Combining Logics

8/6/2019 Combining Logics

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Combining Logics

First published Thu Sep 13, 2007

The subject of combinations of logics is still a young topic of contemporary

logic. Besides the pure philosophical interest offered by the possibility ofdefining mixed logic systems in which distinct operators obey logics of

different nature, there exist also many pragmatical and methodological

reasons for considering combined logics. In fact, the use of formal logic as a

tool for knowledge representation in Computer Science frequently requires the

integration of several logic systems into a homogeneous environment.

Important questions in the philosophy of logic such as: why there are so

many logics instead of just one?(or even, instead of none), as e.g., raised

in Epstein 1995, can be naturally counterposed by several other questions: ifthere are indeed many logics, are they excluding alternatives, or are they

compatible? Is it possible to combine different logics into coherent systems

with the purpose of using them in applications and to shed some light on the

properties of complex logics? Moreover, if we cancompose logics, why

not decompose them? And, if a logic is decomposed into elementary

sublogics, is it possible to recover it by combining such fragments? What kind

of properties of logics can be transferred to their combinations? Questions of

this kind have been only partially tackled in the literature, and reflect

challenges to be confronted 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

o 4.1 Fusion and Productso 4.2 Fibringo 4.3 Algebraic Fibringo

4.4 Possible-Translations Semanticso 4.5 Temporalization, Parameterization and Institutions

5. Lack or excess of interaction: perplexities when combining logics Bibliography Other Internet Resources Related Entries

1. Philosophical and methodological motivations for combining logics


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David Hume generated a popular controversy with his famous passages of A

Treatise of Human Nature (cf. Hume 2000, Book 3, Part 1, Section 1,

paragraph 27) where he noted that sometimes people draw conclusions

involving prescriptive statements of the form ought to be on the basis of

descriptive statements of the form what is. Hume thinks that logic used inthis way involves a dangerous change of subject matter. So, whether or not

ought can be derived from is has become one of the central questions of

ethical theory, the majority of interpreters hold that, for Hume, such a

derivation is impossible.

With our point of view concerning combinations of logics, it is necessary to

investigate the properties of combining deontic and alethic 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 principlewhich specifically connects is and ought. An axiom schemaX,

following Schurz 1991, is a bridge principle iffXcontains at least one

schematic letter which has at least one occurrence within the scope of an

(the deontic obligation operator) and at least one occurrence outside the

scope of any . Thus, for instance,

p p

is a bridge principle representing is-ought that which would appease Hume's

criticism. On the other hand, the much discussed moral principle ought-

implies-can (controversially attributed to I. Kant, see Baumgardt 1946) can

be formalized through another bridge principle:

pp,

where denotes the alethic possibility operator.

Clearly, bridge principles do not solve any philosophical questioning as the

is-ought problem; nonetheless, they contribute to clarify the problem and touncover hidden assumptions. The idea of combining logics lend clarification

to questions of this kind by making clear that, for instance, is and ought are

indeed independent notions. This is elucidated through a formal analysis of

the composition of the logics involved (in this case, alethic and deontic) or by

decomposition of the complex logic (in this case, bimodal) into simpler ones.

In such circumstances, combining logics can be perceived as a tool for

simplifying problems involving heterogeneous reasoning.


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The fact that ought is not conveyed as a predicate, but as a modal operator

ranging over actions or states of affairs, was responsible for the delay of

formal treatments of this centuries-old question.

Such a treatment was only possible after the development of general modallogic. Indeed, what we are dealing with here is a bimodal logic, which is

properly treated only after a deeper understanding of the semantical subtleties

of mixing alethic and deontic logics. Moreover, according to some

philosophers who have argued that it is not possible to link is and ought

(that is, who defend Hume's thesis that no non-trivial is-ought deductions are

possible), it is mandatory to use combinations of first-order, alethic and

deontic logics (cf. e.g., Stuhlmann-Laeisz 1983 and Schurz 1997).

