The subject of combinations of logics is still a young topic incontemporary logic. Besides the pure philosophical interest offered bythe possibility of defining mixed logic systems in which distinctoperators obey logics of different nature, as for instance eroteticlogics (the logical analysis of questions) which require combiningepistemic and deontic logics, there also exist many pragmatic andmethodological reasons for considering combined logics. In fact, theuse of formal logic as a tool for knowledge representation in ComputerScience frequently requires the integration of several logic systemsinto a homogeneous environment.

Important questions in the philosophy of logic such as: “why arethere so many logics instead of just one?” (or even, instead ofnone), as for instance, raised in Epstein 1995, can be naturallycounterposed by several other questions: if there are indeed manylogics, are they excluding alternatives, or are they compatible? Is itpossible to combine different logics into coherent systems with thepurpose of using them in applications and to shed some light on theproperties of complex logics? Moreover, if we cancomposelogics, why notdecompose them? And, if a logic isdecomposed into elementary sublogics, is it possible to recover it bycombining such fragments? What kind of properties of logics can betransferred to their combinations? Questions of this kind have beenonly partially tackled in the literature, and reflect challenges to beconfronted in the evolution of this topic.

• 1. Philosophical and methodological motivations for combining logics
• 2. Splitting logics versus splicing logics
• 3. The importance of language and the presentation of logics
• 4. Methods for combining and decomposing logics
  • 4.1 Fusion and Products
  • 4.2 Fibring (or Fibring by functions)
  • 4.3 Categorial (or Algebraic) Fibring
  • 4.4 Possible-Translations Semantics
  • 4.5 Temporalization, Parameterization and Institutions
  • 4.6 New perspectives on combining logics
• 5. Lack or excess of interaction: perplexities when combining logics
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Philosophical and methodological motivations for combining logics

David Hume generated a popular controversy with his famous passages of“A Treatise of Human Nature” (see Hume 2000: Book 3, Part1, Section 1, paragraph 27) where he noted that sometimes people drawconclusions involving prescriptive statements of the form ‘oughtto be’ on the basis of descriptive statements of the form‘what is’. Hume thinks that logic used in this wayinvolves a dangerous change of subject matter. So, whether or not‘ought’ can be derived from ‘is’ has becomeone of the central questions of ethical theory, and the majority ofinterpreters hold that, for Hume, such a derivation is impossible.

With our point of view concerning combinations of logics, it isnecessary to investigate the properties of combining deontic andalethic logics: in order to perform such a jump from ‘is’to ‘ought’ some authors propose (see e.g., Schurz 1997)that what is necessary is an explicit “bridge principle”which specifically connects ‘is’ and ‘ought’.An axiom schema \(X\), following Schurz (1991), is a bridge principleiff \(X\) contains at least one schematic letter which has at leastone occurrence within the scope of an \(\obcirc\) (the deontic“obligation” operator) and at least one occurrence outsidethe scope of any \(\obcirc\). Thus, for instance, \[p \rightarrow \obcirc p\] is abridge principle representing ‘is-ought’ that which wouldappease Hume’s criticism. On the other hand, the much discussedmoral principle ‘ought-implies-can’ (controversiallyattributed to I. Kant, see Baumgardt 1946) can be formalized throughanother bridge principle: \[\obcirc p \rightarrow \lozenge p,\] where \(\lozenge\) denotes thealethic “possibility” operator.

Clearly, bridge principles do not solve any philosophical questionssuch as the ‘is-ought problem’; nonetheless, theycontribute to clarify the problem and to uncover hidden assumptions.The idea of combining logics lend clarification to questions of thiskind by making clear that, for instance, ‘is’ and‘ought’ are indeed independent notions. This is elucidatedthrough a formal analysis of the composition of the logics involved(in this case, alethic and deontic) or by decomposition of the complexlogic (in this case, bimodal) into simpler ones. In suchcircumstances, combining logics can be perceived as a tool forsimplifying problems involving heterogeneous reasoning.

The fact that ‘ought’ is not conveyed as a predicate, butas a modal operator ranging over actions or states of affairs, wasresponsible for the delay of formal treatments of this centuries-oldquestion.

Such a treatment was only possible after the development of generalmodal logic. Indeed, what we are dealing with here is a bimodal logic,which is properly treated only after a deeper understanding of thesemantical subtleties of mixing alethic and deontic logics. Moreover,according to some philosophers who have argued that it is not possibleto link ‘is’ and ‘ought’ (that is, who defendHume’s thesis that no non-trivial ‘is-ought’deductions are possible), it is mandatory to use combinations offirst-order, alethic and deontic logics (see e.g., Stuhlmann-Laeisz1983 and Schurz 1997).

A. Prior (1960), using the apparatus of contemporary modal logic,tried to characterize the distinction betweennormative andnon-normative sentences in formal terms, which enabled him todefine senses of ‘descriptive content’ versus‘normative content’. A problem, however, occurs with mixedsentences, which have both descriptive and normative components, andPrior comes up with a paradox: wherever we draw the distinctionbetween non-normative and normative sentences, unexpected inferencesfrom non-normative premises to normative conclusions may appear by amere use of laws of classical propositional logic. Consider, forinstance, the following two inferences:

• (1)“Tea-drinkingis common in England. Therefore: Either tea-drinking is common inEngland or all New Zealanders ought to learn to speakLatin”,

formalized as:

• (1′)\(\texttt{d}\,\vdash\, \texttt{d} \vee \obcirc \texttt{s} \).

and

• (2)“Tea-drinkingis common in England, or all New Zealanders ought to learn to speakLatin. Tea-drinking is not common in England. Therefore, all NewZealanders ought to learn to speak Latin”,

formalized as:

• (2′)\(\texttt{d} \vee \obcirc \texttt{s},\, \neg \texttt{d} \,\vdash\,\obcirc \texttt{s}\).

If the mixed sentence \(\texttt{d} \vee \obcirc \texttt{s}\) isconsidered to be normative, then (1) is an example of an‘is-ought’ inference, and if it is considered to benon-normative, then (2) is an example of an ‘is-ought’inference. So, one of them dichotomically represents a violation ofHume’s thesis in Prior’s terms. Prior concluded from thisparadox that Hume’s ‘is-ought thesis’ is simplyfalse (see Prior 1960: 206): onecan simply deriveconclusions which are ethical starting from premises which have noethical character.

Prior recognized however that the inferences involved in the paradoxare ethically irrelevant or trivial, but neither he nor later authorscould find a suitable definition of what it would mean by“ethical irrelevance” or “ethical triviality”attached to an inference (for a discussion on this respect see Weiss2025).

Using the semantics of modal logics, objections against thisconclusion can be raised, as for example in Karmo 1988, in the senseof separating statements between evaluative in some possible worldsand descriptive in others (while keeping their meaning).

By using concepts of combinations of languages and combinations oflogics, G. Schurz (see Schurz 1991; see also Schurz 1997) was able tostate ageneralized Hume’s thesis (GH); asobserved insubsection 4.1, this treatment is in fact a fusion of two modal logics. In(GH) a mixed sentence \(\varphi\) is derived from a set ofpurely descriptive sentences (i.e., sentences free of \(\obcirc\))only if \(\varphi\) is completely \(\obcirc\)-irrelevant (that is,predicates in \(\varphi\) within the scope of \(\obcirc\) can bereplaced by other predicatessalva valididate). Moreover, itis proven that (GH) holds in an alethic-deontic first-orderlogic \(\mathcal{L}\) if, and only if, \(\mathcal{L}\) can beaxiomatized without bridge principles.

The notion of bridge principle lies in the scope of combination oflanguages. In general, many bridge principles can be made explicitwithin modal logic, and will be relevant for analyzing relationshipsamong diverse modalities. For example, if we take necessity\(\square\) and possibility \(\lozenge\) as primitive operators, then\[\lozenge p \to \neg\square\neg p\] is an intuitionistically acceptable bridge principle,while the converse is not.

Besides Hume’s problem, another example of bimodal logic withintrinsic philosophical interest where bridge principles intervene isthe logic of physical and alethic modalities. In this logic, thelanguage permits the expression of two different notions of necessity,namely logical necessity, symbolized by \(\square\), and physicalnecessity, symbolized by \(\boxdot\).

The simplest connection between physical necessity and logicalnecessity that comprises an acceptable philosophical meaning is givenby the following bridge axiom: \[\quad\square p\to\boxdot p\] meaning that logicalnecessity is stronger than physical necessity, that is, anything thatis logically necessary is physically necessary.

The resulting logicKT\(^{\boxdot}\) is axiomatizedby the well-known axioms and rules ofKT for bothmodalities in addition to the bridge axiom above, and is semanticallycharacterized by Kripke frames with two accessibility relations,requiring that the accessibility relation for physical necessity isincluded in the other.

Not only bimodal, but multimodal (also called polymodal) logics, arestandard in the literature. A typical case is the logic of knowledge(or epistemic logic), usually endowed with modal operators\(K_{1},K_{2},\ldots,K_{m}\) representing the knowledge of \(m\)agents (or “knowers”). The formula \(K_{i}\alpha\) means“agent \(i\) knows \(\alpha\)”, and the language is ableto express, for instance, “\(i\) knows that \(j\) does not knowthat \(i\) knows \(p\)” by means of \(K_{i}\neg K_{j}K_{i}p\).No additional mixing principles are mandatory for the combined logicof many agents, but bridge axioms may, of course, be added. A detaileddiscussion about modal systems combining different modalities, as wellas their philosophical interpretations, can be found in the entry onphilosophical aspects of multi-modal logic.

The interest of studying combinations of logics may thus be seen as areflection of the pluralist view in contemporary logical research.Indeed, this kind of bridge axioms can, in principle, connectcompletely distinct logics. Van Benthem (2006) suggests that combininglogics may lead to the emergence of new phenomena, depending on themode of combination, and moreover, it may work as an inspiration (andperhaps as a model) for the study of combining epistemic notions. Heeven suggests that the compartmentalization of logic into subfields as‘modal’, ‘temporal’, ‘epistemic’,‘doxastic’, ‘erotetic’ or‘deontic’ logic has been harmful to philosophicallogic.

Combinations of logics go in the opposite direction of such acompartmentalization by contending that almost any conceptual task tobe analyzed involves immediate reasoning concerning necessity,obligation, action, time, verbal tense, knowledge, belief, etc. From aphilosophical point of view, logical combinations may be the right wayto look at philosophical issues within the theory of causation, ofaction, and so on.

The idea of looking at logic as an entirety avoiding fragmentation isnot new, and philosophers and logicians from Ramón Lull toGottfried W. Leibniz have thought of building schemes where differentlogics or logic-like mechanisms could interact and cooperate insteadof competing. In contemporary terms, the first methods for combininglogics wereproducts of logics (introduced by K. Segerberg(1973) and independently by V. Šehtman (1978)),fusion(introduced by R. Thomason (1984)) andfibring (introduced byD. Gabbay (1996a)), all of them dedicated to combining only modallogics. It is worth noting that M. Fitting (1969) gave early examplesof fusions of modal logics, anticipating the notion of fusion.

