Graph-theoretic Fibring of Logics

Part II - Completeness Preservation

A. Sernadas1 C. Sernadas1 J. Rasga1 M. Coniglio2

1 Dep. Mathematics, Instituto Superior Tecnico, TU LisbonSQIG, Instituto de Telecomunicacoes, Portugal

2 Dep. of Philosophy and CLEState University of Campinas, Brazil

{acs,css,jfr}@math.ist.utl.pt, coniglio@cle.unicamp.br

July 29, 2008

Abstract

A graph-theoretic account of fibring of logics is developed, capital-izing on the interleaving characteristics of fibring. Signatures, interpre-tation structures and deductive systems are defined as enriched graphs.This graph-theoretic view is general enough to accommodate very differ-ent propositional based logics encompassing logics with non-deterministicsemantics, logics with an algebraic semantics, logics with partial seman-tics, substructural logics, among others. Fibring is seen as a universalconstruction in the category of logic systems. Graph-theoretic fibring al-lows the explicit construction of the interpretation structure resulting fromthe fibring of a pair of interpretation structures. Soundness and weak com-pleteness are proved to be preserved under very general conditions. Strongcompleteness is also shown to be preserved under tighter conditions. Inthis setting, the collapsing problem appearing in several combinations oflogic systems is avoided.

1 Introduction

The activity of combining logics offers an important mechanism for modularity,which is an essential ingredient in many applications where logic plays a role.The significance of the combination can be accessed by the preservation ofproperties that the original logics may have. For instance, it is important toknow whether the logic resulting from the combination of complete logics is stillcomplete. Proving preservation results constitute a theoretical challenge and,in many cases, they are obtained only by imposing conditions on the originallogics. It is interesting to observe that combination mechanisms start to play anessential role in contemporary applications, like argumentation theory, spatialand temporal reasoning and information security [13, 6, 14]. In some of thesetopics, the logical context has to be set-up appropriately. It is also worthwhile torefer that several combinations mechanisms have been developed in the contextof modal logic. Examples are fusion [9] and product [11, 12]. Interestingly, even

1

in the specific case of fusion, preservation results are not so easy to prove ascan be seen in [9].

Fibring is a very general mechanism for combining logics that was proposedin [8]. At the language level fibring is an interleaving in the sense that theconstructors of the two logics can be interleaved in the language expressions.The same applies to deduction. In a fibring deduction, we can interleave theapplication of inference rules of the original logics [18, 3]. When developingtechniques for fibring one of the main objectives is not imposing too muchrequirements on the logics being considered. For instance, it would be veryinteresting to be able to fiber modal logic with a paraconsistent logic getting anew logic suitable for paraconsistent modal reasoning.

A starting point for defining fibring is to set-up the notion of logic system.That is, it has to be specified, in a general way, what is a signature, an inter-pretation structure and a deductive system, so that a large class of logics canbe expressed in this context. Herein, we follow [19], where signatures, inter-pretation structures and deductive systems are defined based on the concept ofmulti-graph (m-graph). This graph-theoretic approach goes very well with theinterleaving aspects of fibring and offers a very general, intuitive and uniformway of looking into logic systems and its components.

Another key novelty in the way logic systems are described herein, is thatthere is an abstraction, a lifting, from interpretation structures to signaturesand not (as usual) the other way around. Together with the graph-theoreticapproach, this makes possible to represent logics with very different features,encompassing logics with non-deterministic semantics, logics with algebraicsemantics, logics with partial semantics, substructural logics, among others.Moreover, we believe that this setting can be used in a natural way to definelogic systems for logics with provisos and quantification.

An interesting aspect of graph-theoretic fibring is that, semantically, it isexplicitly defined for each pair of interpretation structures. That is, each in-terpretation structure in the fibring results from a clear and understandableconstruction applied to an interpretation structure of each component. Thiscounts as a positive aspect per se, but also since it makes possible to have inthe fibring a representative of all the models of the original logics (this featurewas not present for instance in [22]). It is also worthwhile to mention thatfibring is seen as the same universal construction at all levels: at the signa-ture level, at the interpretation structure level and at the deduction level. Thismeans that the logic system resulting from fibring is somehow minimal amongthe universe of logic systems considered.

Preservation of soundness and weak completeness are proved herein undervery general conditions, in contrast with strong completeness that is shown tobe preserved but under tighter conditions. These preservation results allow us toconclude that the graph-theoretic semantics we propose for fibring goes handin hand with the usual fibred deduction. It is worthwhile to mention that,to our knowledge, other approaches to fibring were not successful in provingpreservation of weak completeness.

The collapsing problem appearing in several combinations of logic systemsis avoided in this graph-theoretic approach to fibring. Namely, the well known

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collapse when combining classical propositional logic with intuitionistic propo-sitional logics does not appear [7, 20, 2]. This happens since in our settingthe fibred semantics is in a sense an exhaustive interleaving of the semantics ofthe components. Other cases of collapse can also be avoided, namely involvinglogics with non-deterministic semantics (like paraconsistent logics).

The study of graph-theoretic fibring promises nice developments in the nearfuture. An exciting topic is to consider logics with provisos and quantification.Another interesting direction is the study, in this context, of other kinds ofdeductive systems like sequents and labelled deduction. We believe that in thiscase a new sort (for judgements) has to be added in order to create the requiredlevel for deduction. Other preservation results related, for instance, with cutelimination and interpolation are also planned to be addressed.

The structure of the paper is as follows. In Section 2, the graph-theoreticfibring of signatures is defined. In Section 3, after recalling the notions ofinterpretation structure, denotation and satisfaction introduced in [19], fibringof interpretation structures is defined after introducing the notion of graphtransformation. Section 4 is dedicated to fibring of deductive systems. Thedeductive system resulting from fibring is what is expected, although presentedin a different (graph-theoretic) way. In Section 5, fibring of logic systems isdefined by putting together signatures, interpretation systems and deductivesystems. In Section 6, we discuss preservation of soundness by using a generaltheorem proved in [19]. Finally, in Section 7, we investigate preservation ofboth weak and strong completeness. Throughout the paper we use the fibringof classical and intuitionist logics as the running example. We assume a verymoderate knowledge of category theory (the interested reader can consult [15]).

2 Fibring signatures

In this section we investigate fibring at the signature level. We start by recallingthe notion of m-graph, m-graph morphism and signature introduced in [19].

By a multi-graph (in short a m-graph) we mean a tuple

G = (V,E, src, trg)

where V is a set (of vertexes or nodes), E is a set (of m-edges), and src : E → V +

and trg : E → V are maps. And, by a m-graph morphism

h : G1 → G2

we mean a pair of maps hv : V1 → V2 and he : E1 → E2 such that src2 ◦ he =hv ◦ src1 and trg2 ◦he = hv ◦ trg1. As described in the companion paper [19], wedenote by

G+

the category with binary products generated by a m-graph G, and we denoteby h+ : G+

1 → G+2 the functor induced by a m-graph morphism h : G1 → G2.

A language signature or, simply, a signature is a tuple

Σ = (G, π, ♦)

3

where G is a m-graph, and π and ♦ are in V . Nodes in a signature play therole of language sorts. Node π is the propositions sort (the sort of schema for-mulas), and node ♦ is the concrete sort, used as the source sort of propositionalsymbols. The m-edges play the role of constructors for building expressions ofthe available sorts.

Example 2.1 Signature for classical propositional logic.Let Π be a set of propositional symbols. The propositional signature ΣΠ is am-graph G with sorts π and ♦ and the following m-edges:

• p : ♦→ π for each p in Π;

• ¬ : π → π;

• ⊃ : ππ → π.

The m-edges ¬ and ⊃ represent the connectives negation and implication, re-spectively. The m-edge p represents the propositional symbol p. For instancethe morphism

⊃ ◦ 〈⊃ ◦ 〈p,¬ q〉, q〉 : ♦→ π

in G+ represents the formula (p⊃ (¬ q))⊃ q. ∇

We now describe the signature used for intuitionistic propositional logic.

Example 2.2 Signature for intuitionistic propositional logic.Let Π be a set of propositional symbols. The propositional signature with con-junction and disjunction Σ∧,∨Π is a m-graph obtained from ΣΠ by adding them-edges

• ∧,∨ : ππ → π

for representing conjunction ∧ and disjunction ∨. ∇

Given signatures Σ1 and Σ2, a signature morphism

h : Σ1 → Σ2

is a m-graph morphism h : G1 → G2 such that h(π1) = π2 and h(♦1) = ♦2. Wedenote by Sig the category of signatures and their morphisms, and by SigVthe subcategory of Sig whose objects are signatures with sort set V and whosemorphisms h are such that hv = idV .

Before defining fibring of signatures, we show that SigV has coproducts. Itis enough to show the existence of coproducts in this subcategory essentiallyby semantic reasons as it will become clear in Section 3. As we will see in thatsection, this is not a restriction since we can always enrich the signature of alogic system with additional sorts without changing the entailment.

Proposition 2.3 Category SigV has binary coproducts (and so all non-emptyfinite coproducts).

4

ΣΠ

π

¬

NN♦q1 //

Σ∧,∨Π

π ♦q1

mm\\\\

¬

PP

π¬i

cc♦��

q1i

q1c//

¬c

NN

i1

4<

i2

ai

Fib(ΣΠ,Σ∧,∨Π )

Figure 1: Part of the signatures for classical propositional logic, described inExample 2.1 and for intuitionistic propositional logic, described in Example 2.2,and for their fibring, see Example 2.4, containing the m-edges for negation andfor a propositional symbol.

Proof: Let Σ1 and Σ2 be signatures with sort set V . Their coproduct is thetriple

Σ1 ] Σ2 = (Σ, i1, i2)

where Σ = ((V,E, src, trg), π, ♦) is such that:

• E is the coproduct in Set of E1 and E2 with injections ie1 and ie2, respec-tively;

• src ◦ ie1 = src1, src ◦ ie2 = src2, similarly for trg;

• ij = (idV , iej) for j = 1, 2.

It is straightforward to check that the triple is indeed a coproduct. QED

Let Σ1 and Σ2 be signatures in Sig not necessarily with the same set ofsorts. The first step when defining fibring is to enrich the given signatures sothat they have the same set of sorts. To this end, we introduce a Set♦ indexedmap ·(V1,π1,♦1) on signatures, such that

Σ(V1,π1,♦1)2 = ((V,E2, src, trg), π, ♦)

where

• ((V, π, ♦), i1, i2) is the coproduct in Set♦ of (V1, π1, ♦1) and (V2, π2, ♦2);

• src = i2 ◦ src2 and trg = i2 ◦ trg2;

then, the fibring FibSig of signatures is defined as a map from |Sig|2 to |Sig|such that

FibSig(Σ1,Σ2) = Σ(V2,π2,♦2)1 ] Σ(V1,π1,♦1)

2 .

5

Example 2.4 Signature resulting from the fibring of the signatures for classicaland for intuitionistic propositional logic.Consider the signature ΣΠ for classical propositional logic in Example 2.1 andthe signature Σ∧,∨Π for intuitionistic propositional logic in Example 2.2. Then

FibSig(ΣΠ,Σ∧,∨Π ) = ΣΠ ] Σ∧,∨Π

since

• ΣΠ(Vi,πi,♦i) = ΣΠ;

• Σ∧,∨Π

(Vc,πc,♦c) = Σ∧,∨Π ;

where ΣΠ ] Σ∧,∨Π = ((V,E, src, trg), π, ♦) is a signature resulting from the co-product of ΣΠ and Σ∧,∨Π , defined, as follows:

• V = {π, ♦};

• E(♦, π) = {pc : p is a propositional symbol in ΣΠ} ∪ {pi : p is a proposi-tional symbol in Σ∧,∨Π };

• E(π, π) = {¬c,¬i};

• E(ππ, π) = {⊃c,⊃i,∧i,∨i};

• all the other components are empty.

