Transfers between Logics and their Applications

M.E. Coniglio and W.A. CarnielliGroup for Theoretical and Applied Logic

Department of Philosophy and

Center for Logic, Epistemology and the History of Science

State University of Campinas

P.O. Box 6133, 13081-970

Campinas, SP, Brazil

E-mail: {coniglio, carniell}@cle.unicamp.br

Dedicated to the memory of Carlos Alchourron

Abstract

In this paper, logics are conceived as two-sorted first-order structures,and we argue that this broad definition encompasses a wide class of log-ics with theoretical interest as well as interest from the point of view ofapplications. The language, concepts and methods of model theory canthus be used to describe the relationship between logics through mor-phisms of structures called transfers. This leads to a formal frameworkfor studying several properties of abstract logics and their attributes suchas consequence operator, syntactical structure, and internal transforma-tions. In particular, we treat Belief Revision Systems (BRS) as our mainexample, defining the Wide Belief Revision Systems (WBRS’s). This gen-eralization allows us to define BRS’s in an abstract setting for classicaland non-standard logics. We also show how the concept of translationbetween logics can be obtained as a particular case of transfers.

1 Basic motivations

In one of the fundamental books on Belief Revision, P. Gardenfors suggested that“...a theory of epistemic dynamics may be philosophically more fundamentalthan logic and semantics” (cf. [22], pp. 3). However, such a theory has not beentreated in full generality because, for example, Belief Revision Systems (BRS’s)have been too scarcely considered for non-standard logics (see, for instance, [23],[21] and Section 5.3 in [25]), and there is not a clear concept of what a epistemicdynamic is. Abstract logics seem to be the appropriate setup for treating suchquestions. Approaches towards abstract logics, with a categorial flavor, weretaken in [9] and [15]. In this paper, following research started in [14], we prefer

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to use the approach by Brown and Suszko to (propositional) abstract logics inthe sense of the seminal paper [6], because it is more appropriate for connectionswith basic model theory.

One of the important results of adopting this perspective is the definition ofa general notion of attribute of logics: an attribute formalizes any n-ary relationbetween formulas and/or sets of formulas, as exemplified below.

Several familiar concepts can be seen as attributes: a consequence relationin any logic is a relation between sets of formulas and formulas; any binaryconnective is a (functional) relation between ordered pairs of formulas and for-mulas; the operations of expansion, revision and contraction in any BRS are(functional) relations between ordered pairs (formed by sets of formulas andformulas) and sets of formulas (cf. [1, 22]).

Transfers are mappings between abstract logics which preserve attributes.The source and target (propositional) logics connected by transfers can be of dif-ferent kinds: monotonic, non-monotonic, modal, intuitionistic, paraconsistent,substructural, and so on. We show that there is an ample family of logics towhich such methods and concepts are appliable. A precise definition of attributeand of transfer will be given in Definition 3.7.

Translations between logics are special cases of transfers. The concept oftranslation between logics has been used since the thirties (see [16] for example,and also [10] for comments, main results and general properties on translations).Although when used in particular cases with limited scope there is little doubtabout the meaning of a translation, when considered in a broad perspective thequestion “What does a translation really preserve?” acquires another meaning.So, for example if a logic L1 is translated “in the best possible way” into a logicL2, what are the real distinctions between them, if any?

Part of the difficulty is that there is no theory capable of expressing the gen-eral properties of abstract logics and explaining the full meaning of translationsbetween them. In this paper we discuss this problem, and offer a formal frame-work where logics and their translations can be treated and their propertiesexpressed.

It is naively held that if T is a translation between logics L1 and L2, thenthe meaning of L1 is preserved within L2, and our methods will be appropriatefor examining this slogan (cf. Section 5).

Our Bourbakian approach to logics is realized in the very spirit of modeltheory: abstract logics are two-sorted, first-order structures (a sort for set offormulas, and another for formulas), and transfers are morphisms relating suchstructures. Model theory helps us to express, in an powerful and elegant manner,the effects and the meaning of transfers. And it also permits us to prove amongother things, that the class of abstract logics has several remarkable properties:it is closed under ultraproducts, and each abstract logic in our sense admitsa canonical representation where all attributes acquire standard meaning (cf.Theorem 3.5).

The most illustrative example of the scope of our approach concerns BeliefRevision Systems (cf. Section 6). We show how this formalism leads to anample generalization of AGM’s postulates, called Wide Belief Revision Systems

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(WBRS’s), liberating the underlying theory from the constraints of particularlanguages and from certain restrictions such as supraclassicality. Indeed, ourmain application shows that WBRB’s can be conceived for a large class of logicsas long as they satisfy some conditions expressed by abstract forms of AGM’spostulates. One of the important results of this view is that belief revisionoperations can be induced from one logic into another by means of transfers.In practical terms, this means that, if T : L1−→L2 is a transfer satisfyingcertain conditions, then a BRS defined for L2 creates a BRS in L1 (cf. Theorem6.12). In particular, if L1 and L2 are different formulations of the same logic,for instance, two distinct presentations of classical logic, then a given revisionmethod induces a new one by means of a self-translation. This would permitcomparisons between algorithmic aspects of competing BRS’s.

2 Elementary Concepts from Model Theory

The present section recalls some basic definitions and results from (two-sorted)model theory which will be used throughout the paper. The experienced (orimpatient) reader may skip this section. Standard references for model theoryare [13, 4].

The notion of language adequate for our purposes (and which will act asa common meta-language for all the logical systems studied here) needs thefollowing ingredients:

Definition 2.1 A (first-order, two-sorted) language is a set

L = {A1,A2} ∪ P ∪ F ∪ C

where {A1,A2} is the set of basic sorts of L, P is a set of symbols for predicates;F is a set of symbols for functions; and C a the set of constants. �

To simplify matters, we assume that the sort of each predicate symbol P ∈ Pis of form An

1 × Am2 for some n,m ≥ 0. An analogous assumption holds for

function symbols.The languages defined above are interpretable in two-sorted structures as

follows:

Definition 2.2 Let L be a language. A two-sorted structure for L is a 5-upleA = 〈A1, A2,PA,FA, CA〉 where

• A1 and A2 are nonempty sets;

• PA contains, for every P ∈ P of sort An1 ×Am

2 (n,m ≥ 0), a subset PA ofAn

1 ×Am2 ;

• FA contains, for every f : An1×Am

2 −→A1 in F , a map fA : An1×Am

2 −→A1

and, for every g : Ar1 ×As

2−→A2 in F , a map gA : Ar1 × As

2−→A2 (forn,m, r, s ≥ 0);

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• CA contains, for every constant c ∈ C of sort A1, an element cA ∈ A1 and,for every constant d ∈ C of sort A2, an element dA ∈ A2. �

The notion of satisfiability of a formula ϕ(~x; ~y) in a two-sorted structure A withassignments xi 7→ ai ∈ A1 and yj 7→ bj ∈ A2 (1 ≤ i ≤ n and 1 ≤ j ≤ m),denoted as A |= ϕ[~a;~b], is defined recursively as usual. We write (Ai)A = Ai fori = 1, 2.

Remark 2.3 Throughout this paper we will only consider first-order (meta)languages with exactly two sorts (cf. Definition 2.1). The word “structure” willalways mean “first-order, two-sorted structure”. �

Definition 2.4 Let A be a structure and Bi ⊆ Ai = (Ai)A (i = 1, 2) such that:

• if fA ∈ FA, fA : An1 × Am

2 −→X (X ∈ {A1, A2}) and (~x; ~y) ∈ Bn1 × Bm

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then fA(~x; ~y) ∈ Y , where Y = Bi if X = Ai (i = 1, 2);

• CA ⊆ B1 ∪B2.

Then A|〈B1,B2〉, the restriction of A to 〈B1, B2〉, is the structure

A|〈B1,B2〉 = 〈B1, B2,P1,F1, C1〉

whereP1 = {PA ∩Bn

1 ×Bm2 : PA ∈ PA, PA ⊆ An

1 ×Am2 , n,m ≥ 0};

F1 = {(fA)|Bn1 ×Bm

2: fA ∈ FA, Dom(fA) = An

1 ×Am2 , n,m ≥ 0};

C1 = CA.

Any structure B such that B = A|〈B1,B2〉 for some 〈B1, B2〉 as above is calleda substructure of A. �

Definition 2.5 Let A = 〈A1, A2,PA,FA, CA〉 and B = 〈B1, B2,PB,FB, CB〉be two structures for the same language L. A morphism of structures h : A−→Bis a pair h = 〈F1, F2〉 such that:

(i) Fi : Ai−→Bi for i = 1, 2;(ii) if P is a (n+m)-ary predicate symbol of L, then

(→a ;

→b) ∈ PA implies that (F1(a1), ..., F1(an);F2(b1), ..., F2(bm)) ∈ PB

for all→a = (a1, ..., an) ∈ An

1 ,→b= (b1, ..., bm) ∈ Am

2 ;(iii) if c (of sort A1) and d (of sort A2) are constants, then

F1(cA) = cB and F2(dA) = dB;

(iv) if f : An1 ×Am

2 −→A1 is a symbol of function of L, then

F1(fA(→a ;

→b)) = fB(F1(a1), ..., F1(an);F2(b1), ..., F2(bm))

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for all→a = (a1, ..., an) ∈ An

1 and→b= (b1, ..., bm) ∈ Am

2 . An analogous conditionis required if f has codomain of sort A2.A morphism h = 〈F1, F2〉 is an isomorphism if F1 and F2 are both bijec-tive, and “implies that” is substituted by “iff” in condition (ii) above. Wesay that A is embeddable in B, denoted by A ⊆ B, if there exists a morphismh : A−→B (called an embedding) such that h is an isomorphism between Aand B|〈F1[A1],F2[A2]〉, where Fi[Ai] = {Fi(a) : a ∈ Ai} for i = 1, 2. Note that〈F1[A1], F2[A2]〉 satisfies the requirements of Definition 2.4. �

A fundamental result from model theory, known as the Theorem of Isomor-phism, states that isomorphic structures cannot be distinguished by means ofthe underlying language.