Other combination mechanisms followed, such as parameterization andtemporalization, which were more on the side of softwarespecification.

Most of these methods have been encompassed in thealgebraicfibring introduced by A. Sernadas, C. Sernadas and C. Caleiro(1999), which considerably improved the versatility of thesetechniques by means of (universal) categorial constructions, in thisway making it possible to combine wider classes of logics besidesmodal logics.

On the other hand, making heavy use of the language of categorytheory, J. Goguen and R. Burstall introduced the notion ofinstitutions as a kind of abstract model theory devoted toapplications in Computer Science (see Goguen and Burstall 1984, 1992).Institutions are also used as a mechanism for combining logics.

However, combining logics does not only mean synthesizing or composinglogics, but can also yield interesting examples that go in theopposite direction of decomposing logics (seesection 2). A paradigmatic methodology for decomposition is thepossible-translations semantics, a notion proposed inCarnielli 1990 designed to help solve the problem of assigningsemantic interpretations to non-classical logics. Examples ofpossible-translations semantics illustrate how a complex logic can beanalyzed into less complex factors. Other techniques such asdirect union of matrices andplain fibring (seeConiglio and Fernández 2005) can be considered to be methodsfor both composing and decomposing.

All of these methods open the way for a new question in the realm ofcombinations of logics. Is it possible to decompose a given logic intoelementary ones? In other words, are thereprime logicswhich, combined in an appropriate way, may produce all (or part of)the familiar logic systems? This question will be retaken insection 5.

Results on combinations of logics may quickly become too technicalwhen we turn to the combination of higher-order, modal, relevancelogics or non-truth-functional logics, and thus refinements of thenotion of algebraic fibring such asmodulated fibring (see C.Sernadaset al. 2002b) orcryptofibring (see Caleiroand Ramos 2007) may be necessary to solve, for example, somecollapsing problems within combinations of logics (seesection 5). This naturally leads to the use of category theory as a universallanguage and as a tool to deal with such problems. But the fact isthat combinations of logics do not necessarily depend upon any highlytechnical methodology, and even some relatively simple examples can bereally expressive. There is a recognized intersection and interactionbetween philosophy and theoretical computer science, and techniquesfor combining logics are also revealed to be very apt tools forhandling and thus better understanding Kripke models. Having beenintroduced in the domain of philosophical logic, Kripke models areessential in computer science and artificial intelligence as semanticstructures for logics of belief, knowledge, temporal logics, logicsfor actions, etc. Knowledge representation and reasoning may requirecombining several reasoning formalisms, including combinations oftemporal reasoning, reasoning in description logic, reasoning aboutspace and distance, and so on. Logics, combining temporal and modaldimensions, are also becoming a relevant tool in agent-orientedprogramming languages. Other applications of combinations of logicsinclude software specification, knowledge representation,architectures for intelligent computing and quantum computing,security protocols and authentication, secure computation andzero-knowledge proof systems, besides their connections to formalethics and game semantics.

The Belief-Desire-Intention model of agents (BDI) is concerned withthe formal representation of practical reasoning involving action,intention, belief, will, deliberation, goal-driven modeling, etc. Thiskind of reasoning is essential for planning (especially for artificialagents). It is then just natural to think about combining simple modallogics for knowledge, belief, obligation, capability, opportunity,etc., so as to define more robust BDI logics. Governatorietal. 2002 investigates the relationships between BDI logics and aparticular case of Gabbay’sfibring semantics (seesubsection 4.2) calleddovetailing, showing that a (general) logical accountof BDI can be handled by means of dovetailing multimodal logics.

But combinations of logics also work from another perspective: insteadof directly combining logic systems and looking for the interpretationof the resulting system, one can start from a purely mathematicalperspective. In van Benthemet al. 2006, for instance, theauthors introduce the notion of horizontal and vertical topologies onthe product of topological spaces, and show that the modal logic ofproducts of topological spaces with horizontal and vertical topologiescoincides with the fusion ofS4 with itself. Theresulting completeness proof has deep connections with sometopological properties of the real and of the rational numbers.

2. Splitting logics versus splicing logics

It is reasonable to expect that a method for combining logics wouldwork in two opposite directions: On the one hand, a logic that onewants to investigate could be decomposed into factors of lessercomplexity; for instance, a bimodal alethic-deontic logic could bedecomposed into its alethic and deontic fragments. In this case, itwould be relevant to see if the logic under investigation is the leastextension of its factors, or if additional bridge principles wouldhave to be added. This approach, in which a given logic is decomposedinto (possibly) simpler factors, is said to be a process ofsplitting logics. On the other hand, one might be interestedin creating new logic systems where different aspects are integrated,starting from given logics. This demand typically occurs in softwareengineering and security, where knowledge representation, formalspecification and verification of algorithms and protocols have amarked need for working with several logics. In a less pragmaticalscenery, this would be the case if one is interested, for instance, inadding a modal dimension to an intuitionistic or a paraconsistentlogic. Moreover, it is interesting to characterize which properties ofthe factors can be transferred to the combined logic. This directionis said to be a process ofsplicing logics.

The essential distinctions between splicing (in the direction ofsynthesis) and splitting (in the direction of analysis) take intoaccount the intentions one may have in mind, and consequently eachdirection encompasses specifically designed techniques.

The paradigm of splicing logics assumes a bottom-up perspective: Itcombines given logics, synthesizing them, and producing a new one. Thecombined logic should be minimal in some sense, that is, if \({{\calL}}\) is obtained from \({{\cal L}}_1\) and \({{\cal L}}_2\) by somecombination process, it should be expected that (1) \({{\cal L}}\)extends both \({{\cal L}}_1\) and \({{\cal L}}_2\), and (2)\({{\cal L}}\) is aminimal extension of both \({{\calL}}_1\) and \({{\cal L}}_2\). For instance, some methods may require\({{\cal L}}\) to be the least conservative extension of both \({{\calL}}_1\) and \({{\cal L}}_2\). This point will be discussed insection 5.

On the other hand, splitting a logic \({{\cal L}}\) assumes a top-downperspective: Logics are decomposed into (presumably simpler)factors.

It should be stressed that most of the methods for combining logicsfound in the literature are better understood from the splicingperspective, placing prominence on the creation of a logic system fromfamiliar logics. However, some splicing methods such as fusion (seesubsection 4.1) are more usefully regarded as a method of decomposition of logicsinto simpler fragments, and in this way also work in the splittingdirection. Possible-translations semantics (seesubsection 4.4), on the other hand, constitute a typical method within the splittingperspective.

3. The importance of language and the presentation of logics

Suppose that two given logics \({{\cal L}}_1\) and \({{\cal L}}_2\)are to be combined using some technique. It should be obvious that anymethod applied to combine \({{\cal L}}_1\) and \({{\cal L}}_2\) willcreate a new logic \({{\cal L}}\) which contains thesignature (logic symbols such as connectives, quantifiers,propositional variables etc.) ofboth logics, that is,\({{\cal L}}\) will be defined in a mixed language, which allowscombinations of symbols of the underlying languages. That is, acombination of logic systems presupposes the previous combination ofthe respective signatures. This is why the choice of the signature ofthe combined system is as important as the logic itself. For instance,the definition of the language of parameterization is fundamental inorder to obtain the intended combined logic (seesubsection 4.5). Another example is found in Schurz 1991, where the formal treatmentof Hume’s ‘is-ought problem’ (recallsection 1) presupposes careful handling of subtle combinations of languages.

Besides the definition of the appropriate language for the combinedlogic, another important question that immediately arises is: Shouldthe logics \({{\cal L}}_1\) and \({{\cal L}}_2\) (to be combined) bepresented in the same way? In other words: Is it possible to combinelogics defined by different paradigms? For instance, how could onecombine a logic \({{\cal L}}_1\), defined by a sequent calculus, witha logic \({{\cal L}}_2\), represented by a (Hilbert-style) axiomaticsystem? How should the resulting logic \({{\cal L}}\) be represented,as a sequent calculus, as an axiomatic system or as a mixed proofsystem? Consider now another (even worse) situation, when the logic\({{\cal L}}_1\) is described by semantical means (that is, throughsemantic structures such as valuations or Kripke models) whereas thelogic \({{\cal L}}_2\) is presented through a syntactical proofsystem, such as a natural deduction system, sequent calculus or aHilbert-style axiomatization. Could the resulting (combined) logic bebetter presented semantically or syntactically?

This annoyance does not occur in the majority of cases, where thelogics being combined are complete with respect to some kind ofsemantics and are syntactically presented in a homogeneous way. Still,there are logics which are only reasonably presented by syntacticalmeans, or exclusively by semantical means. Such is the case, e.g., ofthe first-order theory of torsion groups, known to benon-axiomatizable, and of incomplete modal logics which are onlypresented in syntactical (proof-theoretical) terms.

A possible solution to the problem of combining heterogeneous logics,which naturally leads us to the deeper question of “what is alogic?”, is to consider a common component of the majority oflogics: their respective consequence relations. Thus, given \({{\calL}}_1\) and \({{\cal L}}_2\) presented in different ways, it is alwayspossible to extract the respective consequence relations and thencombine them (taking, for instance, their supremum in an appropriatelattice of consequence relations). But in this way, the resultinglogic \({{\cal L}}\) is presented in a very abstract form because theonly information available from \({{\cal L}}\) is its consequencerelation, and so the characteristics and particularities of each logiccomponent are definitively lost.

Returning to the first example (combining a sequent calculus with anaxiomatic system), a better solution was proposed in Cruz-Filipeetal. 2008: the idea is to define an abstract formalism for proofsystems, general enough as to encode the main proof mechanisms foundin the literature. Thus, after reformulating \({{\cal L}}_1\) and\({{\cal L}}_2\) as abstract proof systems of this kind, the resultingcombined logic \({{\cal L}}\) is an abstract proof system in which itis possible to recognize the ‘genetic traces’ of theoriginal inference rules of each component within derivations in\({{\cal L}}\). The latter question, namely, how to characterizederivations in a fibred logic in terms of the derivations of thecomponents, was carefully analyzed by S. Marcelinoet al. ina series of papers Marcelinoet al. 2015; Marcelino and Caleiro2016; Marcelino and Caleiro 2017a; and Caleiro and Marcelino, 2022.Seesubsection 4.3.

Despite the above mentioned results on combining heterogeneous logics,it seems more reasonable to combine logics defined in a homogeneousway, and, in fact, this is the case with most of the proposals in theliterature. For instance, the usual combinations of modal logics (asfusion, product and fibring) are performed between systems presentedaxiomatically, or between classes of Kripke models. It is frequent,therefore, to define different categories of logic systems(consequence relations, Hilbert calculi, algebraizable logics etc.)with appropriate morphisms between them, in which the combination (ordecomposition) of logics appear as universal constructions. Algebraicfibring, to be described insubsection 4.3, is a good example of this approach.

