Modulated Fibring and the Collapsing Problem

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Modulated Fibring and the Collapsing Problem

^∗

Cristina Sernadas¹, Jo˜ao Rasga¹ and Walter A. Carnielli²

1 CLC, Departamento de Matem´atica, IST, Portugal E-mail:{css, jfr}@math.ist.utl.pt

2 CLE and IFCH, UNICAMP, Brazil E-mail: [email protected]

Abstract

Fibring is recognized as one of the main mechanisms in combining logics, with great significance in the theory and applications of mathematical logic. However, an open challenge to fibring is posed by the collapsing problem: even when no symbols are shared, certain combinations of logics simply collapse to one of them, indicating that fibring imposes unwanted interconnections between the given logics. Modulated fibringallows a finer control of the combination, solving the collapsing problem both at the semantic and deductive levels. Main properties like soundness and completeness are shown to be preserved, comparison with fibring is discussed, and some important classes of examples are analyzed with respect to the collapsing problem.

1 Introduction

Among the contemporary research on theory and application of logic, the topic of the combination of logics is one of the most interesting. Logicians, philoso- phers and computer scientists are finally emerging from the complexity and the perplexity of isolated logical systems, learning how to capitalize on the intricate characteristics of particular logic systems towards a general manner of inves- tigating the way logics can be combined, the way such combinations can be applied and understanding the general properties (cf. [1]). Among the several approaches for spelling out such combinations, the techniques of fibring (cf.

[9, 10, 11, 16, 18]) have been the most auspicious: the fibring of logics leads to a new logic where not only connectives are mixed, but proof methods are combined.

Although the fibring techniques can be defined in the context of quantifica- tional logic (cf. [17]), even if restricted to the propositional level (avoiding variables, terms, binding operators such as quantifiers, and the subtleties therein), fibring propositional based logics such as modal, intuitionistic and many-valued

∗This work started during a visit by Walter A. Carnielli to CMA at IST, entirely supported byFunda¸cão para a Ciência e a Tecnologia, Portugal. A subsequent visit to CMA by Walter A. Carnielli was also supported byCoordena¸cão de Aperfei¸coamento do Pessoal do Ensino Superior, Brazil.

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logics produces huge amounts of possibilities connected to real applications (in engineering and artificial intelligence) and is open to interesting philosophical interpretations. Furthermore, although most of the work on fibring has been restricted to logics endowed with truth-functional semantics, some steps have been taken towards encompassing logics (like paraconsistent logics [7]) with non truth-functional semantics [3].

Fibring can be presented from a proof-theoretical or from a model-theoretical perspective. From the proof-theoretical point of view, fibring is treated in a natural way over logic systems with a Hilbert-like (axiomatic) deductive style presentation, but this kind of deduction seems more appropriate for meta- mathematical investigation than for real applications, a practice sometimes re- quiring Gentzen (sequent) or tableau style presentation. An appropriate frame- work for fibring natural deduction systems by means of labeled (or annotated) deduction systems is given in [14].

Usually, soundness is preserved under the process of fibring in the sense that the fibring of a family of logics is sound, provided that the components are sound. However, a more difficult problem is to show that completeness is also preserved. This question was solved in [18] for a wide class of (truth-functional) propositional based logics, where it was shown that under certain reasonable requirements (to wit, that the component logics are complete under general frame semantics and endowed with congruence relations) a kind of transfer of completeness can be obtained, guaranteeing that the result of the fibring is complete.

The use of categorial language is very appropriate for defining fibring, because fibring appears as a universal construction in the appropriate category of logic systems [16], emphasizing the canonical nature of fibring. Furthermore, it is often useful to show that certain collections of objects together with certain appropriate transformations make up a category, such as the category of signatures of logics together with arity preserving maps. In this way, the underlying theory is held maximally uniform and general.

However, general as it is, the original notion of fibring is not yet broad enough to accommodate more subtle aspects of combinations of logics, for example to avoid the collapsing problem, which consists of the unexpected collapse of two logics when combined even by unconstrained fibring (no symbols are shared). A simple yet paradigmatic example was provided in [8, 10], where it is shown that, in our terms, the unconstrained fibring (sharing nothing!) of classical and intuitionistic logic collapses into classical logic. The result is that the original notion of fibring is not appropriate for controlling this kind of phenomenon, and in the present paper we extend the notion of fibring to a much more powerful notion of modulated fibring.

The main idea behind modulated fibring at the semantic level is as follows.

In the original fibring (as clearly shown in [18]), even when no symbols are shared, an interconnection is imposed upon the two given logics (to wit, only pairs of models sharing the same algebra of truth values contribute to the resulting logic). In the novel notion of modulated fibring this imposition is relaxed by giving as input to the fibring a translation between the truth value algebras of the two given logics. This translation modulates the result. An

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appropriate choice of the translation recovers the original notion of fibring, but other translations are possible and in the example of [8] a manner of avoiding the collapsing is shown. The modulated fibring is also introduced at the deductive system level leading to some provisos when applying the inference rules.

Therefore, the main goal of this paper is to achieve a mechanism for combining logics both at the semantic and the deductive levels but avoiding when desired the collapsing phenomenon. Preservation of soundness and completeness are also investigated.

Besides the pioneering example of [8], other cases of collapsing are described and for each of them we show how modulated fibring avoids the collapsing under a specific choice of the truth values translation.