A. Prior (1960), using the apparatus of contemporary modal logic, tried to

characterize the distinction between normative andnon-normative sentences in

formal terms, which enabled him to define senses of descriptive content

versus normative content. A problem, however, occurs with mixed

sentences, which have both descriptive and normative components, and Prior

comes up with a paradox: wherever we draw the distinction between non-

normative and normative sentences, there appear inferences from non-

normative premises to normative conclusions by a mere use of laws of

classical propositional logic. Consider, for instance, the following two

inferences:

(1) Tea-drinking is common in England. Therefore: Either tea-drinking is

common in England or all New Zealanders ought to be shot,

formalized as:

(1) dd s,

and

(2) Tea-drinking is common in England, or all New Zealanders ought to be

shot. Tea-drinking is not common in England. Therefore, all New Zealanders

ought to be shot,

formalized as:

(2) d s, d s.

If the mixed sentence d s is considered to be normative, then (1) is anexample of an is-ought inference, and if it is considered to be non-


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normative, then (2) is an example of an is-ought inference. So, one of them

dichotomically represents a violation of Hume's thesis in Prior's terms. Prior

concluded from this paradox that Hume's is-ought thesis is simply false

(cf.Prior 1960, p. 206): one can simply derive conclusions which are ethical

starting from premises which have no ethical character.

Prior recognized however that the inferences involved in the paradox are

ethically irrelevant or trivial, but neither him nor later works could find a

suitable definition of what it would mean by ethical irrelevance or ethical

triviality attached to an inference.

Using the semantics of modal logics, objections against this conclusion can be

raised, as for example in Karmo 1988, in the sense of separating statements

between evaluative in some possible worlds and descriptive in others (while

keeping their meaning).

By using concepts of combinations of languages and combinations of logics,

G. Schurz (cf. Schurz 1991; see also Schurz 1997) was able to state

a generalized Hume's thesis (GH); as observed in Subsection 4.1, this

treatment is in fact a fusion of two modal logics. In (GH) a mixed sentence

is derived from a set of purely descriptive sentences (i.e., sentences free of )

only if is completely -irrelevant (that is, predicates in within the scope

of can be replaced by other predicates salva valididate). Moreover, it is

proven that (GH) holds in an alethic-deontic first-order logicL if, and only

if,L can be axiomatized without bridge principles.

The notion of bridge principle lies in the scope of combination of languages.

In general, many bridge principles can be made explicit within modal logic,

and will be relevant for analyzing relationships among diverse modalities. For

example, if we take necessity and possibility as primitive operators, then

p p

is an intuitionistically acceptable bridge principle, while the converse is not.

Besides Hume's problem, another example of bimodal logic with intrinsic

philosophical interest where bridge principles intervene is the logic of

physical and alethic modalities. In this logic, the language permits the

expression of two different notions of necessity: the logical necessity,

symbolized by , and the physical necessity, symbolized by .


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The simplest connection between physical necessity and logical necessity that

comprises an acceptable philosophical meaning is given by the following

bridge axiom:

p p

meaning that logical necessity is stronger than physical necessity: anything

that is logically necessary is physically necessary.

The resulting logic KT is axiomatized by the well-known axioms and rules

ofKT for both modalities in addition to the bridge axiom above, and is

semantically characterized by Kripke frames with two accessibility relations,

imposing that the accessibility relation for physical necessity is included in the

other.

Not only bimodal, but multimodal (also called polymodal) logics, are standard

in the literature: a typical case is the logic of knowledge (or epistemic logic),

usually endowed with modal operators K1, K2,, Km representing the

knowledge ofm agents (or knowers). The formula Ki means

agent i knows , and the language is able to express, for instance, i knows

thatjdoes not know that i knowsp by means ofKiKjKip. No additional

mixing principles are mandatory for the combined logic of many agents, but

bridge axioms may of course be added.

The interest of studying combinations of logics may thus be seen as a reflex of

the pluralist view of contemporary logical research. Indeed, this kind of bridge

axioms can, in principle, connect completely distinct logics. Van Benthem

(2005) suggests that combining logics may lead to the emergence of new

phenomena, depending on the mode of combination, and moreover, it may

work as an inspiration (and perhaps as a model) for the study of combining

epistemic notions. He even suggests that the compartmentalization of logic

into subfields as modal, temporal, epistemic, doxastic, erotetic or

deontic logic has been harmful to Philosophical Logic.

Combinations of logics go in the opposite direction of such a

compartmentalization: considering that almost any conceptual task to be

analyzed involves immediate reasoning concerning necessity, obligation,

action, time, verbal tense, knowledge, belief, etc.; from a philosophical point

of view, logical combinations may be the right way to look at philosophical

issues within the theory of causation, of action, and so on.