4. Methods for combining and decomposing logics

4.1 Fusion and Products

The method offusion of normal modal logics was introduced byR. Thomason (1984), and constitutes one of the first general methodsfor combining logics. In the original formulation, it combines normalmodal logics presented syntactically and semantically (throughHilbert-style axioms and Kripke semantics, respectively). The maincharacteristics of the method are described in the followingparagraphs.

Consider Kripke models of the form \[\langle W,R,V\rangle\] such that \(W\) is anon-empty set (the set of worlds), \(R\subseteq W\times W\) is abinary relation (the accessibility relation) and \(V: \mathbb{P} \to\wp W\) from the set of propositional variables into the power set of\(W\) is a valuation map. Let \({{\cal L}}_1\) and \({{\cal L}}_2\) betwo propositional normal modal logics defined over the same classicalsignature which contains the connectives \(\neg\) (negation) and\(\to\) (implication). Denote by \(\square_1\) and \(\square_2\) thenecessity operators of \({{\cal L}}_1\) and \({{\cal L}}_2\),respectively. Let \({{\cal M}}_1\) and \({{\cal M}}_2\) be the classesof Kripke models for \({{\cal L}}_1\) and \({{\cal L}}_2\),respectively. Since both logics are normal, it is granted that bothmodalities \(\square_1\) and \(\square_2\) satisfy the axiom \(K\) andthe necessitation rule. Thefusion of \({{\cal L}}_1\) and\({{\cal L}}_2\) is then defined to be the normal bimodal logic\({{\cal L}}\) with two independent boxes \(\square_1\) and\(\square_2\) together with the connectives \(\neg\) (negation) and\(\to\) (implication). The semantics of \({{\cal L}}\) is given by theclass \({{\cal M}}\) of Kripke structures of the form \[\langle W,R_1,R_2,V\rangle\] suchthat \(\langle W,R_1,V\rangle\) and \(\langle W,R_2,V\rangle\) belongto \({{\cal M}}_1\) and \({{\cal M}}_2\), respectively. In otherwords, each structure of the fusion corresponds to a pair of models: amodel \(\langle W,R_1,V\rangle\) for \({{\cal L}}_1\) and a model\(\langle W,R_2,V\rangle\) for \({{\cal L}}_2\) sharing the same setof worlds \(W\). Technically speaking, each structure of the fusionhave, as a reduct, a model of \({{\cal L}}_1\) and a model of \({{\calL}}_2\).

Given a structure \(M=\langle W,R_1,R_2,V\rangle\), the accessibilityrelation \(R_1\) is used to evaluate the box \(\square_1\), whereas\(R_2\) is used to evaluate \(\square_2\). Since the language of\({{\cal L}}\) is freely generated by the union of the signatures of\({{\cal L}}_1\) and \({{\cal L}}_2\), it contains mixed formulas suchas \(\varphi = \square_1(\square_2 p \to p)\). Now, the structure\(M\) satisfies \(\varphi\) above at a world \(w \in W\) if and onlyif, for every \(w_1 \in W\) such that \(w R_1 w_1\), \(M\) satisfies\((\square_2 p \to p)\) at \(w_1\). But this means that, either thereexists \(w_2\) such that \(w_1 R_2 w_2\) and \(w_2 \not\in V(p)\), or\(w_1 \in V(p)\).

What concerns axiomatics, a Hilbert calculus for \({{\cal L}}\) isobtained by joining up the (schema) axioms of both systems. Thus,\({{\cal L}}\) has, among others, two \(K\) axioms, two necessitationrules and just onemodus ponens (because implication \(\to\)is shared). Considering that the language of \({{\cal L}}\) has mixedformulas (as \(\varphi\) above), schema variables occurring in theschema rules of the given logics can now be replaced in \({{\cal L}}\)by mixed formulas. For instance, \(\varphi\) can be derived in\({{\cal L}}\) from the formula \((\square_2 p \to p)\) by anapplication of the necessitation rule for the box \(\square_1\).

An interesting example of fusion appears in Schurz 1991, when analethic-deontic logic is defined by fusing a pure alethic logic with apure deontic logic. This combination is used to analyze Hume’s‘is-ought thesis’ (seesection 1 above) in formal terms. Other intuitively appealing examples offusion are given in the pioneering paper Fitting 1969 by M. Fitting,where alethic and deontic modalities are fused (before the concept offusion had ever been introduced).

Since then fusion has been a much worked theme. Important results arethe applications of fusion to simulations and to the question oftransfer of properties among modal logics. Simulations make thestrength of normal monomodal logics explicit, as they can, in a sense,simulate all modal logics (see Kracht and Wolter 1999). With respectto transfers, the preservation of properties such as completeness,finite modal property, decidability and interpolation by fusion ofmodal logics was extensively studied in Fine and Schurz 1996. Moregeneral and deeper results in the same spirit were obtained in Krachtand Wolter 1991, and a survey of most of those results can be found inKracht and Wolter 1997. These results show the robustness of fusion asa combination method within the scope of modal logics, for fulfillingthe requirement of preserving the properties of the logics beingcombined.

The question of how completeness results (and other model-theoreticalproperties) can be transferred from a propositional modal logic to itsquantificational counterpart, and from a monomodal quantified modallogic to their multimodal combinations by means of fusion, isinvestigated in Schurz 2011. The paper also deals with the question onhow completeness can be transferred from quantified modal logic withrigid designators to the ones with non-rigid designators.

Rasgaet al. 2010 defines a categorical approach of fusion formodal logic systems labeled with truth values, and it is shown thatthe preservation of completeness requires some careful assumptions,while soundness is preserved without further provisos. A wide varietyof logics (besides modal logics) including several non-classicallogics can be treated in this way.

An interesting note is that there is a notable difference betweencombining logics from the syntactical and from the semanticalperspective. For instance, the joining of two Hilbert calculi shouldbe intuitively obtained by simply putting together the axioms andrules of both logics, while the semantical counterpart is not soobviously determined. Regarding this, an alternative to fusion is thefibred semantics (seesubsection 4.2).

Fusion, even if it is a very natural method for combining modallogics, however, is not obviously extendable to combinations ofnon-normal modal logics with normal modal logics. In Roggia 2012,Subsection 4.2.2, the concept of fusion was generalized from normalmodal logics to a broader class of (labeled) logical systems calledrelational LSLTV. This was achieved through a categoricaldescription of fusion (as defined in Rasgaet al. 2010). InChapter 5 of Roggia 2012, an example of the fusion of non-normal modallogics was provided, by combining the non-normal partitionlogic—introduced in Fajardo and Finger 2005—with itself.Despite this development, which is strongly based on the notion oflabeled deduction systems, extending fusion to logical systems of adifferent nature is by no means straightforward. Algebraic fibring,described insubsection 4.3 below, constitutes a generalization of fusion (at the propositionallevel), and generally addresses the problem of combining logics.

Another early method for combining (modal) logics is the so-calledproduct of modal logics. This mechanism, independentlyintroduced in Segerberg 1973 and in Šehtman 1978, isappropriate to represent time-space information. Given two modallogics \({{\cal L}}_1\) and \({{\cal L}}_2\) as above, the product\({{\cal L}}_1 \times {{\cal L}}_2\) is the bimodal logic over themixed signature (endowed with two boxes) characterized by the class ofKripke structures of the form \[\langle W_1\times W_2,S_1,S_2,V_1 \times V_2 \rangle\] defined from Kripke models\(\langle W_1,R_1,V_1\rangle\) and \(\langle W_2,R_2,V_2\rangle\) for\({{\cal L}}_1\) and \({{\cal L}}_2\), respectively. The accessibilityrelations \(S_1,S_2\subseteq ( W_1 \times W_2) \times (W_1 \timesW_2)\) are defined as follows:

• \(\langle u_1,u_2\rangle S_1 \langle w_1,w_2 \rangle\) iff \(u_1 R_1w_1\) and \(u_2=w_2\);

• \(\langle u_1,u_2\rangle S_2 \langle w_1,w_2 \rangle\) iff \(u_2 R_2w_2\) and \(u_1=w_1\);

• \((V_1 \times V_2)(p)=V_1(p) \times V_2(p)\).

A somewhat surprising feature of the product of modal logics is thatsome new interactions between modalities arise. These new validformulas are a sort of bridge principles (recallsection 1). Using the standard notation \(\lozenge_1 \varphi\) for \(\neg\square_1 \neg \varphi\) (and analogously for \(\lozenge_2\)) for thepossibility operator, the following bridge principles are always validin the product logic:

• \((\lozenge_1\lozenge_2 p \to \lozenge_2\lozenge_1 p )\) Commutativity1;

• \((\lozenge_2\lozenge_1 p \to \lozenge_1\lozenge_2 p )\) Commutativity2;

• \((\lozenge_1\square_2 p \to \square_2\lozenge_1 p )\) Church-Rosserproperty 1;

• \((\lozenge_2\square_1 p \to \square_1\lozenge_2 p )\) Church-Rosserproperty 2.

Due to such interactions it is not possible to directly obtain theHilbert calculus for the product of two modal logics, as in the caseof fusion. The bridge principles must be explicitly added to the unionof the original axiomatics in order to ensure completeness.

In general, the axiomatization of products of modal logics is adelicate issue, and some interesting phenomena can arise. Forinstance, the two-dimensional modal product logicS5\(\,\times\,\)S5 has a finiteaxiomatization but, for \(n \geq 3\), the \(n\)-dimensional productlogicS5\(^{n}\) is non-finitely axiomatizable. InKurucz and Marcelino 2012 there were found the first examples ofdecidable two-dimensional products of finitely axiomatizable modallogics, such asK4.3\(\,\times\,\)S5, which fail tobe finitely axiomatizable. These are examples of logics that fall preyto what we may call thefinite crash phenomenon, where thefinite axiomatizability property is destroyed under the action ofproducts.

