ)∗ ≡ (t2θ)∗.(iv) t1 is s1θ

′, t2 is s2θ′ and t1 = t2 is obtained using the conditional equation

(s11 = s21 & . . . & s1k = s2k → s1 = s2) ∈ S• from S• ∪ {( → eq ) : eq ∈E} `CEQ

Σ(C,Ξ) s11θ′ = s21θ

′, . . . , s1kθ′ = s2kθ

′ by CEQ Modus Ponens.Obviously, all the terms are τ -terms and the induction hypothesis impliesthat E∗ `PD

p (s11θ′θ)∗ ≡ (s21θ

′θ)∗, . . . , (s1kθ′θ)∗ ≡ (s2kθ

′θ)∗. Moreover, by

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equational appropriateness, we have {(s11θ′θ)∗ ≡ (s21θ

′θ)∗, . . . , (s1kθ′θ)∗ ≡

(s2kθ′θ)∗} `PD

p (s1θ′θ)∗ ≡ (s2θ

′θ)∗. Therefore, E∗ `PDp (t1θ)∗ ≡ (t2θ)∗. QED

THEOREM 3. Every equationally appropriate logic system is adequate.

Proof: Let L be an equationally appropriate logic system and Γ∪{δ} ⊆ L(C, Ξ).If Γ ²Sg δ then also Γ `Sg δ, i.e., S• ∪ {( → v(γ) = >) : γ ∈ Γ} `CEQ

Σ(C,Ξ)

v(δ) = >, by Theorem 1. Therefore, using the previous lemma, we have that{γ ≡ t : γ ∈ Γ} `PD

p δ ≡ t. Trivial intuitionistic reasoning allows us to concludethat `PD

p ξ1 ≡ (ξ1 ≡ t), and therefore it follows that Γ `PDp δ.

If Γ ²Sl δ then also Γ `Sl δ, i.e., S• ∪ {( → v(ξ1) ≤ v(γσ+1)) : γ ∈ Γ} `CEQΣ(C,Ξ)

v(ξ1) ≤ v(δσ+1), by Theorem 1. Therefore, using the previous lemma, we havethat {(ξ1 ∧ γσ+1) ≡ ξ1 : γ ∈ Γ} `PD

p (ξ1 ∧ δσ+1) ≡ ξ1. Trivial intuitionisticreasoning allows us to conclude that `PD

p (ξ1 ⊃ ξ2) ≡ ((ξ1 ∧ ξ2) ≡ ξ1), andtherefore it follows that {ξ1 ⊃ γσ+1 : γ ∈ Γ} `PD

p ξ1 ⊃ δσ+1. Thus, by Lemma3, we already know that Γσ+1 `PD

d δσ+1 and by structurality, using σ−1, wehave Γ `PD

d δ. QED

The equivalence between adequacy and equational appropriateness for richsystems will be used below for showing that adequacy is preserved by fibring richsystems, but this equivalence may also be useful for establishing the adequacyof logics endowed with a semantics presented by conditional equations. Indeed,it is a much easier task to verify equational appropriateness than to establishadequacy directly.

5.2. Preservation of completeness by fibring

We consider in turn the preservation of soundness and of adequacy by fibring.

THEOREM 4. Soundness is preserved by fibring.

Proof: Let L be the fibring of two sound logic systems L′ and L′′. It is enoughto prove the following: Γ `PD

p δ implies that Γ `Sg δ, and Γ `PDd δ implies

that Γ `Sl δ, by Theorem 1. Moreover, it is enough to prove that Prem(r) `SgConc(r) for every r ∈ P , and Prem(r) `Sl Conc(r) for every r ∈ D. Let r ∈ P .Assume, without loss of generality, that r = h′(r′). Then, by definition of proof,Prem(r′) `P ′D′

p Conc(r′) and, so, Prem(r′) `S′g Conc(r′), by the soundness ofL′. This means that

S′• ∪ {( → v(γ′) = >) : γ′ ∈ Prem(r′)} `CEQΣ(C′,Ξ) v(Conc(r′)) = >

and then, by the uniformness of CEQ under change of notation by h′ (cf.(Meseguer, 1998)), we obtain

h′(S′)• ∪ {( → v(h′(γ′)) = >) : γ′ ∈ Prem(r′)} `CEQ

Σ(C,Ξ) v(h′(Conc(r′))) = >.