The remainder of the paper is structured as follows. Section 2 is dedicated to modulated fibring at the semantic level including the notions of interpretation system and morphism. In this section some interesting examples selected from intuitionistic and many-valued logics are presented. Moreover, it shows how to extract from an interpretation system the (local and global) notions of entailment and establishes some basic results. Finally, introduces the notion of modulated fibring as a pushout in the category of interpretation systems, indicates how to recover the original notion of fibring as a special case, and shows how to set up the the base diagram of the pushout from the intended translation between the truth value algebras. Section 3 concentrates on deductive aspects of modulated fibring starting with Hilbert systems and their morphisms. The modulated fibring at the deductive level appears as a pushout in the category of Hilbert systems. We conclude the section with examples and a comparison with fibring. Section 4 is dedicated to logic systems putting together interpretation systems on one hand and Hilbert systems on the other hand, fibring of logic systems and preservation of soundness. Section 5 concentrates on preservation of completeness namely establishing sufficient conditions. We conclude in Section 6 with some remarks and open problems.

2 Interpretation systems

In this section, we investigate modulated fibring from a semantic point of view.

We start with signatures and proceed in Subsection 2.2 with the notions of interpretation system and morphism between interpretation systems. We conclude the section with several examples that will be used later on. In Subsection 2.3 we introduce (global and local) semantic entailments. Finally, in Subsection 2.4 we define modulated fibring as a pushout in the category of interpretation systems and give several examples showing that it is possible to choose bridges that lead to non-collapsing situations.

2.1 Signatures

We introduce the basic symbols that we need in each signature. We start by identifying the notion of pre-signature.

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Definition 2.1 Apre-signatureis a triple Σ =hC,&,ΞiwhereCis an indexed family of sets over the natural numbers, & is a symbol and Ξ is a set.

Elements of C_k are constructors of arity k, and elements of Ξ are meta- variables. The role of the symbol & will become clear when giving the semantics.

Moreover this symbol is also essential for technical reasons in Section 5.

Definition 2.2 A pre-signaturemorphismh:hC,&,Ξi → hC⁰,&⁰,Ξ⁰iis a pair hh₁, h₂i such that h₁ ={h₁_k}_k∈_N is a family of maps fromC_k toC_k⁰ for every k∈Nand h₂ : Ξ→Ξ⁰ is a map.

Pre-signatures and their morphisms constitute the categorypSig. This category has finite colimits and in particular pushouts.

Definition 2.3 Asignatureis a co-cone in pSig, that is Σ =hC,&,Ξ, Si.

The set S contains the “safe-relevant” morphisms whose destination is hC,&,Ξi. Safety will play an important role in the definition of the entailments by constraining the admissible assignments to meta-variables in the range of safe-relevant morphisms. This is also the reason why the meta-variables are local to signatures which was not the case of fibring in [18].

Definition 2.4 A signaturemorphism h: Σ→Σ⁰ is a co-cone morphism, that is,his a pre-signature morphism such that h◦f ∈S⁰ whenever f ∈S.

Signatures and their morphisms constitute the category Sig. Again this category has finite colimits, in particular pushouts.

2.2 Basic notions

The basic semantic unit is the structure for a signature. Typically in an alge- braic setting, a structure is an algebra.

Definition 2.5 A Σ-structure B = hB,≤, νi is a pre-ordered algebra over C and & with finite meets¹ such that

1. ν₂(&)(b₁, b₂) =b₁ub₂;

2. νk(c)(b1, . . . , bk)∼=νk(c)(d1, . . . , dk) whenever bi∼=di fori= 1, . . . , k.² The elements in B are the truth values (or degrees) and ν_k(c) is the denotation of constructor c of arity k which is an operation in the algebra. The symbol & is the syntactical counterpart of 2-ary meets. Constraint 1. indicates that & behaves like a conjunction (whether or not such symbol is a constructor in the signature). Constraint 2. is congruence requirement: denotations of a constructor on “equivalent” truth values should be “equivalent”.

1In a pre-order, meets are unique up to isomorphism. We use the notation u{b1, . . . , bk} or evenb1u · · · ubk for a choice of one of the meets for{b1, . . . , bk}and >foru ∅. Observe that finite meets exist iff 0-ary and 2-ary meets exist.

2Byb1∼=b2 it is meantb1≤b2andb2≤b1.

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In the fibring as presented in [18], structures were power set algebras based on sets of points (worlds). The more general setting of considering an algebra (not necessarily a power set algebra) also includes logics (like multi-valued logics) whose semantics is not provided in terms of points.

In the sequel we omit the reference to the arity of the constructors and the subscripts in signature morphisms in order to make the notation lighter.

Sometimes we also use~bas a short hand for b₁, . . . , b_k.

Definition 2.6 An interpretation system is a tuple I =hΣ, M, Ai where Σ is a signature,M is a class (of models),A is a map associating to each m∈M a Σ-structureB_m.

The interpretation system could be a pair hΣ,Bi. We include M because one can take the models of the logic at hand and useAto extract the underlying algebras andM also simplifies the notion of interpretation system morphism.

Definition 2.7 An interpretation system morphism h : I → I⁰ is a tuple hˆh, h,h,˙ ¨hi where:

• ˆh: Σ→Σ⁰ is a morphism in Sig;

• h:M⁰ →M is a map;

• h˙ = {h˙_m⁰}_m⁰_∈M⁰ where ˙h_m⁰ : hB_h(m0),≤_h(m0)i → hB_m⁰ 0,≤⁰_m0i is a monotonic map;

• ¨h = {¨h_m⁰}_m⁰_∈M⁰ where ¨h_m⁰ : hB_m⁰ 0,≤⁰_m0i → hB_h(m0),≤_h(m0)i is a monotonic map preserving finite meets;³

such that for everym⁰∈M⁰,~b∈B_h(m^k 0) and~b⁰∈B_m^k0: 1. ¨hm⁰ is left adjoint of ˙hm⁰;

2. ν_m⁰ 0(ˆh(c))(~b⁰)∼=m⁰ h˙_m⁰(ν_h(m⁰₎(c)(¨h_m⁰(~b⁰))) for everyc∈C_k. 4 Recall that ¨h_m⁰ is left adjoint of ˙h_m⁰ iff for every b⁰∈B_m⁰ andb∈B_h(m⁰₎:

b⁰≤_m⁰ h˙_m⁰(¨h_m⁰(b⁰)) and ¨h_m⁰( ˙h_m⁰(b))≤_h(m⁰₎b.