The idea of looking at logic as an entirety avoiding fragmentation is not new,and philosophers and logicians from Ramn Lull to Gottfried W. Leibniz have


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thought of building schemes where different logics or logic-like mechanisms

could interact and cooperate instead of competing. In contemporary terms, the

first methods for combining logics wereproducts of logics(introduced by K.

Segerberg (1973) and independently by V. Sehtman

(1978)),fusion (introduced by R. Thomason (1984)) andfibring (introducedby D. Gabbay (1996a)), all of them dedicated to combining only modal logics.

It is worth noting that M. Fitting (1969) gave early examples of fusion of

modal logics, anticipating the notion of fusion.

Other combination mechanisms followed, such as parameterization and

temporalization, which were more on the side of software specification.

Most of these methods have been encompassed in the algebraic

fibring introduced by A. Sernadas, C. Sernadas and C. Caleiro (1999), which

notoriously improved the versatility of these techniques by means of

(universal) categorial constructions, in this way making it possible to combine

wider classes of logics besides modal logics.

On the other hand, making heavy use of the language of category theory, J.

Goguen and R. Burstall introduced the notion ofinstitutions as a kind of

abstract model theory devoted to applications in Computer Science

(see Goguen and Burstall 1984and Goguen and Burstall 1992). Institutions are

also used as a mechanism for combining logics.

However, combining logics does not only mean synthesizing or composing

logics, but can also yield interesting examples that go in the opposite direction

of decomposing logics (see Section 2). A paradigmatic methodology for

decomposition is the possible-translations semantics, a notion proposed

in Carnielli 1990 designed to help solve the problem of assigning semantic

interpretations to non-classical logics. Examples of possible-translations

semantics illustrate how a complex logic can be analyzed into less complex

factors. Another closely related technique is the nondeterministic

semantics (cf. Avron and Lev 2001 and Avron and Lev 2005), whereas directunion of matrices andplain fibring (cf. Coniglio and Fernndez 2005) can be

considered to be methods for both composing and decomposing.

All of these methods open the way for a new subject in the realm of

combinations of logics: is it possible to decompose a given logic into

elementary ones? In other words, are thereprime logics which, combined in

an appropriate way, may produce all (or part of) the familiar logic systems?

Results on combinations of logics may quickly become too technical when weturn to the combination of higher-order, modal, relevance logics or non-truth-


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functional logics, and thus refinements of the notion of algebraic fibring such

as modulated fibring (cf. Sernadas et al. 2002b) or cryptofibring (cf. Caleiro

and Ramos 2007) may be necessary to solve, for example, some collapsing

problems within combinations of logics (see Section 5). This naturally leads to

the use of category theory as a universal language and as a tool to deal withsuch problems. But the fact is that combinations of logics does not necessarily

depend upon any highly technical methodology, and even some relatively

simple examples can be really expressive. There is a recognized intersection

and interaction between Philosophy and theoretical Computer Science, and

techniques for combining logics also reveal to be a very apt tool for handling

and thus better understanding Kripke models. Having been introduced in the

domain of Philosophical Logic, Kripke models are essential in Computer

Science and Artificial Intelligence as semantic structures for logics of belief,

knowledge, temporal logics, logics for actions, etc. Knowledge representationand reasoning may require combining several reasoning formalisms, including

combinations of temporal reasoning, reasoning in description logic, reasoning

about space and distance, and so on. Logics, combining temporal and modal

dimensions, are also becoming a relevant tool in agent-oriented programming

languages. Other applications of combinations of logics include software

specification, knowledge representation, architectures for intelligent

computing and quantum computing, security protocols and authentication,

secure computation and zero-knowledge proof systems, besides its

connections to formal ethics and game semantics.

2. Splitting logics versus splicing logics

It is reasonable to expect that a method for combining logics would work in

two opposite directions: on the one hand, a logic that one wants to investigate

could be decomposed into factors of lesser complexity; for instance, a bimodal

alethic-deontic logic could be decomposed into its alethic and deontic

fragments. In this case, it would be relevant to see if the logic under

investigation is the least extension of its factors, or if additional bridge

principles would have to be added. This approach, in which a given logic is

decomposed into (possibly) simpler factors, is said to be a process ofsplitting

logics.