The problem of finding a canonical axiomatization for non-finitelyaxiomatizable products of finitely axiomatizable logics was solved inHampson, Kikot, Kurucz and Marcelino 2020, by analyzingtwo-dimensional modal product logics involving the unimodal logic ofthe differenceDiff, introduced by von Wright 1979 asthe logic of ‘elsewhere’. This logic is defined by the setof unimodal formulas that are valid in all the frames \(\langleW,\neq_{W}\rangle\), where \(\neq_{W}\) is the non-identity relationon a non-empty set \(W\); these frames are calleddifferenceframes. The logicDiff can be characterized asthe logic of all the frames \(\langle W,R\rangle\) where \(R\) is apseudo-equivalence relation, that is, where \(R\) is symmetric andpseudo-transitive: if \(R(x,y)\) and \(R(y,z)\) then either \(x=z\) or\(R(x,z)\). Since, in particular, equivalence relations are frames forDiff, the latter is a sublogic ofS5. The paper proves that, despiteDiff being a finitely axiomatizable subsystem ofS5, the two-dimensional product logicDiff\(\,\times\,\)Diff isnon-finitely axiomatizable, and can be axiomatized by infinitely manySahlqvist axioms. LetDiff\(\,\times^{sq}\,\)Diff be the‘square’ version ofDiff\(\,\times\,\)Diff, which ischaracterized by the family of all the products \(\langleU,\neq_{U}\rangle \times \langle V,\neq_{V}\rangle\) of differenceframes such that the sets \(U\) and \(V\) have the same cardinality;henceDiff\(\,\times^{sq}\,\)Diffcontains the logicDiff\(\,\times\,\)Diff. The paperproves that the former is not a finite axiomatic extension of thelatter, and can be axiomatized by adding infinitely many Sahlqvistaxioms. The modal logicsDiff\(\,\times\,\)Diff andDiff\(\,\times^{sq}\,\)Diff are thefirst examples of products of finitely axiomatizable modal logics thatare not finitely axiomatizable, although axiomatizable by explicitinfinite sets of canonical (Sahlqvist) axioms.

As in the case of fusion, the technique of products of logics does notallow a direct generalization to logics other than modal ones.

4.2 Fibring (orfibring by functions)

Thefibred semantics of modal logics was originally proposedin Gabbay 1996a and Gabbay 1996b (see also Gabbay 1999). As in thecase of fusion and products, the mechanism of fibring also applies tomodal logics only. Assume the same notation as insubsection 4.1. Thus, given \({{\cal L}}_1\) and \({{\cal L}}_2\), we start bydefining the fibred language (or the fibring of the languages), whichis the language generated by \(\square_1\), \(\square_2\), \(\neg\)and \(\to\) from the propositional variables. The basic idea is toconsider Kripke models with distinguished (actual) worlds togetherwith two transfer mappings: \(h_1\) from the set of worlds of theclass of models \({{\cal M}}_1\) of \({{\cal L}}_1\) into the class ofmodels \({{\cal M}}_2\) of \({{\cal L}}_2\), and \(h_2\) from the setof worlds of the class of models \({{\cal M}}_2\) of \({{\cal L}}_2\)into the class of models \({{\cal M}}_1\) of \({{\cal L}}_1\). When aKripke model of \({{\cal L}}_1\) has to evaluate a formula of the form\(\square_2 \varphi\) at the actual world \(w_1\), the validitychecking is then transferred to the validity checking of \(\square_2\varphi\) within the Kripke model \(h_1(w_1)\) at its actual world.The evaluation of a formula of the form \(\square_1 \varphi\) within aKripke model of \({{\cal L}}_2\) at the actual world \(w_2\) isperformed analogously, but now using the map \(h_2\).

Thus, the fibring (orfibring by functions, as it is calledin Carnielliet al. 2008) of \({{\cal L}}_1\) and \({{\calL}}_2\) is a normal bimodal logic characterized semantically asfollows: Let

and

be a pair of transfer mappings. For simplicity, we assume that thesets of worlds \(W_m\) of \(m\in {{\cal M}}_1\) are pairwisedisjoints, and the same holds for \({{\cal M}}_2.\) Given \(m\in{{\cal M}}_1 \cup {{\cal M}}_2\), \(w\in W_m\) and a formula\(\varphi\) in the fibred language, the satisfaction of \(\varphi\) in\(\langle h_1,h_2,m,w\rangle\), denoted by \(\langleh_1,h_2,m,w\rangle\Vdash \varphi\), is defined recursively as usualwhenever the main connective of \(\varphi\) is Boolean (\(\neg\) or\(\to\)), or when \(\varphi\) is atomic. For the modalities,satisfaction is defined as follows: Suppose (without loss ofgenerality) that \(m \in {{\cal M}}_1\), and let \(h_1(w)=\langlem_2,w_2\rangle\), with \(m=\langle W_m,R_m,V_m\rangle\) and\(m_2=\langle W_{m_2},R_{m_2},V_{m_2}\rangle\). Then:

• \(\langle h_1,h_2,m,w\rangle\Vdash \square_1\varphi\)
  iff \(\langle h_1,h_2,m,w_1\rangle\Vdash \varphi\), for every \(w_1\)such that \(w R_m w_1\);

• \(\langle h_1,h_2,m,w\rangle\Vdash \square_2\varphi\)
  iff \(\langle h_1,h_2,m_2,w_2\rangle\Vdash \square_2\varphi\)
  iff \(\langle h_1,h_2,m_2,w_3\rangle\Vdash \varphi\), for every\(w_3\) such that \(w_2 R_{m_2} w_3\).

The definition of \(\langle h_1,h_2,m,w\rangle\Vdash\square_i\varphi\) for \(i=1,2\) and \(m \in {{\cal M}}_2\) isanalogous.

Then, \(\langle h_1,h_2\rangle\) satisfies \(\varphi\), denoted by\(\langle h_1,h_2\rangle \Vdash \varphi\), if \(\langleh_1,h_2,m,w\rangle\Vdash \varphi\) for every \(m\in {{\cal M}}_1 \cup{{\cal M}}_2\) and \(w\in W_m\). Finally, \(\varphi\) is valid in thefibred semantics whenever \(\langle h_1,h_2\rangle \Vdash \varphi\)for every pair \(\langle h_1,h_2\rangle\) as above.

For instance, given \(h_1, h_2\) as above, let \(\langleW_2,R_2,V_2\rangle \in {{\cal M}}_2\) and \(w_2\in W_2\) such that\(h_2(w_2)= \langle \langle W_1,R_1,V_1\rangle,w_1\rangle\). Then:

• \(\langle h_1,h_2,\langle W_2,R_2,V_2\rangle,w_2\rangle \Vdash\square_1 \square_2 \neg p\)

• iff \(\langle h_1,h_2,\langle W_1,R_1,V_1\rangle,w_1\rangle \Vdash\square_1 \square_2 \neg p\)

• iff \(\langle h_1,h_2,\langle W_1,R_1,V_1\rangle,w'_1\rangle \Vdash\square_2 \neg p\), for every \(w'_1\) such that \(w_1 R_1w'_1\).

Suppose that \(h_1(w'_1)= \langle \langleW'_2,R'_2,V'_2\rangle,w'_2\rangle\). Then, the latter is valid iff

• \(\langle h_1,h_2,\langle W'_2,R'_2,V'_2\rangle,w'_2\rangle \Vdash\square_2 \neg p\),

for every \(w'_1\) such that \(w_1 R_1 w'_1\); i.e., for every\(w'_1\) such that \(w_1 R_1 w'_1\) and for every \(w''_2\) such that\(w'_2 R'_2 w''_2\), \(\langle h_1,h_2,\langleW'_2,R'_2,V'_2\rangle,w''_2\rangle \Vdash \neg p\). This is equivalentto saying that, for every \(w'_1\) such that \(w_1 R_1 w'_1\) and forevery \(w''_2\) such that \(w'_2 R'_2 w''_2\), \(w''_2 \not\inV'_2(p)\).

With respect to axiomatics, the logics obtained by fibring byfunctions can, in some cases, be axiomatized by the union of theaxioms schemas of the given logics. But some logics may require theaddition of some new bridge principles (mixing rules and axioms) inorder to ensure the preservation of completeness. This may explainsome discrepancy between the approaches of fusion and fibring; thecompleteness of fibring as exposed in Gabbay 1999 does not workexactly as a substitute for more technically intricate completenessproofs as in Kracht and Wolter 1991 and in Fine and Schurz 1996. Formore on this discussion, see Kracht 2004.

The technique of fibring by functions is an interesting alternative tofusion and products, but, as much as its competitors, it cannot beextended to non-modal logics in any obvious way (see Coniglio andFernández 2005 for an adaptation of the method of fibring byfunctions to matrix logics). One reason for the failure of fibring byfunctions to what concerns generalizations is that it is not auniversal construction (in categorial terms). Moreover, the lack of asystematic definition of the axiomatization for the logics obtained byfibring is another negative aspect of this technique. The nextsubsection describes a categorial generalization of fibring whichsolves all the mentioned problems.

4.3 Categorial (or Algebraic) Fibring

In order to overcome the limitations of the original method of fibringas exposed in the last subsection, A. Sernadas and collaboratorspropose, in Sernadaset al. 1999, a general definition offibring using the conceptual tools of category theory. The centralidea of the generalization is simple; suppose that \({{\cal L}}_1\)and \({{\cal L}}_2\) are two propositional logics which are to becombined. Suppose, for simplicity, that no connectives are to beshared, that is, the language of the logic \({{\cal L}}\) to beobtained is the free combination of the connectives of both logics. Incategorial terms, the signature \(C\) of \({{\cal L}}\) is thecoproduct (disjoint union) of the signatures \(C_1\) of \({{\calL}}_1\) and \(C_2\) of \({{\cal L}}_2\), in the underlying category ofsignatures. Then \({{\cal L}}\), which is theleast logicdefined over \(C\) which extends simultaneously \({{\cal L}}_1\) and\({{\cal L}}_2\), is defined as the coproduct of \({{\cal L}}_1\) and\({{\cal L}}_2\) in the underlying category of logics. The minimalityof \({{\cal L}}\) attends a criterion expressed in Gabbay 1999 (seealsosection 5) and also conforms to the ideal of fusing logics, see Kracht andWolter 1991. This combination process, calledunconstrainedfibring, can be generalized, by allowing \(C_1\) and \(C_2\) toshare some connectives. Thus, the logic obtained by this second kindof fibring is defined in a language such that some connectives of\({{\cal L}}_1\) and \({{\cal L}}_2\) are identified. The logicproduced by this operation, calledconstrained fibring,starts by considering two logics \({{\cal L}}_1\) and \({{\cal L}}_2\)over signatures \(C_1\) and \(C_2\), respectively, and a signature\(C_0\) contained in both \(C_1\) and \(C_2\). This signature containsexactly the connectives of \({{\cal L}}_1\) and \({{\cal L}}_2\) whichare to be identified (or shared) throughout the combination process.After computing the unconstrained fibring (that is, the coproduct)\({{\cal L}}_1 \oplus {{\cal L}}_2\) of \({{\cal L}}_1\) and \({{\calL}}_2\), which is defined over the signature \(C_1 \oplus C_2\) (thecoproduct of \(C_1\) and \(C_2\)), a new logic \({{\cal L}}\) isobtained. This logic,the fibring of \({{\cal L}}_1\) and \({{\calL}}_2\) by sharing (orconstrained to) \(C_0\), isobtained from \({{\cal L}}_1 \oplus {{\cal L}}_2\) by identifying twoconnectives (of the same arity) iff both come from the same connectivein \(C_0\). In terms of category theory, it is required that theforgetful functor \(N\) from the category of logics to the category ofsignatures be a cofibration. Then, if \(i_j:{{\cal C}}_0\to C_j\) isthe inclusion morphism (for \(j=1,2\)), \(h_j:C_j \to C_1 \oplus C_2\)is the canonical injection of the coproduct (for \(j=1,2\)) and\(q:C_1 \oplus C_2 \to C\) is the coequalizer of \(h_1 \circ i_1\) and\(h_2 \circ i_2\), then the constrained fibring \({{\cal L}}\) is thecodomain of the cocartesian lifting of \(q\) by \(N\).