This immediately implies h′(Prem(r′)) `Sg h′(Conc(r′)), that is, Prem(r) `SgConc(r). The proof for derivations is similar. QED

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Consider two sound logic systems L′ and L′′ that are both consistent, in thesense that both contain formulae that are not theorems. It may happen that thecorresponding fibred logic system L is no longer consistent. For instance, assumethat L′ corresponds to classical propositional logic plus an additional axiom A,for some proposition A, and that L′′ also corresponds to classical propositionallogic but with additional axiom ¬A. Obviously, L is not consistent. In orderto have both L′ and L′′ sound, it is clear that the corresponding isp’s mustimply v(A) = > and −(v(A)) = >, respectively. Thus, the isp of L mustimply > = ⊥. Therefore, the only interpretation structure presented by theisp of L corresponds to the trivial Boolean algebra, which, as mentioned earlier(just after Definition 4), satisfies any set of conditional equations. This examplemakes clear that Theorem 4 just states the preservation of soundness by fibringand clearly does not imply the preservation of consistency. Preservation ofsoundness is nevertheless very important in itself. Finding sufficient conditionsunder which consistency might be preserved by fibring is beyond the scope ofthis paper.

Finally, we consider the problem of preservation of adequacy by fibring,taking advantage of the technical machinery presented before on the encodingof the meta-logic in the object Hilbert calculus.

LEMMA 5. Richness is preserved by fibring provided that conjunction, dis-junction, implication, true and false are shared.

Proof: It is trivial that the signature and valuation requirements are preservedsince we are sharing conjunction, disjunction, implication, true and false. More-over, it is clear that Modus Ponens is still a derivation rule in the fibring. QED

LEMMA 6. Equational appropriateness is preserved by fibring provided thatconjunction, disjunction, implication, true and false are shared.

Proof: Let L′ and L′′ be equationally appropriate logic systems, and L theirfibring by sharing conjunction, disjunction, implication, true and false. Fromthe previous lemma we already know that L is rich.

Now, let ceq be (t1 = s1 & . . . & tn = sn → t = s) ∈ S•, and θ a groundsubstitution. Clearly, by definition of fibring, ceq must be the translation of aconditional equation in some of the components. Let us assume, without lossof generality, that ceq comes from L′, i.e., ceq is h′(ceq′), where ceq′ is theconditional equation (t′1 = s′1 & . . . & t′n = s′n → t′ = s′) ∈ S′•. Since we knowthat L′ is equationally appropriate, it follows that

{(t′1θ′)∗ ≡ (s′1θ′)∗, . . . , (t′nθ′)∗ ≡ (s′nθ′)∗} `P ′D′

p (t′θ′)∗ ≡ (s′θ′)∗,

where θ′ is the following ground substitution:

− for every i ≥ 1, θ′(xi) = v(ξ2i−1) and θ′(yi) = ξ2i.

By definition of fibring of Hilbert calculi, then, it must be also the case that

{h′((t′1θ′)∗ ≡ (s′1θ′)∗), . . . , h′((t′nθ′)∗ ≡ (s′nθ′)∗)} `PD

p h′((t′θ′)∗ ≡ (s′θ′)∗).

Consider now the substitution σ on schema variables defined by:

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− σ(ξ2i−1) = θ(xi)∗;

− σ(ξ2i) = θ(yi).

Using σ and the structurality of the Hilbert calculus, now, we must also have

{h′((t′1θ′)∗ ≡ (s′1θ′)∗), . . . , h′((t′nθ′)∗ ≡ (s′nθ′)∗)}σ `PD

p h′((t′θ′)∗ ≡ (s′θ′)∗)σ.