A graphical description of part of this signature is presented in Figure 1, where,for the sake of simplicity, only negation connectives as well as one propositionalsymbol for each component are considered. For example the morphism

⊃c ◦ 〈¬i ◦ ¬i ◦ pc, pc〉

represents the formula (¬i(¬ipc))⊃ pc over the fibred signature. ∇

3 Fibring interpretation structures

Before investigating the fibring of interpretation structures we briefly recallsome definitions from [19]. We start by recalling the notion basis, common toboth interpretation structures and deductive systems. A basis over a m-graphG is a pair

(G′, α)

such that G′ is a m-graph (the operations graph) and α : G′ → G is a m-graphmorphism (the abstraction morphism). So, an interpretation structure I overa signature (G, π, ♦) is a triple

(G′, α,D, �)

such that the pair (G′, α) is a basis over G, D ⊆ (αv)−1(π) is a non-empty setand � ∈ (αv)−1(♦). Observe that V ′ is partitioned by α: we denote by V ′v thedomain (αv)−1(v) of values for each v in V . The elements of V ′π are the truth

6

values and the elements of V ′♦ are the concrete values. The elements of the set Dare the distinguished truth values. Given s in V + we denote by V ′+s the subsetof V ′+ consisting of the set ((αv)+)−1(s), that is, {s′ : (αv)+(s′) = s}. The setE′ of m-edges of the operations graph is also partitioned by α: we denote byE′e the set (αe)−1(e) for each e in E. An interpretation structure is a pair

(Σ, I)

where Σ is a signature and I is an interpretation structure over Σ.We now present interpretation structures for classical propositional logic

and intuitionistic propositional logic. Posteriorly, we present in Example 3.4the fibring of these interpretation structures.

Example 3.1 Interpretation structure for classical propositional logic.Consider the signature ΣΠ as introduced in Example 2.1 where Π = {q1, q2, q3}.Let v : {q1, q2, q3} → {0, 1} be a classical valuation such that v(q1) = 1 andv(q2) = v(q3) = 0. The interpretation structure Ic = (G′, α,D, �) over signatureΣΠ corresponding to v is defined as follows:

• G′ is such that1:

V ′ = {0, 1} ∪ {�};E′ = {q′1, q′2, q′3,¬0,¬1,⊃00,⊃01,⊃10,⊃11};src′ and trg′ are such that:

q′1 : �→ 1;q′i : �→ 0 for i = 2, 3;¬v′ : v′ → (1− v′) for each v′ in {0, 1};⊃v′1v′2 : v′1 v

′2 → ((1− v′1) + v′2) for each v′1 and v′2 in {0, 1}.

• α : G′ → G is such that:

αv(0) = π;

αv(1) = π;

αv(�) = ♦;

αe(q′i) = qi for i = 1, 2, 3;

αe(¬v′) = ¬ for each v′ in V ′π;

αe(⊃v′1v′2) = ⊃ for each v′1 and v′2 in V ′π.

• D = {1}. ∇

Example 3.2 Interpretation structure for intuitionistic propositional logic.Consider the signature Σ∧,∨Π as introduced in Example 2.2 where Π = {q1, q2, q3}.Let m = (W,R, v) be the intuitionistic Kripke structure where W = {u1, u2},R = {(u1, u1), (u1, u2), (u2, u2)}, v(q1) = {u2}, v(q2) = {u1, u2}, and v(q3) = ∅.By simplicity we will denote by u2 and u1u2 the sets {u2} and {u1, u2} re-spectively. The interpretation structure Ii = (G′, α,D, �) over signature Σ∧,∨Π

corresponding to m is defined as follows:1Using module 2 arithmetical operations within V ′.

7

• G′ is such that:

V ′ = {∅, u2, u1u2} ∪ {�};E′ = {q′1, q′2, q′3,¬∅,¬u2 ,¬u1u2} ∪ {⊃v′1v′2 : v′1, v

′2 ∈ V ′π} ∪ {∧v′1v′2 :

v′1, v′2 ∈ V ′π} ∪ {∨v′1v′2 : v′1, v

′2 ∈ V ′π};

src′ and trg′ are such that:

q′1 : �→ u2;q′2 : �→ u1u2;q′3 : �→ ∅;¬∅ : ∅ → u1u2;¬u2 : u2 → ∅;¬u1u2 : u1u2 → ∅;⊃v′1v′2 : v′1 v

′2 → u1u2 whenever v′1 ⊆ v′2 for each v′1, v

′2 ∈ V ′;

⊃v′1v′2 : v′1 v′2 → v′2 whenever v′1 6⊆ v′2 for each v′1, v

′2 ∈ V ′;

∧v′1v′2 : v′1 v′2 → v′1 ∩ v′2 for each v′1, v

′2 ∈ V ′;

∨v′1v′2 : v′1 v′2 → v′1 ∪ v′2 for each v′1, v

′2 ∈ V ′.

• α : G′ → G is such that:

αv(b) = π for each b ∈ {∅, u2, u1u2};αv(�) = ♦;

αe(q′i) = qi for i = 1, 2, 3;

αe(¬v′) = ¬ for each v′ ∈ V ′;αe(⊃v′1v′2) = ⊃ for each v′1, v

′2 ∈ V ′;

αe(∧v′1v′2) = ∧ for each v′1, v′2 ∈ V ′;

αe(∨v′1v′2) = ∨ for each v′1, v′2 ∈ V ′.

• D = {u1u2}. ∇

As expected, a morphism between interpretation structures is a pair of mapsrelating their signatures and m-graphs. The m-graphs are related by m-graphtransformations. By a m-graph transformation τ : G2 → G1 we mean a pairdefined as follows:{

τ v : V2 → V1

{τ est : G1(τ v(s), τ v(t))→ G2(s, t)}s,t∈V +

2.

So, an interpretation structure morphism between (Σ1, I1) and (Σ2, I2), is a pair

(h, τ) : (Σ1, I1)→ (Σ2, I2)

where h : Σ1 → Σ2 is a signature morphism and τ : G′2 → G′1 is a m-graphtransformation such that:

• hv ◦ αv1 ◦ τ v = αv

2;

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• he(αe1(e′1)) = αe

2(τ es′2t

′2(e′1)) for e′1 ∈ E′1(τ v(s′2), τ v(t′2)) and s′2, t

′2 ∈ V ′

+

2 ;

• (τ v)−1(D1) ⊆ D2;

• τ v((V ′2)π \D2) ⊆ ((V ′1)π \D1).

As expected the m-graph transformation is contravariant with the signaturemorphism. Moreover, the truth value map τ v is contravariant with the sig-nature morphism. Finally, note that each operation in G′1 can be mapped toseveral operations in G′2, and note that the third condition in the definition ofinterpretation structure morphism is equivalent to saying that if v′1 = τ v(v′2)and v′1 ∈ D1 then v′2 ∈ D2.

We denote by Int the category of interpretation structures and their mor-phisms, and by IntV the subcategory of Int composed by all interpretationstructures with sort set V and morphisms (h, τ) : (Σ1, I1)→ (Σ2, I2) such thathv = idV . Moreover, given a sequence s of pairs, we denote by (s)j the sequenceof all j-components of the pairs in s, for j = 1, 2.

Proposition 3.3 Category IntV has binary coproducts.

Proof: Let (Σ1, I1) and (Σ2, I2) be interpretation structures with sort set V .Their coproduct is

(Σ1, I1) ] (Σ2, I2) = ((Σ, I), (i1, τ1), (i2, τ2))

where

• (Σ, i1, i2) is the coproduct in SigV of Σ1 and Σ2;

• (I, τ1, τ2) is defined as follows:

– I = ((V ′, E′, src′, trg′), α,D, �) where

∗ V ′ is such that:· V ′π = (D1 ×D2) ∪ ((V ′1π \D1)× (V ′2π \D2));· V ′v = V ′1v × V ′2v for each v ∈ V \ {π};

∗ E′(s′, t′) is, for each s′ and t′ in V ′+, the coproduct in Setof E′1((s′)1, (t′)1) and E′2((s′)2, (t′)2) with injections (τ1)es′t′ and(τ2)es′t′ , respectively;∗ src′((τj)es′t′(e

′j)) = s′ and trg′((τj)es′t′(e

′j)) = t′ for j = 1, 2;

∗ α is such that· αv((v′1, v

′2)) = αv

1(v′1);· αe((τj)es′t′(e

′j)) = iej(α

ej(e′j)) for j = 1, 2;

∗ D = D1 ×D2;∗ � = (�1, �2);

– τ vj ((v′1, v

′2)) = v′j and (τj)e = {(τj)es′t′}s′,t′∈V ′+ for j = 1, 2.

It is straightforward to show that τ1 and τ2 are m-graph transformations. Notethat i1 and i2 are signature morphisms by definition. Furthermore, the tuple(ij , τj) is a interpretation structure morphism for j = 1, 2 since

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• αv1(v′1) = αv((v′1, v

′2));

• iej(αej(e′j)) = αe((τj)es′t′(e

′j)) for e′j ∈ E′j((τj)v(s′), (τj)v(t′));

• (τ v1 )−1(D1) ⊆ D. Let v′ = (v′1, v

′2) ∈ (τ v

1 )−1(D1). Then τ v1 (v′) = v′1 ∈ D1

and so, by definition of D, v′ ∈ D. Similarly for (τ v2 )−1(D2) ⊆ D;

• τ v1 (V ′π \D) ⊆ V ′1π \D1. Let (v′1, v

′2) ∈ V ′π \D. By definition of D, we know

that v′1 /∈ D1. Similarly for τ v2 (V ′π \D) ⊆ V ′2π \D2.

It remains to show the universal property of the coproduct. Let (Σ3, I3) be ainterpretation structure in IntV and (g1, σ1) : (Σ1, I1)→ (Σ3, I3) and (g2, σ2) :(Σ2, I2) → (Σ3, I3) interpretation structure morphisms. Consider the tuple(g, σ) such that:

• g = (idV , ge) where ge is the unique morphism in Set such that ge1 = ge◦ ie1

and ge2 = ge ◦ ie2;

• σ = (σv, σe) where σv(v′3) = (σv1(v′3), σv

2(v′3)) and

σes′3t

′3((τj)eσv(s′3)σv(t′3)(e

′j)) = (σe

j)s′3t′3(e′j)

for every e′j ∈ E′j((τj)v(σv(s′3)), (τj)v(σv(t′3))).

1. (g, σ) is an interpretation structure morphism from (Σ, I) to (Σ3, I3).

(a) (σv1(v′3), σv

2(v′3)) ∈ V ′. For instance, assume that αv3(v′3) = π and v′3 ∈ (V ′3 \

D3). Since αv1 ◦ σv

1 = αv3, then σv

1(v′3) ∈ (V ′1)π. Moreover, σv1(v′3) ∈ (V ′1)π \D1,

since σv1((V ′3)π \D3) ⊆ (V ′1)π \D1.

(b) τ vj ◦ σv = σv

j for j = 1, 2. Indeed τ vj (σv(v′3)) = τ v

j ((σv1(v′3), σv

2(v′3))) = σvj (v′3).

(c) σe is a family of well defined maps. Let s′3, t′3 ∈ V ′3

+. Assume thate′ ∈ E′(σv(s′3), σv(t′3)). Then e′ = (τj)eσv(s′3),σv(t′3)(e

′j) for some j = 1, 2. We now

show that src′j(e′j) = σv

j (s′3) and trg′j(e

′j) = σv

j (t′3). Indeed, src′j(e

′j) = τ v

j (σv(s′3)).By (b), τ v

j (σv(s′3)) = σvj (s′3) as we wanted to show. Similarly for trg′j .

(d) gv ◦ αv ◦ σv = αv3. Indeed, gv(αv(σv(v′3))) = gv(αv((σv

1(v′3), σv2(v′3)))) by

definition of σv, which is equal to αv((σv1(v′3), σv

2(v′3))) since gv = idV . Sinceαv((σv

1(v′3), σv2(v′3))) = αv

j(σvj (v′3)) for j = 1, 2 and αv

j(σvj (v′3)) = αv

3(v′3) (due tothe fact that σj is a m-graph transformation), the thesis follows.