In order to exemplify the technique of categorial fibring (withoutentering into technical details), suppose that \({{\cal L}}_1\) and\({{\cal L}}_2\) are two modal logics defined through Hilbert calculiover the same signatures \(C_1\) and \(C_2\) of subsections4.1 and4.2, respectively, such that both logics contain the rules ofmodusponens and necessitation. Then \(C_1 \oplus C_2\) consists of twonegations \(\neg_1\) and \(\neg_2\), two implications \(\to_1\) and\(\to_2\) and two boxes \(\square_1\) and \(\square_2\). Theunconstrained fibring \({{\cal L}}_1 \oplus {{\cal L}}_2\) of \({{\calL}}_1\) and \({{\cal L}}_2\) is, therefore, the Hilbert calculus over\(C_1 \oplus C_2\) defined by joining up the axiom schemas andinference rules of both calculi. This logic has, among other axiomsand inference rules, two versions ofmodus ponens (one foreach implication) as well as two versions of the necessitation rule(one for each box). It should be noted that, by using a fixed set ofschema variables for writing the axioms and rules of every calculus,the calculi obtained by fibring are also formed by schematic axiomsand inference rules. Thus, for instance, in the rule ofmodusponens in \({{\cal L}}\): \[\frac{\xi_1 \hspace{1cm} (\xi_1 \to_1 \xi_2)}{\xi_2}\] the schema variables\(\xi_1\) and \(\xi_2\) can be replaced by mixed formulas. Instancessuch as \[\frac{\neg_2 p \hspace{1cm} (\neg_2 p \to_1 \square_2(q \to_2 \square_1 r))}{\square_2(q \to_2 \square_1 r)}\] are new, because the formulas \(\neg_2 p\) and\(\square_2(q \to_2 \square_1 r)\) do not belong to the language of\({{\cal L}}_1\). Analogous replacements apply, of course, to otherinference rules and axioms of \({{\cal L}}_1 \oplus {{\calL}}_2\).

Continuing with this example, suppose now that we want to share (oridentify) both negations, as well as both implications, which is areasonable move when, for instance, these connectives are classical.In such case \((\varphi \to_1 \neg_2 \psi)\) would represent the sameproposition as \((\varphi \to_2 \neg_1 \psi)\).

In order to do this, the signature \(C_0\) just containing \(\neg\)and \(\to\) is taken into consideration, and so \(\neg_1\) isidentified with \(\neg_2\) in \(C_1 \oplus C_2\), as well as \(\to_1\)is identified with \(\to_2\). The resulting signature is \(C\), whichjust contains the connectives \(\neg\), \(\to\), \(\square_1\) and\(\square_2\). In the resulting logic \({{\cal L}}\), defined over\(C\), there is now just one rule ofmodus ponens:

However, there remain two necessitation rules, since there are stilltwo boxes in \(C\). The resulting \({{\cal L}}\) is thus the fibringof \({{\cal L}}_1\) and \({{\cal L}}_2\) constrained by \(C_0\). Thisprocedure precisely coincides with fusion of modal logics. The noveltyhere is that this technique applies to a broad class of logics, whichare not necessarily restricted to (normal) modal logics, as in thecase of fusion.

Constrained and unconstrained fibring, being categorial, are universalconstructions, and so enjoy well-defined and theoretically predictableformal properties. Profiting from universal constructions, in order tohandle algebraic fibring, it is enough to define appropriatecategories of signatures and logic systems. Indeed, thesamefibring construction (coproduct or cocartesian lifting) can beperformed indifferent categories of logic systems. This is aremarkable advantage of the categorial perspective for fibring. Thereare several proposals in the literature devoted to combining logicspresented in different ways by means of algebraic fibring such aspropositional Hilbert calculi, sequent and hypersequent calculi,first-order modal logics, higher-order modal logics,non-truth-functional logics, logics semantically presented throughordered algebras (encompassing generalized Kripke models) etc.

An important question connected to combination of logics (and, inparticular, to algebraic fibring) is the preservation ofmetaproperties such as completeness, interpolation etc. For instance,when \({{\cal L}}_1\) and \({{\cal L}}_2\) are complete logic systemspresented both semantically and syntactically, under which conditionis their fibring also complete? In this regard, Zanardoet al.2001 and Sernadaset al. 2002a give a partial solution to thisquestion. The preservation of soundness and completeness with respectto the technique ofimporting logics (seesubsection 4.6) was proved in Rasgaet al. 2013. Besides, in a series ofpapers (Marcelinoet al. 2015, Marcelino and Caleiro 2016,Marcelino and Caleiro 2017a, and Caleiro and Marcelino, 2022), S.Marcelinoet al. study the preservation of decidability byunconstrained fibring (that is, by algebraic fibring without sharingconnectives). The key result is a general characterization of thederivations in the system obtained by fibring in terms of thederivations in each component. The complexity of the decisionprocedures was also analyzed in these papers. Thus, it was shown thatthe decision problem for the fibred logic can be reduced polynomiallyto the worst decision problem of the given logics. This means that, inparticular, if the decision problems for two given logics \({{\calL}}_1\) and \({{\cal L}}_2\) are both in a complexity class\(\mathcal{C}\) then the decision problem for the unconstrainedfibring \({{\cal L}}_1 \oplus {{\cal L}}_2\) is also in\(\mathcal{C}\), provided that \(\mathcal{C}\) contains the basiccomplexity classP (also known as PTIME) and is closed forcomposition with polynomials. Additional results on decidability andcomplexity of combined logics can be found in Caleiro and Marcelino,2022. Other examples of preservation of metaproperties are provided bythe mechanism for combining logics known asmeet combination oflogics, introduced by Sernadaset al. 2012 (seesubsection 4.6). Specifically, the preservation of soundness and completeness (seeSernadaset al. 2012), Craig interpolation (see C. Sernadas etal. 2013 ) and admissibility of rules (see Rasgaet al. 2016)have been obtained for this method. On the other hand, transferresults have been extensively studied in the case of fusion of modallogics, as already mentioned insubsection 4.1.

Besides the studies concerning preservation of metaproperties byfibring mentioned above, the semantics of fibred logics is not so easyto determine from the semantics of the components. That is, there isno natural correspondence between the models of the component logicsand the class of models which characterizes the least logic in themixed language which extends simultaneously the given logics, that is,their fibring. This question is far from being simple; for instance,in Marcelino and Caleiro 2017a it was shown that the act of fibringtwo logics, each of one being characterized by a single finite matrix,can produce a logic which is uncharacterizable by a single matrix(even infinite). In Marcelino and Caleiro 2017b a first step towardsthe solution of this problem was given, by characterizing thesemantics of unconstrained fibring of logics presented by means ofnondeterministic matrix semantics. Nondeterministic matrices(or Nmatrices) generalize logical matrices by allowing the connectivesto be interpreted by multivalued truth-functions instead of ordinarytruth-functions. Nmatrices were formally introduced in Avron and Lev2001 (see also Avron and Lev 2005) to provide a semantical account forlogics uncharacterizable by a single finite matrix. However, the useof Nmatrices for logics was already considered by several authors(Rescher 1962; Ivlev 1973, 1985, 1988, 2024 and Kearns 1981 in thecontext of modal logics; and Crawford and Etherington 1998). By usingtwo different notions of product of Nmatrices (strict-product and\(\omega\)-power), in Marcelino and Caleiro 2017b it is shown that thedisjoint fibring of two propositional logics, each of one presented bya single Nmatrix, is characterized by a single Nmatrix obtained fromthe given ones by using these products. This encompasses the disjointfibring of logics presented by matrices, since every logical matrixis, in particular, a non-deterministic matrix. In Caleiro andMarcelino 2024 these products were extended in order to define thefibring of two logics by sharing connectives. This goal was obtainedby considering logics characterized by partial Nmatrices (PNmatrices,for short), i.e., Nmatrices in which some entries of the truth-tablescan be empty sets. However, when dealing with standard Tarskian logics(which, by definition, have single-conclusion consequence relations)it is required to consider \(\omega\)-powers of the correspondingPNmatrices, which are infinite structures.

An alternative proposal to the problem of combining two logicscharacterized by (finite-valued) Nmatrices by means of fibring (evenby sharing connectives) was given in Coniglio 2025. That proposal isbased on the notion ofswap structures, which is ananalytical way to describe Nmatrices. In its simpler form, a swapstructure is a finite-valued Nmatrix in which its domain is formed by\((n+1)\)-tuples (calledsnapshots) describing the state of aformula \(\varphi\) in terms of 0-1 values assigned to \(\varphi,\psi_1(\varphi), \ldots, \psi_n(\varphi)\), where each \(\psi_i(p)\)is a formula depending on a single propositional variable \(p\). Themultioperators of the swap structures are described by means ofBoolean functions or relations on the components of the snapshots. InConiglio 2025 it is shown that, under certain conditions (includingthat the logics being combined contain propositional classical logic),the constrained fibring of two logics characterized by finite-valuedswap structures is a logic also characterized by a finite-valued swapstructure naturally obtained from the given ones by concatenating theconditions on the snapshots, in a process calledsuperposition ofsnapshots.

The relationship between fusion and algebraic fibring deserves somecomments. When restricted to modal propositional logics, fusion is aparticular case of algebraic fibring in the category of interpretationsystems, where logics are presented through ordered algebras, becauseit is enough to consider interpretation systems defined over power setalgebras induced by Kripke models. At the syntactical level, fusion isalso a particular case of algebraic fibring in the category of Hilbertcalculi, in the realm of propositional signatures. As much asfirst-order modal logics are concerned, the approaches diverge, mainlybecause there are different semantical accounts for treatingfirst-order modalities. For instance, the semantical approach to modalfirst-order logics by Sernadaset al. 2002a in the context ofalgebraic fibring differs from that of Kracht and Kutz (2002) in thecontext of fusion.

The fact that algebraic fibring generalizes (at least at thepropositional level) the fusion of modal logics makes the formermethod become very natural and useful. Moreover, the universality ofthe construction allows one to define algebraic fibring in verydifferent logical contexts (categories of logics), such asnon-truth-functional logics, higher-order logics, sequent calculi etc.As it will be shown insection 5, the different notions of morphisms between logics affect the strengthof the logics obtained by algebraic fibring in the differentcategories of logic systems. For general accounts of algebraic fibringsee, for instance, Caleiroet al. 2005 and Carniellietal. 2008.

4.4 Possible-Translations Semantics

The methods for combining logics described above adhere to thesplicing methodology in that they are used to combine logics creatingnew systems which extend the given logics.

