Marcelo E. Coniglio
Institute of Philosophy and the Humanities (IFCH), and Centre for Logic, Epistemology and the History of Science (CLE) University of Campinas
(UNICAMP), Campinas, Brazil
Dedicated to Edelcio G. de Souza on the occasion of his 60th birthday
Abstract
We propose in this paper a generalization of fibring of propo- sitional logic systems within the framework of category theory.
Specifically, we generalize the categorial construction of fibring by using arbitrary colimits and limits. We prove that limits and col- imits preserve completeness (under reasonable conditions) in the category of propositional Hilbert calculi endowed with general al- gebraic semantics, generalizing a well-known result in the literature of combining logics.
1 Introduction
Combining logics is still a young subject in contemporary logic.1 It offers a natural philosophical interest, given the possibility of defining mixed logic systems in which the logical operators satisfy laws of differ- ent nature. Besides the interest coming from Philosophy, there are also pragmatical and methodological reasons which justify considering com- bined logics. For instance, in Computer Science it can be required the integration of several logic systems into a homogeneous environment, in the context of knowledge representation.
Several interesting questions in the philosophy of logic naturally arise concerning this topic: by asuming a pluralist position, are there logics which are incompatible? Is it possible to combine different logics by producing new coherent logic systems? If it is possible to compose logics, would it also be possible to decompose them? By decomposing a given logic into several fragments, would it be possible to recover the original
1For general references on combining logics see [13] and [14].
logic by combining such fragments? What kind of metaproperties of given logics can be transferred to their combinations?
As observed in [12], an illustrative and early example of combin- ing logics can be found in the well-known “ought-implies-can” thesis attributed to I. Kant, according to which, if an agent ought to do an act, then it has to be logically possible to do it. This problem could be analyzed from the perspective of combining logics, specifically con- nected to accepting properties of combining deontic and alethic logics.
In formal terms, the “ought-implies-can” thesis concerns sentences of the form Op → ♦p, where O represents the deontic “obligatory” op- erator, the diamond ♦ denotes the alethic “possibly” operator, and p is a proposition (representing an action). Thus, this principle means that if an action is obligatory then it must be possible. According to another interpretations, what Kant allegedly believed is that we can- not be obliged to do something if we are not capable of acting in that way. This would be formalized by the contrapositive of the previous for- mula, namely ¬♦p → ¬Op, meaning that “cannot-implies-has no duty to”. Formulas involving modalities of different kind (or, more generally, connectives from different logics) naturally arise when combining logics, and are called bridge principlesin [12].2
The first methods for combining logic systems wereproducts of logics, independently introduced by K. Segerberg in [27] and by V. ˇSehtman in [33]; fusion, introduced by R. Thomason in [32]; and fibring, intro- duced by D. Gabbay in [24], Observe that all of these methods were defined exclusively to combine modal logics. It should be mentioned that M. Fitting in 1969 already gave early examples of fusion of modal logics, anticipating the notion of fusion (see [23]).
Other combination mechanisms where afterwards introduced: in the context of formal software specification, M. Finger and D. Gabbay in- troducedtemporalization([22]), which was generalized in [9] towards the method called parameterization.
All of these methods are designed for creating new logic systems from given ones, with the aim of integrating different aspects of them.
This situation can appear in Computer Science, for instance in software engineering and security. Specifically, in formal specification and veri- fication of algorithms and protocols it is useful and convenient to work
2This name has been introduced in the literature to denote a statement that binds factualities to norms, which appears in the context of David Hume’s “is-ought problem”.
with several logics. This direction (or approach) to combining logic is what is called in [11] a process of splicing logics. In the terminology introduced in [14], it would be a synthesis process, in which a logic is synthesized from given ones.
However, it would be reasonable to expect that a method for com- bining logics would work in the opposite directions: hence, a logic that one wants to investigate could be decomposed into factors of lesser com- plexity. To give an example, a bimodal alethic-deontic logic could be decomposed into its alethic and deontic fragments. It would be inter- esting to see whether the given logic is the least extension of its factors, or if additional bridge principles would have to be added in order to recover the original system, a problem which has been analyzed in [16].
This approach to combining logics, in which a given logic is decomposed into (possibly) simpler factors, is what is called in [11] a process ofsplit- ting logics. According to the terminology introduced in [14], it would correspond to an analysis process, in which a given logic is analyzed into simpler components. An important method for splitting logics is possible-translations semantics, introduced by W. Carnielli in [10].
Many of the early splicing methods for combining logics mentioned above have been generalized by the categorial(a.k.a. algebraic) notion of fibring introduced by A. Sernadas, C. Sernadas and C. Caleiro in [28].
Indeed, this framework dramatically improved the scope of these tech- niques by means of (universal) categorial constructions. From this, it is possible to combine wider classes of logics besides modal logics, see for instance [28, 34, 30, 31, 7, 18, 21, 8, 16, 19, 29, 17].
The interplay between the general framework of category theory and abstract logic has shown to be extremely useful. For instance, E. de Souza, A. Costa-Leite and D. Dias consider in [20] a functor between categories of suitable logics in order to provide, in an uniform way, a natural paraconsistent expansion of any given logic. In the realm of combining logics, the work of A. Sernadas and his collaborators in for- malizing fibring for several classes of logic systems by using category theory clarifies the fact that fibring can be seen as a particular kind of colimit in the category in which the fibred logics are represented.