Proposition 2.6 If 〈F1, F2〉 : A−→B is an isomorphism, then

A |= ϕ[a1, ..., an; b1, ..., bm] iff B |= ϕ[F1(a1), ..., F1(an);F2(b1), ..., F2(bm)]

for all formulas ϕ(x1, ..., xn; y1, ..., ym) with at most x1, ...., xn and y1, ..., ym asfree variables (of sort A1 and A2, respectively), for all (a1, ..., an) ∈ An

1 and forall (b1, ..., bm) in Am

2 . In particular, A |= σ iff B |= σ for all L-sentences σ. �

A notion stronger than embedding is that of elementary embedding. Briefly, itstates that a substructure A of B is elementary if any formula with parametersin A, which is true in A, is also true in B (with the same parameters). Formally:

Definition 2.7 Let 〈F1, F2〉 : A−→B be a morphism. We say that 〈F1, F2〉 isan elementary embedding if, for any formula ϕ(x1, ..., xn; y1, ..., ym) and for anytuples ~a = (a1, ..., an) ∈ An

1 , ~b = (b1, ..., bm) ∈ Am2 , the following is true:

A |= ϕ[~a;~b] iff B |= ϕ[F1(a1), ..., F1(an);F2(b1), ..., F2(bm)].

Equivalently: 〈F1, F2〉 is an embedding such that for any formulas of the form(∃x0)ϕ(x0, x1, ..., xn; y1, ..., ym) and (∃y0)ψ(x1, ..., xn; y0, y1, ..., ym) (where x0

and y0 have sort A1 and A2, respectively), and for any tuples ~a = (a1, ..., an) ∈An

1 , ~b = (b1, ..., bm) ∈ Am2 , the following is true:

• if B |= (∃x0)ϕ[F1(a1), ..., F1(an);F2(b1), ..., F2(bm)], there exists a0 ∈ A1

such that B |= ϕ[F1(a0), F1(a1), ..., F1(an);F2(b1), ..., F2(bm)];

• if B |= (∃y0)ψ[F1(a1), ..., F1(an);F2(b1), ..., F2(bm)], there exists b0 ∈ A2

such that B |= ψ[F1(a1), ..., F1(an);F2(b0), F2(b1), ..., F2(bm)]. �

Definition 2.8 Let A1 = 〈A1, A2,P1,F1, C1〉 and A2 = 〈A1, A2,P2,F2, C2〉 bestructures. We say that A2 expands A1, or A2 is an expansion of A1, if P1 ⊆ P2,F1 ⊆ F2, and C1 ⊆ C2. This means that A2 has “more structure” than A1,therefore the language L2 of A2 has more symbols than the language L1 of A1,and we say that A1 is the reduct of A2 to L1. �

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When expanding structures by adding new distinguished elements (constants)we can give names to each object of the structure in the extended language.This is a useful tool for constructing models.

Definition 2.9 Let A = 〈A1, A2,PA,FA, CA〉 be a structure, and L the lan-guage of A. Let LA1,A2 be the language obtained from L by adding new symbolsof constants

{ca : a ∈ A1} ∪ {db : b ∈ A2},

where each ca has sort A1, each db has sort A2 and ca 6= cb if a 6= b, and da 6= db

if a 6= b. Consider the obvious expansion of A to a LA1,A2-structure

AA1,A2 = 〈A1, A2,PA,FA, CA ∪A1 ∪A2〉.

The diagram of A, denoted ∆A, is the set of LA1,A2-sentences ∆A = ∆+A ∪∆−

A ,where

∆+A = {σ : σ is an atomic LA1,A2-sentence and AA1,A2 |= σ},

∆−A = {σ : σ is ¬α for some atomic LA1,A2-sentence α, and AA1,A2 |= σ}.

�

Proposition 2.10 Let A and B be structures for L. Then A is embeddable inB iff B can be expanded to a model of the diagram of A. 2

3 Logics as structures

In this section we show how model-theoretic concepts can express useful prop-erties of abstract logics and transfers between them. An important notion tobe defined is that of attribute of logics, as mentioned in Section 1: as we shallsee, attributes will be the symbols (elements) of our formal meta-language. Wefirst define a basic meta-language and its extensions.

Definition 3.1 The basic language of abstract logics is the first-order two-sorted language L given by

{form,Sform} ∪ {ε,`} ∪ {d, s} ∪ {0}

where {form,Sform} is the set of basic sorts of L, ε and ` are symbols forpredicates of sort form×Sform and Sform× form, respectively; d : Sform×Sform−→Sform and s : form−→Sform are symbols for functions; and 0 is aconstant of sort Sform. �

If τ is a term of sort form and Υ,Ξ are terms of sort Sform, we will write τ ε Υ,Υ ` τ and Υ d Ξ in the place of ε(τ,Υ), ` (Υ, τ) and d(Υ,Ξ), respectively.Variables of sort form and Sform will be denoted by w, x, y, z and W,X, Y, Z(occasionally with subscripts), respectively.

The concept of abstract (propositional) logic is now introduced as the modelsof the above basic language.

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Definition 3.2 An abstract logic L is a two-sorted structure for the basic lan-guage L of the form

L = 〈A,P, εL,`L,dL, sL,0L〉

satisfying the following set of axioms in L:

[Ax1] (∀X)(∀Y )(X = Y ↔ (∀x)(x ε X ↔ x ε Y ));

[Ax2] (∀x)(∀y)(y ε s(x) ↔ y = x);

[Ax3] (∀X)(∀Y )(∀x)(x ε X d Y ↔ ((x ε X) ∨ (x ε Y )));

[Ax4] (∀x)¬(x ε 0);

[Ax5] (∀X)(∃Y )(∀x)(x ε Y ↔ ¬(x ε X));

[Ax6] (∀X)(∃Y )(∀x)(x ε Y ↔ X ` x).

The set of axioms {[Ax1], ..., [Ax6]} will be denoted by Ax . If L′ is an extensionof L, and L is a structure for L′, we say that L is an abstract logic for thelanguage L′ if L restricted to L is an abstract logic. If a logic L is a substructureof another logic L′, we say that L is a sublogic of L′. �

The set Ax introduces the minimal amount of set theory which permits usto refer to single formulas, “sets” of formulas, (finite) “unions” and the “emptyset”. So, [Ax1] expresses “extensionality of sets”, and axioms [Ax2], [Ax3], [Ax4]say, respectively, that s(x) is a “singleton”, XdY is the “union” of X and Y , and0 is the “empty set”. Axiom [Ax5] introduces the “complement of a set” and[Ax6], the only non set-theoretical in character, establishes that for each “set”X there exists the “set of consequences of X”. The reason we use quotationmarks is because these axioms do not exclude the existence of non-standardmodels. For example, the set P is not necessarily a subset of the powerset ℘(A)of A, or even if P ⊆ ℘(A), the interpretation of the singletons s(x) might notbe real singletons, and so on.

The following notion of standard models captures the desired intention aboutmodels.

Definition 3.3 Let L′ be an extension of L. A standard abstract logic for L′

is an abstract logic L for L′ such that P ⊆ ℘(A) = {Γ : Γ ⊆ A}; εL ⊆ A×P isthe (set-theoretic) membership relation; dL : P × P−→P is the (set-theoretic)join operation ∪; sL : A−→P is given by sL(a) = {a} for all a ∈ A; and 0L isthe empty set ∅. �

Remark 3.4 By elementary model-theoretic arguments it follows, from thedefinitions above, that L is a standard abstract logic for L′ if and only if itsatisfies the following conditions:

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• P ⊆ ℘(A) is a Boolean algebra w.r.t. the set-theoretic operations suchthat ∅, A ∈ P ;

• {a} ∈ P for all a ∈ A;

• if Γ ∈ P then {a ∈ A : Γ `L a} ∈ P . �

Throughout the rest of the paper we will denote a standard abstract logic overL simply as 〈A,P,`L〉, omitting the symbols corresponding to set-theoreticaloperations. If L′ extends L by adding sets P, F and C of symbols for predicates,symbols for functions and constants, respectively, then a standard abstract logicL over L′ will be denoted as L = 〈A,P, {`L} ∪ PL,FL, CL〉.

The following result shows that it suffices to consider standard abstractlogics (up to isomorphisms). It establishes a sort of representation theoremfor abstract logics, and guarantees that the standard models are the only oneswe need to take into account. In other words, the meta-properties of abstractlogics, expressed by formulas of L′, are characterized by standard models.

Theorem 3.5 (Representation Theorem for Abstract Logics) Let L =〈A,P, {εL,`L} ∪ PL, {dL, sL} ∪ FL, {0L} ∪ CL〉 be an abstract logic over anextension L′ of L. Then, there exists a standard abstract logic L′ over L′ whichis isomorphic to L.