(e) ge(αe(e′)) = αe3(σe

s′3t′3(e′)) for e′ ∈ E′(σv(s′3), σv(t′3)) for s′3, t

′3 ∈ V ′3

+. As-sume that e′ = (τj)eσv(s′3)σv(t′3)(e

′j) with e′j ∈ E′j(τ

vj (σv(s′3)), τ v

j (σv(t′3))). Thenαe

3(σes′3t

′3(e′)) = αe

3((σej)σv

j(s′3)σv

j(t′3)(e′j)), by definition of σe, which is equal to

gej(α

ej(e′j)) since 〈gj , σj〉 is a m-graph transformation. On the other hand, by

definition of ge, gej(α

ej(e′j)) = ge(iej(α

ej(e′j))). Finally, the thesis follows, since

iej(αej(e′j)) = αe((τj)eσv(s′3)σv(t′3)(e

′j)).

(f) (σv)−1(D) ⊆ D3. Let v′3 ∈ (σv)−1(D). Then σv(v′3) = (σv1(v′3), σv

2(v′3)) ∈ D.Without loss of generality, assume that σv

1(v′3) ∈ D1. Hence v′3 ∈ (σv1)−1(D)

and so v′3 ∈ D3.

10

(g) σv(V ′3π \ D3) ⊆ V ′π \ D. Let v′3 ∈ V ′3π \ D3. Then σv1(v′3) ∈ V ′1π \ D1 and

σv2(v′3) ∈ V ′2π \D2 and so (σv

1(v′3), σv2(v′3)) = σv(v′3) ∈ V ′π \D.

2. (g, σ) ◦ (ij , τj) = (gj , σj) for j = 1, 2.

(a) g ◦ ij = gj . That is, gv ◦ ivj = gvj and ge ◦ iej = ge

j . The first equality followsdirectly since all the maps are identities on V . The second equality follows bythe universal property of the coproduct.

(b) σvj = τ v

j ◦ σv. This follows by 1(b) above.

(c) (σej)s′3t′3 = σe

s′3t′3◦ τ e

σv(s′3)σv(t′3). The thesis follows since E′j(σvj (s′3), σv

j (t′3)) is

E′j((τj)v(σv(s′3)), (τj)v(σv(t′3))) by (c) above.

3. Uniqueness of (g, σ). Assume that there is a m-graph transformation (g, σ)such that (g, σ) ◦ (ij , τj) = (gj , σj) for j = 1, 2.

(a) gv = gv since both are the identity on V .

(b) ge = ge. This follows from the universal property.

(c) σv = σv. Let v′3 ∈ V ′3 and σv(v′3) = (u′1, u′2). Then σv(v′3) = (σv

1(v′3), σv2(v′3))

using the definition of σv. So (σv1(v′3), σv

2(v′3)) = (τ v1 (σv(v′3)), τ v

2 (σv(v′3))) by com-mutativity. By definition of τ v

j , (τ v1 (σv(v′3)), τ v

2 (σv(v′3))) = (u′1, u′2) and hence

σv(v′3) = σv(v′3).

(d) σes′3t

′3

= σes′3t

′3. Note that σv = σv. Let e′ ∈ E′(σv(s′3), σv(t′3)). Then

e′ = (τj)eσv(s′3)σv(t′3)(e′j) with e′j ∈ E′j(τ

vj (σv(s′3)), τ v

j (σv(t′3))). Then σes′3t

′3(e′) =

(σej)σv

j(s′3)σv

j(t′3)(e′j) = σe

s′3t′3(e′). QED

In order to be able to define fibring for any pair of interpretation structures,and not only for structures with the same set of sorts, we define a Set♦ indexedmap ·(V1,π1,♦1) that enriches each interpretation structure with the sorts only inthe other structure, in such a way that satisfaction is maintained:

• (Σ2, I2)(V1,π1,♦1) = (Σ(V1,π1,♦1)2 , I

(V1,π1,♦1)2 ) where:

– ((V, π, ♦), i1, i2) is the coproduct in Set♦ of (V1, π1, ♦1) and (V2, π2, ♦2);

– I(V1,π1,♦1)2 = ((V ′, E′2, src

′, trg′), (αv, αe2), D2, �2) is such that:

∗ V ′ = V ′2 ∪ {v1′ : v1 ∈ V1 \ V2};

∗ αv : V ′ → V is such that· αv(v′) = i2(αv

2(v′)) for v′ in V ′2 ;· αv(v1

′) = i1(v1) otherwise;∗ src′ : E′2 → V ′+ and trg′ : E′2 → V ′ coincide with src′2 and trg′2;

so, the fibring FibInt of interpretation structures is defined as a map from |Int|2to |Int| such that:

FibInt((Σ1, I1), (Σ2, I2)) = (Σ1, I1)(V2,π2,♦2) ] (Σ2, I2)(V1,π1,♦1).

11

(ΣΠ, Ic) (Σ∧,∨Π , Ii)FibInt((ΣΠ, Ic), (Σ∧,∨Π , Ii))

〈ic,τc〉 +3 〈ii,τi〉ks

'&%$ !"#1 0

¬0uu

¬1

55

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αc

OO

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OO

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¬

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55

¬1u1u20∅

ss

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OO(�,�)

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pp

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ll

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Figure 2: Part of the interpretation structures and signatures for classical logic,described in Example 3.1, for intuitionistic logic, described in Example 3.2, andfor their fibring, see Example 3.4, containing the m-edges for negation and fora propositional symbol.

For the sake of illustration we now describe the interpretation structureresulting from the fibring of the interpretation structure for classical logic in-troduced in Example 3.1 with the interpretation structure for intuitionistic logicintroduced in Example 3.2.

Example 3.4 Interpretation structure resulting from the fibring of classicaland intuitionistic interpretation structures.Consider the interpretation structures for classical logic introduced in Exam-ple 3.1 and for intuitionistic logic introduced in Example 3.2. Then

FibInt((ΣΠ, Ic), (Σ∧,∨Π , Ii)) = (ΣΠ, Ic) ] (Σ∧,∨Π , Ii)

since

• (ΣΠ, Ic)(Vi,πi,♦i) = (ΣΠ, Ic);

12

• (Σ∧,∨Π , Ii)(Vc,πc,♦c) = (Σ∧,∨Π , Ii);

where (Σ, I) = (ΣΠ, Ic) ] (Σ∧,∨Π , Ii) is defined, according to Proposition 3.3, asfollows:

• Σ = ((V,E, src, trg), π, ♦) where:

– V = {π, ♦};– E(π, π) = {¬c,¬i};– E(ππ, π) = {⊃c,⊃i,∧i,∨i};– E(♦, π) = {q1c, q2c, q3c, q1i, q2i, q3i};– all the other components are empty;

• I = ((V ′, E′, src′, trg′), α,D, (�, �)) where:

– V ′ = {(0, ∅), (0, u2), (1, u1u2)};– E′((0, u2), (1, u1u2)) = {¬0u21(u1u2)

};– E′((0, u2), (0, ∅)) = {¬u20∅0};– E′((1, u1u2), (0, u2)) = {¬1(u1u2)0u2

};– E′((0, ∅), (1, u1u2)) = {¬∅0(u1u2)1

,¬0∅1(u1u2)};

– E′((1, u1u2), (0, ∅)) = {¬(u1u2)1∅0 ,¬1(u1u2)0∅};– E′((�, �), (0, ∅)) = {q10∅

};– E′((�, �), (0, u2)) = {q10u2

, q1u20};

– αv((0, ∅)) = αv((0, u2)) = αv((1, u1u2)) = π;

– αe(q10∅) = αe(q10u2

) = q1c;

– αe(q1(u1u2)1) = q1i;

– αe(¬0∅1(u1u2))=αe(¬0u21(u1u2)

)=αe(¬1(u1u2)0∅)=αe(¬1(u1u2)0u2)=¬c;

– αe(¬∅0(u1u2)1) = αe(¬u20(u1u2)1

) = αe(¬(u1u2)1∅0) = ¬i;– D is {(1, u1u2)}.– similarly for the implication, disjunction, conjunction and the re-

maining propositional symbols;

A graphical description of the interpretation structure (Σ, I) can be seen inFigure 2. For the sake of simplicity, only negation connectives are consideredas well as one propositional symbol for each component. ∇

It is worthwhile to point out that enrichment does not change satisfaction.

Proposition 3.5 The interpretation structure (Σ2, I2) and (Σ2, I2)(V1,π1,♦1) sat-isfy the same formulas.

13

We omit the proof of the proposition above since it follows straightforwardly.We now recall the concepts of assignment, denotation and satisfaction. An

assignment ρ for an interpretation structure I over a signature Σ is a family{ρs}s∈V + such that ρs contained in V ′+s is a concatenation of basic sets andρ♦ = {�}. The denotation of a path w : s→ t over G† at I and ρ, denoted by

[[w]]Iρ

is a concatenation of basic sets contained in V ′+t , inductively defined on thecomplexity of the path w as follows:

• [[εs]]Iρ is ρs;

• [[pv1...vmi w1]]Iρ is ([[w1]]Iρ)i where v1, . . . , vm are in V ;

• [[〈w1, . . . , wn〉w0]]Iρ is [[w1w0]]Iρ . . . [[wnw0]]Iρ;

• [[ew1]]Iρ is trg′(E′e([[w1]]Iρ,−)) for e in E.

For instance, for evaluating ew1 over I and ρ, we start by evaluating w1 gettinga set of values. For each value s′ we pick up all the m-edges in G′ with sources′ and which are mapped into e. Finally, the envisaged denotation is obtainedby taking the collection of targets of such edges. The denotation [[w]]Iρ of amorphism w in G+ over I and ρ is defined as

[[w]]Iρ = [[w]]Iρ.

Note that the denotation of a concrete formula does not depend on the assign-ment ρ. Therefore, if ϕ is a concrete formula we write [[ϕ]]I to refer to thedenotation of ϕ in I.

Example 3.6 Consider the interpretation structure in Example 3.4 for thelogic resulting from the fibring of classical propositional logic and intuitionisticpropositional logic. Then

• [[((¬i(¬i q1i))⊃i q1i)]]I = {(0, u2), (1, u1u2)} since

[[ε♦]]I = {�}

[[q1i]]I = trg′(E′q1i([[ε♦]]I ,−))

= trg′(E′q1i({�},−))= trg′({q1u20

: �→ (0, u2)})= {(0, u2)}

[[¬i q1i]]I = trg′(E′¬i([[q1i]]

I ,−))= trg′(E′¬i((0, u2),−))= trg′({¬u20∅0 : (0, u2)→ (0, ∅),¬0u21(u1u2)

: (0, u2)→ (1, u1u2)}= {(0, ∅), (1, u1u2)}

[[¬i ¬i q1i]]I = {(0, ∅), (1, u1u2)}

14

[[((¬i(¬i q1i))⊃i q1i)]]I = {(0, u2), (1, u1u2)}.

∇

We recall now the notion of satisfaction. A schema formula ϕ is said to besatisfied by I and ρ, written as

I, ρ ϕ

whenever [[ϕ]]Iρ is non-empty and is contained in D. Moreover, we say I satisfiesϕ, written as

I ϕ

whenever I, ρ ϕ for every assignment ρ over I. Satisfaction is extended tosets of schema formulas as expected: I, ρ Γ if I, ρ γ for each γ ∈ Γ, andsimilarly for sequences of schema formulas: I, ρ ϕ1 . . . ϕn if I, ρ ϕi fori = 1, . . . , n.

Example 3.7 Consider Example 3.6 for denotation of ((¬i(¬i q1i))⊃iq1i) in theinterpretation structure resulting from the fibring of interpretation structuresfor classical and intuitionistic logics. Then

I 6 ((¬i (¬iq1i))⊃i q1i)

since [[((¬i(¬i q1i)) ⊃i q1i)]]I = {(0, u2), (1, u1u2)} is not contained in D. This

example shows that the intuitionistic implication does not collapse into theclassical implication. Therefore, the graph-theoretic fibring of interpretationstructures retains the intuitionistic character of the component. ∇

Thus, the example above shows that the fibring of classical propositionallogic with intuitionistic propositional logic does not collapse into classical logic.Otherwise the interpretation structure above would satisfy ((¬i(¬i q1)) ⊃i q1)showing that intuitionistic connectives behave like classical connectives. Similarexamples of non collapse can be given even for fibring involving paraconsistentlogics (with a non-deterministic semantics).