As a consequence, ˙h_m⁰ also preserves meets for everym⁰ ∈ M⁰. Observe that

¨hm⁰( ˙hm⁰(b))∼=_h(m⁰₎bwhenever ¨hm⁰ is surjective. Moreover, ν_m⁰ 0(&⁰)( ˙h_m⁰(b₁),h˙_m⁰(b₂))∼=⁰_m⁰ h˙_m⁰(ν_h(m⁰₎(&)(b₁, b₂)).

The map h is expected to be contravariant. The family of maps ˙hm⁰ and

¨h_m⁰ indicate that we need to represent the truth values of B_h(m⁰₎ in the truth values ofB_m⁰ 0 and vice versa. Clause 1. states constraints that the maps should

3Observe that a map preserves finite meets iff preserves 0-ary and 2-ary meets.

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fulfill. Clause 2. indicates that denotations of constructors fromC in a model m⁰ can be given for any truth values inB⁰_m0 by using the two maps.

The morphism between interpretation systems presented in [18] is a particular case of the one in Definition 2.7 with ˙h_m⁰ = id_B⁰

m0, ¨h_m⁰ = id_B_h(m0) and hence,B_h(m⁰₎=B_m⁰ 0, etc.

Prop/Definition 2.8 Interpretation systems and their morphisms constitute the category Int.

Some of the examples we consider are many-valued logics. For more details about these logics see [2, 13]. In all examples the signature is as follows: Σ = hC,&,Ξ, Siwheret∈C0(in general inC0 we also have propositional symbols), C1 = {¬}, C2 = {∧,∨,⇒}, C_k = ∅ for all k ≥ 3, & is ∧, Ξ = {ξ_i : i ∈ N} and S =∅. Thus the interpretation systems in the examples only differ in the semantic part, that is inM andA.

Example 2.9 Propositional interpretation system.

• M is the class of all pairs m = hB, Vi where B= hB,u,t, ,>,⊥i is a Boolean algebra andV :C₀ →B is a map such that V(t) =>;

• A(m) =hB,≤, νi where – b1≤b2 iff b1ub2 =b1;

– ν₀(c) =V(c),ν₁(¬) = ,ν₂(∧) =u, and ν₂(∨) =t;

– ν₂(⇒) =λb₁b₂.( b₁)tb₂. 4

Example 2.10 Intuitionistic interpretation system.

• M is the class of all pairs m = hB, Vi where B= hB,u,t,A,⊥,>i is a Heyting algebra andV :C₀ →B such that V(t) =>;

– ν0(c) =V(c),ν2(∧) =uand ν2(∨) =t;

– ν₂(⇒) =A;

– ν₁(¬) =λb.bA⊥. 4

Example 2.11 (3-valued) G¨odel interpretation system.

G¨odel logics were introduced as approximations to intuitionistic logic, and extended the propositional intuitionistic calculus.

• M is the class of all pairs m= hB, Vi whereB =hB,u,t,A, ,⊥,>i is a 3-valued G¨odel algebra⁴ and V :C₀ →B such thatV(t) =>;

4Recall that the typical 3-valued G¨odel algebra hasB={⊥,1/2,>}and operations and Aare defined as follows: b= 1 wheneverb= 0 and 0 otherwise, andb1 Ab2 is>ifb1≤b2

andb2 otherwise.

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• A(m) =hB,≤, νi where – b₁≤b₂ iff b₁ub₂ =b₁;

– ν₀(c) =V(c),ν₂(∧) =u,ν₂(∨) =t and ν₁(¬) = ;

– ν(⇒) =A. 4

Example 2.12 (3-valued) Lukasiewicz interpretation system.

Lukasiewicz logics, introduced in the twenties, were the first logics introducing a third truth value, designed to express linguistic modalities outside the scope of classical logic, like the possible (contingent) future.

• Mis the class of all pairsm=hB, ViwhereB=hB,⊕, ,⊥iis a 3-valued multi-valued algebra⁵ and V :C₀→B is a map;

– ν0(c) =V(c),ν1(¬) = ,ν2(∧) =u andν2(∨) =t;

– ν₂(⇒) =A. 4

2.3 Satisfaction and entailment

The objective of this section is to introduce the notion of entailment. As in other papers on fibring (e.g. [16]) we have two entailments: global entailment corresponding to proof and local entailment corresponding to derivation. We start by defining the languages over a given signature.

Definition 2.13 The set L(Σ) of Σ-formulae is the free algebra over C,&,Ξ taking the elements of Ck ask-ary operations, & as a 2-ary operation and the elements of Ξ as 0-ary operations. We denote by L(C,&) the subset of L(Σ) composed byground formulae, that is formulae without meta-variables.

We need the notion of assignment for defining the denotation of formulae and entailments. Assignments that give special values to schema variables that come from safe-relevant morphisms are referred to as safe.

Let s: ˘Σ → Σ be a signature morphism and B a Σ-structure. Then, B(s) is the smallest subalgebra of B for signature s( ˘Σ). Observe that B_m⁰ 0(ˆh) ⊆ h˙_m⁰(B_h(m⁰₎) whenever ˆh∈S⁰ and his an interpretation system morphism.

Definition 2.14 Anassignmentover a Σ-structureBis a mapα: Ξ→B. The assignmentαis said to besafefor a set of formulae Γ⊆L(Σ) iffα(s( ˘ξ))∈B(s) for everys: ˘Σ→Σ in S and s( ˘ξ)∈Γ.