Proof: For each Γ ∈ P let Γθ = {a ∈ A : a εL Γ}. Since L satisfies [Ax1]then: Γθ = ∆θ implies that Γ = ∆ for all Γ,∆ ∈ P . Let P = {Γθ : Γ ∈ P};then P ⊆ ℘(A) and θ : P−→P , θ(Γ) = Γθ is a bijection. Since L is a model forthe axioms in Ax , then:

(sL(a))θ = {a} for all a ∈ A;

(Γ dL ∆)θ = Γθ ∪∆θ for all Γ,∆ ∈ P ;

(0L)θ = ∅.

Define now a standard abstract logic L′ = 〈A,P , {`L′} ∪ PL′ ,FL′ , CL′〉 over L′

as follows. The relation `L′ ⊆ P × A is given by: Γθ `L′ a iff Γ `L a for allΓ ∈ P and a ∈ A. For all n,m ≥ 0 consider the bijection

θn,m : Pn ×Am−→Pn ×Am, θn,m(~Γ;~a) = ((Γ1)θ, ..., (Γn)θ; a1, ..., am).

For all RL ∈ PL (corresponding to a symbol for relation R ∈ P ⊆ L′) suchthat RL ⊆ Pn × Am define the relation RL′ ⊆ P

n × Am as RL′ = θn,m[RL].Then PL′ = {RL′ : R ∈ P}. For all fL : Pn × Am−→X in FL (correspondingto a symbol for function f ∈ F ⊆ L′) such that X ∈ {A,P} define the mapfL′ : P

n ×Am−→Y (where Y = A if X = A and Y = P if X = P ) as follows:

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fL′ =

fL ◦ (θn,m)−1 if X = A

θ ◦ fL ◦ (θn,m)−1 if X = P.

The constants in C ⊆ L′ are defined in L′ as a particular case of the equationabove (when n = m = 0). Consequently FL′ = {fL′ : f ∈ F} and CL′ = {cL′ :c ∈ C}. It can immediately be seen that 〈idA, θ〉 : L−→L′ is an isomorphism.�

Definition 3.6 We call the standard abstract logic L′ isomorphic to L, as de-fined in Theorem 3.5, the canonical representation of L. �

From Proposition 2.6, if L′ is the canonical representation of L, then

L |= ϕ[a1, ..., an; Γ1, ...,Γm] iff L′ |= ϕ[a1, ..., an; (Γ1)θ, ..., (Γm)θ]

for all formula ϕ(x1, ..., xn;X1, ..., Xm) in L′. In particular, L and L′ satisfythe same sentences in the language L′. From now on, we will only deal withstandard abstract logics.

We are now in a position to give a precise meaning to the concepts of attributeof a logic and of transfers between logics, hinted in Section 1:

Definition 3.7 Let L′ be an extension of L.(i) An attribute of an abstract logic L over L′ is just an element of L′.(ii) Let Li be abstract logics over L′ such that formLi = Ai and SformLi = Pi

(i = 1, 2). A transfer from L1 into L2 is a morphism 〈T, T∗〉 : L1−→L2 suchthat

T∗(Γ) = {T (a) : a ∈ Γ} ( = T [Γ]) for all Γ ∈ P1.

An isomorphic transfer between L1 and L2 is called a L-homeomorphism fromL1 to L2. A transfer 〈T, T∗〉 is conservative if it is a morphism satisfying “iff”instead of “implies that” in condition (ii) of Definition 2.5. We say that 〈T, T∗〉is an elementary transfer if it is an elementary embedding from L1 into L2.Finally, if L′ = L then a transfer is called a translation. �

Remark 3.8 Note that in the definition above we are assuming that both L1

and L2 are standard. In general, a morphism 〈T, T∗〉 : L1−→L2 is a transfer if,for all a2 ∈ A2 and Γ ∈ P1, the following is true:

a2εL2T∗(Γ) iff a2 = T (a1) for some a1 ∈ A1 such that a1εL1Γ.

If θi : Pi−→Pi is defined as in the proof of Theorem 3.5 (i = 1, 2) then it isequivalent to say that θ2(T∗(Γ)) = T [θ1(Γ)]. �

It should be clear that any element of L′ expresses a relation between formulas(or sets of formulas) and formulas (or sets of formulas) as mentioned in the firstsection. On the other hand, we see that transfers between logics are specialcases of morphisms between structures.

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It is worth observing that, if L′ = L and 〈T, T∗〉 : L1−→L2 is a transfer (thatis, a translation), then

Γ `L1 a implies that T [Γ] `L2 T (a),

and, if 〈T, T∗〉 is conservative, the following holds:

Γ `L1 a iff T [Γ] `L2 T (a),

that is, we recapture the usual definitions of translation, conservative translationand L-homeomorphism introduced in [16] and [19]. We return to the topic oftranslations between logics in Section 5.

It is also to be noted that the parameter T∗ is derived from T in our definitionof transfer (by Remark 3.8, T∗ = θ−1

2 ◦ T ◦ θ1 in the general case); for thisreason, from now on, we will denote a transfer 〈T, T∗〉 simply as T . On theother hand, the image under a conservative transfer T : L1−→L2 of a logic L1

such that P1 = ℘(A1), is also a logic. For, if T is conservative then T (L1)satisfies axiom [Ax6] of Definition 3.2. On the other hand, if Γ ∈ P1 thenT [A1] − T [Γ] = T [A1 − T−1(T [Γ])] such that A1 − T−1(T [Γ]) ∈ P1, and thus[Ax5] of Definition 3.2 is also satisfied by T (L1) (and these axioms involvingexistential quantifiers are the only critical ones in Ax ). Note that T is a L-homeomorphism iff T is a bijective conservative transfer. This kind of transferis very important because it determines that the logics involved are the same(from the point of view of the language L′). In other words, the logics related bytransfers of this kind are indiscernible by sentences in the language L of logics.

Proposition 3.9 Let T : L1−→L2 be a L-homeomorphism. Then for anyformula ϕ(x1, ..., xn;Y1, ..., Ym), every (a1, ..., an) ∈ An

1 and (Γ1, ...,Γm) in Pm1 ,

L1 |= ϕ[a1, ..., an; Γ1, ...,Γm] iff L2 |= ϕ[T (a1), ..., T (an);T [Γ1], ..., T [Γm]],

where ϕ has at most x1, ..., xn (of sort form) and Y1, ..., Ym (of sort Sform) asfree variables. In particular, for each sentence σ in the language L,

L1 |= σ iff L2 |= σ.

Proof: An immediate consequence of Proposition 2.6. �

The definition of abstract logic does not impose any conditions on the con-sequence relation `. If we set CL(Γ) := {a ∈ A : Γ `L a} then CL : P−→Pis defined to be the closure (or consequence) operator associated with L (seeDefinition 3.11 below). Recall that Tarski’s well-known axioms for consequenceoperators C : ℘(X)−→℘(X) are the following(cf. [16]):

[C1] Γ ⊆ C(Γ) for all Γ ⊆ X;[C2] Γ ⊆ ∆ implies C(Γ) ⊆ C(∆) for all Γ,∆ ⊆ X;[C3] C(C(Γ)) ⊆ C(Γ) for all Γ ⊆ X.

In order to express in the language L not only such concepts, but several others,we introduce the following abbreviations:

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Definition 3.10

• Y1 ⊆ Y2 for (∀x)(x ε Y1 → x ε Y2) (“Y1 is included in Y2”);

• Ent(Y1, Y2) for (∀x)(x ε Y2 → Y1 ` x) (“Y1 entails Y2”);

• Str(Y1, Y2) for (∀x)(Y2 ` x → Y1 ` x) (“Y1 is logically stronger thanY2”);

• Con(Y1, Y2) for (∀x)(x ε Y2 ↔ Y1 ` x) (“Y2 is the set of consequences ofY1”);

• Th(Y ) for Con(Y, Y ) (“Y is a closed theory”);

• Eq(x, y) for (∀Y )(Con(s(x), Y ) ↔ Con(s(y), Y )) (“x and y are conse-quentially equivalent”). �

Note that [Ax6] can now be written as

(∀X)(∃Y )Con(X,Y ).

We can now express Tarski’s axioms as follows:

[A1] (∀Y )Ent(Y, Y );[A2] (∀Y1)(∀Y2)(Y1 ⊆ Y2 → Str(Y2, Y1));[A3] (∀Y1)(∀Y2)(Con(Y1, Y2) → Str(Y1, Y2)).

As a consequence of Proposition 3.9, if T is a L-homeomorphism from L1 to L2,then L1 satisfies [A1]-[A3] iff L2 satisfies [A1]-[A3]. That is, L1 is a logic (in thesense of [16]) iff L2 is a logic in that sense (provided that Pi = ℘(Ai), i = 1, 2).

In general, properties of logics will be expressed by means of sentences in L,that is, by theories T over L. For example, the sentence

[TH!] (∀Y )(∃Y1)(Con(Y, Y1) ∧ (∀Y2)(Con(Y, Y2) → Y1 = Y2))

is true in L and then Eq(x, y) is an equivalence relation over the set A. Twoadditional examples are as follows:

Definition 3.11 Let L be a standard abstract logic where A = formL andP = SformL.(a) The consequence operator associated to L is the map CL : P−→P given by

CL(Γ) = {a ∈ A : Γ `L a}.

We say that L is Tarskian if L satisfies axioms [A1], [A2], [A3] (or, equivalently,if CL satisfies properties [C1], [C2], [C3]).(b) The equivalence relation ≡L over A associated to L is defined as follows:

a ≡L b iff CL({a}) = CL({b}). �

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Clearly the map CL is obtained by the interpretation of Con(X,Y ) and ≡L isobtained by interpreting the formula Eq(x, y). As another example of applica-tion of the language L, a theory T in L may contain the sentence

(∀Y1)(∀Y2)(∀x)(∀y)[((Y1 ` x) ∧ (Y2 d s(x) ` y)) → (Y1 d Y2 ` y)]

expressing the usual cut rule, which we call CUT property. Therefore, L1 hasCUT iff L2 has CUT, provided that there exists a L-homeomorphism betweenthem.