But, in fact, a straightforward inductive proof allows us to conclude thath′((u′θ′)∗)σ = (h′(u′)θ)∗ for every term u′ of sort τ over Σ(C ′, Ξ), and therefore

{(t1θ)∗ ≡ (s1θ)∗, . . . , (tnθ)∗ ≡ (snθ)∗} `PDp (tθ)∗ ≡ (sθ)∗.

Thus, L is equationally appropriate. QED

THEOREM 5. Given two rich and complete logic systems, their fibring whilesharing conjunction, disjunction, implication, true and false is also complete.

Proof: The preservation of soundness is immediate consequence of Theorem 4.The preservation of adequacy is a consequence of the previous lemma and theequivalence between equational appropriateness and adequacy for rich systems.In fact, let L′ and L′′ be two rich and complete logic systems and let L be theirfibring while sharing conjunction, disjunction, implication, true and false. ByTheorem 2, L′ and L′′ are also equationally appropriate and thus so is L, byLemma 6. Finally, by Theorem 3, L is adequate. QED

EXAMPLE 10. By fibring while sharing conjunction, disjunction, implication,true and false the logic systems LC1 and LKD we obtain a new modal para-consistent logic system LC1⊕KD that is complete. Observe that if we add toLC1⊕KD:

− ( → v(y◦1) ≤ v((Ly1)◦)) as a valuation axiom;

− 〈∅, ξ◦1 ⊃ (Lξ1)◦〉 as an axiom in the Hilbert calculus;

we still obtain a complete logic system that is equivalent to the system CD1 of

(da Costa and Carnielli, 1988) both at the proof-theoretic and the semanticlevels. 4

6. Self-fibring versus truth-functionality

In this section we address the problem of fibring two copies of the same logic(self-fibring) and show that, contrarily to what happens in the case of a truth-functional logic, in the case of a logic with non-truth-functional semantics thereis no collapse of unshared symbols. This construction is illustrated within thecontext of the Cn hierarchy of paraconsistent systems (da Costa, 1963).

It is not difficult to see that, if S is a truth-functional isp (that is, all theconnectives are truth-functional derived connectives) then the self-fibring S⊕S

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of S with itself, without sharing of connectives (just sharing the propositionalsymbols in C0), produces a copy of S where each connective appears duplicate.In fact, if c ∈ Ck (k > 0) and c′ is its duplicate then vA([[c(t1, ..., tk)]]

ρA) is

equal to vA([[c′(t1, ..., tk)]]ρA) for every interpretation A, assignment ρ over A

and terms t1, .., tk of sort φ. Additionally, if c ∈ C0 is a constant symbol, suchas t or f , then vA([[c]]A) is also equal to vA([[c′]]A). Of course, this propertycan be extended to every rich and complete logic system L while sharing C0,conjunction, disjunction and implication in the self-fibring L⊕L. In such casesthe formulae c(t1, ..., tk) and c′(t1, ..., tk) will be equivalent in L ⊕ L, even if cand c′ are not explicitly shared.

On the other hand, if S is an isp with some non-truth-functional connectivec, then vA([[c(t1, ..., tk)]]

ρA) and vA([[c′(t1, ..., tk)]]

ρA) do not necessarily coincide in

the models of S⊕S and, consequently, the formulae c(t1, ..., tk) and c′(t1, ..., tk)are not necessarily equivalent in L ⊕ L, unless c and c′ are explicitly shared.

As a concrete instance, consider the isp S of Example 1, representing thesemantics of the paraconsistent calculus C1 (cf. (da Costa, 1963)). If we performthe fibring S⊕S of S with itself, while sharing the symbols in C0, then we obtaintwo families of connectives: {∧,∨,⊃,¬} and {∧′,∨′,⊃′,¬′}. A model for S ⊕Sgives a valuation map vA such that

vA([[t1]t2]]ρA) = vA([[t1]′t2]]