On the other hand, one might be interested in creating new logic systems

where different aspects are integrated, starting from given logics. This demand

typically occurs in software engineering and security: knowledge

representation, formal specification and verification of algorithms and

protocols have a marked need for working with several logics. In a less

pragmatical scenery, this would be the case if one is interested, for instance, in


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adding a modal dimension to an intuitionistic or a paraconsistent logic.

Moreover, it is interesting to characterize which properties of the factors can

be transferred to the combined logic. This direction is said to be a process

ofsplicing logics.

The essential distinctions between splicing (in the direction of synthesis) and

splitting (in the direction of analysis) take into account the intentions one may

have in mind, and consequently each direction encompasses specifically

designed techniques.

The paradigm of splicing logics assumes a bottom-up perspective: it combines

given logics, synthesizing them, and producing a new one. The combined

logic should be minimal in some sense: that is, ifL is obtained

from L1

and L2

by some combination process, it should be expected that:

(1) Lextends both L1 and L2; and (2) L is a minimal extension of

both L1 and L2. For instance, some methods may require L to be the least

conservative extension of both L1 and L2. This point will be discussed in

Section 5.

On the other hand, splitting a logic L assumes a top-down perspective: logics

are decomposed into (presumably simpler) factors.

It should be stressed that most of the methods for combining logics found in

the literature are better understood from the splicing perspective, placingprominence on the creation of a logic system from familiar logics. However,

some splicing methods such as fusion (see Subsection 4.1) are more usefully

regarded as a method of decomposition of logics into simpler fragments, and

in this way also work in the splitting direction. Possible-translations semantics

(see Subsection 4.4), on the other hand, constitute a typical method within the

splitting perspective.

3. The importance of language and the presentation of logics

Suppose that two given logics L1 and L2 are to be combined using some

technique. It should be obvious that any method applied to

combine L1 and L2 will create a new logic L which contains

the signature (logic symbols such as connectives, quantifiers, propositional

variables etc.) ofboth logics: L will be defined in a mixed language, which

allows combinations of symbols of the underlying languages. That is, a

combination of logic systems presupposes the previous combination of the

respective signatures. This is why the choice of the signature of the combined

system is as important as the logic itself. For instance, the definition of thelanguage of parameterization is fundamental in order to obtain the intended


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combined logic (see Subsection 4.5). Another example is found in Schurz

1991, where the formal treatment of Hume's is-ought problem (recall

Section 1) presupposes careful handling of subtle combinations of languages.

Besides the definition of the appropriate language for the combined logic,another important question that immediately arises is: should the

logics L1 and L2 (to be combined) be presented in the same way? In other

words: is it possible to combine logics defined by different paradigms? For

instance, how could one combine a logic L1, defined by a sequent calculus,

with a logic L2, represented by a (Hilbert-style) axiomatic system? How

should the resulting logic L be represented: as a sequent calculus, as an

axiomatic system or as a mixed proof system? Consider now another (even

worse) situation: the logic L1 is described by semantical means (that is,

through semantic structures such as valuations or Kripke models) whereas thelogic L2 is presented through a syntactical proof system, such as a natural

deduction system, sequent calculus or a Hilbert-style axiomatization. Could

the resulting (combined) logic be better presented semantically or

syntactically?

This annoyance does not occur in the majority of cases, where the logics being

combined are complete with respect to some kind of semantics and are

syntactically presented in an homogeneous way. However, it may happen that

the logics are found in peculiar ways; for instance, linear logic and other

substructural logics have no usual consequence relations because derivations

are exclusively displayed by using multisets or sequences of formulas.

Combinations of such logics with usual modal logics, for instance, are not so

obvious, although both are complete.

Still, there are logics which are only reasonably presented by syntactical

means, or exclusively by semantical means. Such is the case, e.g., of the first-

order theory of torsion groups, known to be non-axiomatizable, and of

incomplete modal logics which are only presented 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 a logic?, is to consider a

common component of the majority of logics (but still excluding certain

substructural logics): their respective consequence relations. Thus,

given L1 and L2 presented in different ways, it is always possible to extract the

respective consequence relations and then combine them (taking, for instance

their supremum in an appropriate lattice of consequence relations). But in this

way, the resulting logic L is presented in a very abstract way: the onlyinformation available from L is its consequence relation, and so the


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characteristics and particularities of each logic component are definitively

lost.