We now define the concept of sublogic generated by given sets:

Definition 3.12 Let L = 〈A,P,`〉 be a standard abstract logic over L, and let∅ 6= A0 ⊆ A and ∅ 6= P0 ⊆ P . The sublogic of L generated by 〈A0, P0〉 is thestandard abstract logic L1 = 〈A1, P1,`1〉 over L where A1 = A0 ∪ (

⋃P0),

P1 =⋂{W ⊆ P(A1) ∩ P : P0 ⊆ W and W satisfy the three

conditions stated in Remark 3.4}

and `1 = ` ∩ (P1 × A1). If A0 and P0 are finite sets, then the logic L0 is saidto be finitely generated. �

Finally, we show the following:

Proposition 3.13 Let L be an abstract logic, and let T be a theory over L.Then L is embeddable in an abstract logic which is a model of T iff every finitelygenerated sublogic of L can be embedded in an abstract logic which is a modelof T.

Proof: The “only if” part is immediate. For the “if” part, we will adapt theproof of Corollary 2.1.9 (cf. [13]). Let Σ0 be a finite subset of (T∪Ax )∪∆L. Wewant to prove that Σ0 has a model. Let {ca1 , ..., can

} and {dΓ1 , ..., dΓm} be the

constants of LA,P occurring in Σ0 having sort form and Sform, respectively(cf. Definition 2.9); if n = 0 or m = 0 we can add (ca1 = ca1) or (dΓ1 = dΓ1) toΣ0 without affecting satisfiability. Consider the sublogic L0 of L generated by〈{a1, ..., an}, {Γ1, ...,Γm}〉 (cf. Definition 3.12). Then L0 is a finitely generatedsublogic of L, and therefore L0 can be embedded in an abstract logic L1 whichmodels T, by hypothesis. By Proposition 2.10, L1 can be expanded to a modelL2 of ∆L0 , therefore L2 is a model of (T∪Ax )∪∆L0 and then it is an abstractlogic. Thus, L2 is a model of Σ0, because Σ0 ⊆ (T∪Ax )∪∆L0 . By compactnessof first-order classical logic (cf. [13]) the set (T∪Ax )∪∆L has a model L3. Thereduct L′ of L3 to L is a model of T ∪ Ax which can be expanded to a modelof ∆L (the logic L3). By Proposition 2.10 again we see that L is embeddable inL′, a logic which is a model of T. 2

4 Some Properties of Abstract Logics

In this section we show some basic properties of logics regarded as structures,taking profit from model-theoretic constructions. Some properties are just il-lustrative of the high degree of generality of our approach as, for example, the

12

result showing that the class of (standard) abstract logics is closed under ultra-products. Some other properties of transfers will be relevant for applications,such as Corollary 4.9 below.

We begin by proving a version of the well-known Los’s Ultraproduct Theoremadapted to the class of standard abstract logics. The next definition recalls theusual concepts of filter, ultrafilter and ultraproduct.

Definition 4.1 (a) Let I be a nonempty set. A filter over I is a set D ⊆ ℘(I)such that:

[F1] I ∈ D;[F2] R,S ∈ D implies R ∩ S ∈ D;[F3] R ∈ D and R ⊆ S ⊆ I implies S ∈ D.

If D 6= ℘(I) and D is maximal, then D is an ultrafilter. Thus D is an ultrafilteriff D is a filter satisfying: ∅ 6∈ D, and R ∈ D or (I −R) ∈ D for all R ⊆ I.

(b) Let F = {Ai : i ∈ I} be a family of sets, and D an ultrafilter over I. Theultraproduct of F module D is the set ΠDF = {fD : f ∈ Πi∈IAi}, where

fD = {g ∈ Πi∈IAi : f ∼D g} (f ∈ Πi∈IAi), and

f ∼D g iff {i ∈ I : fi = gi} ∈ D (f, g ∈ Πi∈IAi).

Let’s observe that ∼D is an equivalence relation.

(c) Let F = {Ai : i ∈ I} be a family of two-sorted structures over the same lan-guage L1, with Ai = 〈Ai, Bi,Pi,Fi, Ci〉 for all i ∈ I, and let D be an ultrafilterover I. The ultraproduct of F over D is the structure A = 〈A,B,PA,FA, CA〉for L1 where, for every predicate symbol P ∈ P, every function symbol F ∈ Fand every constant c ∈ C:

A = ΠD{Ai : i ∈ I}, B = ΠD{Bi : i ∈ I};

(f1D, ..., f

nD; g1

D, ..., gmD ) ∈ PA iff {i ∈ I : (f1

i , ..., fni ; g1

i , ..., gmi ) ∈ PAi} ∈ D;

FA(f1D, ..., f

nD; g1

D, ..., gmD ) = 〈FAi

(f1i , ..., f

ni ; g1

i , ..., gmi ) : i ∈ I〉D;

cA = 〈cAi: i ∈ I〉D. �

The next result is the Fundamental Theorem of Ultraproducts (or Los’ The-orem), which characterizes structures obtained as the ultraproduct of givenstructures.

Theorem 4.2 Let F = {Ai : i ∈ I} be a family of two-sorted structures overthe same language L1, and let A be the ultraproduct of F over an ultrafilter

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D ⊆ ℘(I). ThenA |= σ iff {i ∈ I : Ai |= σ} ∈ D

for every L1-sentence σ. �

If we conceive the elements of D as “large” subsets of I, the FundamentalTheorem of Ultraproducts 4.2 says that the ultraproduct A satisfies a sentenceσ iff σ is satisfied by a “large” number of structures Ai.

We now prove the important fact that the class of standard abstract logicsis closed under ultraproducts.

Theorem 4.3 (Ultraproducts of Standard Abstract Logics) Consider afamily F = {Li : i ∈ I} of standard abstract logics over an extension L′ of L.Then the ultraproduct L of F over an ultrafilter D ⊆ ℘(I) is a standard abstractlogic over L′.

Proof: Since each logic Li satisfies the set Ax of axioms (see Definition 3.2)then the ultraproduct L of F over D also satisfies Ax . In fact, if σ ∈ Ax then{i ∈ I : Li |= σ} = I ∈ D, thus L |= σ, using Theorem 4.2. This shows that Lis an abstract logic, and the result follows from Theorem 3.5. �

Corollary 4.4 If L is the ultraproduct of the family {Li}i∈I over D, then

L |= σ iff {i ∈ I : Li |= σ} ∈ D

for every L′-sentence σ.

Proof: Immediate from Theorem 4.2. �

This means that the ultraproduct logic L satisfies a property σ (expressed inthe language L′) iff the property σ is verified by a subset of logics Li that is“large” (in some sense).

Some additional results from model theory can be obtained to our generalapproach to logics as structures. All the new results (which appear as corollaries)are immediate, and the proofs are omitted.

Definition 4.5 Let ϕ be a formula of a two-sorted language L′.(a) ϕ is a Π0

1-formula if ϕ = (∀x1 · · · ∀xn)ψ, where each xi is a variable (of sortA1 or sort A2) and ψ has no quantifiers.(b) ϕ is a Π0

2-formula if ϕ = (∀x1 · · · ∀xn)(∃y1 · · · ∃ym)ψ, where each xi and yj

is a variable (of sort A1 or sort A2) and ψ has no quantifiers.(c) ϕ is a positive formula if ϕ has no → and ¬ symbols. �

Proposition 4.6 Let T be a closed theory. If T is axiomatizable by Π01-

formulas, then T is preserved by submodels. �

Corollary 4.7 Let L be a logic. If L satisfies a set T of Π01-formulas, then

every sublogic L′ of L also satisfies T. �

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Proposition 4.8 Let T be a consistent closed theory. Then, T is preserved byhomomorphisms (that is: A |= T implies that B|〈F1[A1],F2[A2]〉 |= T for every〈F1, F2〉 : A−→B) iff T has a positive set of axioms. 2

The following corollary deduced from Proposition 4.8 establishes an importantproperty of transfers: the content of positive sentences is preserved under trans-fers.

Corollary 4.9 Let L be a logic such that formL = A and SformL = P =℘(A). Given a transfer T : L−→L′, then

L |= ϕ[a1, ..., an; Γ1, ...,Γm]

implies thatL′ |= ϕ[T (a1), ..., T (an);T [Γ1], ..., T [Γm]]

for every positive formula ϕ, every (a1, ..., an) ∈ An and every (Γ1, ...,Γm) ∈ Pm.Therefore, if a logic L such that P = ℘(A) satisfies a set T of positive properties,then, for every conservative transfer T : L−→L′, the image T (L) of L under Talso satisfies T. �

The additional assumption of T being conservative in the result above guaran-tees that T (L) is a logic (cf. comments after Definition 3.7).

We close this rather technical section by observing that Los’s Theorem canbe viewed as an interesting tool for combining logics where each logic Li can beseen as a local agent and the ultraproduct L represents the global logic of allagents, in the spirit of so-called society semantics (cf. [11]). The other resultsin this section demonstrate how opportune the study of abstract logics is withinthe framework of model theory.

5 First Application: Universal Logic and Trans-lations

As one of the main applications we show how general properties of universallogic (in the sense of Beziau, [5]) can be treated from our vantage viewpoint.Universal logics are abstract logics with no reference to connectives, guardingthe same relation to concrete logics as universal algebra to concrete algebras.In a certain sense, our toolkit offers a formalization to universal logic.