ρA) for ] ∈ {∧,∨,⊃},

because those connectives are truth-functional. On the other hand, vA([[¬t]]ρA)does not coincide necessarily with vA([[¬′t]]ρA). For example, let v1 and v2 be twoC1-bivaluations such that v1(p1) = v1(¬p1) = 1 and v2(p1) = 1, v2(¬p1) = 0.Moreover, assume that in fact v1(p) = v2(p) for every propositional symbolp ∈ C0. We can thus define an interpretation A of S ⊕ S as follows:

− Aτ = {0, 1} (with its usual Boolean algebra structure);

− vA restricted to the fragment {∧,∨,⊃,¬} coincides with v1;

− vA restricted to the fragment {∧′,∨′,⊃′,¬′} coincides with v2;

− vA extended to the mixed language C ⊕ C ′ is obtained from v1 and v2

by using the same techniques used in the proof of Proposition 6.1 in(Carnielli and Coniglio, 1999), namely, vA([[¬ t]]ρA) = 1 iff vA([[t]]ρA) = 0iff vA([[¬′ t]]ρA) = 1 for all assignments ρ and all mixed terms t of sort φ.

The interpretation A satisfies: vA([[¬p1]]A) 6= vA([[¬′p1]]A), showing that ¬and ¬′ do not collapse. Considering the fibring at the logic system level, weobtain, by the completeness preservation Theorem 5, that ¬t1 and ¬′t1 are notequivalent formulae (unless they are both theorems).

The example above shows that the fibring of Cn with itself produces, for everyn ≥ 1, two disjoint copies of Cn (as the same argument can be applied to thewhole hierarchy of paraconsistent calculi Cn). We exclude C0(which is just theClassical Propositional Calculus) since in this case the self -fibring will collapsewith C0, because this system is truth-functional. On the other hand, the fibringof Cn with Cm, with m > n, produces a new paraconsistent system with two

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paraconsistent negations, ¬n and ¬m, whose axioms correspond to adding theaxioms of Cn and of Cm, and whose interpretations are given by maps which aresimultaneously Cn and Cm valuations. It is an open question whether or not thereexists a formula (in the language of Cm) which encodes in Cm the paraconsistentnegation ¬n, for m > n (this, in fact, happens with the classical negation, whichis representable in every calculi Cn). If this question has a positive answer, thenthe fibring of Cn and Cm (without sharing the negation) will be equivalentto Cm, for m > n. Of course, the fibring of Cn and Cm (while sharing thenegation) will be equivalent to Cn. If we generalize this argument, including inthe object signature C an unary symbol ¬n for every n ≥ 0, then the infinitefibring of the whole hierarchy Cn, without sharing the negations ¬n, will producea new paraconsistent system with infinitely many paraconsistent negations.If negations are shared in the fibring, the result coincides with the ClassicalPropositional Calculus C0. It is worth remarking that, although we concentratemost of the time on paraconsistent calculi because these are excellent examplesof interesting non-truth-functional logic systems, it is clear that our treatmentis fully general.

7. Concluding remarks

The first main contribution of this paper is a general semantics for two basicforms of fibring propositional-based logics encompassing systems with possiblynon-truth-functional valuations. We should stress that the present approach isnot just an adaptation of previous work but it involves the conceptual break-through of dropping the widely accepted principle of truth-functionality. Thisgoal is achieved by recognizing that such valuations can be represented insome appropriate meta-logic and by developing new techniques based on thisrepresentation. In this case, since it suited our needs, we have used condi-tional equational logic as the meta-logic. However, it must be clear that wecould have adopted, instead, any other meta-logic where non-truth-functionalvaluation semantics could be defined. Although restricted to systems with afinitary propositional base, the proposed semantics deals with a wide variety oflogics from paraconsistent to modal, many-valued and intuitionistic systems. Inthis setting, fibring appears as a universal construction within the underlyingcategory, generalizing previous results for truth-functional systems (Sernadaset al., 1999).

It should be stressed that the two basic forms of fibring we consider (un-constrained fibring and constrained fibring by sharing some symbols) appearas “operations” on the class of logics at hand. Usually, in applications, theseoperations are not enough to obtain the envisaged logic: a fine tuning of theresulting logic may be necessary, namely by adding interaction axioms, like itis illustrated in Example 6.