In fact, in the ‘classical’ approach to categorial fibring, there are two possibilities for fibring logics: to perform a free (or unconstrained) fib- ring, without sharing of connectives, or to perform aconstrainedfibring, by sharing some connectives. In terms of category theory, the former is characterized as a coproduct in the underlying category of logic systems.
The latter is constructed by a cocartesian lifting from the category of signatures. In most cases it is possible to substitute the cocartesian lift- ing by a pushout in the underlying category of logic systems (cf. [34]).
Therefore, fibring two logics, from the point of view of category theory, consists basically of coproducts or pushouts in the category in which the logic systems are represented.
The main goal of this paper is to extend the standard notion of cat- egorial fibring to arbitrary colimits and limits in appropriate categories of logic systems. In particular, when the category of logic systems is composed by systems with both semantic and syntactic consequence re- lations, it is desirable to preserve completeness through the combination process. This important question will be addresed here.
As a first approach to the question of extending fibring to other cate- gorial constructions, in this paper we concentrate our attention in three categories of logic systems: Hil, Int, and Lsp, which were introduced in [34]. The first consists of (propositional) Hilbert calculi in which the notions of local (derivations) and global (proofs) inferences are distin- guished. The second consists of the semantic (algebraic) counterpart of Hilbert calculi: the category of interpretation systems, which generalizes Kripke frames. The third category consists of logic system presentations (l.s.p.’s) where the objects are simultaneously a Hilbert calculus as well as an interpretation system. Clearly, a l.s.p. is interesting when both semantic and syntactic entailments coincide.
The framework for representing logics proposed in [34], both at the proof-theoretical level (by means of Hilbert calculi) and at the semanti- cal level (by means of interpretation systems) is cleary oriented to modal logics. Indeed, as mentioned above, two notions of semantical entailment are considered: the global ones, which deal with global truths (formu- las valid in every state or world), and the local ones, whose trueness is preserved pointwise (from world/state to world/state). Accordingly, the Hilbert calculi have two kind of inference rules: the proof rules, which represent theoremhood, and the derivation rules, apt to deal with infer- ences from premisses. This difference is clear in the context of modal logics, which are usually presented in terms of global semantics and theoremhood instead of local semantics and derivations from premisses.
The kind of semantical structures adopted in [34] is formed by alge- bras defined over subsets of the powerset ℘(U) of a given universe U. This framework is slightly more general than the one proposed in [28], in which the algebras are defined exclusively over the powerset ℘(U) of
U. It is worth noting that, in [14, Chapter 3], general algebras were considered instead of algebras defined over powersets.
In [34], general conditions which guarantee the preservation of com- pleteness of l.s.p.’s by fibring were found. The main result of this paper states that the same conditions ensure the preservation of completeness by arbitrary colimits. On the other hand, it is shown that completeness is preserved by arbitrary limits. These stimulating results suggest that it is possible to obtain complex systems of logics through specifications by diagrams, taking limits or colimits, and completeness will be preserved under reasonable assumptions.
2 The category of signatures
Propositional signatures constitute the formal basis to describe terms in abstract algebras, as well as propositional languages for logic systems.
We begin by briefly analyze the category Sig of propositional-based signatures, on which all the logic systems considered in this paper are based. We assume that the reader is familiar with the (very) basic notions from category theory, such as limits, colimits and functors.3 In what follows, Set (Cls, respectively) denotes the category of sets (classes, resp.) and functions between them.
Definition 2.1 (Propositional signatures). Apropositional signature is a denumerable family of sets C = {Ck}k∈N such that Ck∩Ci =∅ for every i6=k. A signature morphism h:C−→C0 between signatures is a familyh={hk}k∈Nof functions hk:Ck−→Ck0. The compositionh0◦h: C−→C00 of two signature morphisms h : C−→C0 and h0 : C0−→C00 is given by the family{h0k◦hk}k∈N. For any signatureCthe identity arrow idC :C−→C is given by the family{idCk}k∈N, whereidCk :Ck→Ck is the indentity function.
Observe that propositional signatures and their morphisms, with composition and identity morphisms as described in Definition 2.1, con- stitute the comma category Set/N, which is (small) complete and co- complete. That is, it contains the limits and colimits of every (small) diagram.
In order to describe propositional logic systems and their combi- nations, it will be convenient to consider schema formulas. This will
3Good general references to category theory are [25] and [26].
be done by following the approach introduced by A. Sernadas et al.
in [28], by using a set of variables (called schema variables) which acts as metavariables for denoting arbitary (concrete) propositional formu- las. Observe that schema variables are not the same as propositional variables: while the latter are concrete formulas (usually called atomic fomulas), the former are not formulas, but schema formulas. The in- tended meaning of a schema formula is that it represents an arbitary formula of a given language. The distinction between schema formulas and formulas is useful in order to describe process for combining logics, but this is also interesting for describing propositional logic systems: it avoids, for instance, the necessity of using the uniform substitution in- ference rule in the context of Hilbert calculi (see Definition 3.4 below).4 Definition 2.2 (The category Sig of propositional signatures over Ξ).
Let us fix from now on a denumerable set Ξ ={ξk : k∈N}of symbols called schema variables. The category Sig of signatures over Ξ is the full subcategory of the category of signatures described in Definition 2.1, by considering as objects signatures C such that Ck∩Ξ = ∅ for every k∈N.
It is easy to see thatSigis (small) complete and cocomplete. This is a fundamental feature of Siggiven that the limits and colimits in all the categories of logics to be considered along this paper are based on limits and colimits, respectively, of the underlying signatures over Ξ. When there is no risk of confusion, given a signature morphism h={hk}k∈N, the subscript k in hk will be omitted. Clearly a morphismh :C−→C0 is monic iff each hk is an injective map, and it is epic iff each hk is a surjective map.