Recall here the abbreviations introduced in Definition 3.10.

Definition 5.1 Consider the following properties of abstract logics (cf. [5]):

[A] (∀x)(s(x) ` x) (Auto-deductivity);

[M∞] (∀Y1)(∀Y2)(∀x)[((Y1 ` x) ∧ (Y1 ⊆ Y2)) → Y2 ` x](Infinite monotonicity);

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[Y∞] (∀Y1)(∀Y2)(∀x)[(Ent(Y1, Y2) ∧ (Y2 ` x)) → Y1 ` x](Infinite syllogism);

[AMY] ≡def [A] ∧ [M∞] ∧ [Y∞];

[H∞] (∀Y1)(∀Y2)(∀Y3)(∀x)[(Ent(Y1, Y2)∧ (Y2 d Y3 ` x)) → Y1 d Y3 ` x](Infinite Herz’s law);

[N] (∀Y1)(∀x)[(Y1 ` x) ↔ (∀Y2)(Ent(Y2, Y1) → (Y2 ` x))](Normal law). �

Proposition 5.2(i) [A1] and [Y∞] imply [M∞].(ii) If [A1] holds then: [Y∞] is equivalent to [H∞].(iii) [N] is equivalent to [AMY].

Proof: By adapting the proofs of these results found in [5] to our first-orderformalism. �

It is interesting to notice that the axioms [Y∞] and [A1] are sufficient to en-code the metaproperties of Hilbert calculi, and they imply that the logics areTarskian. In fact, these two properties are equivalent to Tarski’s axioms, in thecontext of logics.

Proposition 5.3(a) [Y∞] ∧ [A1] ` [A2] ∧ [A3]. Therefore, every logic L satisfying [Y∞] and[A1] is Tarskian.(b) [A2] ∧ [A3] ∧ [Ax6] ` [Y∞]. Therefore, every Tarskian logic L must satisfy[Y∞] ∧ [A1].(c) A logic L is Tarskian iff it satisfies [Y∞] and [A1].

Proof: Straightforward (remembering that any logic satisfies axiom [Ax6]). �

Now we can define logics presented as Hilbert calculi.

Definition 5.4 Let C = {Ck}k∈N be a (propositional-based) signature whereeach Ck is a set of connectives of arity k (cf. [27]). Connectives in C0 are calledpropositional symbols. Let LC be the language obtained from L and C, consider-ing each f ∈ Ck as a symbol of function f : formk−→form. In particular, eachp ∈ C0 is a constant of sort form. A propositional Hilbert calculus specificationis a pair 〈C,R〉 where C is a (propositional-based) signature and R is a set ofsentences in the language LC of the form

(∀)(s(ξ1, ..., ξn) ` ξ) and (∀)(0 ` ξ),

where (∀)Φ denotes the universal closure of the formula Φ, and s(ξ1, ..., ξn)denotes the term s(ξ1) if n = 1 and (· · · s(ξ1) d · · ·) d s(ξn) if n ≥ 2. Given a

16

specification E = 〈C,R〉 we obtain the standard logic LE = 〈A,P,`LE;CLE

〉over LC defined as follows:

• A is the free algebra generated by CLE= {(CLE

)k}k∈N (where cLE6= c′LE

whenever c 6= c′);

• P = ℘(A);

• `LE=

⋂{`1 ⊆ P ×A : 〈A,P,`1;CLE

〉 |= Ax ∪R ∪ {[A1], [Y∞]}}.

Given Γ ∪ {α} ⊆ A we say that α is provable from Γ in LE if

LE |= (Y1 ` x1)[Γ, α]. �

Observe that, by Proposition 5.2(i), LE |= [M∞]. Moreover, by Proposition5.3(c), LE is Tarskian. It is immediate that E = 〈C,R〉 induces, in fact, apropositional Hilbert calculus HE defined as follows:

• the language of HE is the free algebra generated by C (that is, the set ofLC-closed terms of sort form);

• (schema) axioms of HE : all the (schema) formulae ξ, where (∀)(0 ` ξ) isin R;

• (schema) rules of inference of HE : all the (schema) rules of the form

ξ1, ..., ξnξ

where (∀)(s(ξ1, ..., ξn) ` ξ) ∈ R.

In the schema formulae above, the free variables are interpreted as schema(that is, arbitrary) formulae in the language of HE . For instance, the LC-term(x1 ⇒ (x2 ⇒ x1)) must be read in HE as a C-schema axiom where x1 and x2

range over arbitrary C-formulae. As intended, the standard logic LE codifiesthe Hilbert calculus HE in a natural way (see Proposition 5.5 below). If K∪{ξ}is a set of LC-closed terms of sort form then we assume the following notation:ξLE will denote the interpretation in LE of ξ, and KLE = {ξ′LE : ξ′ ∈ K}. Asusual, K `HE

ξ will means that there exists a proof in HE of ξ from the set Kof hypothesis.

Proposition 5.5 Let K ∪ {ξ} be a set of LC-closed terms of sort form. ThenK `HE

ξ iff LE |= (Y1 ` x1)[KLE , ξLE ].

Proof: Suppose that K `HEξ. By induction on the lenght n of a proof in HE of

ξ from K we will prove that LE |= (Y1 ` x1)[KLE , ξLE ]. If n = 1, suppose thatξ is an instance ξ′(ξ1, ..., ξk) of an schema axiom ξ′(x1, ..., xk), where x1, ..., xk

are all the free variables of the LC-term ξ′ and ξ1, ..., ξk are C-formulae (thatis, LC-closed terms of sort form). Since (∀)(0 ` ξ′) ∈ R and LE |= R thenLE |= (0 ` x1)[ξLE ], thus LE |= (Y1 ` x1)[KLE , ξLE ] because LE is Tarskian.

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If ξ ∈ K the result is obvious, because LE is Tarskian. Assume that the resultis true for any proof of lenght ≤ n, and suppose that ξ is proved in HE fromK in n+ 1 steps. If ξ is an instance of an axiom or ξ ∈ K the proof is as above.If ξ is ξ′(ξ1, ..., ξk), obtained from ξ′1(ξ1, ..., ξk), ...., ξ′m(ξ1, ..., ξk) by an instanceof a rule

ξ′1(x1, ..., xk), ..., ξ′m(x1, ..., xk)ξ′(x1, ..., xk)

in HE then LE |= (Y1 ` x1)[KLE , ξ′i(ξ1, ..., ξk)LE ] for i = 1, ...,m, by inductionhypothesis. But LE |= (Y1 ` x1)[{ξ′1(ξ1, ..., ξk)LE , ..., ξ′m(ξ1, ..., ξk)LE}, ξLE ],since LE |= R. Therefore we obtain that LE |= (Y1 ` x1)[KLE , ξLE ] be-cause LE is Tarskian. The converse is obvious, because 〈A,P,`1;CLE

〉 |=Ax ∪R ∪ {[A1], [Y∞]}, where `1= {〈K ′LE , ξ′

LE 〉 : K ′ `HEξ′}. �

We close this section by applying our concepts to the theory of translationsbetween logics. In [16] and [19] it is claimed that conservative translations (seeDefinition 3.7) preserve the meaning of logics, without the stronger assumptionof being an isomorphism. The next two examples show that this is not truein all cases; the herein introduced concept of elementary translation (again, seeDefinition 3.7) is an intermediate notion between conservative translation andisomorphism, which allow us to characterize the transference of “meaning oflogics” (or, using our terminology, to transfer attributes of abstract logics) in amuch better way.

The first example shows that the concept of conservative translation by itself,without the requirement of injectivity, is not sufficiently apt to preserve intrinsicproperties of logics: any trivial logic can be conservatively translated into anyother logic which merely satisfies {β} ` β for some formula β.

Example 5.6 Let L1 and L2 be two logics such that:

• Γ `L1 α for all Γ ∈ P1 and any α ∈ A1;

• {β} `L2 β for some β ∈ A2.

Then T (x) := β for all x ∈ A1 is a conservative translation between L1 andL2. In particular, this holds if P1 = ℘(A1) and L2 is Tarskian. Therefore, thetriviality of L1 is “conservatively” codified in L2 by the pair {β} `L2 β. �

The example above also shows that trivial theories are not preserved by conser-vative translations: it suffices to add the extra requirement that {β} 6`L2 α forsome α ∈ A2.

The next example shows that, if the target logic does not satisfy [Y∞], thenthe equivalence between formulas is not preserved by non-surjective conservativetranslations. Therefore, conservative translations do not work well outside therealm of Tarskian logics.

Example 5.7 Let T : L1−→L2 be a conservative translation. Then the equiv-alence between formulas of L1 is not necessarily preserved by T . Indeed, it is

18

possible to exhibit formulas α and β such that T (α) `L2 T (δ) iff T (β) `L2 T (δ)for all L1-formula δ but T (α) `L2 γ and T (β) 6`L2 γ for some L2-formula γ,provided that T is not surjective and L2 is not Tarskian. Indeed, suppose thatA1 = {α, β}, P1 = ℘(A1) and Γ `L1 x iff Γ 6= ∅. Then α and β are equivalentin L1. Let A2 = {a, b, c}, P2 = ℘(A2) and

`L2= {〈{a}, x〉 : x ∈ A2} ∪ {〈{b}, b〉, 〈{b}, a〉} ∪ {〈{c}, c〉}.