5Recall that the operations⊗,u,tandAare defined as abbreviations: b1⊗b2= ( b1⊕ b2),b1Ab2= ( b1)⊕b2,b1tb2= (b1⊗( b2))⊕b2,b1ub2= (b1⊕( b2))⊗b2and>= ⊥ The typical multi-valued algebra with three elements isB ={⊥,1/2,>}and operations andAare defined as: b= 1−bandb1Ab2is 1/2 for the pairsb1= 1/2 andb2=⊥,b1=>

andb2= 1/2, is⊥for the pairb1=>,b2=⊥and is>otherwise.

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Safe assignments show the relevance of having the componentSin signatures and will be relevant when defining the entailment.

Definition 2.15 The interpretation of formulae over a Σ-structure B and an assignmentα is a map [[.]]^B_α :L(Σ)→B inductively defined as follows:

• [[c]]^B_α =ν(c), wheneverc∈C₀;

• [[ξ]]^B_α =α(ξ), wheneverξ ∈Ξ;

• [[c(γ1, . . . , γk)]]^B_α = ν(c)([[γ1]]^B_α, . . . ,[[γk]]^B_α), whenever k ∈ N, c ∈ Ck and γ₁, . . . , γ_k∈L(Σ).

A formula γ is globally satisfied by B and a safe assignment α for γ, written Bαγ, iff [[γ]]^B_α ∼=>. A formulaγ is locally satisfiedbyB, a safe assignment α forγ and b∈B, writtenBαbγ, iff b≤[[γ]]^B_α.

In the context of an interpretation system, we can use [[γ]]^m_α instead of [[γ]]^A(m)α . Moreover, we write mα γ and mαb γ whenever B_mα γ and B_mαbγ, respectively. Observe that local satisfaction of a formula at a truth value b indicates that a formula is at least as true as b. And we say that an assignment is over a modelmiff α: Ξ→B_m.

Definition 2.16 A formula δ is a p-semantic consequence of a finite set of formulae Φ, written Φ p δ, iff, for every model m and safe assignment α for Φ∪{δ},mαδ whenevermαϕfor everyϕ∈Φ. A formulaδis ap-semantic consequence of a set of formulae Γ, written Γ p δ, iff there is a finite set Φ contained in Γ such that Φp δ.

Definition 2.17 A formula δ is a d-semantic consequence of a finite set of formulae Φ, written Φd δ, iff mαb δ whenever mαb ϕ for every ϕ∈ Φ, m ∈ M, safe assignment α over m for Φ∪ {δ} and b ∈ B_m. A formula δ is a d-semantic consequence of a set of formulae Γ, written Γd δ, iff there is a finite set Φ contained in Γ such that Φdδ.

Proposition 2.18 Let Φ be a finite set of formulae and δ a formula. Then Φ d δ iff u_ϕ∈Φ[[ϕ]]^m_α ≤ [[δ]]^m_α for every model m ∈ M and safe assignment α over m for Φ∪ {δ}.

A signature morphism ˆh can be extended to a map ˆh^∗ between formulae:

ˆh^∗(c) = ˆh(c) for c ∈ C0, ˆh^∗(ξ) = ˆh(ξ), ˆh^∗(ϕ1&ϕ2)) = ˆh^∗(ϕ1)&⁰ˆh^∗(ϕ2) and ˆh^∗(c(ϕ1, . . . , ϕ_k)) = ˆh(c)(ˆh^∗(ϕ1), . . . ,ˆh^∗(ϕ_k)). Below, ˆh is used for the map ˆh^∗. We show below that p and d semantic entailments are preserved by some kind of morphisms. Before giving the result we need a lemma relating denotations of formulae in one signature with their counterparts in another signature.

Lemma 2.19 Let h :I → I⁰ be an interpretation system morphism such that

¨h_m⁰ is surjective for every m⁰ ∈M⁰ and α⁰ is an assignment over m⁰. Then:

• [[ˆh(ξ)]]^m_α0⁰ ∼=m⁰ h˙m⁰([[ξ]]^h(m

0)

h(α⁰)) whenever α⁰ is safe for ˆh(ξ) and ˆh∈S⁰;

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• [[ˆh(γ)]]^m_α0⁰ ∼=m⁰ h˙_m⁰([[γ]]^h(m_h(α0⁰)⁾), for every γ including at least a connective fromC where h(α⁰)(ξ) = ¨h_m⁰(α⁰(ˆh(ξ))).

Proposition 2.20 Let h :I → I⁰ be an interpretation system morphism such that ¨h_m⁰ is surjective for every m⁰ in M⁰ and hˆ ∈ S⁰ whenever Γ∪ {δ} has meta-variables. Then (1) ˆh(Γ) ⁰p ˆh(δ) whenever Γ p δ and (2) ˆh(Γ)⁰_d ˆh(δ) whenever Γdδ.

Proof: Observe that if α⁰ is a safe assignment over m⁰ for ˆh(ϕ) then the as- signmenth(α⁰) over h(m⁰) as defined in Lemma 2.19 is safe forϕ.

We only show claim (2). Let m⁰ be in M⁰ and α⁰ be an assignment over m⁰ safe for ˆh(Γ∪ {δ}). Assume Γ d δ. Then there exists a finite set Φ of Γ withu_ϕ∈Φ[[ϕ]]^m_α ≤[[δ]]^m_α for every modelm inM and assignmentα overmsafe for Φ∪ {δ}. So u_ϕ∈Φ[[ˆh(ϕ)]]^m_α0⁰ ∼= u_ϕ∈Φh˙_m⁰([[ϕ]]^h(m_h(α0⁰)⁾) = ˙h_m⁰(u_ϕ∈Φ[[ϕ]]^h(m_h(α0⁰)⁾) ≤ h˙m⁰([[δ]]^h(m

0)

h(α⁰))∼= [[ˆh(δ)]]^m_α0⁰. Therefore ˆh(Γ)dˆh(δ). QED As we shall see in Section 3, in the modulated fibring the morphisms that relate interpretation systems do have the required properties.