Definition 2.3 (Schema formulas). Let C be a signature in Sig. Let CΞ be the propositional signature obtained from C by adding Ξ toC0.5 The set of schema formulas over C is the CΞ-algebra freely generated by C0∪Ξ, which will be denoted by L(C,Ξ). That is, L(C,Ξ) is the least CΞ-algebra satisfying:
C0∪Ξ⊆L(C,Ξ);
4In [16] the notion of schema variables for schema formulas was generalized to schema variables for contexts in formal sequent calculi, in order to consider their combination by fibring. This technique was extended in [17] to fibring of formal hypersequent calculi.
5Observe thatCΞ is not an object inSig, given that (CΞ)0∩Ξ = Ξ6=∅.
ifc∈Ck (fork >0) andγ1, . . . ,γk∈L(C,Ξ) thenc(γ1, . . . , γk)∈ L(C,Ξ).
The set of formulas over C (which is the C-algebra freely generated by C0) will be denoted by L(C).
As usual, a morphismh:C−→C0 induce a (unique) function bh:L(C,Ξ)−→L(C0,Ξ)
defined inductively as follows:
bh(ξ) =ξ ifξ ∈Ξ;
bh(c) =h0(c) if c∈C0;
bh(c(γ1, . . . , γk)) =hk(c)(bh(γ1), . . . ,bh(γk)), for c∈Ck and k >0.
2.1 Limits in Sig
Recall that a (small) diagram in a category C is a pair D = hO,Mi, where O = {Oi}i∈I is a small (possibly empty) family of objects of C, and M is a (possibly empty) set of morphisms in C contained in S
i,j∈IC(Oi, Oj).
LetD=h{Ci}i∈I,Mi be a diagram inSig. The limit ofD inSig is hC,{hi}i∈Ii where
Ck = {(ci)i∈I ∈ Q
i∈ICki : if Ci h //Cj is in M then cj = hk(ci)} for all k∈N.
hi:C−→Ci,hik((ci)i∈I) =ci.
In particular, the terminal object in Sig is 1 given by 1k = {∗k} for every k∈ N. For any signature C, the unique morphism !C :C−→1 is given by !C(c) = ∗k if c ∈ Ck. The product C of a family {Ci}i∈I is given by Ck =Q
i∈ICki. 2.2 Colimits in Sig
LetD=h{Ci}i∈I,Mibe a diagram inSig. For eachk∈Nconsider the set Ck =S
i∈ICki × {i} and the equivalence relation: (c, i)∼k (c0, j) iff there existf1, . . . ,fn,g1, . . . ,gm inM(possiblyn= 0 orm= 0) with Dom(fn) = Dom(gm) and there exists c∈Dom(fkn) =Dom(gkm) such
that c= (fk1◦ · · · ◦fkn)(c) andc0 = (gk1◦ · · · ◦gmk)(c). For (c, i) in Ck let (c, i)/∼k be its equivalence class under ∼k and let Ck =Ck/∼k be the quotient of Ck under∼k. The colimit of D inSigis hC,{hi}i∈Ii where
C ={Ck}k∈N;
hi:Ci−→C,hik(c) = (c, i)/∼k for all i∈I,k∈Nand c∈Cki. Thereforehi((f1◦ · · · ◦fn)(c)) andhj((g1◦ · · · ◦gm)(c)) always coincide inCk, whenever f1, . . . ,fn,g1, . . . , gm belongs toM andn, m≥0.
3 The category of Hilbert calculi
In this section we analyze the categoryHilof propositional-based Hilbert calculi as defined in [34]. In this context there exist two kinds of infer- ences: the local entailment and the global entailment. The latter uses proof rules (those in the set P below) while the former uses derivation rules (those in the set D below) plus proof rules applied to theorems.
These two kinds of inferences appears frequently in complex inference systems such as modal logic and first-order logic, in which thenecessita- tionrule or the generalizationrule only can be applied to theorems (see for instance [28, 34]). On the other hand, in some cases (propositional classical logic, for instance) there is no distinction between proofs and derivations. A detailed discussion on local and global inferences can be found in [14, Section 3.1].
From now on,℘fin(X) denotes the set of finite subsets of a given set X.
Definition 3.1. A Hilbert calculusis a triple hC, P, Di such that C is a signature, P ⊆℘fin(L(C,Ξ))×L(C,Ξ) andD⊆P ∩((℘fin(L(C,Ξ))\
∅)×L(C,Ξ)).
As observed at the beginning of Section 2, elements in Ξ play the rˆole of “arbitrary” formulas, which can be replaced throughsubstitution maps σ : Ξ−→L(C,Ξ) when an inference rule is applied (see Defini- tion 3.2 below). Observe that any substitution σ : Ξ−→L(C,Ξ) can be extended to a unique endomorphism ¯σ :L(C,Ξ)−→L(C,Ξ) defined inductively as follows:
σ(ξ) =¯ σ(ξ) if ξ∈Ξ;
σ(c) =¯ cifc∈C0;
σ(c(γ¯ 1, . . . , γk)) =c(¯σ(γ1), . . . ,σ(γ¯ k)), for c∈Ck and k >0.
Definition 3.2. Let Γ∪ {δ} ⊆L(C,Ξ) be a set of schema formulas.