Then T (α) := a, T (β) := b is a conservative translation such that T (α) andT (β) are inequivalent in L2. �

Why do conservative translations fail? Example 5.7 shows that, considering theformula φ(x, z) given by (∃y)((s(x) ` y) ∧ (s(z) 6` y)), then

L2 |= φ(x, z)[T (α), T (β)],

but the witness y 7→ c lies outside the image of T . A similar phenomenonoccurs with the case mentioned after Example 5.6: if the formula ψ(X) is givenby (∃y)(X 6` y) then L2 |= ψ(X)[T (Γ)] and again the witness y 7→ α doesnot belong to the image of T . The key is that, although L1 is isomorphic toT (L1), it is not an elementary substructure of L2, that is, T is not an elementaryembedding (cf. Definition 2.7).

What is missing, as mentioned above, is a still finer definition of translation(or, more generally, of transfer) which is stronger than a conservative translationand weaker than an isomorphism; this is suggested by our model-theoreticalsetting throughout the notion of elementary transfer (cf. Definition 3.7). In theparticular case where L′ = L we obtain the notion of elementary translation.

Recall that (see definitions 3.7 and 2.7) an elementary transfer between logicsL1 and L2 is a morphism T : L1−→L2 satisfying the following property: for anyformula ϕ(x1, ..., xn;X1, ..., Xm) in any extension language L′ of L and for anytuples ~a = (a1, ..., an) ∈ An

1 , ~Γ = (Γ1, ...,Γm) ∈ Pm1 , the following holds:

L1 |= ϕ[~a; ~Γ] iff L2 |= ϕ[T (a1), ..., T (an);T [Γ1], ..., T [Γm]].

Equivalently: T is an embedding such that for any formulas of the form

(∃x0)ϕ(x0, x1, ..., xn;X1, ..., Xm) and

(∃X0)ψ(x1, ..., xn;X0, X1, ..., Xm)

and for any tuples ~a = (a1, ..., an) ∈ An1 , ~Γ = (Γ1, ...,Γm) ∈ Pm

1 , the followingis true:

• if L2 |= (∃x0)ϕ[T (a1), ..., T (an);T [Γ1], ..., T [Γm]], there exists a0 ∈ A1

such that L2 |= ϕ[T (a0), T (a1), ..., T (an);T [Γ1], ..., T [Γm]];

• if L2 |= (∃X0)ψ[T (a1), ..., T (an);T [Γ1], ..., T [Γm]], there exists Γ0 ∈ P1

such that L2 |= ψ[T (a1), ..., T (an);T [Γ0], T [Γ1], ..., T [Γm]].

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This means that the logic L1 is “faithfully transferred” within L2 in the follow-ing sense: any existential property of the form (∃x)φ or (∃X)ψ which is satisfiedby L1 it is also satisfied by L2 throughout T (L1), that is, there exists a “wit-ness” in L2 of the form T (a) or T [Γ], respectively. Two important instances ofthe above definition are the following:

φ(x) given by (∃X)(X ` x) and ψ(X) given by (∃x)¬(X ` x).

If L1 satisfies φ(x) with parameter a ∈ A1 then it is also satisfied by T (L1) andconsequently by L2 with parameter T (a). Conversely, if L2 satisfies φ(x) withparameter T (a), then there exists a set ∆ ∈ P2 such that ∆ `L2 T (a), but ∆is not necessarily of the form T [Γ] for some Γ ∈ P1. In the second case, thesatisfiability of ψ(X) with parameter Γ in L1 is equivalent to the satisfiability ofψ(X) in T (L1) with parameter T [Γ], which implies satisfiability in L2 with thesame parameter. But the converse is not true, as seen in the example above: itis possible to have

T [Γ] `L2 T (a) for all a ∈ A1 but T [Γ] 6`L2 b for some b ∈ A2.

Elementary transfers (in particular, elementary translations) avoid such prob-lems, and we believe that they give a more precise answer to the question men-tioned in Section 1: “What does a translation preserve?”. A translation is“good” if it preserves the existential requirements of elementary translations:the more requirements are preserved, the “better” the translation. In particular,a basic requirement is to preserve trivialization, in the sense that

T [Γ] `L2 T (α) for all α ∈ A1 implies that T [Γ] `L2 β for all β ∈ A2.

In this sense, the well-known, already classical, translations of Godel, Glivenkoand Gentzen from classical logic into intuitionistic logic are “good”, since theypreserve the bottom. This suffices, in view of the following proposition.

Proposition 5.8 Let L1 and L2 be two (standard) logics over a language L′

containing a constant ⊥ ∈ C such that Li |= (∀x)(s(⊥) ` x) (i = 1, 2). Supposethat L2 satisfies the CUT property (cf. paragraph after Definition 3.11) andlet T : L1−→L2 be a transfer. Then, for any Γ ∈ P1, the following is true:

T [Γ] `L2 T (α) for all α ∈ A1 implies that T [Γ] `L2 β for all β ∈ A2.

Proof: Let β ∈ A2. Since T (⊥L1) = ⊥L2 and T [Γ] `L2 T (⊥L1) then T [Γ] `L2

⊥L2 . But {⊥L2} `L2 β and L2 satisfies CUT, so T [Γ] `L2 β. �

The above discussions suggest that we may think about a wide range of increas-ingly “better” notions of translations, depending on the amount of existentialformulas that are preserved.

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6 Second Application: Wide Belief Revision Sys-tems

In this section we analyze the theory of belief revision (cf. [1]) under our per-spective. A Belief Revision System (BRS) deals with changes of epistemicalstates under certain criteria of rationality, cf. [22]. Such criteria of rationalityhave to do with epistemic dynamics against a logical background, usually takento be classical logic, perhaps expanded by some modalities.

However, there is no reason to refrain from considering BRS’s based on logicsother than classical: we call the approach to belief changes based on classical ornon-standard logics Wide Belief Revision Systems (WBRS’s). Abstract logicsare thus the natural environment for treating WBRS’s.

Definition 6.1 Let LB be the language obtained from L by adding to the sig-nature of L the following symbols of function :

+ : Sform× form−→Sform;

∗ : Sform× form−→Sform;

− : Sform× form−→Sform. �

Definition 6.2 A standard abstract logic

L = 〈A,P,`L,+L,−L, ∗L〉

for the language LB is a Wide Belief Revision System (shortly, a WBRS) ifP = ℘(A) and it satisfies Tarski’s axioms [A1], [A2] and [A3], plus the followingaxioms (see Definition 3.10):

Expansion:

[K+] (∀Y )(∀x)Con(Y d s(x),+(Y, x))

Contraction:

[K − 1] (∀Y )(Th(Y ) → (∀x)Th(−(Y, x)))

[K − 2] (∀Y )(Th(Y ) → (∀x)(−(Y, x) ⊆ Y ))

[K − 3] (∀Y )(Th(Y ) → (∀x)(Y 6` x → − (Y, x) = Y ))

[K − 4] (∀Y )(Th(Y ) → (∀x)(−(Y, x) ` x → 0 ` x))

[K − 5] (∀Y )(Th(Y ) → (∀x)(Y ⊆ + (−(Y, x), x)))

21

[K − 6] (∀Y )(Th(Y ) → (∀x)(∀y)(Eq(x, y) → − (Y, x) = −(Y, y)))

Revision:

[K ∗ 1] (∀Y )(Th(Y ) → (∀x)Th(∗(Y, x)))

[K ∗ 2] (∀Y )(Th(Y ) → (∀x)(∗(Y, x) ` x))

[K ∗ 3] (∀Y )(Th(Y ) → (∀x)(∗(Y, x) ⊆ + (Y, x)))

[K ∗ 4] (∀Y )(Th(Y ) → (∀x)((∃y)(Y d s(x) 6` y) → ∗ (Y, x) = +(Y, x)))

[K ∗ 5] (∀Y )(Th(Y ) → (∀x)((∃y)(s(x) 6` y) → (∃y)(∗(Y, x) 6` y))

[K ∗ 6] (∀Y )(Th(Y ) → (∀x)(∀y)(Eq(x, y) → ∗ (Y, x) = ∗(Y, y))) �

Remark 6.3 It should be clear to anyone familiar with the elements of thetheory of belief change that the axioms above constitute our formulation ofthe so called AGM’s postulates in a slightly general form (cf. [1, 22]). We nowcomment on each axiom. Axiom [K+] is usually written as K+α = C(K∪{α}).In the contraction and revision axioms, we add the condition Th(Y ) in all cases,reflecting the fact that AGM’s postulates are restricted to closed theories (beliefsets). Axioms [K − 1] and [K ∗ 1] (called closure) state that contraction andrevision produce belief sets. The content of axioms [K − 2] and [K ∗ 3] (calledinclusion) as well as [K− 3] (vacuity), [K− 4] and [K ∗ 2] (success), and [K− 5](recovery), are just reformulations of the original postulates. In axioms [K − 6]and [K ∗ 6] (equivalence) we substitute the condition ` ϕ ↔ ψ by the moregeneral C({ϕ}) = C({ψ}).