4 Fibring deductive systems

In this section we define fibring of deductive systems. As we will see, thedeductive system resulting from the fibring is the expected one, in the sensethat it includes all the axioms and inference rules of the components.

Recall, from [19], that by a meta-signature we mean a tuple

Φ = (Σ,>,R)

where Σ = (G, π, ♦) is a language signature such that

GΦ = (V Φ, EΦ, srcΦ, trgΦ)

is a m-graph extending G with

15

• V Φ = V ;

• EΦ = E ∪ R where R = {Rn :n︷ ︸︸ ︷

π . . . π → π}n>0;

and > is a set {>s : s→ π}s∈V + .Each Rn is a symbolic expression for representing inference rules with n

premises. The edge >s is called s-verum and is important to represent, in oursetting, axioms. An axiom is the target of a unary rule whose antecedent isa verum schema formula. Meta-signatures can be related by morphisms. Wedenote by G> the m-graph obtained by enriching G with the m-edges >s : s→π.

A morphism between meta-signatures h from Φ1 = (Σ1,>1,R1) to Φ2 =(Σ2,>2,R2) is a m-graph morphism h : GΦ1

> → GΦ2> such that its restriction hG1

to G1 is a signature morphism from Σ1 to Σ2, he(>s) = >hv(s) and he((R1)n) =(R2)n for each n > 0. As seen for interpretation structures we assume that, forfibring, the signatures have the same set V of sorts. We denote by MSigV thecategory of meta-signatures with the same set V os sorts and their morphisms.We now show that meta-signatures with the same set of sorts can be composed.

Proposition 4.1 Category MSigV has binary coproducts.

Proof: Let (Σ1,>1,R1) and (Σ2,>2,R2) be meta-signatures denoted by Φ1 andΦ2 respectively, with the same set V of sorts. Their coproduct is the triple

Φ1 ] Φ2 = ((Σ1 ] Σ2,>,R), i1, i2)

where (Σ1 ] Σ2, i1G1, i2G2

) is a coproduct in SigV of Σ1 and Σ2. It is straight-forward to check that the triple is indeed a coproduct. QED

We say that a morphism w of G+

> is in G+ whenever there is a path u overG† and u = w. We may denote a schema formula of G+

> not in G+ as a verumschema formula. Given morphisms w1 : s → s1 and w2 : s1 → s2 of G+

> inG+ it is straightforward to see that w2 ◦ w1 is also in G+. Moreover given themorphism >s : s→ π of G+

> it is straightforward to see that for any u : s→ s1

in G+

> the morphism >s ◦ u is also not in G+.By a deductive system over a meta-signature Φ we mean a basis (G′′, β) over

GΦ where G′′ is such that

• V ′′ is the class of morphisms of G+

> whose target is in V ;

• E′′(w1 : s→ v1 . . . wn : s→ vn, w : s→ v), for w in G+, contains, amongothers, the m-edges e : v1 . . . vn → v of E such that w = e ◦ 〈w1, . . . , wn〉in G+;

• E′′(w1 : s1 → v1 . . . wn : sn → vn, w : s→ v) = ∅ whenever w is not in G+

or si 6= s for some i = 1, . . . , n, or wi is not in G+ and n 6= 1;

and β is such that

• βv(w : s→ v) = v;

16

• βe(e : (w1 : s → v1 . . . wn : s → vn) → (w : s → v)) = e if e is in E andw = e ◦ 〈w1, . . . , wn〉;

• βe(f ′) ∈ R otherwise.

The first clause on E′′ imposes the inclusion of the constructor morphismsthat make commutative the obvious diagrams, as it is usually done in categoricallogic. As stated in the last clause of βe all the other m-edges correspond toinference rules. All the m-edges corresponding to inference rules must have aspremises and conclusion, expressions with the same source, and with target π.The same source condition is imposed by the second and the third clause of thedefinition of E′′ and is crucial for defining instantiation as we will see below.They must have target π by definition of β and of R.

By a deductive system D we mean a triple

(Φ, G′′, β)

such that Φ is a meta-signature and (G′′, β) is a deductive system over Φ.

Example 4.2 Deductive system for classical propositional logic.Consider the well known Hilbert axiomatization of classical propositional logicwith three axiom schemas and Modus Ponens. This axiomatization can berepresented as the deductive system Dc = (ΦΠ, G

′′, β) such that:

• ΦΠ is the meta-signature (ΣΠ,>,R) where ΣΠ is the propositional signa-ture (G, π, ♦) introduced in Example 2.1;

• G′′ has, besides the mandatory m-edges for connectives, the following onesfor rules:

– m-edge ax1 : >ππ → ax1 such that ax1 is (ξ ⊃ (ξ′ ⊃ ξ)) : ππ → πwhere ξ is pππ1 and ξ′ is pππ2 ;

– m-edge ax2 : >πππ → ax2 such that ax2 is ((ξ ⊃ (ξ′ ⊃ ξ′′)) ⊃ ((ξ ⊃ξ′)⊃ (ξ ⊃ ξ′′))) : πππ → π where ξ is pπππ1 , ξ′ is pπππ2 and ξ′′ is pπππ3 ;

– m-edge ax3 : >ππ → ax3 such that ax3 is (((¬ ξ)⊃ (¬ ξ′))⊃ (ξ′⊃ ξ)) :ππ → π where ξ is pππ1 and ξ′ is pππ2 ;

– m-edge MP : pππ1 ⊃ → pππ2 ;

• β : G′′ → GΦΠ is such that:

– βe(axi) = R1 for i = 1, 2, 3;

– βe(MP) = R2. ∇

Example 4.3 Deductive system for intuitionistic propositional logic.Consider the well known Hilbert axiomatization of intuitionistic propositionallogic with axiom schemas and Modus Ponens. This axiomatization can berepresented as the deductive system Di = (ΦΠ, G

′′, β) such that:

• ΦΠ is the meta-signature (Σ∧,∨Π ,>,R) where Σ∧,∨Π is the intuitionisticpropositional signature (G, π, ♦) introduced in Example 2.2;

17

• G′′ has, besides the mandatory m-edges for connectives, the followingones:

– m-edge ax1 : >ππ → ax1 such that ax1 is (ξ ⊃ (ξ′ ⊃ ξ)) : ππ → πwhere ξ is pππ1 and ξ′ is pππ2 ;

– m-edge ax2 : >πππ → ax2 such that ax2 is ((ξ ⊃ (ξ′ ⊃ ξ′′)) ⊃ ((ξ ⊃ξ′)⊃ (ξ ⊃ ξ′′))) : πππ → π where ξ is pπππ1 , ξ′ is pπππ2 and ξ′′ is pπππ3 ;

– m-edge ax3 : >ππ → ax3 such that ax3 is (ξ⊃ (ξ′⊃ (ξ ∧ ξ′)) : ππ → πwhere ξ is pππ1 and ξ′ is pππ2 ;

– m-edge ax4 : >ππ → ax4 such that ax4 is ((ξ ∧ ξ′)⊃ ξ) : ππ → π and((ξ ∧ ξ′)⊃ ξ′) : ππ → π where ξ is pππ1 and ξ′ is pππ2 ;

– m-edge ax5 : >ππ → ax5 such that ax5 is (ξ ⊃ (ξ ∨ ξ′)) : ππ → π and(ξ′ ⊃ (ξ ∨ ξ′)) : ππ → π where ξ is pππ1 and ξ′ is pππ2 ;

– m-edge ax6 : >πππ → ax6 such that ax6 is ((ξ ⊃ ξ′′) ⊃ ((ξ′ ⊃ ξ′′) ⊃((ξ ∨ ξ′)⊃ ξ′′))) : πππ → π where ξ is pπππ1 , ξ′ is pπππ2 and ξ′′ is pπππ3 ;

– m-edge ax7 : >ππ → ax7 such that ax7 is ((ξ ⊃ ξ′) ⊃ ((ξ ⊃ (¬ ξ′)) ⊃(¬ ξ))) : ππ → π where ξ is pππ1 and ξ′ is pππ2 ;

– m-edge ax8 : >ππ → ax8 such that ax8 is (ξ ⊃ ((¬ ξ)⊃ ξ′)) : ππ → πwhere ξ is pππ1 and ξ′ is pππ2 ;

– m-edge MP : pππ1 ⊃ → pππ2 ;

• β : G′′ → GΦΠ is such that:

– βe(axi) = R1 for i = 1, . . . , 8;

– βe(MP) = R2. ∇

A deductive system morphism from the deductive system (Φ1, G′′1, β1) to

(Φ2, G′′2, β2) is a pair (h, g) where h is a meta-signature morphism from Φ1 to

Φ2 and g is a m-graph morphism from G′′1 to G′′2 such that:

• gv(w1) is (he)+(w1) for each vertex w1 of V ′′1 ;

• hv ◦ βv1 = βv

2 ◦ gv;

• he ◦ βe1 = βe

2 ◦ ge.

Let Ded be the category of deductive systems and their morphisms andDedV the full subcategory of Ded composed by all deductive systems with thesame set V of sorts.

Proposition 4.4 Category DedV has coproducts.

Proof: Let D1 = (Φ1, G′′1, β1) and D2 = (Φ2, G

′′2, β2) be deductive systems in

DedV . Their coproduct is

D1 ] D2 = (((Σ,>,R), G′′, β), (i1, j1), (i2, j2))

such that

18

• ((Σ,>,R), i1, i2) is a coproduct in MSigV of Φ1 and Φ2. So Σ = (G, π, ♦)is Σ1 ] Σ2 and (Σ, i1G1> , i2G2>) is a coproduct in SigV of Σ1 and Σ2;

• the restriction of j1e and j2e to (β1

e)−1(R) and (β2e)−1(R) respectively,

are injections from those sets to their coproduct in Set;

• G′′ = (V ′′, E′′, src′′, trg′′) is such that,

– V ′′ is the class of morphism of G+

> whose target is in V

– E′′(w1 : s1 → v1 . . . wn : sn → vn, w0 : s0 → v0) contains:

∗ e : v1 . . . vn → v0 in E, whenever w0 = e ◦ 〈w1, . . . , wn〉 in G+;∗ j1e(f1), if f1 ∈ (β1

e)−1(R1)(u1 . . . un, u0) and (i1G1>)+(ui) is wifor i = 0, . . . , n; similarly for f2 ∈ (β2

e)−1(R2);

• βv(w : s→ v) = v;

• βe(e : (w1 : s → v1 . . . wn : s → vn) → (w : s → v)) = e if e is in E andw = e ◦ 〈w1, . . . , wn〉;

• βe(j1e(r1)) = Rn if r1 ∈ (R1)n and βe(j2(r2)) = Rn if r2 ∈ (R2)n;

• jv1(w1) = (ie1)+(w1) for each vertex w1 in V ′′1 , and similarly for j2;

• j1e(e1) = i1e(e1) if e1 is in E1 and similarly for j2.

It is straightforward to show that (i1, j1) and (i2, j2) are deductive system mor-phisms.

Let D3 be a deductive system in DedV and (h1, g1) : D1 → D3 and (h2, g2) :D2 → D3 deductive system morphisms. Consider the pair (h, g) such that:

• h = (idV , he) is the unique morphism in MSig such that h1 = h ◦ i1 andh2 = h ◦ i2;

• gv(w) = (he)+(w) for each vertex w of V ′′;

• ge(i1e(e1)) = g1e(e1) and ge(i2e(e2)) = g2

e(e2);

• ge(j1e(f1)) = g1e(f1) when f1 ∈ (β1

e)−1(R1);

• ge(j2e(f2)) = g2e(f2) for f2 ∈ (β2

e)−1(R2).

(1) h is a meta-signature morphism from Φ to Φ3 by definition.

(2) g is a m-graph morphism from G′′ to G′′3 since:

(a) gv(w) = (he)+(w) for each object w of G′′.

(b) hv ◦ βv = βv3 ◦ gv. The thesis follows since hv(βv(w : s → v)) = hv(v) and

βv3(gv(w)) = βv

3((he)+(w) : (hv)+(s)→ hv(v)) = hv(v).