(1) We say that δ is provable from Γ in the Hilbert calculus hC, P, Di, denoted by Γ `phC,P,Di δ, if there exists a finite sequence γ1, . . . , γn in (L(C,Ξ) such that γn isδ and, for every 1≤i≤n, either
γi∈Γ, or
there exists a rule h∆, ψi inP and a substitution map
σ : Ξ−→L(C,Ξ) such that ¯σ(∆)⊆ {γ1, . . . , γi−1} and ¯σ(ψ) =γi. We say that δ isprovablein hC, P, Di if ∅ `phC,P,Diδ.
(2) We say that δ is derivable from Γ in the Hilbert calculus hC, P, Di, denoted by Γ `dhC,P,Di δ, if there exists a finite sequence γ1, . . . , γn in (L(C,Ξ) such that γn isδ and, for every 1≤i≤n, either
γi∈Γ, or
δi is provable inhC, P, Di, or
there exists a rule h∆, ψi inD and a substitution map
σ : Ξ−→L(C,Ξ) such that ¯σ(∆)⊆ {γ1, . . . , γi−1} and ¯σ(ψ) =γi. Example 3.3 (Modal Hilbert calculi). LetC be amodal signaturesuch that C0 = V AR = {pn : n ∈ N} (a denumerable set of propositional variables); C1 ={¬,};C2 ={ ⇒ };Cn=∅ forn >2.
(1) A Hilbert calculus for the alethic modal systemKisHK =hC, PK, DKi such that PK consists of the following rules:6
h∅, ξ1 ⇒(ξ2⇒ξ1)i
h∅,(ξ1⇒(ξ2⇒ξ3))⇒((ξ1⇒ξ2)⇒(ξ1⇒ξ3))i
h∅,(¬ξ1 ⇒ξ2)⇒((¬ξ1⇒ ¬ξ2)⇒ξ1)i
h∅,(ξ1 ⇒ξ2)⇒(ξ1⇒ξ2)i
h{ξ1},{ξ1}i
h{ξ1,(ξ1⇒ξ2)}, ξ2i
6As usual, we will adopt infix notation, writing (ψ⇒ϕ), or evenψ⇒ϕ, instead of ⇒(ψ, ϕ). Moreover, we will write¬ψ andψinstead of¬(ψ) and(ψ).
and DK ={ h{ξ1,(ξ1⇒ξ2)}, ξ2i }.
(2) A Hilbert calculus for the standard deontic system KD is HKD = hC, PKD, DKDisuch thatPKD =PK∪{ h∅,ξ1⇒ ¬¬ξ1i }andDKD = DK.
Definition 3.4. The categoryHilofHilbert calculiis defined as follows:
Objects: Hilbert calculihC, P, Di.
Morphisms: A morphism h : hC, P, Di−→hC0, P0, D0i in Hil is given by a morphism h:C−→C0 inSig such that:
– hΓ, δi ∈P implies thatbh(Γ)`phC0,P0,D0ibh(δ);
– hΓ, δi ∈D implies thatbh(Γ)`dhC0,P0,D0ibh(δ).
Composition and identity arrows: Inherited from Sig.
Definition 3.5. The forgetful functorN:Hil−→Sigis given by N(hC, P, Di) =C and N(h) =h.
Proposition 3.6. The forgetful functorNhas a left adjointN:Sig−→Hil.
Proof. For every signature C consider the Hilbert calculus N(C) = hC,∅,∅i. ThenhidC,N(C)iisN-universal forC. For ifH0is a Hilbert cal- culus andh:C−→N(H0) is a morphism inSigthenh∗ =h:N(C)−→H0 is the unique morphism in Hilsuch that
C idC//
h ##
N(N(C))
N(h∗)
N(H0) commutes inSig.
This means that, if Hi = hCi, Pi, Dii is a Hilbert calculus and fi : C−→Ci is a monic in Sig (for i= 1,2) then the pushout in Hil of the diagram
N(C)
f1
||
f2
""
H1 H2
is the fibring of H1 and H2 constrained by fi : C−→Ci (i = 1,2) (cf.
[6] and [34]). Thus, unconstrained fibrings (when C = ∅) are coprod- ucts, and constrained fibrings are pushouts (cf. [28] and [34]). We will prove now the existence in Hil of arbitrary (small) limits and colimits, extending the notion of fibring in Hil.
3.1 Limits in Hil
As discussed in Section 1, there is a way of decomposing a logic system by a splitting or an analysis process. This kind of process could be rep- resented by a limit in the corresponding category of logics. In particular, this could be done in the category Hilof Hilbert calculi. With this aim in mind, in this section it will be proven that Hil is (small) complete, by showing how to construct the limit of any given diagram.
LetD=h{Hi}i∈I,Mibe a diagram inHil, whereHi=hCi, Pi, Dii for eachi∈I. The limit ofDinHilishH,{hi}i∈IiwhereH=hC, P, Di is defined as follows:
hC,{hi}i∈Ii is the limit inSig of h{Ci}i∈I,Mi;
P = {hΓ, δi ∈ ℘fin(L(C,Ξ))×L(C,Ξ) : bhi(Γ) `pH
i bhi(δ) for all i∈I};
D={hΓ, δi ∈ (℘fin(L(C,Ξ))\ ∅)×L(C,Ξ) : bhi(Γ)`dH
i bhi(δ) for all i∈I}.
Observe thatD⊆P, henceH is indeed a Hilbert calculus.
Proposition 3.7. The pair hH,{hi}i∈Ii defined above is the limit in Hil of D.