2.4 Modulated fibring of interpretation systems

The idea is that each model in the modulated fibring of I⁰ and I⁰⁰ will be a pairhm⁰, m⁰⁰iwherem⁰ is a model ofI⁰ andm⁰⁰ is a model ofI⁰⁰. Moreover the truth values in the algebra ofhm⁰, m⁰⁰ishould be the union of the truth values in the algebras of m⁰ and m⁰⁰. However, for denotations of formulae we need some relationship between the truth values ofm⁰ and m⁰⁰ for everym⁰ and m⁰⁰. Such a relationship is established by a bridge.

Definition 2.21 A bridge between interpretation systems I⁰ and I⁰⁰ is a diagram β = hf⁰ : ˘I → I⁰, f⁰⁰ : ˘I → I⁰⁰i in Int such that ˆf⁰, ˆf⁰⁰, ˙f_m⁰ 0 and ˙f_m⁰⁰00

are injective maps and ¨f_m⁰ 0 and ¨f_m⁰⁰00 are surjective maps for everym⁰∈M⁰ and m⁰⁰∈M⁰⁰, respectively.

Before defining modulated fibring, we introduce an auxiliary category and two functors.

Prop/Definition 2.22 The category poFam has pushouts. The objects are families of pre-orders with finite meets of the form P = {hP_i,≤_ii}_i∈I and the morphisms h : {hP_i,≤_ii}_i∈I → {hP_i⁰0,≤⁰_i0i}_i⁰_∈I⁰ are pairs hh,hi˙ such that h : I⁰ →I is a map andh˙ ={h˙_i⁰ :P_h(i⁰₎→P_i⁰0}_i⁰_∈I⁰ is a family of monotonic maps.

Proof: Let β=hhf⁰,f˙⁰i: ˘P →P⁰,hf⁰⁰,f˙⁰⁰i: ˘P →P⁰⁰i. Then the pairhhg⁰,g˙⁰i: P⁰ → P,hg⁰⁰,g˙⁰⁰i : P⁰⁰ → Pi, where P = {hP_i,≤_ii}_i∈I, I = {hi⁰, i⁰⁰i : f⁰(i⁰) = f⁰⁰(i⁰⁰), i⁰ ∈ I⁰, i⁰⁰ ∈I⁰⁰},g⁰(hi⁰, i⁰⁰i) = i⁰,g⁰⁰(hi⁰, i⁰⁰i) = i⁰⁰, and hhP_i,≤_ii,g˙⁰_i,g˙⁰⁰_ii is a pushout of ˙f_g⁰0(i) and ˙f_g⁰⁰00(i) in the category of pre-orders with finite meets for

eachi∈I, is a pushout ofβ inpoFam. QED

Let Sg:Int→ Sigbe the functor such that Sg(I) = Σ andSg(h) = ˆh and poF : Int→ poFam be the functor such that poF(I) = {hB_m,≤_mi}_m∈M and poF(h) =hh,hi. We are now ready to show that the category˙ Inthas pushouts.

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Prop/Definition 2.23 The modulated fibring of interpretation systemsI⁰ and I⁰⁰ by a bridge β is a pushout of β in Int.

Proof: Letβ =hf⁰ : ˘I → I⁰, f⁰⁰ : ˘I → I⁰⁰i. Consider hg⁰ :I⁰→ I, g⁰⁰:I⁰⁰→ Ii defined as follows:

• hˆg⁰ : Σ⁰ →Σ,ˆg⁰⁰: Σ⁰⁰→Σi is a pushout inSigof Sg(β);

• hhg⁰,g˙⁰i : poF(I⁰) → poF(I),hg⁰⁰,g˙⁰⁰i :poF(I⁰⁰) → poF(I)i is a pushout inpoFamof poF(β);

• hB_hm⁰_,m⁰⁰_i,≤_hm⁰_,m⁰⁰_ii= (poF(I))_hm⁰_,m⁰⁰_i;

• A(hm⁰, m⁰⁰i) =hB_hm⁰_,m⁰⁰_i,≤_hm⁰_,m⁰⁰_i, νhm⁰,m⁰⁰ii;

• ¨g_hm⁰ 0,m⁰⁰i( ˙g_hm⁰ 0,m⁰⁰i(b⁰)) =b⁰;

• ¨g_hm⁰ 0,m⁰⁰i( ˙g_hm⁰⁰ 0,m⁰⁰i(b⁰⁰)) = ˙f_m⁰ 0( ¨f_m⁰⁰00(b⁰⁰));

• ¨g_hm⁰ 0,m⁰⁰i( ˙g_hm⁰ 0,m⁰⁰i(b⁰)u_hm⁰_,m⁰⁰_ig˙_hm⁰⁰ 0,m⁰⁰i(b⁰⁰)) =

¨

g_hm⁰ 0,m⁰⁰i( ˙g_hm⁰ 0,m⁰⁰i(b⁰))u⁰_m0 ¨g⁰_hm0,m⁰⁰i( ˙g⁰⁰_hm0,m⁰⁰i(b⁰⁰));

• ν_hm⁰_,m⁰⁰_i(ˆg⁰(c⁰))(~b) = ˙g_hm⁰ 0,m⁰⁰i(ν_m⁰ 0(c⁰)(¨g⁰_hm0,m⁰⁰i(~b)));

• ¨g_hm⁰⁰ 0,m⁰⁰i and ν_hm⁰_,m⁰⁰_i(ˆg⁰⁰(c⁰⁰)) defined in a similar way.