(c) he ◦ βe = βe3 ◦ ge. There are two cases to consider: (i) he(βe(i1e(e1))) =

he(i1e(e1)) and βe3(ge(i1e(e1))) = βe

3(g1e(e1)) = h1

e(βe1(e1)) = h1

e(e1) and sothe result follows by definition of h. Similarly for e2 in E2. (ii) Let r1 ∈

19

(β1e)−1(R1n). Then he(βe(j1e(r1))) = he(Rn) = (R3)n and βe

3(ge(j1e(r1))) =βe

3(g1e(r1)) = he

1(βe1(r1)) = he

1(R1n) = (R3)n.

(3) (h, g)◦(i1, j1) = (h1, g1). Commutativity for the first argument holds by def-inition. On the other hand, gv(j1v(w1)) = (he)+(jv1(w1)) = (he)+((ie1)+(w1)) =(he

1)+(w1) = gv1(w1). Moreover, (i) given e1 in E1, ge(je1(e1)) = ge(ie1(e1)) =

ge1(e1); and (ii) given f1 ∈ (β1

e)−1(R1), ge(je1(f1)) = g1e(f1) by definition.

(4) (h, g) ◦ (i2, j2) = (h2, g2). Similar to case (3).

(5) Uniqueness of (h, g). Let (h′, g′) be a deductive system morphism fromD1∪D2 to D3 such that (h′, g′)◦(i1, j1) = (h1, g1) and (h′, g′)◦(i2, j2) = 〈h2, g1〉.Then h′v = idV . Since h′e◦ i1 = he then h′e = he. Moreover g′v(w) = gv(w) sinceg′v(w) = (h′e)+(w) = (he)+(w) = gv(w) by definition of deductive system mor-phism. Let e1 be in E1. Then g′e(je1(e1)) = ge

1(e1) = ge(i1e(e1)) = ge(j1e(e1)).Let f1 ∈ (β1

e)−1(R1). Then g′e(j1e(f1)) = g1e(f1) = ge(j1e(f1)). QED

As we did for signatures and interpretation structures, we only consider thecoproduct of deductive systems over signatures with the same set of sorts. So,in order to define the unconstrained fibring of deductive systems over signatureswith different sets of sorts, we have to enrich each deductive system with thesorts of the other system. For that we define a Set♦ indexed map ·(V1,π1,♦1) ondeductive systems as follows:

• ((Σ2,>2,R2), G′′2, β2)(V1,π1,♦1) = ((Σ2(V1,π1,♦1),>2,R2), G′′, (βv, βe

2)) with:

– Σ(V1,π1,♦1)2 = (G, π, ♦);

– G′′ = (V ′′, E′′2 , src′′, trg′′) is such that

∗ V ′′ is the class of morphisms of G+

> whose target is in V ;∗ src′′ : E′′2 → V ′′+, trg′′ : E′′2 → V ′′ coincide with src′′2 and trg′′2,

respectively;∗ βv : V ′′ → V is such that βv(w : s→ v) = v.

The fibring FibDed of deductive systems is a map from |Ded|2 to |Ded|such that:

FibDed(D1,D2) = D1(V2,π2,♦2) ] D2

(V1,π1,♦1).

We stress that although presented in a different way, we get the expecteddeductive system for fibring. Observe that in derivations the application of therules from the components is interleaved.

Example 4.5 Deductive system resulting from the fibring of the classical andintuitionistic propositional deductive systems.Consider the deductive system for classical propositional logic described in Ex-ample 4.2 and the deductive system for intuitionistic propositional logic intro-duced in Example 4.3. The deductive system (Φ, G′′, β) corresponding to theirfibring is as follows:

• Φ is the coproduct in MSigV of the meta-signatures for both components;

20

Dc DiFibDed(Dc,Di)〈i1,j1〉 +3 〈i2,j2〉ks

β1

OO

β2

OO

π

¬,R1

NN♦q1 // oo R2 π

♦q1

eeLLLL

¬,R1

BB oo R2

π oo R2♦��

q1i

q1c//

¬c,R1,¬i

\\

i119

i2

fn

β

OO

j1

GO

j2

OWππ

π>ππ���

� ππ

π

ax1����ax1 //

...ππ

π>ππ���

� ππ

π

ax3����ax3 //

ππ

π

pππ1 ���� ππ

π

⊃���� ππ

π

pππ2����

MP//

s

π

s

π

ϕ����

¬ ◦ ϕ����

¬//

. . .

ππ

π>ππ���

� ππ

π

ax1����ax1 //

...ππ

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� ππ

π

ax8����

ax8 //

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pππ1 ���� ππ

π

⊃���� ππ

π

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s

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s

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π

axc1����ax1 // . . .

ππ

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� ππ

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π

pππ1 ���� ππ

π

⊃c ���� ππ

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pππ1 ���� ππ

π

⊃i ���� ππ

π

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ϕc����

¬c ◦ ϕc����

¬c//

s

π

s

πϕi ���

�¬i ◦ ϕi���

�¬i

//. . .

Figure 3: Part of the deductive systems and meta-signatures for classical logic,described in Example 4.2, for intuitionistic logic, described in Example 4.3, andfor their coproduct, see Example 4.5.

• G′′ has, besides the mandatory m-edges for connectives, the following onesfor rules:

– m-edge ax1 : >ππ → axc1 such that axc1 is (ξ ⊃c (ξ′ ⊃c ξ)) : ππ → πwhere ξ is pππ1 and ξ′ is pππ2 ;

– m-edge ax2 : >πππ → axc2 such that axc2 is ((ξ⊃c (ξ′⊃c ξ′′))⊃c ((ξ⊃cξ′)⊃c (ξ⊃c ξ′′))) : πππ → π where ξ is pπππ1 , ξ′ is pπππ2 and ξ′′ is pπππ3 ;

– m-edge ax3 : >ππ → axc3 such that axc3 is (((¬c ξ)⊃c (¬c ξ′))⊃c (ξ′⊃cξ)) : ππ → π where ξ is pππ1 and ξ′ is pππ2 ;

– m-edge ax4 : >ππ → axi4 such that axi4 is (ξ ⊃i (ξ′ ⊃i ξ)) : ππ → πwhere ξ is pππ1 and ξ′ is pππ2 ;

– m-edge ax5 : >πππ → axi5 such that axi5 is ((ξ ⊃i (ξ′ ⊃i ξ′′))⊃i ((ξ ⊃iξ′)⊃i (ξ⊃i ξ′′))) : πππ → π where ξ is pπππ1 , ξ′ is pπππ2 and ξ′′ is pπππ3 ;

– m-edge ax6 : >ππ → axi6 such that axi6 is (ξ⊃i(ξ′⊃i(ξ∧iξ′)) : ππ → πwhere ξ is pππ1 and ξ′ is pππ2 ;

21

– m-edge ax7 : >ππ → axi7 such that axi7 is ((ξ ∧i ξ′) ⊃i ξ) : ππ → πand ((ξ ∧i ξ′)⊃i ξ′) : ππ → π where ξ is pππ1 and ξ′ is pππ2 ;

– m-edge ax8 : >ππ → axi8 such that axi8 is (ξ ⊃i (ξ ∨i ξ′)) : ππ → πand (ξ′ ⊃i (ξ ∨i ξ′)) : ππ → π where ξ is pππ1 and ξ′ is pππ2 ;

– m-edge ax9 : >πππ → axi9 such that axi9 is ((ξ ⊃i ξ′′)⊃i ((ξ′ ⊃i ξ′′)⊃i((ξ ∨i ξ′) ⊃i ξ′′))) : πππ → π where ξ is pπππ1 , ξ′ is pπππ2 and ξ′′ ispπππ3 ;

– m-edge ax10 : >ππ → axi10 such that axi10 is ((ξ ⊃i ξ′) ⊃i ((ξ ⊃i(¬i ξ′))⊃i (¬i ξ))) : ππ → π where ξ is pππ1 and ξ′ is pππ2 ;

– m-edge ax11 : >ππ → axi11 such that axi11 is (ξ ⊃i ((¬i ξ) ⊃i ξ′)) :ππ → π where ξ is pππ1 and ξ′ is pππ2 ;

– m-edge MPc : pππ1 ⊃c → pππ2 ;

– m-edge MPi : pππ1 ⊃i → pππ2 ;

• β : G′′ → GΦ is such that:

– βe(axi) = R1 for i = 1, . . . , 11;

– βe(MPc) = βe(MPi) = R2.

A graphical representation of part of this deductive system can be seen inFigure 3. ∇

5 Fibring of logic systems

A logic system has three components: the signature, the interpretation systemand the deduction system. All of them are defined in terms of m-graphs. Thelanguage and derivation are described in the induced categories. A logic systemis a triple

L = (Σ, I,D)

such that:

• I = (Σ, I) is an interpretation system;

• D = (Φ, G′′, β) is a deductive system;

where Φ is a meta-signature over Σ. We now recall the notion of semanticentailment. Given an interpretation system I = (Σ, I) and a set Γ ∪ {ϕ} ofschema formulas over Σ, we say that Γ entails ϕ in I, written as

Γ �I ϕ,

whenever I Γ implies I ϕ for every I in I. Similarly, we define entailmentover sequences of schema formulas as follows: ~γ �I ~ϕ whenever I ~γ impliesI ~ϕ for every I in I. The logic system L is said to be sound

if Γ �I ϕ whenever Γ `D ϕ

22

for a set Γ ∪ {ϕ} of formulas and is said to be strong complete if the converseholds. The logic system L is said to be weak complete

if `I ϕ whenever �D ϕ.

Observe that a strong complete logic system is also weak complete but not theother way around in general. Finally, an interpretation structure I in I is saidto be sound for a rule r in D,

if I, ρ CONC(r) whenever I, ρ proper(ANT(r))

for every assignment ρ over I, where the map proper(·) when applied to a se-quence ~ϕ of schema formulas in G+

> returns the subsequence of schema formulasthat are in G+. When I is sound for all the rules in D we say that I is soundfor D. These schema formulas are called proper. The logic system L is said tobe sound for a deductive rule r in D, if all its interpretation structures over itssignature are sound for r.

Example 5.1 Logic system for classical propositional logic.The tuple (ΣΠ, Ic, Dc) where

• ΣΠ is the signature for classical logic introduced in Example 2.1;

• Ic is the class of all interpretation structures over ΣΠ sound for Dc;

• Dc is the deductive system for classical logic introduced in Example 4.2;

is a logic system, denoted by Lc, for classical propositional logic. Note thatthe interpretation structure Ic for classical logic introduced in Example 3.1 issound for Dc as it is straightforward to show. So Ic is in Ic. ∇

Example 5.2 Logic system for intuitionistic propositional logic.The tuple (Σ∧,∨Π , Ii,Di) where

• Σ∧,∨Π is the signature intuitionistic logic introduced in Example 2.2;

• Ii is the class of all interpretation structures over Σ∧,∨Π sound for Di;

• Di is the intuitionistic deductive system for introduced in Example 4.3

is a logic system, denoted by Li, for intuitionistic propositional logic. Note thatthe interpretation structure Ii for intuitionistic logic introduced in Example 3.2is sound for Di as it is straightforward to show. So Ii is in Ii. ∇

We denote by Log the class of all logic systems. The fibring Fib of logicsystems is a map from Log2 to Log such that:

Fib(L1,L2) = (Σ, I,D)

where

• Σ = FibSig(Σ1,Σ2);

23

• I is composed by the interpretation structures I over Σ such that the pair(Σ, I) = FibInt((Σ1, I1), (Σ2, I2)) for each I1 ∈ I1 and I2 ∈ I2;

• D = FibDed(D1,D2).

We now explain why there is no collapse, semantically, in the logic systemresulting from the graph-theoretic fibring of the logic systems for classical andintuitionistic logics presented above.