As mentioned insection 2, there is a converse direction, namely the splitting methodology inwhich a given logic system is decomposed into other systems. Thepossible-translations semantics (in short,PTS),introduced in Carnielli 1990 (see also Carnielli 2000), is one of thefew supporters of this viewpoint.

The notion ofPTS was originally defined as an attempt toendow certain logics with recursive and palatable semanticinterpretations. Actually, several paraconsistent logics which are notcharacterizable by finite matrices can be characterized by suitablecombinations of many-valued logics. The idea of the decomposition isquite natural. Given a logic \({{\cal L}}\), presented as a pair\({{\cal L}}=\langle C,\vdash_{{\cal L}}\rangle\) in which \(C\) is asignature and \(\vdash_{{\cal L}}\) is a consequence relation, afamily of translations \(f_i:L(C) \to L(C_i)\) (for \(i\in I\)) istaken into consideration. Here, \(L(C)\) and \(L(C_i)\) denote thealgebra of formulas defined by the signature \(C\) and \(C_i\),respectively. Recall that a translation from a logic \({{\cal L}}\)into a logic \({{\cal L}}'\) is a mapping \(f\) between the respectivesets of formulas which preserve derivability, that is: \(\Gamma\vdash_{{\cal L}}\varphi\) (in the source logic \({{\cal L}}\))implies that \(f(\Gamma) \vdash_{{{\cal L}}'} f(\varphi)\) (in thetarget logic \({{\cal L}}'\)).

A pair \(P=\langle \{{{\cal L}}_i\}_{i\in I}, \{f_i\}_{i\inI}\rangle\) as above is called apossible-translations framefor \({{\cal L}}\). We say that \(P\) is apossible-translations semantics for \({{\cal L}}\) if, forevery \(\Gamma\cup\{\varphi\}\subseteq L(C)\), \[\Gamma \vdash_{{\cal L}}\varphi \\textrm{ iff } \ f_i(\Gamma) \vdash_{{{\cal L}}_i} f_i(\varphi), \\textrm{ for every } i \in I.\] This meansthat checking derivability in \({{\cal L}}\) is equivalent to checkingderivability in every factor logic \({{\cal L}}_i\) through thetranslations. In many cases, the factor logics \({{\cal L}}_i\) arepresented by finite matrices. Since the length of a formula is finite,it is enough to test a finite number of translations in order todetermine if a formula of \({{\cal L}}\) is valid in \({{\cal L}}\).Thus, checking the validity of a formula of \({{\cal L}}\) isequivalent to performing a finite number of finitary tests. Thisdecidability property is of real advantage when the original logic\({{\cal L}}\) is not characterizable by finite matrices. Forinstance, in Carnielli 2000 (see also Marcos 1999) the well-knowhierarchy \(\{{{\cal C}}_n\}_{n\in{\mathbb{N}}}\) of paraconsistentlogics of N. da Costa, formed by logics which cannot be characterizedby finite matrices, is represented by means of aPTS whosefactors are presented through finite matrices; this grants a decisionprocedure for each logic \({{\cal C}}_n\).

In order to exemplify the concept ofPTS as a splittingmethodology, consider the paraconsistent logicbC,introduced in Carnielli and Marcos 2002. This logic is, in particular,alogic of formal inconsistency (LFI), inthe sense that there exists a unary connective \(\circ\) expressingthe consistency of a formula. Thus, from \(\varphi\) and\(\neg\varphi\) does not follow, in general, an arbitrary formula\(\psi\). However, \(\{\varphi, \neg\varphi, \circ\varphi\}\) entailsany formula \(\psi\). The signature \(C\) ofbCconsists of a paraconsistent negation \(\neg\), a consistency operator\(\circ\), and classical connectives \(\wedge,\vee,\to\). It wasproved in Carnielliet al. 2007 thatbC, andmany other logics of formal inconsistency extending it, cannot becharacterized by finite matrices. Nonetheless,bC isdecomposed into several copies of a three-valued logic by means ofpossible-translations as follows. Consider the signature\(C_1=\{\neg_1,\neg_2,{\circ}_1,{\circ}_2, {\circ}_3,\wedge,\vee,\to\}\) consisting of two negations, three consistencyoperators, a conjunction, a disjunction and an implication. Let \(M\)be the matrix over \(C_1\) with domain \(\{T,t,F\}\) displayed below,where \(\{T,t\}\) is the set of designated values.

Let \(\{f_i\}_{i\in I}\) be the family of all the mappings \(f:L(C)\toL(C_1)\) satisfying clauses \((tr0)\), \((tr1)\), \((tr2)\), \((tr3)\)and \((tr4)\) below.

• (tr0)\(f(p)= p \\textrm{ for } p \textrm{ a propositional variable;}\)
• (tr1)\(f(\neg\varphi)\in \{\neg_1 f(\varphi), \neg_2 f(\varphi)\}\);
• (tr2)\(f(\varphi\#\psi)= (f(\varphi)\#f(\psi)), \textrm{ for }\# \in \{\land,\lor,\rightarrow\}\);
• (tr3)\(f({\circ}\varphi)\in \{{\circ}_1 f(\varphi), {\circ}_2f(\varphi), {\circ}_3 f(\varphi)\}\);
• (tr4)if\(f(\neg\varphi)=\neg_2 f(\varphi)\) then\(f({\circ}\varphi)={\circ}_1 f(\varphi)\).

The family of mappings \(\{f_i\}_{i\in I}\) can be shown to define aPTS which characterizesbC in a decidableway (see Carnielliet al. 2008). As an example, it can beeasily checked that \(\varphi\wedge\neg\varphi\to\neg{\circ}\varphi\)is a theorem ofbC: just consider all its finitelymany translations and test that all of them are three-valuedtautologies. On the other hand,\((\neg(\varphi\wedge\neg\varphi)\to{\circ}\varphi)\) is not a theoremofbC, which can be promptly verified by showing thatat least one of its translations is not a tautology using thethree-valued tables above. For an alternativePTScharacterization ofbC and related logics see Marcos2008.

This example shows that a non-truth functional connective, such as theparaconsistent negation \(\neg\) or the consistency operator \(\circ\)ofbC, can be mimicked by interpreting it (viatranslations) into different truth-functional connectives. The idea ofinterpreting (or decomposing) a connective into simpler ones can berelated to the notion of non-deterministic matrix semantics proposedby Avron and Lev, and mentioned insubsection 4.3.

Indeed, in Carnielli and Coniglio 2005 it is shown thatnon-deterministic matrices can be simulated by appropriatepossible-translations semantics. In particular, the familiar matrixsemantics are a particular case of possible-translations semantics, aswell as the historical examples of translations between logics foundin the literature. These facts evidence that possible-translationssemantics is a widely applicable conceptual tool for decomposinglogics.

4.5 Temporalization, Parameterization and Institutions

Apart from the logical and philosophical import of combining logics,there is a genuine interest in developing applications based on thesetechniques. One of the main areas interested in the methods forcombining logics is software specification. Certain techniques forcombining logics were developed almost exclusively with the aim ofapplying them to this area. In this section some of these methods willbe briefly mentioned, namely,temporalization,parameterization andinstitutions.

Temporalization was introduced in Finger and Gabbay 1992 (see alsoFinger and Gabbay 1996), and generalized in Caleiroet al. 1999towards the method called parameterization.

Parameterization, in rough terms, consists of replacing the atomicpart of a given logic \({{\cal L}}\) by another logic \({{\cal L}}'\).Thus, the logic \({{\cal L}}\) is theparameterized logic;the atomic part is theformal parameter and the logic\({{\cal L}}'\) is theparameter logic. Formally, a mixedsignature based on the signature of \({{\cal L}}\) is considered, towhich the formulas of \({{\cal L}}'\) are added as constants. In theparticular case of temporalization, the parameter logic is a temporallogic. In turn, it can be proven that parameterization is a particularcase of constrained fibring (recallsubsection 4.3).

The method can be explained by means of a simple example taken fromCarnielliet al. 2008: consider a propositional modal logic\({{\cal L}}\), to be parameterized with first-order logic \({{\calL}}_{fol}\) in order to describe the dynamics of data bases. Theresulting logic is defined in a language whose formulas are obtainedby replacing propositional constants in formulas of \({{\cal L}}\) byfirst-order formulas. That is, modalities can be freely used, butquantifiers cannot be applied to modal formulas (other propositionalconnectives such as negation and implication are shared).

The semantic structures for the new logic are Kripke structures wherethe valuation for propositional constants is replaced by a kind of“zooming in” mapping (in the sense of Blackburn and deRijke 1997) associating a first-order semantic structure together witha fixed assignment for individual variables to each state.

The deductive system for the new logic is formed by the rules of bothlogics. The rules of \({{\cal L}}\) can be instantiated with formulasof the parameterized language, but the rules of first-order logic canonly be applied to pure first-order formulas.

One important difference between parameterization (in particular,temporalization) and constrained fibring is the degree of symmetry:the parameterized language and inference rules are intrinsicallyasymmetric, while this is not the case of constrained fibring.

Institutions were introduced by J. Goguen and R. Burstall (see Goguenand Burstall 1984, 1992) as a kind of “abstract modeltheory” for computer science, and are adequate for developingconcepts of specification languages such as structuring ofspecifications and implementation.

The theory of institutions is mainly applicable to softwarespecification defined by multiple logical systems (see, for instance,Diaconescu and Futatsugi 2002). Thus, under an abstract view ofsoftware development, different components of the same program can bespecified using different formalism in a heterogeneous setting. Thisis formalized by the use of institutions and morphisms between them(see, for instance, Tarlecki 2000). A problem concerning institutionmorphisms is that formulas involving connectives from different logicsbeing combined are not allowed. A solution to this problem wasproposed in Goguen and Burstall 1986 and Mossakowski 1996, by usingthe so-calledparchments andparchmentmorphisms.

4.6 New perspectives on combining logics

Despite the fact that algebraic fibring is suitable for combining anample class of logic systems, some kinds of logics, namelysubstructural logics such as linear logic, and logics equipped with anondeterministic semantics, lie outside the scope of this combinationmethod. Moreover, at the semantical level, algebraic fibring, by itsown nature, does not make possible to keep representatives of all themodels of the original logics (which leads to thecollapsingproblem of fibring, seesection 5).

With the aim of still enlarging the range of application of algebraicfibring, so as to make it able to deal with substructural logics andwith logics endowed with nondeterministic semantics, as well as tocombine pointwise models of each logic, a formalism for representinglogics and their combinations based on the general notion ofmulti-graphs (or, for short, m-graphs) was proposed by A. Sernadas andhis collaborators (see Sernadaset al. 2009a, 2009b).Multi-graphs are graphs where the source of each edge is a finitesequence of nodes (instead of a unique node). Concerning signatures,the nodes of the m-graph are seen as sorts and the m-edges are seen aslanguage constructors. From the semantical viewpoint, nodes aretruth-values and m-edges are relations between truth-values. Finally,concerning deductive systems, nodes are language expressions, andm-edges are inference rules. The fibring of logics described bym-graphs (a.k.a.graph-theoretic fibring) is defined bypointwise combining models of each combined logic, in contrast to theusual notion of semantic fibring in which entire classes of models arecombined. This allows one to avoid the collapsing problem (see thenext section) in a very natural way.