We have to check that hI, g⁰, g⁰⁰i is a pushout in Int of f⁰ and f⁰⁰. For this purpose we considerm⁰ ∈M⁰ and m⁰⁰ ∈M⁰⁰ and for the sake of simplification will omit the subscripts involving bothm⁰ and m⁰⁰. Moreover we will consider thatf⁰(m⁰) =f⁰⁰(m⁰⁰) = ˘m.

1. ¨g⁰ (also ¨g⁰⁰) is well defined. On one hand, ¨g⁰( ˙g⁰( ˙f⁰(˘b))) ∼= ˙f⁰(˘b) using the definition of g⁰ and on the other hand, ¨g⁰( ˙g⁰⁰( ˙f⁰⁰(˘b))) ∼= ˙f⁰( ¨f⁰⁰( ˙f⁰⁰(˘b))) ∼= ˙f⁰(˘b) using the same definition and surjectivity of ¨f⁰⁰.

2. ¨g⁰ (also ¨g⁰⁰) is a monotonic map.

Observe that≤is lf p(∆, D₀) whereD₀ includes:

• g˙⁰(≤⁰) and ˙g⁰⁰(≤⁰⁰);

• the pairs ˙g⁰(b⁰)ug˙⁰⁰(b⁰⁰)≤g˙⁰(b⁰) for everyb⁰ and b⁰⁰;

• the pairs ˙g⁰(b⁰)ug˙⁰⁰(b⁰⁰)≤g˙⁰⁰(b⁰⁰) for everyb⁰ andb⁰⁰;

• b≤g˙⁰(b⁰)ug˙⁰⁰(b⁰⁰) whenever b≤g˙⁰(b⁰),b≤g˙⁰⁰(b⁰⁰) and bis ˙g⁰( ˙f⁰(˘b));

• g˙⁰(b⁰₁)ug˙⁰⁰(b⁰⁰₁) ≤ g˙⁰(b⁰₂)ug˙⁰⁰(b⁰⁰₂) whenever ˙g⁰(b⁰₁) ≤ g˙⁰(b⁰₂) and ˙g⁰⁰(b⁰⁰₁) ≤

˙ g⁰⁰(b⁰⁰₂);

and ∆ :℘B² →℘B²is such that ∆(D) is the one-step transitive closure. There- fore ∆ is extensive and monotonic. We prove that ¨g⁰(b1) ≤⁰ ¨g⁰(b2) whenever b1≤b2∈∆^µ(D0) by induction.

Base: µ= 0.

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(i) Assume that b₁ and b₂ are either ˙g⁰(b⁰₁) and ˙g⁰(b⁰₂) for some b⁰₁, b⁰₂ ∈ B⁰ or

˙

g⁰⁰(b⁰⁰₁) and ˙g⁰⁰(b⁰⁰₂) for someb⁰⁰₁, b⁰⁰₂ ∈B⁰⁰. Then ¨g⁰(b1)≤⁰ ¨g⁰(b2) by definition of≤ and using the fact that ¨g⁰ and ¨g⁰⁰ are surjective.

(ii) b₁ is ˙g⁰(b⁰)ug˙⁰⁰(b⁰⁰) and b₂ is ˙g⁰(b⁰). Then ¨g⁰(b₂) = b⁰ and ¨g⁰(b₁) is b⁰ u⁰ f˙⁰( ¨f⁰⁰(b⁰⁰)) and sob⁰u⁰f˙⁰( ¨f⁰⁰(b⁰⁰))≤⁰ b⁰.

(iii) b₁ is ˙g⁰( ˙f⁰(˘b)) = ˙g⁰⁰( ˙f⁰⁰(˘b)) and b₂ is ˙g⁰(b⁰)ug˙⁰⁰(b⁰⁰) with ˙f⁰(˘b) ≤⁰ b⁰ and f˙⁰⁰(˘b) ≤⁰⁰ b⁰⁰ (therefore ˘b≤˘f¨⁰⁰(b⁰⁰)). Then ¨g⁰( ˙g⁰( ˙f⁰(˘b)))∼=⁰ f˙⁰(˘b) and ¨g⁰( ˙g⁰⁰(b⁰⁰))∼=⁰ f˙⁰( ¨f⁰⁰(b⁰⁰)). Hence ˙f⁰(˘b)≤⁰b⁰ and ˙f⁰(˘b)≤⁰f˙⁰( ¨f⁰⁰(b⁰⁰)).

(iv) b1 is ˙g⁰(b⁰₁) ug˙⁰⁰(b⁰⁰₁) and b2 is ˙g⁰(b⁰₂) ug˙⁰⁰(b⁰⁰₂) with ˙g⁰(b⁰₁) ≤ g˙⁰(b⁰₂) and

˙

g⁰⁰(b⁰⁰₁) ≤ g˙⁰⁰(b⁰⁰₂). So b⁰₁ ≤⁰ b⁰₂, b⁰⁰₁ ≤⁰⁰ b⁰⁰₂ and ˙f⁰( ¨f⁰⁰(b⁰⁰₁)) ≤⁰ f˙⁰( ¨f⁰⁰(b⁰⁰₂)). Then

¨

g⁰( ˙g⁰(b⁰₁)) ≤⁰ g¨⁰( ˙g⁰(b⁰₂)) and ¨g⁰( ˙g⁰⁰(b⁰⁰₁)) ≤⁰ g¨⁰( ˙g⁰⁰(b⁰⁰₂)). Therefore ¨g⁰( ˙g⁰(b⁰₁))u⁰

¨

g⁰( ˙g⁰⁰(b⁰⁰₁))≤⁰g¨⁰( ˙g⁰(b⁰₂))u⁰g¨⁰( ˙g⁰⁰(b⁰⁰₂)).

Step: µ=+ 1.