Example 5.3 Logic system resulting from the fibring of the logic systems forclassical and intuitionistic propositional logic.Consider the logic system Lc for classical propositional logic, described in Ex-ample 5.1, and the logic system Li for intuitionistic propositional logic, de-scribed in Example 5.2. Their fibring is the logic system Lc+i = Fib(Lc,Li) =(Σc+i, Ic+i,Dc+i) where

• Σc+i = FibSig(ΣΠ,Σ∧,∨Π );

• Ic+i is composed by the interpretation structures I over Σc+i such that(Σc+i, I) = FibInt((ΣΠ, I1), (Σ∧,∨Π , I2)) for each I1 ∈ Ic and I2 ∈ Ii;

• Dc+i = FibDed(Dc,Di);

and so the interpretation structure described in Example 2 resulting from thefibring of interpretation structures for classical and intuitionistic logic is also inIc+i. Hence, taking into account Example 3.7, 6�Ic+i (¬i(¬i q1i))⊃i q1i. ∇

6 Soundness preservation

We start by investigating a property of the interpretation structure morphismthat is important in the proof of the preservation of soundness. In particular,the injection morphisms from the components to the fibring enjoy this property.

An interpretation structure morphism (h, τ) : (Σ1, I1) → (Σ2, I2) is said tobe non-creative whenever for each e1 in E1 and e′2 : s′2 → t′2 in E′2h(e1) there ise′1 in E′1e1 such that τ e

s′2t′2(e′1) = e′2. Moreover, given an interpretation structure

morphism (h, τ) : (Σ1, I1) → (Σ2, I2) and an assignment ρ2 over I2, we denoteby

ρ2(h,τ)

the assignment over I1 such that (ρ2(h,τ))s1 = τ v((ρ2)hv(s1)).

The next result relates denotations in different interpretation structuresrelated by a non-creative interpretation structure morphism.

Lemma 6.1 Given a non-creative interpretation structure morphism (h, τ) :(Σ1, I1)→ (Σ2, I2), a morphism w1 of G1

+, and an assignment ρ2 over I2,

[[w1]]I1ρ2(h,τ)

= τ v([[h+(w1)]]I2ρ2).

24

Proof: We show equivalently that [[w1]]I1ρ2(h,τ)

= τ v([[h†(w1)]]I2ρ2) by inductionon the complexity of w1:

- w1 is εs1 . Then [[w1]]I1ρ2(h,τ)

= (ρ2(h,τ))s1 = τ v((ρ2)hv(s1)) = τ v([[εhv(s1)]]

I2ρ2) =τ v([[h†(w1)]]I2ρ2).

- w1 is pv1...vni w10. Then [[w1]]I1ρ2

(h,τ)

= [[pv1...vni w10]]I1ρ2

(h,τ)

= ([[w10]]I1ρ2(h,τ)

)i =(τ v([[h†(w10)]]I2ρ2))i = τ v(([[h†(w10)]]I2ρ2)i) = τ v([[ph

v(v1)...hv(vn)i h†(w10)]]I2ρ2) =

τ v( [[h†(pv1...vni w10)]]I2ρ2);

- w1 is 〈w11, . . . , w1n〉w10. Then [[w1]]I1ρ2(h,τ)

= [[〈w11, . . . , w1n〉w10]]I1ρ2(h,τ)

=[[w11w10]]I1ρ2

(h,τ)

. . . [[w1nw10]]I1ρ2(h,τ)

which, by induction hypothesis, is equal toτ v([[h†(w11w10)]]I2ρ2) . . . τ v([[h†(w1nw10)]]I2ρ2) equal to the set τ v([[h†(w11w10]]I2ρ2

. . . [[h†(w1nw10)]]I2ρ2) = τ v([[〈h†(w11), . . . , h†(w1n)〉h†(w10)]]I2ρ2) as we wanted toshow;

- w1 is ew10. So [[w1]]I1ρ2(h,τ)

= [[ew10]]I1ρ2(h,τ)

= trg′1(E′1e([[w10]]I1ρ2(h,τ)

,−)) =trg′1(E′1e(τ

v([[h†(w10)]]I2ρ2),−)) = τ v(trg′2(E′2h(e)([[h†(w10)]]I2ρ2 ,−))) since (h, τ)

is non-creative, which is τ v([[h†(ew10)]]I2ρ2). QED

The relationship established for denotations can be extended to satisfactionin a similar way.

Proposition 6.2 Given a non-creative interpretation structure morphism (h, τ) :(Σ1, I1)→ (Σ2, I2), a morphism ϕ1 in G1

+, and an assignment ρ2 over I2,

I2, ρ2 h+(ϕ1) whenever I1, ρ2(h,τ) ϕ1.

Proof: Assume that I1, ρ2(h,τ) ϕ1. Henceforth [[ϕ1]]I1ρ2

(h,τ)

⊆ D1. Then(τ v)−1([[ϕ1]]I1ρ2

(h,τ)

) ⊆ D2 since (h, τ) is an interpretation structure morphismand so (τ v)−1(D1) ⊆ D2. So, by Lemma 6.1, (τ v)−1(τ v([[h+(ϕ1)]]I2ρ2)) ⊆ D2.Hence [[h+(ϕ1)]]I2ρ2 ⊆ D2 and so I2, ρ2 h+(ϕ1). QED

The next result states the converse of the previous proposition under differ-ent conditions.

Proposition 6.3 Given a interpretation structure morphism (h, τ) : (Σ1, I1)→(Σ2, I2) such that τ v(D2) ⊆ D1, a schema formula ϕ1 in G1

+, and an assignmentρ2 over I2,

I1, ρ2(h,τ) ϕ1 whenever I2, ρ2 h+(ϕ1).

Proof: Assume that I2, ρ2 h+(ϕ1). Henceforth [[h+(ϕ1)]]I2ρ2 ⊆ D2. Thenτ v([[h+(ϕ1)]]I2ρ2) ⊆ D1 by hypothesis. Hence [[ϕ1]]I1ρ2

(h,τ)

⊆ D1 by Lemma 6.1,and so I1, ρ2

(h,τ) ϕ1. QED

Recall from [19] the result stating that: A logic system is sound if it is soundfor its deductive rules.

25

Theorem 6.4 Soundness is preserved by the fibring of logic systems sound forits rules.

Proof: Let L1 and L2 be logic systems sound for the rules. Note that therules of FibDed(D1,D2) are the image by the respective signature morphismsof the rules in D1 and in D2, respectively. We show that Fib(L1,L2) is soundfor its inference rules. Let (Σ, I) = (Σ1, I1)(V2,π2,♦2) ] (Σ2, I2)(V1,π1,♦1) be aninterpretation structure in the fibring and ρ an assignment over I. Let r1 be arule in D1. Assume that I, ρ i1

+(proper(ANT(r1))). Then, by Proposition 6.3,I1

(V2,π2,♦2), ρ(i1,τ1) proper(ANT(r1)). Hence, I1, ρ(i1,τ1) proper(ANT(r1)) by

Proposition 3.5 and so, since L1 is sound for its rules, I1, ρ(i1,τ1) CONC(r1).

Again by Proposition 3.5, I1(V2,π2,♦2), ρ(i1,τ1) CONC(r1). Finally, by Propo-

sition 6.2, I, ρ i1+(CONC(r1)) since the morphism from (Σ1, I1)(V2,π2,♦2) to

(Σ, I) is non-creative. The proof for a rule in D2 follows a similar reasoning sowe omit it. QED

Basically, Theorem 6.4 says that soundness is in almost all cases preserved byfibring, since normally a logic system does not contain interpretation structuresthat are not sound for its rules. Soundness is useful for establishing the non-collapse of intuitionistic into classic connectives, deductively, in Lc+i. Recallthat in Example 5.3 we already concluded that there is not, semantically, sucha collapse in Lc+i.

Example 6.5 Soundness of the logic system resulting from the fibring of thelogic systems for classical and intuitionistic propositional logic.The logic system Lc+i presented in Example 5.3 resulting from the fibring ofthe logic system Lc for classical propositional logic and of the logic system Lifor intuitionistic propositional logic is sound. So, 6`Dc+i (¬i(¬i q1i))⊃i q1i since6�Ic+i (¬i(¬i q1i))⊃i q1i. ∇

Taking into account Theorem 6.4, it is straightforward to see that the fibringof the logic systems composed by the deductive systems described in [19] andwith all the interpretation structures sound for the rules, is sound (the caseof the relevance logic has to be worked further due to the different notion ofderivation). Moreover, it is also straightforward to see that there is also nocollapse of the entailment and of the consequence relations of one system intoanother.

7 Completeness preservation

The goal of this section is to investigate preservation of completeness. Preser-vation of completeness means that the logic system resulting from the fibringis complete whenever the component logic systems are complete. Usually com-pleteness is not preserved unless we impose more properties on the componentsof the logic systems at hand. That is, in most cases, it is not possible to provecompleteness preservation for all possible complete component logic systems.We start by defining the canonical interpretation structure induced by a deduc-tive system.

26

Canonical structure

Let D1 and D2 be deductive systems with the same set of sorts, and Γ a set offormulas in G+ where G is G1]G2. The canonical structure SΓ

D2(D1) generated

by D1 over D2 and Γ, is such that:

• SΓD2

(D1) = (Σ1, (G′1, α1, D1, �1)) where

– G′1 = 〈V ′1 , E′1, src′1, trg′1〉 is such that

∗ V ′1 are the morphisms of G+ whose target is an element of V ;∗ E′1(w1 . . . wn, w) is composed by all the m-edges e in E1 such

that w = e ◦ 〈w1, . . . , wn〉 is in G+;

– αv1(e : s→ v) = v and αe

1(c) = c;

– D1 = {w ∈ V ′1 : Γ `D1]D2 w};– ◦1 is the morphism id♦ in G+.

In the sequel we will write SD2(D1) for S∅D2(D1), and will assume that G means

the m-graph G1 ] G2 (the disjoint union of the two m-graphs). We now showthat the SD2(D1) is sound for the deductive rules in D1.

Lemma 7.1 Given deductive systems D1 and D2 with the same set of sorts, aset Γ of formulas in G+, a path w : s → t over G†, and an assignment ρ overSΓD2

(D1), [[w]]SΓD2

(D1),ρ = w ◦ ρs.

Proof: The proof follows by induction on the complexity of w:

- w is εs. Then [[w]]SΓD2

(D1),ρ = ρs = ids ◦ ρs = εs ◦ ρs = w ◦ ρs;

- w is ps1i w1. Then [[w]]SΓD2

(D1),ρ = [[ps1i w1]]SΓD2

(D1),ρ = ([[w1]]SΓD2

(D1),ρ)i =(w1 ◦ ρs)i = ps1i ◦ w1 ◦ ρs = w ◦ ρs;

- w is 〈w1, . . . , wn〉w0. Hence [[w]]SΓD2

(D1),ρ = [[w1w0]]SΓD2

(D1),ρ. . . [[wnw0]]S

ΓD2

(D1),ρ =(w1 ◦ w0 ◦ ρs) . . . (wn ◦ w0 ◦ ρs) = 〈w1, . . . , wn〉 ◦ w0 ◦ ρs as we wanted to show;

- w is ew1. Therefore [[w]]SΓD2

(D1),ρ = trg′(E′e([[w1]]SΓD2

(D1),ρ,−)) = trg′(E′e(w1 ◦

ρs,−)) = e ◦ w1 ◦ ρs = w ◦ ρs. QED

The importance of the canonical structure is that in the context of thisstructure it is possible to relate satisfaction with derivation.

Lemma 7.2 Given deductive systems D1 and D2 with the same set of sorts,and a set Γ ∪ {ϕ : s → π} of schema formulas in G+, Γ `D1]D2 ϕ ◦ ρs if andonly if SΓ

D2(D1), ρ ϕ, for every assignment ρ over SΓ

D2(D1).

Proof: Let ρ be an assignment over SΓD2

(D1). Then Γ `D ϕ ◦ ρs if and only if,

by Lemma 7.1, Γ `D [[ϕ]]SΓD2

(D1),ρ iff [[ϕ]]SΓD2

(D1),ρ ⊆ D iff SΓD2

(D1), ρ ϕ. QED

We now state a useful lemma that when the proper premises of a rule arederivable, then the conclusion of the rule is also derivable. We omit its proofsince it constitutes a particular case of Lemma 6.6 in [19].