Proof. Note that each hi : H−→Hi is in fact a morphism in Hil such that h◦hi =hj for all h :Hi−→Hj in M. Let H0 =hC0, P0, D0i be a Hilbert calculus and letgi :H0−→Hifor everyi∈I such thath◦gi=gj for all h :Hi−→Hj inM. Since C is the limit in Sig of h{Ci}i∈I,Mi there exists an unique morphismh0 :C0−→CinSigsuch thathi◦h0 =gi for all i∈I, therefore
L(C0,Ξ) bh0 //
bgi %%
L(C,Ξ)
bhi
L(Ci,Ξ)
commutes inSetfor alli∈I. LethΓ, δi ∈P0. Sincegi :H0−→HiinHil then bgi(Γ) `pH
i bgi(δ) for each i ∈I, that is, bhi(bh0(Γ)) `pH
i bhi(bh0(δ)) for eachi∈I. By definition ofP we infer thatbh0(Γ)`pH bh0(δ). Analogously, ifhΓ, δi ∈D0 thenbh0(Γ)`dH bh0(δ). Hence h0:H0−→H is a morphism in Hil such that
H0 h0 //
gi H
hi
Hi
commutes in Hil for all i ∈ I. Clearly h0 is unique, by the universal property of C, thusH is the limit in Hil ofD.
Corollary 3.8. Let D be a diagram in Hil and let H = hC, P, Di be its limit with morphisms hi, for i∈ I. Let Γ∪ {δ} ⊆L(C,Ξ) be a set of schema formulas. Then Γ `pH δ iffbhi(Γ) `pH
i bhi(δ) for all i ∈I, and Γ`dH δ iffbhi(Γ)`dH
i bhi(δ) for alli∈I.
Limits of Hilbert calculi can be interesting, as the following examples show.
Example 3.9. As a particular case of the construction of limits given in the proof of Proposition 3.7, the product of a family {Hi}i∈I inHil is hH,{hi}i∈Ii whereH =hC, P, Di is defined as follows:
hC,{hi}i∈Ii is the product inSigof {Ci}i∈I;
P = {hΓ, δi ∈ ℘fin(L(C,Ξ))×L(C,Ξ) : bhi(Γ) `Hpi bhi(δ) for all i∈I};
D={hΓ, δi ∈ (℘fin(L(C,Ξ))\ ∅)×L(C,Ξ) : bhi(Γ)`Hdi bhi(δ) for all i∈I}.
In particular, the terminal object in Hilis H1=h1, P1, D1i where 1 is the terminal signature in Sig (see the end of Subsection 2.1) and
P1=℘fin(L(1,Ξ))×L(1,Ξ);
D1= (℘fin(L(1,Ξ))\ ∅)×L(1,Ξ).
For any H =hC, P, Di, the unique morphism !H :H−→H1 is given by
!C :C−→1.
Remark 3.10. From the perspective of combining logics, a product of Hibert calculi could be seen as a splitting of the product into its factors, the canonical projections beingtranslationsbetween them. The relationship between products of logics and splitting logics by means of traslations between them was already investigated in [4] (see also [5]).
The latter establishes a link between category theory and the splitting method of possible-translations semantics mentioned in Section 1.
Example 3.11. Consider the well-known hierarchy of da Costa’s para- consistent calculi C ={Cn}n∈N, where C0 is propositional classical logic (see, for instance, [15]). By the very definition, T h(C0) ⊃T h(C1)⊃ · · ·, whereT h(Cn) denotes the set of theorems ofCn. An interesting question is how to axiomatize the so-called limit of the hierarchy C, a calculus Clim such thatT h(Clim) =T
n∈NT h(Cn) (see [15]). However, it is inter- esting to note that in factClim appears inHilas the limit ofC(together with the respective embeddings). Consider for each n∈Nthe signature C0n = {pnk : k ∈ N}, C1n = {¬n}, C2n = {⇒n,∧n,∨n}, and Ckn = ∅ if k > 2, as well as the morphism gn : Cn+1−→Cn, g0n(pn+1k ) = pnk, g1n(¬n+1) =¬n andg2n(cn+1) =cnfor all cn+1∈C2n+1. Define a Hilbert calculus Hn = hCn, Pn, Dni corresponding to Cn (identifying local and global inferences, cf. [7]). Thus gn :Hn+1−→Hn is a morphism in Hil and D = h{Hn}n∈N,{gn}n∈Ni is a diagram with limit H = hC, P, Di such that C0 ={pk : k ∈ N}, C1 ={¬}, C2 = {⇒,∧,∨}, Ck =∅ if k > 2, with the obvious morphisms hn :H−→Hn, c7→ cn. Clearly the limitH representsClim (compare with Corollary 7.3 in [11]).
3.2 Colimits in Hil
By duality, and taking into account the discussion in Section 1, it makes sense to consider now the other way of combining logics, by a splicing or a synthesis process. Such a process could be represented by a colimit in the corresponding category of logics, generalizing so the notion of categorial fibring. In particular, this process could be done in the category Hil.
Then, as a natural counterpart of what was done in Subsection 3.1, in this section it will be proven thatHil is (small) cocomplete. The proof will be constructive, that is, by showing how to construct the colimit of any given diagram.
LetD=h{Hi}i∈I,Mibe a diagram inHil, whereHi=hCi, Pi, Dii for each i ∈ I. The colimit of D in Hil is hH,{hi}i∈Ii where H = hC, P, Di is defined as follows:
hC,{hi}i∈Ii is the colimit inSigof h{Ci}i∈I,Mi;
P ={hbhi(Γ),bhi(δ)i ∈℘fin(L(C,Ξ))×L(C,Ξ) : i∈I and hΓ, δi ∈ Pi};
D = {hbhi(Γ),bhi(δ)i ∈ (℘fin(L(C,Ξ))\ ∅)×L(C,Ξ) : i ∈ I and hΓ, δi ∈Di}.