Letb be such that b1 ≤b, b ≤b2 ∈D. By the induction hypothesis ¨g⁰(b1) ≤⁰

¨

g⁰(b) and ¨g⁰(b)≤⁰ ¨g⁰(b2) and so by transitivity of≤⁰ we have ¨g⁰(b1)≤⁰g¨⁰(b2).

Step: µ is a limit ordinal. Straightforward.

3. The preservation of meets by ¨g⁰ and ¨g⁰⁰ is again straightforward.

4. f¨⁰(¨g⁰(b)) ∼= ¨f⁰⁰(¨g⁰⁰(b)): Let b be ˙g⁰(b⁰). Then ¨f⁰(¨g⁰( ˙g⁰(b⁰))) ∼= ¨f⁰(b⁰) and f¨⁰⁰(¨g⁰⁰( ˙g⁰(b⁰)))∼= ¨f⁰⁰( ˙f⁰⁰( ¨f⁰(b⁰))) and so ¨f⁰⁰( ˙f⁰⁰( ¨f⁰(b⁰)))∼= ¨f⁰(b⁰) since ¨f⁰⁰ is surjective. The other cases follow straightforwardly.

5. ν(ˆg⁰( ˆf⁰(˘c)))(~b)∼= ˙g⁰(ν⁰( ˆf⁰(˘c))(¨g⁰(~b)))∼= ˙g⁰( ˙f⁰(˘ν(˘c)( ¨f⁰(¨g⁰(~b)))))∼=

˙

g⁰⁰( ˙f⁰⁰(˘ν(˘c)( ¨f⁰⁰(¨g⁰⁰(~b)))))∼= ˙g⁰⁰(ν⁰⁰( ˆf⁰⁰(˘c))(¨g⁰⁰(~b)))∼=ν(ˆg⁰⁰( ˆf⁰⁰(˘c)))(~b).

6. ¨g⁰ is left adjoint of ˙g⁰ (¨g⁰⁰ is left adjoint of ˙g⁰⁰).

(i) b ≤ g˙⁰(¨g⁰(b)): consider the case of b being ˙g⁰⁰(b⁰⁰): b⁰⁰ ≤⁰⁰ f˙⁰⁰( ¨f⁰⁰(b⁰⁰)), then ˙g⁰⁰(b⁰⁰) ≤ g˙⁰⁰( ˙f⁰⁰( ¨f⁰⁰(b⁰⁰))), hence ˙g⁰⁰(b⁰⁰) ≤ g˙⁰( ˙f⁰( ¨f⁰⁰(b⁰⁰))) and so ˙g⁰⁰(b⁰⁰) ≤

˙

g⁰(¨g⁰( ˙g⁰⁰(b⁰⁰))). (ii) ¨g⁰( ˙g⁰(b⁰))≤b⁰: straightforward.

7. Universal property. Let h⁰ : I⁰ → I⁰⁰⁰ and h⁰⁰ : I⁰⁰ → I⁰⁰⁰ be interpretation system morphisms such thath⁰◦f⁰ =h⁰⁰◦f⁰⁰.

Existence. ˆhis the unique morphism inSigsuch that ˆh◦gˆ⁰ = ˆh⁰and ˆh◦ˆg⁰⁰= ˆh⁰⁰; h = hh⁰, h⁰⁰i; ˙h is the unique morphism in poFam such that ˙h◦g˙⁰ = ˙h⁰ and h˙ ◦g˙⁰⁰ = ˙h⁰⁰; and ¨h_m⁰⁰⁰(b⁰⁰⁰) =^def g˙⁰_h(m000)(¨h⁰_m000(b⁰⁰⁰)) ug˙_h(m⁰⁰ 000)(¨h⁰⁰_m000(b⁰⁰⁰)). So,

¨

g⁰(¨h(b⁰⁰⁰))∼= ¨g⁰( ˙g⁰(¨h⁰(b⁰⁰⁰)))u¨g⁰( ˙g⁰⁰(¨h⁰⁰(b⁰⁰⁰)))∼= ¨h⁰(b⁰⁰⁰)uf˙⁰( ¨f⁰⁰(¨h⁰⁰(b⁰⁰⁰)))∼= ¨h⁰(b⁰⁰⁰)u f˙⁰( ¨f⁰(¨h⁰(b⁰⁰⁰)))∼= ¨h⁰(b⁰⁰⁰). We can also conclude that ¨his monotonic and preserves finite meets and that ¨his left adjoint to ˙h. ν⁰⁰⁰(ˆh(ˆg⁰(c⁰)))(~b⁰⁰⁰)∼=ν⁰⁰⁰(ˆh⁰(c⁰))(~b⁰⁰⁰)∼= h˙⁰(ν⁰(c⁰)(¨h⁰(~b⁰⁰⁰)))∼= ˙h( ˙g⁰(ν⁰(c⁰)(¨g⁰(¨h(~b⁰⁰⁰)))))∼= ˙h(ν(ˆg⁰(c⁰))(¨h(~b⁰⁰⁰))).

Uniqueness. Assume thatk:I → I⁰⁰⁰ is a morphism such that k◦g⁰ =h⁰ and k◦g⁰⁰ =h⁰⁰. We want to show thatk=h that is ¨k= ¨h. We start by showing that ¨k(b⁰⁰⁰) = ˙g⁰(¨g⁰(¨k(b⁰⁰⁰)))ug˙⁰⁰(¨g⁰⁰(¨k(b⁰⁰⁰))). Assume that ¨k(b⁰⁰⁰) = ˙g⁰(b⁰). Note that ˙g⁰(b⁰) ≤ g˙⁰( ˙f⁰( ¨f⁰(b⁰))) ∼= ˙g⁰⁰( ˙f⁰⁰( ¨f⁰(b⁰))) ∼= ˙g⁰⁰(¨g⁰⁰( ˙g⁰(b⁰))). Then ¨k(b⁰⁰⁰) =

˙

g⁰(¨g⁰(¨k(b⁰⁰⁰)))ug˙⁰⁰(¨g⁰⁰(¨k(b⁰⁰⁰))). The other cases follow in a straightforward manner. Since ¨g⁰(¨k(b⁰⁰⁰)) = ¨h(b⁰⁰⁰) and ¨g⁰⁰(¨k(b⁰⁰⁰)) = ¨h(b⁰⁰⁰) thenk=h. QED Examples and the collapsing problem We give some examples of modulated fibring namely showing how the collapse can be avoided. We start by a description of the most common collapse and then give a result stating how the

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bridge can be chosen to avoid the collapse when no constructors are shared.