The second main contribution of this paper is the completeness preservationtheorem that generalizes to possibly non-truth-functional logics the result es-tablished in (Zanardo et al., 2001). This new result is obtained using a different

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technique exploiting the properties of conditional equational logic where therequirements on the valuations are specified.

As an example of application of our techniques, the use of fibring for combin-ing paraconsistent logics with other logics is illustrated by recovering the modalparaconsistent logic CD

1 of da Costa and Carnielli (da Costa and Carnielli, 1988)as a fibring plus an additional interaction axiom. We believe that fibring is avery natural way of establishing new combined systems involving non-truth-functional logics. This approach is more widely applicable than it appears tobe: a large family of logics (including many-valued and intuitionistic) admitsbivalued non-truth-functional semantics (cf. (Beziau and da Costa, 1994)).Whenever that happens, the completeness preservation theorem can then beused for establishing the completeness of the result as long as the given logicsare complete and fulfill the requirements of richness. As another example, thefibring of a logic with itself is examined within the context of the Cn hierarchyof paraconsistent systems (da Costa, 1963).

In short, our approach offers a framework formalizing the minimal meta-mathematical requirements that are sufficient to express a large variety of logicsystems, possibly non-truth-functional; such representations of logic systemsconstitute a category and they can be combined by means of fibring (that is,through universal constructions in the category). We have also proved that thecompleteness of such logic systems is preserved by fibring under certain reason-able assumptions on the logic systems, guaranteeing the important property ofnon-destructiveness of our fibring constructions.

Note that, although we have adopted Hilbert calculi as the proof-theoreticnotion of logic, it would have been possible to consider other well knownformalisms. Namely tableaux systems, as in (Beckert and Gabbay, 1998), orsystems of natural deduction, as in (Rasga et al., 2002).

Other lines of research are obvious, towards relaxing the assumptions of thispaper. For instance, we may want to work with more general object logics (e.g.,predicate logics), or with a more general meta-logic (e.g., disjunctive conditionalequational logic), or with an even more general universe of truth-values (e.g.,involving less or extra generators), or with weaker richness requirements andstill obtain completeness preservation by fibring. This line of work is even moreimportant as it should help to solve a small but annoying technicality related toour notion of truth-functional connective. In fact, as mentioned with respect toExample 2, the modality L is not truth-functional according to our definition.As we have explained, it would be very easy to make it truth-functional (as itshould) by adding a modal operator ¤ to the meta-signature. It is importantthat we solve this lack of expressiveness in order to raise the distinction betweenthis modality L and the paraconsistent negation ¬ of Example 1 that, on thecontrary, is well known to be non-truth-functional in an essential way.

Still other lines of research are related to more general forms of fibring,namely heterogeneous forms of fibring where we want to combine two (or more)logics that are defined in quite different forms (either at the deductive systemlevel or at the semantic level).

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Acknowledgments

The authors are grateful to Joao Rasga and Joao Marcos for many valuablediscussions on these and related topics. Thanks are also due to the anony-mous referees for suggesting several improvements on an earlier version ofthe paper. This work is a natural sequel of the work presented in (Coniglioet al., 2000), which started during a visit by the third author to CMA atIST, entirely supported by Fundacao para a Ciencia e a Tecnologia, Portu-gal. A subsequent visit to CMA by the second and third authors was alsosupported by Coordenacao de Aperfeicoamento do Pessoal do Ensino Superior ,Brazil. The third author also acknowledges financial support of his postdoc atIST under the grant 01/1045-0 of Fundacao de Amparo a Pesquisa do Es-tado de Sao Paulo, Brazil. The work of the first, fourth and fifth authorswas partially supported by Fundacao para a Ciencia e a Tecnologia, Por-tugal, namely via the ProbLog (PRAXIS/P/MAT/10002/1998) and FibLog(POCTI/2001/MAT/37239) Projects, also supported by EU FEDER.

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Fibring Non-Truth-Functional Logics: Completeness Preservation

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