Clearly D⊆P, and so H is indeed a Hilbert calculus.
Proposition 3.12. hH,{hi}i∈Ii defined above is the colimit in Hil of D.
Proof. Note that each hi : Hi−→H is in fact a morphism in Hil, by definition of H. Moreover, if h :Hi−→Hj is inM then hj ◦h =hi in Sig and then, by definition of Hil, the diagram
Hi hi //
h H
Hj
hj
OO
commutes in Hil. Let H0 = hC0, P0, D0i be a Hilbert calculus and let gi :Hi−→H0 for every i∈ I such that gj◦h =gi for all h :Hi−→Hj in M. Since C is the colimit in Sig of h{Ci}i∈I,Mi there exists an unique morphism h0 : C−→C0 in Sig such that h0 ◦ hi = gi for all i ∈ I, therefore bh0◦bhi = bgi for all i ∈ I. Let hbhi(Γ),bhi(δ)i ∈ P such that hΓ, δi ∈ Pi. Since gi :Hi−→H0 in Hil then gbi(Γ) `pH0 bgi(δ), that is, bh0(bhi(Γ)) `pH0 bh0(bhi(δ)). Analogously, if hbhi(Γ),bhi(δ)i ∈D such that hΓ, δi ∈Dithenbh0(bhi(Γ))`Hd0 bh0(bhi(δ)), thush0 :H−→H0 is a morphism inHil such that
Hi g
i //
hi H0
H
h0
OO
commutes in Hil for all i ∈ I. Clearly h0 is unique, by the universal property of C, thusH is the colimit inHil of D.
Corollary 3.13. The categoryHilis (small) complete and cocomplete.
4 The category of interpretation systems
In this section we analyze the categoryInt of Interpretation systems as defined in [34]. This is the semantic counterpart of Hil.
Definition 4.1. Let C be a signature. A C-structure is a triple S = hU,B, νisuch thatUis a non-empty set,Bis a non-empty subset of℘(U) and ν = {νk}k∈N is a family of maps νk : Ck−→B(Bk). We denote by Str(C) the class of allC-structures. IfhU,B, νi,hU0,B0, ν0i ∈Str(C) we say that they are isomorphicif there exists a bijection f :U−→U0 such that f(B) = B0 and f(νk(c)(~b)) =νk0(c)(f(~b)) for all k∈N,c∈Ck and
~b∈ Bk (here,f(~b) denotes (f(b1), . . . , f(bk)) whenever~b= (b1, . . . , bk)).
In this case we writehU,B, νi ∼=hU0,B0, ν0i.
Example 4.2 (KripkeC-structures). Recall the modal Hilbert calculi K and KD defined over the modal signature C in Example 3.3. Let m = hW, R, Vi be a Kripke model, that is: W is a non-empty set (of worlds) andR⊆W×W is a relation overW (theaccessibility relation).
Then, m induces a C-structure Sm = hW, ℘(W), νi called Kripke C- structure, which is defined as follows:
ν0(p) =V(p) for every propositional variablep∈V AR;
ν1(¬)(b) =W \bfor every b⊆W;
ν1()(b) ={w∈W : wRw0 implies thatw0∈b, for everyw0∈W} for every b⊆W;
ν2(⇒)(b, b0) = (W \b)∪b0 for everyb, b0 ⊆W.
If m is a Kripke model for KD (that is, R is serial, meaning that for every w ∈ W there exists some w0 ∈ W such that wRw0) then the induced C-structure Sm is called a KripkeC-structure for KD.
Definition 4.3. The categoryIntofInterpretation systemsis given by:
Objects: Triples hC, M, Ai where C is a signature, M is a class and A : M−→Str(C) is an injective map. We also assume that hC, M, Ai is closed under isomorphic images, disjoint unions and subalgebras, that is:
– if A(m) ∼= S for S ∈ Str(C) and m ∈ M then there exists m0∈M such thatS =A(m0);
– assume the notation A(m) =hUm, Bm, νmi for eachm ∈M, and let N be a subset of M such that Un ∩ Un0 = ∅ for every n 6= n0 in N. Then there exists m ∈ M such that Um = S
n∈NUn, Bm = {b ∈ ℘(Um) : b∩Un ∈ Bn for all n ∈ N} and, for every k ∈ N, c ∈ Ck and ~b ∈ Bmk, νkm(c)(~b) =S
n∈Nνkn(c)(b1∩Un, . . . , bk∩Un);
– let m ∈ M and suppose that B ⊆ Bm is a νm-subalgebra of Bm, that is: B is closed under every operation νkm for all k ∈ N. Then there exists m0 ∈ M such that Um0 = Um, Bm0 =B andνkm0(c) =νkm(c)|Bk.
Morphisms: A morphism hh, Fi:hC, M, Ai−→hC0, M0, A0i inInt is given by a morphism h : C−→C0 in Sig and a function F : M0−→M such that:
– UF(m0) =Um0 andBF(m0)=Bm0 for everym0 ∈M0;
– νkF(m0)(c) =νkm0(hk(c)) for everyk∈N,c∈Ck and m0 ∈M0.
Composition: hh, Fi ◦ hh0, F0i=hh◦h0, F0◦Fi.