As an immediate application of this graph-theoretic setting, thepreservation of the finite-model property by graph-theoretic fibringwas proved in Coniglioet al. 2011. Since (under reasonableconditions) the finite model property entails decidability, thisresult is particularly useful.

Another application of the graph-theoretic account of logic is thedefinition of an asymmetric combination technique calledimportinglogics (see Rasgaet al. 2012). Temporalization, as wellas its generalization,modalization (see Fajardo and Finger2003 and Finger and Weiss 2002), are particular cases of importinglogics. Under this approach, the combined language is endowed with anexplicit constructor calledimporting connective whichtransforms formulas of the imported logic into formulas of theimporting logic. This is the main difference between the technique ofimporting logics and the related technique of parameterization (whichalso generalizes temporalization, seesubsection 4.5). Semantically, each model of the resulting logic obtained by themethod of importing logics is a pair composed of a model of theimporting logic and a model of the imported logic, plus theinterpretation of the importing connective. In Rasgaet al.2014 a new formulation of algebraic fibring, calledbiporting, was introduced, which turns out to be equivalentto the original one. From this, it is possible to prove that someparticular cases of importing, like temporalization, modalization andglobalization, are subsumed by fibring.

A new mechanism for combining logics calledmeet combination oflogics was firstly presented in C. Sernadaset al. 2013and additionally developed in Rasgaet al. 2016. This techniqueis based on the idea of melding or pairing connectives (of the samearity) of two given logics being combined. The melded connectives ofthe resulting logic inherit the common properties enjoyed in bothlogics, instead of the union of their properties, as it occurs in thecase of the shared connectives of constrained fibring. The idea ofpairing connectives within a single logic was already explored inSernadaset al. 2011a and 2011b, towards identifying commonproperties of any given pair of connectives of the same arity.

5. Lack or excess of interaction: perplexities when combining logics

Up to this point, several techniques for composing logics have beendescribed and exemplified. Are these processes appropriate forcomposing, without surprises, any pair of logics? In other words,given a pair of logics (presented in a homogeneous way), are theycomposable in a meaningful way? Does the composition makephilosophical sense? As pointed out by Schurz 1991, it is conceivablethat some multimodal logics obtained as a combination of modal logicsby adding arbitrarily chosen bridge principles could be worthless.

From the technical point of view there is an important problemconcerning composition of logics known ascollapsing problem,firstly identified in Gabbay 1996b, where he pointed to the difficultyof the fibring methodology in combining classical and intuitionisticpropositional logics, CPL and IPL, without avoiding the collapsing ofthe intuitionistic connectives into classical ones. An explicitexhibition of this collapsing was given in del Cerro and Herzig 1996,where they study a logic called C + J combining classical logic (C)and intuitionistic logic (J). By defining the semantics of C + J in anatural way, starting from a possible-worlds semantics for J andadding the usual interpretation for classical implication and negationwithin any possible world, they obtain, on the axiomatic side, thesurprising result that the Hilbert axiomatization related to thatsemantics, attained by simply joining the axioms of C and J plus somesupplementary interaction axioms (involving connectives from both),collapses into classical logic. They solve the problem by modifyingthe intuitionistic weakening axiom schema in a suitable way.Previously, Humberstone 1979 had already investigated the question ofcombining intuitionistic and classical logics (independent fromfibring techniques), obtaining a completeness proof based on canonicalmodels.

Basically, the phenomenon pointed up by Gabbay arises, because bothimplications collapse, and then intuitionistic implication becomesclassic. From the (algebraic) semantical point of view, it happensthat the models of the fibred logic are Heyting algebras which aresimultaneously Boolean algebras: evidently the algebras collapse tothe Boolean ones. From the point of view of proof theory, the problemappears as a direct consequence of the metaproperty calledDeduction Metatheorem (DMT): let \(\to_1\) and \(\to_2\) bethe intuitionistic and the classical implications, respectively. Then\[\begin{array}{lll}\Gamma,\varphi\vdash\psi & \textrm{ iff }& \Gamma\vdash(\varphi\to_1\psi)\\\Gamma,\varphi\vdash\psi & \textrm{ iff }& \Gamma\vdash(\varphi\to_2\psi)\\\end{array}\] Thus, the following argument applies (see Gabbay 1996b):\[\begin{array}{ll}(\varphi\to_1\psi)\vdash(\varphi\to_1\psi) & \textrm{(Axiom)}\\(\varphi\to_1\psi),\,\varphi\vdash\psi & \textrm{(DMT for } \rightarrow_1) \\(\varphi\to_1\psi)\vdash(\varphi\to_2\psi) & \textrm{(DMT for } \rightarrow_2) \\\end{array}\] A similar argument shows that\((\varphi\to_2\psi)\vdash(\varphi\to_1\psi)\). That is, classical andintuitionistic implications collapse in the combined logic.

It is worth noting that the previous arguments depart from the verystrong assumption that the metaproperty DMT is preserved in thecombined logic. As we shall see below, this is not the case foralgebraic fibring, unless a stronger notion of morphism between logicsis adopted.

Some alternatives to avoid the collapsing problem were proposed in theliterature. In C. Sernadaset al. 2002b other examples ofcollapse were presented, and a solution to the problem was proposed bymeans of a controlled notion of algebraic fibring calledmodulatedfibring. In turn, Caleiro and Ramos 2007 proposed anotherextension of fibring at the semantical level by means of a techniquecalledcryptofibring. Using this tool, a conservativeextension of classical and intuitionistic logic is possible. Asmentioned insubsection 4.6, the graph-theoretic fibring leads to an additional solution to thisproblem. Another solution to the collapsing problem was obtained bymeans of the meet combination of logics (seesubsection 4.6). Given that the combined connectives inherit the common propertiesenjoyed in both logics, the interaction between the component logicswithin the resulting logic is minimized, which allows to overcome thecollapsing problem. In Toyooka and Sano 2024, the path of Humberstone1979 and del Cerro and Herzig 1996 was retaken, investigating acombined system of intuitionistic and classical propositional logicfrom proof-theoretic viewpoints. Based on the semantic treatment ofHumberstone and del Cerro and Herzig, the paper proposed a sequentcalculus G(C + J) for the combined logic. By imposing restrictions onthe right rule for intuitionistic implication, it was shown that thecalculus G(C + J) enjoys cut elimination and Craig interpolation,which entails the decidability of this combination.

The problem of combining classical and intuitionistic logic wasanalyzed, from a philosophical and proof-theoretical perspective, byPrawitz 2015, when he introduced the important notion ofecumenical logics. According to him, it would be possible todefine a combined system in which both kind of reasoning could beperformed in a non-conflictive way. Thus, an intuitionistic reasonercould recognize and tolerate the validity of an instance \(A \vee_c\neg A\) of excluded middle with respect to the classical disjunction\(\vee_c\). In turn, a classical logician should accept that \(A\vee_i \neg A\) is not valid, where \(\vee_i\) denotes theintuitionistic disjunction. Moreover, according to Prawitz 2015,

The classical logician is not asserting what the intuitionisticlogician denies. For instance, the classical logician asserts \(A\vee_c \neg A\) to which the intuitionist does not object; he objectsto the universal validity of \(A \vee_i \neg A\), which is notasserted by the classical logician.[Prawitz 2015, p. 29]

Based on this perspective, he presented a natural deduction calculusfor a first-order, non-collapsing combination of classical andintuitionistic logic, in a language containing the classical logicalconstants \(\vee_c, \to_c\) and \(\exists_c\), the intuitionisticlogical constants \(\vee_i, \to_i\) and \(\exists_i\), and theconstants \(\bot, \neg, \wedge\) and \(\forall\) that are common toboth logics. He also considered predicates with subscripts \(i\) and\(c\). It is important to notice that classical logic is recoveredjust at the level of theorems and so, in particular,modusponens does not hold (as an inference rule) for the classicalimplication \(\to_c\). This means that his system representsintuitionist logic as a host logic, in which it is possible toconsider classical claims in a way that, for instance, \(A \vee_c B\)is a representation, under the intuitionistic perspective, of\(\neg(\neg A \wedge \neg B)\), while \(A \to_c B\) is expressed as\(\neg(A \wedge \neg B)\). A single-conclusion sequent calculus forecumenical first-order logic was proposed in Pimentelet al.2021. It is worth noting that Girard 1993 already introduced a(non-collapsing) sequent calculus (calledunified calculus,LU) for classical, intuitionistic and linear logics.

Extension of Prawitz’s ideas to the modal context were proposedin the literature. Marinet al. 2021 introduced a pure calculusfor ecumenical modalities, by means of a nested system with astoup, using a suitable notion of polarities for ecumenicalformulas. Following Prawitz’s notion of ecumenical logic, whilethere is only a necessity connective \(\square\), there are twopossibility operators, the intuitionistic \(\lozenge_i\) and theclassical \(\lozenge_c\). In this setting, an intuitionistic modallogic plays the role of the host, which is then expanded withclassical connectives. In turn, Rasga and C. Sernadas 2024 defined anon-collapsing combination of (classical) modal logicS4 with propositional intuitionistic logic. Theirapproach is different from the previous one, since in this case thehost of the combination process is classical modal logicS4. Besides this, they adopt a simplerproof-theoretic framework, through a standard Gentzen calculus. Inthis case, conjunction, disjunction and bottom are shared connectives,while implication and the necessity operator have distinctintuitionistic and classical versions, inducing two differentnegations in terms of bottom and the respective implications. It isproven that the calculus enjoys cut elimination, and it is sound andcomplete with respect to pure Kripke structures forS4, while accommodating intuitionistic connectives.The combined system conservatively extends both intuitionistic andS4 modal logic.

Independently of the collapsing problem of classical andintuitionistic implication, in Beziau 2004 it was observed that byputting together the sequent rules for classical conjunction and therules for classical disjunction, the resulting sequent calculus will(unexpectedly) prove the distributivity between conjunction anddisjunction. The same phenomenon occurs if we join the (two-valued)valuation clauses for classical conjunction with the valuation clausesfor classical disjunction. However, this is avoided by consideringalgebraic fibring in the usual categories (Hilbert calculi orconsequence relations) with translations between logics as morphisms:the logic obtained is the logic of lattices, which does not satisfydistributivity (see Beziau and Coniglio 2005, 2011).