Definition 2.24 In the modulated fibringhg⁰ :I⁰ → I, g⁰⁰:I⁰⁰→ Ii of I⁰ and I⁰⁰ by a bridge β,I⁰⁰ collapses toI⁰ iff there is a bijection jk:C_k⁰⁰ →C_k⁰ for all k∈IN such that

• ˆg⁰(Γ⁰) p ˆg⁰(ϕ⁰) iff ˆg⁰⁰(j⁻¹(Γ⁰)) p gˆ⁰⁰(j⁻¹(ϕ⁰)) iff Γ⁰ ⁰p ϕ⁰ and ˆg⁰(Γ⁰) d

ˆ

g⁰(ϕ⁰) iff ˆg⁰⁰(j⁻¹(Γ⁰))dˆg⁰⁰(j⁻¹(ϕ⁰)) iff Γ⁰ ⁰_dϕ⁰;

• there is a set Γ⁰⁰⊆L(Σ⁰⁰) and a formula ϕ⁰⁰ ∈L(Σ⁰⁰) such that Γ⁰⁰ 2⁰⁰_p ϕ⁰⁰ and ˆg⁰⁰(Γ⁰⁰) p gˆ⁰⁰(ϕ⁰⁰) or there is a set Γ⁰⁰ ⊆ L(Σ⁰⁰) and a formula ϕ⁰⁰ ∈ L(Σ⁰⁰) such that Γ⁰⁰2⁰⁰_dϕ⁰⁰ and ˆg⁰⁰(Γ⁰⁰)dˆg⁰⁰(ϕ⁰⁰).

We now define a specific bridge that leads to a non-collapsing situation whenever there is no sharing of constructors.

Proposition 2.25 Let I⁰,I⁰⁰ be interpretation systems such that t⁰ ∈C₀⁰, t⁰⁰∈ C₀⁰⁰, C⁰ and C⁰⁰ are in one to one correspondence and β a bridge such that C˘0 = {˘t}, C˘_k = ∅ for all k 6= 0, Ξ =˘ ∅, S˘ = ∅ M˘ = {m},˘ B_m_˘ = {>},˘ id_Σ⁰ ∈ S⁰, id_Σ⁰⁰ ∈ S⁰⁰, f⁰(m⁰) = f⁰⁰(m⁰⁰) = ˘m and f¨_m⁰ 0(b⁰) = ¨f_m⁰⁰00(b⁰⁰) = ˘> for every m⁰ ∈M⁰, m⁰⁰∈M⁰⁰, b⁰∈B_m⁰ 0 and b⁰⁰∈B_m⁰⁰00. Then the modulated fibring hg⁰ :I⁰ → I, g⁰⁰:I⁰⁰→ Ii of I⁰ and I⁰⁰ by β does not collapse.

Proof: For every model m⁰⁰ ∈M⁰⁰ all the pairs hm⁰, m⁰⁰i with m⁰ ∈M⁰ are in the modulated fibring. Therefore if Γ⁰⁰ 2⁰⁰p ϕ⁰⁰ then ˆg⁰⁰(Γ⁰⁰) 2p ˆg⁰⁰(ϕ⁰⁰) for every Γ⁰⁰ and ϕ⁰⁰ and if Γ⁰⁰2⁰⁰_dϕ⁰⁰ then ˆg⁰⁰(Γ⁰⁰)2dgˆ⁰⁰(ϕ⁰⁰) for every Γ⁰⁰ and ϕ⁰⁰. QED We say in this case that the interpretation system obtained is the unconstrained modulated fibringofI⁰andI⁰⁰. Thus, we can use this “universal” bridge for defining the modulated fibring whenever we do not want any symbols shared which is the case in most situations. Observe that in C₀⁰ and C₀⁰⁰ we can have propositional symbols.

Proposition 2.25 shows that for all cases of unconstrained modulated fibring (that is, only the verum is shared) it is possible to avoid the collapsing problem. Since idΣ⁰ ∈S⁰,idΣ⁰⁰ ∈S⁰⁰ using Proposition 2.20 we guarantee that the entailments of the component logics will be entailments in the modulated fibring. Observe also that the requirementid_Σ⁰ ∈S⁰,id_Σ⁰⁰∈S⁰⁰does not change the entailments of I⁰ and I⁰⁰. This requirement just prepares the interpretation systems for the combination. We can now instantiate Proposition 2.25 for several cases.

Example 2.26 Modulated fibring of propositional and intuitionistic logics. By choosing an adequate bridge as the one in Proposition 2.25 we can avoid the collapsing between propositional logicI⁰ and intuitionistic logic I⁰⁰. Intuition- istic logic collapses into propositional logic when the formula ((¬(¬ϕ))⇔ϕ) becomes valid which is not the case. Observe that in the modulated fibring,

˙

g⁰(B_m⁰ 0) is a Boolean algebra “equivalent” to B_m⁰ 0 and ˙g⁰⁰(B⁰⁰_m00) is a Heyting

algebra “equivalent” toB_m⁰⁰00. 4