The novelty of our presentation appears in axioms [K ∗4] and [K ∗5]. Axiom[K ∗ 4] (preservation) states, in a general form, that if the new information ϕ isconsistent with the current belief set K, then the revision of K with ϕ coincideswith the expansion of K with ϕ. It is worth nothing that, in our formulation,negation is immaterial. That is, contraction does not presuppose any form ofnegation in the language. This is a big difference from the original postulate for[K ∗ 4]:

If ¬ϕ 6∈ K, then K + ϕ ⊆ K ∗ ϕ

where an occurrence of classical negation is essential.Finally, in [K ∗5] (consistency), we just postulate one direction, as the other

is a consequence of [K ∗ 2] and the Tarskian assumption of monotonicity. Theother direction, originally formulated as

If K ∗ ϕ is trivial (i.e., coincides with the set of all formulas) then ` ¬ϕ,

22

presupposes again classical negation and its explosive character. Our approach,however, does not need any kind of negation, much less classical negation. Westate that if the new information does not entail everything, then the revisedbelief set K ∗ ϕ is not trivial. Thus, the way is open for considering WBRS’sfor paraconsistent and other non-standard logics (cf. [2, 3, 12, 18]). �

The theory of BRS’s assumes additional conditions over the consequenceoperator CL: to sum up, CL must be supraclassical, that is, if LCL and Cldenote the language and the consequence operator of the classical propositionallogic, respectively, then Cl(Γ) ⊆ CL(Γ) for every set of formulas Γ over LCL.Although we do not need such requirements, as noted above, they can be readilydescribed in the formalism as follows.

Definition 6.4 Let C = {Ck}k∈N be a (propositional-based) signature (cf. Def-inition 5.4). Assume that:

• {pi : i ∈ N} ∪ {⊥,>} ⊆ C0

• ∼∈ C1

• {&, ⇒ ,t, ⇔ } ⊆ C2.

The language LCB is defined as LB ∪ LC . �

Definition 6.5 A standard abstract logic

L = 〈A,P,`L, CL,+L,−L, ∗L〉

for the language LCB is a Supraclassical Belief Revision System (in short, a SBRS)

if it is a WBRS (thus, P = ℘(A)) in which A is the free algebra generated byCL and L satisfies the axioms below.

• (∀x)(∀y)Eq(x&y,∼ (x⇒ ∼ y))

• (∀x)(∀y)Eq(x t y, (∼ x) ⇒ y)

• (∀x)(∀y)Eq(x⇔ y,∼ ((x⇒ y) ⇒ ∼ (y ⇒ x)))

• 0 ` >

• (∀x)(s(⊥) ` x)

• (∀x)(∀y)(0 ` (x⇒ (y ⇒ x)))

• (∀x)(∀y)(∀z)(0 ` ((x⇒ (y ⇒ z)) ⇒ ((x⇒ y) ⇒ (x⇒ z))))

• (∀x)(∀y)(0 ` (((∼ y) ⇒ (∼ x)) ⇒ (((∼ y) ⇒ x) ⇒ y)))

• (∀x)(∀y)(s(x⇒ y, x) ` y) �

23

It is clear that these additional axioms refer to the standard axiomatizationfor classical propositional logic.

There are some extra postulates for revision in AGM’s approach, which referto iterate revision (cf. [22]). We can also express them as follows:

Definition 6.6 We say that L is a Full Belief Revision System (in short, L isa FBRS) if it is a SBRS satisfying the additional axioms below.

[K ∗ 7] (∀Y )(∀x)(∀y)(Th(Y ) → Str(∗(Y, x) d s(y), ∗(Y, x&y)))

[K ∗ 8] (∀Y )(∀x)(∀y)(Th(Y ) ∧ ¬((∼ y) ε ∗ (Y, x)) →Str(∗(Y, x&y), ∗(Y, x) d s(y))) �

We can now take profit from all the previous results in this paper on abstractlogics and transfers, introducing a new tool for the theory of belief change: theidea of induced WBRS’s, accomplished by means of transfers.

Definition 6.7 Let L1 and L2 be two WBRS’s given by

Li = 〈Ai, ℘(Ai),`i,+i,−i, ∗i〉

for i = 1, 2. A WBRS-transfer between L1 and L2 is any transfer T : L1−→L2.If L1 and L2 are SBRS’s (resp. FBRS’s) then a transfer T : L1−→L2 is calleda SBRS-transfer (resp. a FBRS-transfer). �

According to Definition 6.7, a WBRS-transfer T : L1−→L2 is a transfer suchthat, for every Γ ∪ {α} ⊆ A1:

• T [+1(Γ, α)] = +2(T [Γ], T (α));

• T [∗1(Γ, α)] = ∗2(T [Γ], T (α));

• T [−1(Γ, α)] = −2(T [Γ], T (α)).

The only additional condition that a WBRS-transfer will impose on the under-lying translation is to be closed, that is, to preserve closed theories (cf. [16]).

Definition 6.8 Let T : L1−→L2 be a transfer. We say that T is closed if

T [Γ] = CL2(T [Γ])

for every Γ ⊆ A1 such that Γ = CL1(Γ). �

Note that, if `1 is Tarskian, then: Γ = CL1(Γ) iff Γ = CL1(∆) for some ∆ ⊆ A1.

Proposition 6.9 Let T : L1−→L2 be a WBRS-transfer.(a) If Γ = CL1(Γ) 6= ∅, then

T [Γ] = CL2(T [Γ]).

(b) If CL1(∅) 6= ∅ or CL2(∅) = ∅, then T is a closed transfer.

24

Proof: (a) Let α ∈ Γ = CL1(Γ); then Γ = CL1(Γ) = CL1(Γ∪{α}) = +1(Γ, α).Since T is a WBRS-transfer, we obtain

T [Γ] = T [+1(Γ, α)] = +2(T [Γ], T (α))

= CL2(T [Γ] ∪ {T (α)}) = CL2(T [Γ])

as desired.(b) If Γ = CL1(∅) 6= ∅, then the result follows by item (a). On the other hand,if CL2(∅) = ∅, then CL1(∅) = ∅ (because T is a transfer). Since T [∅] = ∅ we getthat T is closed from item (a). �

As an immediate consequence of Proposition 6.9(b) we obtain the following.

Corollary 6.10 Let T : L1−→L2 be a SBRS-transfer. Then T is a closedtransfer. �

According to Definition 6.7, a SBRS-transfer T : L1−→L2 is a transfer suchthat:

• T (pL1) = pL2 for every p ∈ C0;

• T (fL1(a1, ..., ak)) = fL2(T (a1), ..., T (ak)) for every f ∈ Ck (k > 0) and forevery a1, ..., ak ∈ A1.

In particular, T (>L1) = >L2 , T (⊥L1) = ⊥L2 , and, for every a, b ∈ A1:

• T (∼L1 a) = ∼L2 T (a);

• T (a&L1b) = T (a)&L2T (b);

• T (a⇒L1 b) = T (a) ⇒L2 T (b);

• T (a tL1 b) = T (a) tL2 T (b);

• T (a⇔L1 b) = T (a) ⇔L1 T (b).

This means that the translation underlying a SBRS-transfer is homophonic (cf.[18]). In the terminology of [6], a SBRS-transfer is a logical morphism. FromProposition 5.8, we see that any SBRS-transfer is a “good” translation, becauseit preserves triviality. We can use this characteristic of this kind of transfers toinduce a BRS in a logic L1 from an embedding T : L1−→L2 into a WBRS L2,as we shall see below.

Definition 6.11 (BRS induction) Let L1 and L2 be two abstract logics overan extension LC of L, where C is a propositional-based signature, and letT : L1−→L2 be an embedding. Suppose that L+

2 is an expansion of L2 to

25

a WBRS over LCB with operations +2, −2 and ∗2. Define in L′

2 = T (L1) =(L2)|〈T [A1],℘(T [A1])〉 the following operations:

+2′ : ℘(T [A1])× T [A1]−→℘(T [A1]), +2

′(Γ, a) = CL′2(Γ ∪ {a});

−2′ : ℘(T [A1])× T [A1]−→℘(T [A1]), −2

′(Γ, a) = −2(Γ, a) ∩ T [A1];

∗2′ : ℘(T [A1])× T [A1]−→℘(T [A1]), ∗2′(Γ, a) = ∗2(Γ, a) ∩ T [A1].

The expansion of L′2 to LC

B with these operations is called the restriction of L+2

to T (L1), and it is denoted by T (L1)+. The WBRS induced from L+2 and T is

the expansion L+1 of L1 to LC

B given by the following operations:

+1 : ℘(A1)×A1−→℘(A1), +1(Γ, a) = CL1(Γ∪{a}) = T−1(+′2(T [Γ], T (a)));

−1 : ℘(A1)×A1−→℘(A1), −1(Γ, a) = T−1(−′2(T [Γ], T (a)));

∗1 : ℘(A1)×A1−→℘(A1), ∗1(Γ, a) = T−1(∗′2(T [Γ], T (a))). �

Theorem 6.12 Let L1, L2, C, L+2 and T : L1−→L2 as in Definition 6.11.

Assume that there exist ⊥ ∈ C0 such that Li |= (∀x)(s(⊥) ` x) (i = 1, 2). ThenL+

1 and T (L1)+ are WBRS’s over LCB isomorphic via the obvious extension of

T to LCB .