Identity arrows: idhC,M,Ai=hidC, idMi.
Remark 4.4 (Modal interpretation systems). Recall the modal sig- nature C introduced in Example 3.3, as well as the notion of Kripke C-structure given in Example 4.2. In [34, Proposition 4.8] it was shown that every C-structure in the interpretation systemhC, M, Ai given by the closure by isomorphic images, disjoint unions and subalgebras of a given class M of Kripke C-structures is isomorphic to a Kripke C- structure. Such interpretation system will be called themodal interpre- tation system generated by M. Thus, for every m ∈ M there exists a Kripke model m0 such that A(m) is the KripkeC-structure Sm0 (up to isomorphisms).
Definition 4.5. The forgetful functorO:Int−→Sigis given byO(hC, M, Ai) =C and O(hh, Fi) =h.
Proposition 4.6. The forgetful functorOhas a left adjoint O:Sig−→Int.
Proof. Given C defineO(C) =hC, Str(C), idStr(C)i. Then hidC,O(C)i is O-universal for C. For ifhC0, M0, A0i is an interpretation system and
h:C−→C0 is a morphism inSigthen there exists an unique morphism h∗:O(C)−→hC0, M0, A0iinIntsuch that the diagram below commutes inSig.
C idC //
h %%
O(O(C))
O(h∗)
O(hC0, M0, A0i)
In fact h∗ = hh, Fi such that F :M0−→Str(C) is given as follows: let m0 ∈ M0 and A0(m0) =hUm0,Bm0, νm0i. Then F(m0) =hUm0,Bm0, νm0i such that νmk0(c) =νkm0(h(c)) for everyk∈N andc∈Ck.
This means that, ifhCi, Mi, Aii is an interpretation system andfi : C−→Ci is a monic in Sig (for i= 1,2) then the pushout in Int of the diagram
O(C)
f1∗
xx
f2∗
&&
hC1, M1, A1i hC2, M2, A2i is the fibring ofhC1, M1, A1i and hC2, M2, A2i constrained by
fi :C−→Ci, fori= 1,2 (cf. [6] and [34]). Thus, unconstrained fibrings (i.e., whenC=∅) are coproducts, and constrained fibrings are pushouts (cf. [34]). We will prove in the next subsection the existence in Int of arbitrary (small) limits and colimits, extending so the notion of fibring inInt.
4.1 Limits in Int
As it was done in Subsection 3.1 for Hilbert calculi, in this section it will be proven that the categoryInt of interpretation systems is (small) complete. According to the discussions above, limits constitute the cate- gorial realization of a splitting or an analysis process of combining logics, in this case involving logics presented semantically as interpretation sys- tems.
LetD=h{hCi, Mi, Aii}i∈I,Mi be a diagram inInt and let M1 = {h : hh, Fi ∈ M for some F},M2 ={F : hh, Fi ∈ Mfor some h}.
Consider the pair hhC, M, Ai,{hhi, Fii}i∈Ii defined as follows:
hC,{hi}i∈Ii is the limit inSig of h{Ci}i∈I,M1i;
hM,{Fi}i∈Ii is the colimit inCls ofh{Mi}i∈I,M2i;
A : M−→Str(C) is a map such that, if i ∈ I, m ∈ Mi and Ai(m) =hUm,Bm, νmithenA(Fi(m)) =hUm,Bm, νFi(m)iis given by
νkFi(m)((ci)i∈I) =νkm(ci) for all k∈Nand (ci)i∈I ∈Ck. Proposition 4.7. LetD be a diagram and consider
hhC, M, Ai,{hhi, Fii}i∈Ii as defined above. The limit of D in Int is hhC, M/∼, Ai,{hhi, Fii}i∈Ii where
M/∼ is the quotient ofM under the equivalence relationFi(m)∼ Fj(m0) iffA(Fi(m)) =A(Fj(m0));
A(Fi(m)/∼) =A(Fi(m)); and
Fi(m) =Fi(m)/∼.
Proof. First we must prove thatAis well-defined. Suppose thatFi((F1◦
· · ·◦Fr)(m)) =Fj((G1◦· · ·◦Gs)(m)) inM for somehh1, F1i, . . . ,hhr, Fri, hk1, G1i, . . . ,hks, Gsi ∈ M and m∈Dom(Fr) =Dom(Gs) =Mi1. Let m= (F1◦ · · · ◦Fr)(m)∈Mi,m0 = (G1◦ · · · ◦Gs)(m)∈Mj and (ci)i∈I ∈ Ck. By definition of Int we have that, letting c = (hr◦ · · · ◦h1)(ci) and c0 = (ks◦ · · · ◦k1)(cj) then νkm(ci) = νkm(c) and νkm0(cj) = νkm(c0).
Since (ci)i∈I ∈Ckthenc=ci1 =c0, thereforeνkm(ci) =νkm0(cj) and thus A is well-defined. Clearly A is well-defined, and it is injective. More- over, hC, M/∼, Ai is closed under isomorphic images, disjoint unions and subalgebras, therefore it is an interpretation system. If (ci)i∈I ∈Ck and m∈ Mi then νkFi(m)((ci)i∈I) =νkm(ci) = νkm(hi((ci)i∈I)), therefore hhi, Fii:hC, M/∼, Ai−→hCi, Mi, Aii is a morphism inInt such that
hC, M/∼, Aihh
i,Fii//
hhj,Fji ''
hCi, Mi, Aii
hh,Fi
hCj, Mj, Aji
commutes in Int for each hh, Fi : hCi, Mi, Aii−→hCj, Mj, Aji in M.