27

Lemma 7.3 For every deductive rule r in D1, set of formulas Γ and expressionu in G+, Γ `D1]D2 CONC(r) ◦ u whenever Γ `D1]D2 proper(ANT(r)) ◦ u.

In the following two propositions we show that SΓD2

(D1) is sound for therules in D1.

Proposition 7.4 For every deductive rule r in D1, set of formulas Γ in G+,and assignment ρ over SΓ

D2(D1), SΓ

D2(D1), ρ CONC(r) whenever SΓ

D2(D1), ρ

proper(ANT(r)).

Proof: Let proper(ANT(r)) be ϕ1 . . . ϕn. Assume SΓD2

(D1), ρ proper(ANT(r)).Then Γ `D1]D2 ϕi ◦ ρs, by Lemma 7.2, for i = 1, . . . , n. As a consequence,Γ `D1]D2 CONC(r) ◦ ρs, by Lemma 7.3, and so, SΓ

D2(D1), ρ CONC(r), by

Lemma 7.2. QED

Weak completeness

We start by investigating preservation of weak completeness (no hypothesesare present). With this purpose in mind, we need to understand better thedenotation of a formula in the fibring of two canonical structures.

Lemma 7.5 Given deductive systems D1 and D2 sharing the same set ofsorts V , a set Γ of formulas in G+ and a concrete path w over G†, (w, w) ∈[[w]]S

ΓD2

(D1)]SΓD1

(D2) if trg†(w) is an element of V .

Proof: The proof follows by induction on the complexity of w:

- w is ε♦. The thesis follows since (w, w) is (id♦, id♦) and [[w]]SΓD2

(D1)]SΓD1

(D2) =ρ♦ = {�} = {(�1, �2)} = {(id♦, id♦)};

- w is ps1i w1. Note that w1 is not ε♦, since otherwise s1 is ♦, which is not possiblesince the source of a projection has length at least 2. Hence, the path w1 ends ina tuple as it is straightforward to see. So, assume that w1 is 〈w11, . . . , w1n〉w0.Then (w, w) = (ps1i ◦w1, p

s1i ◦w1) = (ps1i ◦ 〈w11, . . . , w1n〉◦w0, p

s1i ◦ 〈w11, . . . , w1n〉◦

w0) = (w1i ◦ w0, w1i ◦ w0) and [[w]]SΓD2

(D1)]SΓD1

(D2) = [[ps1i w1]]SΓD2

(D1)]SΓD1

(D2) =

([[w1]]SΓD2

(D1)]SΓD1

(D2))i = ([[w11w0]]SΓD2

(D1)]SΓD1

(D2). . . [[w1nw0]]S

ΓD2

(D1)]SΓD1

(D2))i =[[w1iw0]]S

ΓD2

(D1)]SΓD1

(D2) and so the thesis follows by induction hypothesis;

- w is ew1. Observe that [[w]]SΓD2

(D1)]SΓD1

(D2) = trg′(E′e([[w1]]SΓD2

(D1)]SΓD1

(D2),−)).

The thesis follows since trg′(E′e((w1, w1),−)) ⊆ trg′(E′e([[w1]]SΓD2

(D1)]SΓD1

(D2),−))

by induction hypothesis, and (w, w) = (e ◦ w1, e ◦ w1) ∈ trg′(E′e((w1, w1),−)).QED

Recall from [19], the notion of representative which plays an important rolein the proof of completeness for a single logic system. A logic system containsa representative of the canonical structure over a set Γ when it contains aninterpretation structure IΓ such that

• IΓ ϕ implies SΓ(D) ϕ;

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• IΓ Γ;

for every formula ϕ and set of formulas Γ in G+.The next result shows that SD2(D1) ] SD1(D2) is a representative of the

canonical structure for the fibring of deductive systems D1 and D2.

Proposition 7.6 Given deductive systems D1 and D2 sharing the same set ofsorts and a formula ϕ of formulas in L(Σ1 ] Σ2), then

SD2(D1) ] SD1(D2) ϕ implies S(D1 ] D2) ϕ.

Proof: Assume that SΓD2

(D1) ] SΓD1

(D2) ϕ. Then [[ϕ]]SΓD2

(D1)]SΓD1

(D2) is aset of distinguished truth values in SΓ

D2(D1) ] SΓ

D1(D2). On the other hand, by

Lemma 7.5, (ϕ,ϕ) ∈ [[ϕ]]SΓD2

(D1)]SΓD1

(D2). Therefore ϕ is distinguished in bothSΓD2

(D1) and SΓD1

(D2). Hence, Γ `D1]D2 ϕ and so ϕ is also distinguished in

SΓ(D1 ] D2). So SΓ(D1 ] D2) ϕ since [[ϕ]]SΓ(D1]D2) = {ϕ} using Lemma 6.4

of [19]. QED

In order to show preservation of weak completeness, we use the followingresult proved in Theorem 6.9 of [19]: A logic system is weakly complete if itcontains a representative of the canonical structure over the empty set.

Theorem 7.7 Weak completeness is preserved by fibring of logic systems, eachcontaining the canonical structure over the other and the empty set.

Proof: Let L1 and L2 be such that SD2(D1) ∈ I1 and SD1(D2) ∈ I2. SoSD2(D1) ] SD1(D2) is in I. Using Proposition 7.6, we can conclude thatSD2(D1) ] SD1(D2) is a representative of the canonical structure generated byD1 ] D2 and ∅. So the fibring has a representative of that deductive systemfor the empty set. Hence, invoking Theorem 6.9 of [19], we can conclude thatL1 ] L2 is weakly complete. QED

We omit the proof of the next result since it follows straightforwardly byTheorem 7.7 and Proposition 7.4.

Corollary 7.8 Weak completeness is preserved by fibring logic systems con-taining all the interpretation structures that are sound with respect to the rules.

Corollary 7.8 basically says that in almost all cases the logic system resultingfrom the fibring is at least weak complete. Similarly to Theorem 6.4, this hap-pens since a logic system, normally, does not contain interpretation structuresthat are not sound for its rules.

Example 7.9 Weak completeness for the logic system resulting from the fibringof the logic systems for classical and intuitionistic propositional logics.The logic system Lc+i presented in Example 5.3, resulting from the fibring ofthe logic system Lc for classical propositional logic and of the logic system Lifor intuitionistic propositional logic, is weak complete. ∇

29

Taking into account Corollary 7.8, it is straightforward to see that the fibringof the logic systems composed by the deductive systems described in [19] andwith all the interpretation structures sound for the rules, is weak complete (thecase of the relevance logic has to be worked further due to the different notion ofderivation). So, for instance, the fibring of the logic systems for paraconsistentand modal logic T is weak complete as well as the fibring of the logic systemsfor paraconsistent and intuitionistic propositional logics, as well as the fibringof the logic systems for equational and paraconsistent propositional logics.

Strong completeness

Preservation of strong completeness is a harder task in finding sufficient condi-tions for the component logics. A key concept is the notion of a truth consistentinterpretation structure.

Given an interpretation structure I = 〈G′, β, �〉 over a signature Σ, a setS ⊆ V ′π is truth consistent whenever S ⊆ D or S ∩D = ∅. The interpretationstructure I is truth consistent if for every e : v1 . . . vn → π in E and Si containedin V ′vi such that Si is truth consistent when vi is π, for i = 1, . . . , n, thentrg′(E′e(S1 . . . Sn,−)) is truth consistent.

Lemma 7.10 Let I be a truth consistent interpretation structure over a sig-nature Σ, ρ an assignment over I, w : s→ v1 . . . vn a path in G† with v1, . . . , vnin V and [[w]]I,ρ = S1 . . . Sn with Si contained in V ′si for i = 1, . . . , n. Then Siis truth consistent when si is π, for all i = 1, . . . , n.

Proof: The proof follows by induction on the complexity of w:

w is εs. Then [[w]]I,ρ = ρs which satisfies the requirement.

w is ps1i w1. Then [[w]]I,ρ = ([[w1]]I,ρ)i which is truth consistent by inductionhypothesis.

w is 〈w1, . . . , wn〉w0 and wi : s0 → si for i = 1, . . . , n and w0 : s → s0. Then[[w]]I,ρ = [[w1w0]]I,ρ . . . [[wnw0]]I,ρ. So the result follows straightforwardly byinduction hypothesis.

w is ew1. Then [[w]]I,ρ is trg′(E′e([[w1]]I,ρ,−)). The result follows immediately,using the induction hypothesis on [[w1]]I,ρ since I is truth consistent. QED

The following notion is the deductive counterpart of the truth consistencypresented above. It resembles congruence as was adopted in previous works like[22]. But is that work the component logic systems had to share implicationwhich is not the case herein.

A deductive system D is said to be componentwise congruent for Γ wheneverΓ `D ϕi iff Γ `D ψi for i = 1, . . . , n implies

Γ `D c(ϕ1, . . . , ϕn) iff Γ `D c(ψ1, . . . , ψn)

for each m-edge c in Σ.The following result relates componentwise congruence with truth consis-

tency over the canonical structures.

30

Lemma 7.11 Let D1 and D2 be deductive systems. If D1 ]D2 is component-wise congruent then SΓ

D2(D1) is truth consistent.

Proof: Assume that D1]D2 is componentwise congruent. Let e : v1 . . . vn → πin E and Si contained in V ′vi such that Si is truth consistent when vi is π,for i = 1, . . . , n. Let ϕ1, ϕ2 ∈ trg′(E′e(S1 . . . Sn,−)). Assume that ϕ1 is adistinguished value. Let ϕi1, . . . , ϕin be such that ϕi = e ◦ (ϕi1 . . . ϕin) fori = 1, 2. Observe that, for each j = 1, . . . , n, either ϕ1j , ϕ2j are distinguishedvalues or none is. So either Γ `D1]D2 ϕ1j and Γ `D1]D2 ϕ2j or neither holds, forj = 1, . . . , n. Using the fact that D1 ] D2 is componentwise congruent, eitherΓ `D1]D2 ϕ1 and Γ `D1]D2 ϕ2 or neither holds. Hence Γ `D1]D2 ϕ2 and so ϕ2

is distinguished as well. QED

The following result establishes the preservation of truth consistency by thefibring of canonical structures for the component logics.

Lemma 7.12 Let D1 and D2 be deductive systems. If SΓD2

(D1) and SΓD1

(D2)are truth consistent then so is SΓ

D2(D1) ] SΓ

D1(D2).

Proof: Let e : v1 . . . vn → π in E and Si contained in V ′vi such that Siis truth consistent when vi is π, for i = 1, . . . , n. Let (ϕ1, ϕ2), (ψ1, ψ2) ∈trg′(E′e(S1 . . . Sn,−)). Assume that (ϕ1, ϕ2) is a distinguished value. Then ϕ1

and ϕ2 are distinguished elements in SΓD2

(D1) and SΓD1

(D2), respectively. As-sume, with no loss of generality, that e ∈ E1. So ϕ1 = e ◦ (ϕ11 . . . ϕ1n) andψ1 = e ◦ (ψ11 . . . ψ1n) where (ϕ1i, δ2i) ∈ Si for some δ2i and (ψ1i, θ2i) ∈ Si forsome δ1i and θ2i, i = 1, . . . , n. Since Si is truth consistent then either (ϕ1i, δ2i)and (ψ1i, θ2i) are distinguished values in SΓ

D2(D1) ] SΓ

D1(D2) or none is. So

either ϕ1i and ψ1i are distinguished values in SΓD2

(D1) or none is. So eitherϕ1 and ψ1 are distinguished values in SΓ

D2(D1) or none is. Therefore ψ1 is a

distinguished element in SΓD2

(D1) and so (ψ1, ψ2) is a distinguished value inSΓD2

(D1) ] SΓD1

(D2). QED

The next result shows that SΓD2

(D1) ] SΓD1

(D2) is a representative of thecanonical structure for the fibring of deductive systems D1 and D2, providingthat the fibring of deductive systems is componentwise congruent.