Proof: Let L′2 = T (L1) = (L2)|〈T [A1],℘(T [A1])〉. Then L′

2 is isomorphic to L1

via T , by hypothesis. Note that the consequence operator of L′2 is given by

CL′2(Γ) = CL2(Γ)∩ T [A1], and L′

2 is Tarskian. We first show that L′2 equipped

with the extra operations +2′, −2

′ and ∗2′ satisfies the axioms of WBRS. Theexpansion operator +′

2 works well by the very definition. It is easy to see thatthe contraction operator −′

2 satisfies the required axioms; in particular, [K − 6]is satisfied because −2(Γ, a) ⊆ Γ for all Γ ⊆ A2. The restriction operator ∗′2 alsoworks fine: the only axioms which need some analysis are the last two, axioms[K ∗ 5] and [K ∗ 6]. Let Γ∪ {a} ⊆ A1. In order to prove that [K ∗ 5] is satisfiedby ∗′2, suppose that T [Γ] is L′

2-closed; consequently it is also L2-closed. If thereexists b ∈ A1 such that {T (a)} 6`L′

2T (b) then, since T (b) ∈ A2, there exists

d ∈ A2 such that ∗2(T [Γ], T (a)) 6`L2 d, because ∗2 satisfies [K ∗5]. Suppose that∗′2(T [Γ], T (a)) = T [A1]; then T [A1] ⊆ ∗2 (T [Γ], T (a)). Since T [A1] `L2 T (a1)for all a1 ∈ A1 then, by Proposition 5.8, we infer that T [A1] `L2 a2 for alla2 ∈ A2. In particular, T [A1] `L2 d. But then ∗2(T [Γ], T (a)) `L2 d, bymonotonicity of `L2 , a contradiction. Thus there exists a1 ∈ A1 such that∗′2(T [Γ], T (a)) 6`L′

2T (a1) and then ∗′2 satisfies [K ∗ 5]. Axiom [K ∗ 6] is satisfied

by ∗′2 because, since L2 is Tarskian, then from the equivalence of T (a) and T (b)in L′

2 we infer the equivalence of T (a) and T (b) in L2. This shows that T (L1)+

is a WBRS. The rest of the proof is immediate, because T−1 is an isomorphism.�

26

As an immediate application of Theorem 6.12, we obtain a method for definingBRS’s for non-classical logics which are translatable into classical logic. Given aBRS LCL for classical logic with reduct L2 to L{⊥}, and given an abstract logicL1 over L{⊥} such that there exists an embedding T : L1−→L2, then we candefine a new BRS L+

1 for the logic L1 constructed as in Definition 6.11 usingTheorem 6.12. As a concrete example, we show how to induce a BRS in therelatedness logics R and S, introduced by Epstein (cf. Chapters III and IV in[18]).

Example 6.13 The propositional logics R and S formalize the notion of sub-ject matter of propositions, offering a very flexible theory of relatedness depen-dent on context. Relatedness should not be confused with relevance: dogs andducks may be related if we are discussing living creatures, but if we were dis-cussing mammals and birds, we might take them as unrelated. By means of thecompleteness theorem for R (where relatedness is not necessarily symmetric)and S (where relatedness is symmetric), it is possible to show that every R-and S- tautology is a classical tautology and that several properties of classicalimplication are not valid in relatedness logics, as for example

α⇒ (β ⇒ α),

∼α⇒ (α⇒ β),

(α&∼α) ⇒ β,

α⇒ (β ⇒ β),

((α⇒ β)&(β ⇒ γ)) ⇒ (α⇒ γ).

Tableau systems for S and R are treated in [7]. In [18], Chapter X, an injectiveconservative translation from R into classical propositional logic (CPL) is givenas follows. Assume that R is defined over the signature CR in which

CR0 = {pi : i ∈ N};

CR1 = {∼};

CR2 = { ⇒ }.

On the other hand, consider a presentation of CPL over a signature CCPL suchthat

CCPL0 = {pi : i ∈ N} ∪ {di,j : i ∈ N and j ∈ N};

27

CCPL1 = {∼};

CCPL2 = { ⇒ ,t,&}.

For all formulas α and β of R, consider the formulas Φβα and Ψβ

α of CPL given by

Φβα =

∨{di,j : pi occurs in α and pj occurs in β};

Ψβα =

∨{di,i t ∼di,i : pi occurs in α and in β}.

Assume that the association of the parenthesis in the disjunctions of the formu-las above is previously fixed. The map T : For(R)−→For(CPL) given by

T (pi) = pi (for all i ∈ N);

T (∼α) = ∼T (α);

T (α⇒ β) = (T (α) ⇒ T (β))&(Φβα tΨβ

α)

is an injective conservative translation. It is not difficult to see that the lan-guage of R can be augmented by adding a new symbol ⊥ in CR

0 and the extraaxioms

{⊥} ` α;0 ` (⊥ ⇒ (α⇒ α));0 ` (α⇒ (⊥ ⇒ ⊥));

for all formula α, and the translation T extended by T (⊥) = ⊥. Combining thistranslation with the process of induction of BRS’s through transfers describedin Definition 6.11, each BRS defined in CPL algorithmically induces anotherBRS in R. A similar procedure can be applied to S.

As a simple example, consider Γ = {p1 ⇒ p2, p2 ⇒ p3}, K = CR(Γ), K ′ =CCPL(Γ) and α = ∼(p1 ⇒ p3). Then, in any BRS defined over CPL the revisionprocedure of K ′ by α is different from the expansion of K ′ by α, because ∼αis classically inferred from Γ. On the other hand, in the logic R the situationis different: in this case, the revision of K by α coincides with expansion. Thisis made clear by means of the above mentioned transfer T . For, since R isisomorphic to R′ = T (R) then

T [K] = CR′(T [K]) = CR′(T [Γ]) = CCPL(T [Γ]) ∩ T [For(R)]

(cf. Definition 6.11). Thus

T [K] = CCPL({(p1 ⇒ p2)&d1,2, (p2 ⇒ p3)&d2,3}) ∩ T [For(R)]

and then

∗′(T [K], T (α)) = ∗′(T [K],∼((p1 ⇒ p3)&d1,3)) = +′(T [K],∼((p1 ⇒ p3)&d1,3))

28

because ∼∼((p1 ⇒ p3)&d1,3) is no longer inferred from T [K] in CPL. There-fore, the BRS induced in R from T shows that revision of K by α coincideswith expansion. �

It is not our purpose to enter into details of potential applications of the methodintroduced here. We just remark that the most interesting cases concern trans-lations from a non-standard logic L into CPL, because in such cases we can takeprofit from the well-known BRS procedures for CPL, which will be inheritedby L.

There are numerous examples of translations in the literature arriving intoCPL. Besides R and S above, one of the most promising cases is the (para-consistent and three-valued) logic LFI1 (cf. [12]) which has been applied tothe theory of databases. LFI1 coincides with CLuNs (cf. [2]) and with J3

(cf. [17]). It is interesting to remark that a system equivalent to these appearsin [24] under the name Φv, but with a different motivation. Translations fromCLuNs into CPL are given in [2], where it is also shown that CLoN, CLaN,CLuN, CLoNs, CLaNs, CLoNv, CLaNv, CLuNv are also translatable intoCPL.

A somewhat similar construction, for the particular case in which L1 isthe modal logic K and L2 is the first-order classical logic, was independentlyproposed by [21] using correspondence theory, writing the semantical conditionsfor modalities in first-order logic and adding some extra axioms. Although thiscould be seen, by and large, as a kind of translation between logics, this hasnothing to do with our method. Moreover, they presuppose the existence ofBRS’s for first-order logic, which is not usual. The road taken here promises tolead much farther.

In the converse direction, translations from CPL into non-standard logicscould also be very convenient for BRS’s, in particular when paraconsistent logicsare taken into account. The idea is that paraconsistent logics are the naturaldomain to perform belief revision, and from such a perspective, LFI1 (op. cit.)is a well-suited candidate: if we had BRS’s defined on a logic L of this kind,using the transfer techniques of Theorem 6.12 we could perform the revision inL and subsequently translate it back to CPL, obtaining classical BRS’s.

7 Concluding remarks

We have introduced a formal framework in which abstract logics and their trans-fers can be analysed and (hopefully) better understood. Our definitions andmain results show that several properties of abstract logics can be expressedand their properties proven in our formalism. We have shown, as a particularapplication, that the class of abstract logics is closed under ultraproducts.

As a consequence of our approach, classical results of model theory canbe used to establish important (meta)properties of transfers between abstractlogics, helping to assign a philosophical meaning to the concept of translation.

We have also shown that using our approach to logic as first-order structures,

29

it is possible to represent the general theory of revision of belief sets, withimportant consequences for further research in this field.

Finally, as an additional topic where the setting of abstract logics can beused with profit, it is worth noting that the important method of combinationof logics known as fibring (cf. [20]) is also expressible in the proposed formalism.

The process of fibring propositional calculi studied by Zanardo, Sernadasand Sernadas (cf. [27]) can be described as follows: the unconstrained fibring oftwo logics L1 and L2 over 〈C1, R1〉 and 〈C2, R2〉 (cf. Definition 5.4) is definednaturally as L = L1∨L2, the logic obtained from the specification 〈C,R〉, whereC = C1 tC2 is the (disjoint) union of C1 and C2 with injections i1 and i2, andR = i1(R1) ∪ i2(R2), where ij(Rj) denotes the representation of Rj within LC

according to ij (j = 1, 2). It is easy to see that L1 ∨ L2 is the least logic Lover LC satisfying Li ⊆ L|LCi for i = 1, 2, that is: if Li ⊆ L′

|LCi (i = 1, 2) thenL ⊆ L′. Constrained fibring, obtained by sharing connectives (cf. [27]), can betreated in an analogous way.

The generalization for arbitrary families of logics is not difficult. Thoughdetails are not given here, several properties of fibring could also be formalizedin the present framework. This is of significance, because in this approach thefibring of a family of logics coincides with the supremum of the family, and hencefibring turns out to be a purely (first-order) model-theoretical construction.Methods of model theory could, again, be used with profit to obtain preservationresults.

To summarize, this paper opens research directions in at least three ar-eas. By investigating translations, combinations of logics and belief change viamodel theory, a robust foundation for those subjects is provided, and surprisingapplications in theoretical computer science are anticipated.

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