Returning to the first example (combining a sequent calculus with an

axiomatic system), a better solution was proposed in Cruz-Filipe et al. 2005:the idea is to define an abstract formalism for proof systems, general enough

as to encode the main proof mechanisms found in the literature. Thus, after

reformulating L1 and L2 as abstract proof systems of this kind, the resulting

combined logic L is an abstract proof system in which it is possible to

recognize the genetic traces of the original inference rules of each

components within derivations in L.

Despite these results on combining heterogeneous logics, it seems more

reasonable to combine logics defined in an homogeneous way, and, in fact,

this is the case with most of the proposals in the literature. For instance, the

usual combinations of modal logics (as fusion, product and fibring) are

performed between systems presented axiomatically, 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 (or

decomposition) of logics appear as universal constructions. Algebraic fibring,

to be described in Subsection 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 by R. Thomason

(1984), and constitutes one of the first general methods for combining logics.

In the original formulation, it combines normal modal logics presented

syntactically and semantically (through Hilbert-style axioms and Kripke

semantics, respectively). The main characteristics of the method are described

in the following paragraphs.

Consider Kripke models of the form

W,R, V

such that Wis a non-empty set (the set of worlds),R WWis a binary

relation (the accessibility relation) and V: Wfrom the set of

propositional variables into the power set ofWis a valuation map.

LetL

1 andL

2 be two propositional normal modal logics defined over the sameclassical signature which contains the connectives (negation) and


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(implication). Denote by 1 and 2 the necessity operators ofL1 and L2,

respectively. Let M1 and M2 be the classes of Kripke models for L1 andL2,

respectively. Since both logics are normal, it is granted that both modalities

1 and 2 satisfy the normality axiom Kand the necessitation rule.

Thefusion ofL1 and L2 is then defined to be the normal bimodal logic L withtwo independent boxes 1 and 2 together with the connectives (negation)

and (implication). The semantics ofL is given by the class Mof Kripke

structures of the form

W,R1,R2, V

such that W,R1, V and W,R2, V belong to M1 and M2, respectively. In

other words, each structure of the fusion corresponds to a pair of models: a

model W,R1, V for L

1and a model W,R

2, V for L

2sharing the same

set of worldsW. Technically speaking, each structure of the fusion has, as a

reduct, a model ofL1 and a model ofL2.

Given a structureM= W,R1,R2, V , the accessibility relationR1 is used to

evaluate the box 1, whereasR2 is used to evaluate 2. Since the language

ofL is freely generated by the union of the signatures ofL1 and L2, it contains

mixed formulas such as = 1(2pp). Now, the structureMsatisfies

above at a world w Wif and only if, for every w1 Wsuch

thatwR1w1,Msatisfies (2pp) at w1. But this means that, either there

exists w2 such that w1R2w2 and w2V(p), or w1 V(p).

As concerns axiomatics, a Hilbert calculus for L is obtained by joining up the

(schema) axioms of both systems. Thus, L has, among others, two Kaxioms,

two necessitation rules and just oneModus Ponens (because implication is

shared). Considering that the language ofL has mixed formulas (as above),

schema variables occurring in the schema rules of the given logics can now be

replaced in L by mixed formulas. For instance, can be derived in L from the

formula (2pp) by an application of the necessitation rule for the box 1.

An interesting example of fusion appears in Schurz 1991, when an alethic-

deontic logic is defined by fusing a pure alethic logic with a pure deontic

logic. This combination is used to analyze Hume's is-ought thesis (see

Section 1 above) in formal terms. Other intuitively appealing examples of

fusion are given in the pioneering paper Fitting 1969, where alethic and

deontic modalities are fused (before the concept of fusion had ever been

introduced).

Fusion has since then been a much worked theme. Important results are theapplications of fusion to simulations and to the question of transfer of


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properties among modal logics. Simulations make the strength of normal

monomodal logics explicit, as they can, in a sense, simulate all modal logics

(see Kracht and Wolter 1999). With respect to transfers, the preservation of

properties such as completeness, finite modal property, decidability and

interpolation by fusion of modal logics was extensively studied in Fine andSchurz 1996. Several results in the same spirit were also obtained in Kracht

and Wolter 1991, although they stressed fusion of infinitely many modal

logics. A survey of most of those results can be found in Kracht and Wolter

1997. These results show the robustness of fusion as a combination method

within the scope of modal logics, for fulfilling the requirement of preserving

the properties of the logics being combined.