Proposition 7.13 Given deductive systems D1 and D2 sharing the same setof sorts, and a set Γ ∪ {ϕ} of formulas in G+, then

• SΓD2

(D1) ] SΓD1

(D2) ϕ implies SΓ(D1 ] D2) ϕ;

• SΓD2

(D1) ] SΓD1

(D2) Γ

assuming that D1 ] D2 is componentwise congruent for Γ.

Proof:(1) SΓ

D2(D1) ] SΓ

D1(D2) ϕ implies SΓ(D1 ] D2) ϕ. The proof of this impli-

cation mimics the proof of Proposition 7.6 so we omit it.

31

(2) SΓD2

(D1) ] SΓD1

(D2) Γ. Let γ ∈ Γ. Then Γ `D1]D2 γ. So γ is adistinguished truth value in both SΓ

D2(D1) and SΓ

D1(D2) and so (γ, γ) is a

distinguished truth value in SΓD2

(D1) ] SΓD1

(D2). By Lemma 7.5, (γ, γ) ∈[[γ]]S

ΓD2

(D1)]SΓD1

(D2) and so all the truth values of [[γ]]SΓD2

(D1)]SΓD1

(D2) are dis-tinguished, since SΓ

D2(D1) ] SΓ

D1(D2) is truth consistent by Lemma 7.12 and

Lemma 7.11. Hence, SΓD2

(D1) ] SΓD1

(D2) γ. QED

We now introduce the well known notion of a maximal set of formulas. Alogic system L has maximals if for each consistent set Γ of formulas and formulaϕ such that Γ 6`D ϕ, there is a consistent set Γ extending Γ such that Γ 6`D ϕand Γ ∪ {ψ} `D ϕ for every ψ /∈ Γ. We call Γ a maximal set for Γ and ϕ.

Theorem 7.14 A logic system with maximals containing a representative ofthe canonical structure over a maximal set for each set of formulas and formulais strong complete.

Proof: Let Γ be a set of formulas and ϕ a formula. Assume that Γ 6`D ϕ.Then there is a maximal set Γ for Γ and ϕ. Let IΓ ∈ I be an interpretationstructure representing SΓ(D). Then, SΓ(D) Γ and SΓ(D) 6 ϕ and so IΓ 6 ϕand IΓ Γ. Therefore, Γ 6�I ϕ since IΓ 6 ϕ and IΓ Γ and IΓ ∈ I. QED

We are now ready to provide a sufficient condition for the preservation ofstrong completeness by fibring.

Theorem 7.15 Strong completeness is preserved by fibring of logic systemssuch that:

• the fibring has maximals, for each set of formulas Γ and formula, enjoyingcomponentwise congruence for Γ;

• each component contains the canonical structure over the other and thosemaximal sets.

Proof: Let Γ be a set of formulas and ϕ a formula. Then there is a maximal setΓ for Γ and ϕ. Moreover, D1]D2 is componentwise congruent for Γ. Moreover,by hypothesis, SΓ

D2(D1) ∈ I1 and SΓ

D1(D2) ∈ I2. Using Proposition 7.13, we

can conclude that SΓD2

(D1) ] SΓD1

(D2) ∈ I is a representative of the canonicalstructure SΓ(D1 ] D2). Finally, by Theorem 7.14, we can conclude that thefibring is complete. QED

The conditions in Theorem 7.15 are not so general as the ones we obtainedfor the preservation of soundness and weak completeness. Therefore, less logicsystems can be fibred with preservation of strong completeness. Nevertheless,preservation of strong completeness holds, among others, in the fibring of modallogics at least as stronger as T .

The preservation of soundness and completeness ensures that the graph-theoretic semantics developed herein is adequate with respect to the envisaged

32

deductive system. Moreover, it provides a criterion for checking in a semanticway that other combination approaches are sound and complete with respect tothe widely recognized deductive system for fibring. For instance, the combina-tion approaches described in [21, 2]. We can further outline this idea. Considertwo logics each one characterized by a class of models M ′ and M ′′, respec-tively. Let (Σ′, I ′,D′) and (Σ′′, I ′′,D′′) be graph-theoretic logic systems forthe same logics whose graph-theoretic fibring is sound and complete. Assumethat m′ tm′′ is the model resulting from combining m′ ∈ M ′ and m′′ ∈ M ′′.One can ask whether the combined logic characterized by the class of modelsM = {m′ tm′′ : m′ ∈ M ′,m′′ ∈ M ′′} is sound and complete with respect tothe deductive system for fibring. The results in this paper help to provide ananswer to this question when the following conditions are satisfied:

• for every m ∈M there are I ′ ∈ I ′ and I ′′ ∈ I ′′ such that I ′ ] I ′′ satisfiesthe same formulas as m;

• for every I ′ ∈ I ′ and I ′′ ∈ I ′′ there is m ∈ M such that m satisfies thesame formulas as I ′ ] I ′′.

The first condition guarantees that the combination is sound with respect tothe deductive system for fibring. For instance, assume that `D ϕ. Let m ∈ Mbe an arbitrary model. Then, by the first condition, there are I ′ ∈ I ′ andI ′′ ∈ I ′′ such that I ′ ] I ′′ satisfies the same formulas as m. By soundness of thegraph-theoretic approach we conclude that I ′ ] I ′′ ϕ and so m satisfies ϕ asdesired.

The second condition guarantees that the combination is complete withrespect to the deductive system for fibring. For instance, assume that ϕ is validin M . It is enough to show that ϕ is valid in the graph-theoretic semanticsresulting from fibring. Let I ′ ∈ I ′ and I ′′ ∈ I ′′. Then, by the second condition,there is m ∈M such that m and I ′ ] I ′′ satisfies the same formulas. So, I ′ ] I ′′satisfies ϕ as wanted.

8 Concluding remarks

In the sequel of [19], we developed a graph-theoretic account of fibring of logicsystems. The approach is general enough to cover logics presented in severalguises. It encompasses logics with a non-deterministic semantics as well assubstructural logics like relevance logic. Moreover, it covers logics with an alge-braic semantics. We proved preservation of soundness and weak completenessunder very general assumptions. Preservation of strong completeness requiresa tighter context. To the best of our knowledge, results on weak completenesswere not proved before in previous work on fibring.

Another key feature of our approach is that the components of each interpre-tation structure are abstracted to the components of the signature (through am-graph morphism). This contrasts to the usual (concrete) approach of givingdenotations of components of signatures in terms of components of interpre-tation systems. It is worthwhile to point out that the graph-theoretic setting

33

together with this feature provides a way to avoid well known collapses [7] ina very natural way different from previous works on the topic [20, 2]. Forinstance, in the case of the modulated fibring, the collapse was avoided by re-stricting instantiation in derivation which is not the case in the graph-theoreticapproach.

Several extensions of this work are worthwhile to pursue. The first stepshould be to investigate fibring of logics with quantification and whose infer-ence rules have provisos. Another obvious extension is to consider (labelled)sequent-like calculi instead of Hilbert calculi. Results in this area will pavethe way to analyze preservation results about cut elimination, interpolation,quantifier elimination and decidability, among others. For the latter topic wehope to capitalize on the results obtained in [4, 10, 16] for interpolation andgeneral characterizations of logics with cut elimination [17]. Another directionof future research is to investigate how fibring of protoalgebraic and (weakly)algebraizable logics can be accommodated in the graph-theoretic setting [1, 5].We also believe that the graph-theoretic approach should be explored in thefield of fusion of modal logics, namely to cope with modal logics endowed withgeneral semantics and having different valuation sets.

Acknowledgments

This work was partially supported by FCT and EU FEDER, namely via Quant-Log PPCDT/MAT/55796/2004 Project, KLog PTDC/MAT/68723/2006 Proj-ect, QSec PTDC/EIA/67661/2006 Project and under the GTF (Graph The-oretic Fibring) initiative of IT. Marcelo Coniglio acknowledges support fromFAPESP, Brazil, namely via Thematic Project 2004/14107-2 (“ConsRel”), andby an individual research grant from CNPq, Brazil.

References

[1] W. J. Blok and D. Pigozzi. Algebraizable logics. Memoirs of the AmericanMathematical Society, 77(396), 1989.

[2] C. Caleiro and J. Ramos. From fibring to cryptofibring. A solution to thecollapsing problem. Logica Universalis, 1(1):71–92, 2007.

[3] W. A. Carnielli, M. E. Coniglio, D. Gabbay, P. Gouveia, and C. Sernadas.Analysis and Synthesis of Logics - How To Cut And Paste Reasoning Sys-tems, volume 35 of Applied Logic. Springer, 2008.

[4] W. A. Carnielli, J. Rasga, and C. Sernadas. Preservation of interpolationfeatures by fibring. Journal of Logic and Computation, 18(1):123–151,2008.

[5] J. Czelakowski and R. Jansana. Weakly algebraizable logics. The Journalof Symbolic Logic, 65(2):641–668, 2000.

34

[6] A. S. d’Avila Garcez, D. M. Gabbay, and L. C. Lamb. Value-based ar-gumentation frameworks as neural-symbolic learning systems. Journal ofLogic and Computation, 15(6):1041–1058, 2005.

[7] L. Farinas del Cerro and A. Herzig. Combining classical and intuitionisticlogic. In Frontiers of Combining Systems, volume 3, pages 93–102. KluwerAcademic Publishers, 1996.

[8] D. Gabbay. Fibring Logics, volume 38 of Oxford Logic Guides. OxfordUniversity Press, 1999.

[9] D. M. Gabbay, A. Kurucz, F. Wolter, and M. Zakharyaschev. Many-dimensional Modal Logics: Theory and Applications, volume 148 of Studiesin Logic and the Foundations of Mathematics. North-Holland PublishingCo., 2003.

[10] D. M. Gabbay and L. Maksimova. Interpolation and Definability, vol-ume 46 of Oxford Logic Guides. Oxford University Press, 2005. Modal andintuitionistic logics.

[11] D. M. Gabbay and V. B. Shehtman. Products of modal logics. I. LogicJournal of the IGPL. Interest Group in Pure and Applied Logics, 6(1):73–146, 1998.

[12] D. Gabelaia, A. Kurucz, F. Wolter, and M. Zakharyaschev. Products of‘transitive’ modal logics. The Journal of Symbolic Logic, 70(3):993–1021,2005.

[13] J. Y. Halpern, R. van der Meyden, and M. Y. Vardi. Complete axiom-atizations for reasoning about knowledge and time. SIAM Journal onComputing, 33(3):674–703 (electronic), 2004.

[14] R. Kontchakov, A. Kurucz, F. Wolter, and M. Zakharyaschev. Spatiallogic+temporal logic=? In Handbook of Spatial Logics, pages 497–564.Springer, 2007.

[15] S. Mac Lane. Categories for the Working Mathematician, volume 5 ofGraduate Texts in Mathematics. Springer-Verlag, New York, second edi-tion, 1998.

[16] L. L. Maksimova. A method for proving interpolation in paraconsistent ex-tensions of minimal logic. Algebra i Logika. Institut Diskretnoı Matematikii Informatiki, 46(5):627–648, 665, 2007.

[17] J. Rasga. Sufficient conditions for cut elimination with complexity analysis.Annals of Pure and Applied Logic, 149(1-3):81–99, 2007.

[18] A. Sernadas, C. Sernadas, and C. Caleiro. Fibring of logics as a categorialconstruction. Journal of Logic and Computation, 9(2):149–179, 1999.

35

[19] A. Sernadas, C. Sernadas, J. Rasga, and M. Coniglio. Graph-theoreticfibring of logics: Part I - Completeness. Preprint, SQIG - IT and IST -TU Lisbon, 1049-001 Lisboa, Portugal, 2008. Submitted for publication.

[20] C. Sernadas, J. Rasga, and W. A. Carnielli. Modulated fibring and thecollapsing problem. Journal of Symbolic Logic, 67(4):1541–1569, 2002.

[21] A. Tarlecki, R. M. Burstall, and J. A. Goguen. Some fundamental algebraictools for the semantics of computation. III. Indexed categories. TheoreticalComputer Science, 91(2):239–264, 1991.

[22] A. Zanardo, A. Sernadas, and C. Sernadas. Fibring: Completeness preser-vation. Journal of Symbolic Logic, 66(1):414–439, 2001.

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