This situation, in which new interaction rules between the connectivesarise, is arguably undesirable. In fact, it contradicts a basiccriterion of fibring (and also of fusion), as expressed in Gabbay1999: given logic systems \({{\cal L}}_1\) and \({{\cal L}}_2\), thecombination of \({{\cal L}}_1\) and \({{\cal L}}_2\) should be thesmallest logic system in the combined language which is a conservativeextension of both \({{\cal L}}_1\) and \({{\cal L}}_2\).

Indeed, the distributivity problem and the collapsing problem are twoinstances of the same phenomenon of emergence of unexpectedinteractions (such as bridge principles) between connectives caused bycombination processes. In the case of combination of conjunction withdisjunction, the distributive law emerges. This interaction is due tothe combination process and appears without any apparent reason. Inturn, the collapsing problem is a limit case of interactions, becausethe interderivability between classical and intuitionistic implication(nothing else than two interaction laws between different connectives)is also spontaneously created by the combination process.

It can be argued that the combined logics are excessively strong insuch cases, because they derive too many propositions in the newcombined language.

On the other hand, the opposite (or dual) situation may also beproblematic: suppose, to help intuition, that the logic of classicalnegation is combined with the logic of classical disjunction. Theselogics can be presented, for instance, axiomatically (through Hilbertcalculi) or semantically, say, through valuations over \(\{0,1\}\)(that is, by means of classical truth-tables). The semanticalpresentation of the logic of classical negation consists of the set ofall valuations over \(\{0,1\}\) satisfying the following clause:\[v(\neg\varphi)=0 \textrm{ iff } v(\varphi)=1.\] On the other hand, the logic of classical disjunction canbe characterized by the set of all valuations over \(\{0,1\}\) suchthat: \[v(\varphi \vee \psi)=0 \textrm{ iff } v(\varphi)=0 \textrm{ and } v(\psi)=0.\]

As a consequence, the combined logic of negation and disjunction(which can be defined as the logic over \(\neg\) and \(\vee\)characterized by the valuations over \(\{0,1\}\) satisfying bothclauses above) validates \((\varphi \vee \neg \varphi)\), and soclassical logic is recovered. This is the result obtained by thecombination method calleddirect union of matrices,introduced in Coniglio and Fernández 2005. However, ifalgebraic fibring is considered in categories such as those of Hilbertcalculi or consequence relations, the combination between the logic ofnegation and the logic of disjunction results in a logic defined over\(\neg\) and \(\vee\) which is weaker than classical logic. Inparticular, the interaction law \((\varphi \vee \neg \varphi)\) is nolonger valid. That is, an arguably desirable interaction between theconnectives is lost in the combination process, and classical logicexpressed over \(\neg\) and \(\vee\) cannot be recovered from itsfragments, as long as algebraic fibring in these categories of logicsis used.

Another example of the same kind is the following: the algebraicfibring between the logic of classical negation \(\neg\) and the logicof classical implication \(\to\) regarded in the categories above doesnot recover classical logic expressed over \(\neg\) and \(\to\).Indeed, the resulting logic system, defined over \(\neg\) and \(\to\),cannot validateEx Contradictione Sequitur Quodlibet whenpresented as an axiom: \[\not\vdash(\varphi \to(\neg\varphi \to \psi)).\]

Interestingly enough,Ex Contradictione Sequitur Quodlibet,presented as a derivation, holds in the fibred logic: \[\varphi, \neg\varphi \vdash \psi.\]

Observe that \((\varphi \to(\neg\varphi \to \psi))\) is an interactionrule between the connectives of the logics being combined which cannotbe obtained by algebraic fibring in the categories underconsideration. (However, this principle can be recovered, e.g., bydirect union of matrices.) If one is interested in recovering a logicfrom its fragments, this result is disappointing.

Concerning the study of the expressive power of the combination offragments of classical logic mentioned above, both at the axiomaticlevel, as well as considered as subalgebras of the 2-valued Booleanalgebra, Rautenberg obtained several interesting results in Rautenberg1981. One of his main contributions was the proof that any 2-valuedmatrix can be effectively axiomatized. In turn, the question ofcombination of fragments of classical logic by fibring wasadditionally investigated in Caleiroet al. 2019, based on thedescription of the 2-valued clones made in Post 1941, on theaxiomatization procedures for 2-valued matrices introduced inRautenberg 1981, and on the results on fibring (N)matrices presentedin Marcelino and Caleiro 2017b (seesubsection 4.3). In particular, the paper studied the conditions under which thefibring of the Hilbert calculi of disjoint fragments of classicallogic recovers the union of the fragments. Several examples ofcombinations by fibring of the logics of classical connectives wereanalyzed, obtaining their characterizations by (N)matrices, as well asthe additional inference rules (bridge principles) needed forrecovering the corresponding fragment of classical logic. Amongseveral examples, it was shown that the fibring of two copies of thelogic of classical conjunction collapses into the logic of a singleconjunction. On the other hand, neither the combination of two copiesof the logic of the classical negation nor the combination of twocopies of the logic of the classical disjunction produces an analogouscollapse. The fibring of two copies of the logic of classical negationis not finite-valued, but can be characterized by a 5-valued Nmatrix;and the fibring of two copies of the logic of classical disjunctioncannot be characterized by a single finite-valued Nmatrix. Byanalyzing once again the combination by fibring of the logic ofclassical disjunction and the logic of classical conjunction (that is,the least conservative extension of both logics) it was shown that thecombined logic cannot be characterized by a single finite-valuedNmatrix. The same happens with the fibring of the logic of classicaldisjunction and the logic of classical negation, thus showing that thefibred logics are strictly subclassical in both cases.

These examples, as well as other along the same lines, suggest aproblem dual to that of collapsing and distributivity betweenconjunction and disjunction, namely, some expected interaction lawsfail to be created by some combination processes.

In such cases, it could be said that the combined logics are too weakbecause they are unable to derive certain intended propositions in thenew combined language.

What could be expected when combining logics? Strong logics(guaranteeing, for instance, that a logic can be recovered from itsfragments) or weaker ones (in which undesirable interactions betweenconnectives are blocked)?

The examples above are evidence against and in favor of bothsituations. In order to avoid del Cerro and Herzig’s collapsingproblem, a careful splicing process should be expected, consequentlyeliminating the interaction between the two implications describedabove. On the other hand, if one wants to recover, say, classicallogic from some of its fragments, a more liberal splicing processwould be more appropriate as some intended interactions betweenconnectives of both logics would be recovered.

With respect to the distributivity problem when combining conjunctionand disjunction, the choice of method is also not determined:distributivity could be a desired feature if we adopted the viewpointof recovering a logic from its fragments. In this case, a combinationmethod defining a stronger logic (such as the direct union ofmatrices) would be more appropriate than, for instance, algebraicfibring of Hilbert calculi. But if, as argued in Beziau 2004,distributivity is regarded as an intruder, then a more cautiousprocess would be recommended; algebraic fibring of Hilbert calculiwould be more appropriate in this case. To sum up, the choice of themost adequate combining process depends upon what one wants toobtain.

At this point, it is convenient to notice that the question aboutwhether or not interactive principles are generated when combiningmodal logics, is intrinsically related to Hume’s ‘is-oughtproblem’ discussed insection 1. Indeed, as proven in Schurz 1991, it is possible to obtain nontrivial‘is-ought’ deductions in the combination of alethic anddeontic logics provided that some bridge principles are allowed. Thevalidity of bridge principles as \(\obcirc \varphi \rightarrow\lozenge \varphi\) is nothing else than interaction rules betweenconnectives of the logics being combined. Such principles enjoy asimilar conceptual status as the distributivity laws betweenconjunction and disjunction, or as the collapsing example mentionedabove. Thus, in order to satisfy Hume’s thesis, a combinationprocess generating logics without interactions should be preferred. Onthe other hand, a combination process allowing the creation ofinteractions between the connectives could grant bridge principlesviolating Hume’s thesis.

Finally, it is noteworthy to observe that algebraic fibring does notintrinsically forbid the emergence of interactions between connectivesof the logics being combined. In fact, the notion of morphism in thecategory of logic systems being employed is the key to creating orblocking interactions. In order to exemplify this assertion, considerthe case of the failure to recover classical logic from its\(\{\neg\}\)-fragment and \(\{\vee\}\)-fragment by algebraic fibring.From a proof-theoretic perspective, the key reason for this failure isthat the rule \[\tag{* }\frac{\Gamma, \varphi\vdash \psi \;\quad\; \Delta, \neg\varphi\vdash \psi}{\Gamma, \Delta\vdash \psi}\] of the logic of classical negation is notpreserved by algebraic fibring in categories of logic systems havingtranslations between logics as morphisms (recallsubsection 4.4), such as the category of Hilbert calculi or consequence relations.

When considering algebraic fibring of classical implication withclassical negation in those categories, the missing rule is theDeduction Metatheorem: \[\tag{**} \frac{\Gamma,\varphi\vdash\psi}{\Gamma\vdash(\varphi\to\psi)}.\]

Categories of logic systems having logic translations as morphisms aresuch that the canonical injections of the coproduct are just inclusionmappings. Then, given two logics \({{\cal L}}_1\) and \({{\calL}}_2\), the only rules of these logics which are preserved byalgebraic fibring are those of the form: \[\Gamma \vdash \varphi.\]

On the other hand, suppose that we are dealing with a category oflogic systems in which the preservation of rules such as \((*)\) or\((**)\) above is required by the very notion of morphism. Thus, if alogic \({{\cal L}}\) is obtained as a combination of two other systems\({{\cal L}}_1\) and \({{\cal L}}_2\) then the rules of \({{\calL}}_1\) and \({{\cal L}}_2\) would be faithfully transferred to\({{\cal L}}\). This is the proposal of Coniglio 2007, in whichalgebraic fibring in categories of sequent calculi is investigated,taking into account a notion of morphism which preserves logical rulesof the form \[\textit{If }\;\; \Gamma_1\vdash\varphi_1 \;\;\textit{ and … and }\;\; \Gamma_n\vdash\varphi_n \;\;\textit{ then }\;\;\Gamma\vdash\varphi.\]

In such categories, when a logic system is embedded into a larger oneby algebraic fibring, any rule as above, which can be considered as ameta-theorem of the logic, is preserved by the canonical injections.This is why this process is calledmeta-fibring. From thecategorial point of view, the process is the same as for algebraicfibring, the only difference being that the notion of morphism isstronger.

In Coniglio and Figallo 2015, Coniglio and Figallo extended the ideaof meta-fibring (that is, algebraic fibring of formal sequent calculi)tohypersequents, which allows preserving even strongermeta-properties of the logics being combined. Hypersequents, which arefinite multisets of ordinary sequents, constitute a naturalgeneralization of the proof-method of sequents and turn out to be asuitable tool for presenting cut-free Gentzen-type formulations forseveral non-classical logics.

These examples illustrate the advantages of using category theory andtheir tools for defining combination procedures as universalconstructions. The same construction (in this case, algebraic fibring)can be performed in categories of logic systems with differentfeatures obtaining, as a consequence of this, stronger or weaker logicsystems.

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