Now consider a family ofInt-morphismshgi, Gii:hC0, M0, A0i−→hCi, Mi, Aii for i ∈ I such that hh, Fi ◦ hgi, Gii = hgj, Gji for each hh, Fi :
hCi, Mi, Aii−→hCj, Mj, Aji inM. SincehC,{hi}i∈Iiis the limit inSig of h{Ci}i∈I,M1i there exists in Sig an unique morphism h : C0−→C given by hk(c) = (gik(c))i∈I, such that hi◦h = gi for all i ∈ I. Since hM,{Fi}i∈Ii is the colimit in Cls ofh{Mi}i∈I,M2i there exists inCls an unique map G : M−→M0 given by G(Fi(m)) = Gi(m), such that G◦Fi =Gi for all i∈I. Suppose now thatA(Fi(m)) =A(Fj(m0)) = hU,B, νi. Then
(∗) νkm(ci) =νk((ci)i∈I) =νkm0(cj) for all k∈Nand (ci)i∈I∈Ck. By definition of morphism in Int we have that A0(Gi(m)) = hU,B, ν1i and A0(Gj(m0)) = hU,B, ν2i such that νk1(c) = νkm(gki(c)) and νk2(c) = νkm0(gjk(c)). Buthk(c) = (gik(c))i∈I ∈Ck thus, using (∗) above, νk1(c) = νk2(c) for all c ∈ C0k, that is, A0(Gi(m)) = A0(Gj(m0)). Since A0 is injective we infer that Gi(m) = Gj(m0), therefore the function G : M/∼−→M0 given by G(Fi(m)/∼) = Gi(m) is well-defined, and then hh, Gi : hC0, M0, A0i−→hC, M/∼, Ai is a morphism in Int which com- mutes in Int all the diagrams below.
hC0, M0, A0i hh,Gi//
hgi,Gii ''
hC, M/∼, Ai
hhi,Fii
hCi, Mi, Aii
Clearly hh, Gi is the unique morphism with this property, therefore the pair hhC, M/∼, Ai,{hhi, Fii}i∈Ii constitutes the limit of D.
4.2 Colimits in Int
Now, the results obtaned in the previous subsection will be dualized.
In [34, Prop/Definition 3.9] it was defined (and proved the existence of) the constrained and unconstrained fibring of two given interpretation systems hCi, Mi, Aii (for i = 1,2). In Proposition 3.13 of that paper it was provided a categorial characterization of unconstrained fibring, namely, when no connective is shared by the two given interpretation sys- tems: it corresponds to the coproduct ofhC1, M1, A1iandhC2, M2, A2i in the category Int. If some connectives are to be shared, then the un- constrained fibring, presented in categorial terms, is obtained through the coequalizer inIntof a suitable pair of parallel arrows (see [34, Propo- sition 3.14]). Taking into consideration that coproducts and coequalizers
are special cases of colimits, it is a natural question whether these con- structions can be generalized to arbitrary colimits, as we already done for the category Hil. This should provide new ways of combining in- terpretation systems by a splicing or synthesis process, as observed in Subsection 3.2 for Hilbert calculi. We will prove in this section thatInt is (small) cocomplete, showing how to construct the colimits.
LetD=h{hCi, Mi, Aii}i∈I,Mibe a diagram inIntand letM1and M2 given by
M1={h : hh, Fi ∈ M for some function F}, M2 ={F : hh, Fi ∈ M for someSig-morphismh}.
Consider the pair hhC, M, Ai,{hhi, Fii}i∈Ii defined as follows:
hC,{hi}i∈Ii is the colimit inSigof the diagram h{Ci}i∈I,M1i;
lethM ,{Fi}i∈Iibe the limit inClsof the diagramh{Mi}i∈I,M2i;
then M = {(mi)i∈I ∈ M : Umi = Umj and Bmi = Bmj for all i, j∈I}, andFi :M−→Miis the function given byFi((mi)i∈I) = Fi((mi)i∈I) =mi for alli∈I;
A : M−→Str(C) is the map defined as follows: let (mi)i∈I ∈ M and Ai(mi) = hUmi,Bmi, νmii for each i ∈ I; then A((mi)i∈I) = hUmi,Bmi, ν(mi)i∈Ii where
νk(mi)i∈I(hik(c)) =νkmi(c) for allk∈N,i∈I andc∈Cki. Proposition 4.8. LetD be a diagram. ThenhhC, M, Ai,{hhi, Fii}i∈Ii as defined above is the colimit of D inInt.
Proof. First we need to prove that A is well-defined and it is injective.
Suppose that hi((f1◦ · · · ◦fr)(c)) =hj((g1 ◦ · · · ◦gs)(c)) in Ck where c ∈ Cki1 and Dom(fr) = Ci1 = Dom(gs). Let c = (f1 ◦ · · · ◦fr)(c), c0 = (g1◦· · ·◦gs)(c),m= (Fr◦· · ·◦F1)(mi) andm0 = (Gs◦· · ·◦G1)(mj).
By definition of Intwe have thatνkmi(c) =νkm(c), andνkmj(c0) =νkm0(c).
But (mi)i∈I ∈M, thusm=mi1 =m0 and thenνkmi(c) =νkmj(c0), there- fore Ais well-defined. If A((mi)i∈I) =A((m0i)i∈I) then, by definition of A,
Ai(mi) =A((mi)i∈I)|hi =A((m0i)i∈I)|hi =Ai(m0i)