An interesting note is that there is a notorious difference between combining

logics from the syntactical and from the semantical perspective. For instance,the joining of two Hilbert calculi should be intuitively obtained by simply

putting together the axioms and rules of both logics, while the semantical

counterpart is not so obviously determined. Regarding this, an alternative to

fusion is the fibred semantics (see Subsection 4.2).

Fusion, as a natural method for combining modal logics as it can be, however,

is not obviously extendable to combinations of non-normal modal logics with

normal modal logics. Moreover, fusion is specifically designed for combining

modal logics, and cannot be extended in an obvious way to logics of a

different nature. Algebraic fibring, described in Subsection 4.3 below,

constitutes a generalization of fusion (at the propositional level), and generally

solves the question of combining logics.

Another early method for combining (modal) logics is the so-calledproduct of

modal logics. This mechanism, independently introduced in Segerberg

1973 and in Sehtman 1978, is appropriate to represent time-space information.

Given two modal logics L1 and L2 as above, the product L1 L2 is the bimodal

logic over the mixed signature (endowed with two boxes) characterized by the

class of Kripke structures of the form

W1W2, S1, S2, V1V2

defined from Kripke models W1,R1, V1 and W2,R2, V2 for L1 and L2,

respectively. The accessibility relations S1, S2 (W1 W2) (W1 W2) are

defined as follows:

u1, u2 S1 w1, w2 iffu1R1w1 and u2=w2; u1, u2 S2 w1, w2 iffu2R2w2 and u1=w1; (V1V2)(p) = V1(p) V2(p).


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A somewhat surprising feature of the product of modal logics is that some

new interactions between modalities arise. These new valid formulas are a sort

of bridge principles (recall Section 1). Using the standard notation 1 for

1 (and analogously for 2) for the possibility operator, the following

bridge principles are always valid in the product logic:

12p21p Commutativity 1

21p12p Commutativity 2

12p21p Church-Rosser property 1

21p12p Church-Rosser property 2

Due to such interactions it is not possible to directly obtain the Hilbertcalculus for the product of two modal logics, as in the case of fusion. The

bridge principles must be explicitly added to the union of the original

axiomatics in order to ensure completeness.

As in the case of fusion, this technique does not allow a direct generalization

to logics other than modal ones.

4.2 Fibring

Thefibred semantics of modal logics was originally proposed by D. Gabbay

(1996a and 1996b) (see also Gabbay 1999). As in the case of fusion and

products, the mechanism of fibring also applies to modal logics only. Assume

the same notation as in Subsection 4.1. Thus, given L1 and L2, we start by

defining the fibred language (or the fibring of the languages), which is the

language generated by 1, 2, and from the propositional variables. The

basic idea is to consider Kripke models with distinguished (actual) worlds

together with two transfer mappings: h1 from the set of worlds of the class of

models M1 ofL1 into the class of models M2 ofL2, and h2 from the set of

worlds of the class of models M2 ofL2 into the class of models M1 ofL1.

When a Kripke model ofL1 has to evaluate a formula of the form 2 at the

actual world w1, the validity checking is then transferred to the validity

checking of2 within the Kripke model h1(w1) at its actual world. The

evaluation of a formula of the form 1 within a Kripke model ofL2 at the

actual world w2 is performed analogously, but now using the map h2.

Thus, the fibring by functions ofL1 and L2 is a normal bimodal logic

characterized semantically as follows: let


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h1: m M1Wm m M2 { m, w : w Wm}

and

h2: m M2Wm m M1 { m, w : w Wm}

be a pair of transfer mappings. For simplicity, we assume that the sets of

worlds Wm ofm M1 are pairwise disjoints, and the same holds for M2.

Given m M1 M2, w Wm and a formula in the fibred language, the

satisfaction of in h1,h2,m,w , denoted by h1,h2,m,w , is defined

recursively as usual whenever the main connective of is Boolean ( or ),

or when is atomic. For the modalities, satisfaction is defined as follows:

suppose (without loss of generality) that m M1, and let h1(w) = m2, w2 ,

with m = Wm,Rm, Vm and m2= Wm2,Rm2, Vm2 . Then:

h1, h2, m, w 1iff h1, h2, m, w1 , for every w1 such that wRmw1;

h1, h2, m, w 2iff h1, h2, m2, w2 2

iff h1, h2, m2, w3 , for every w3 such that w2Rm2w3.

The definition of h1, h2, m, w i for i=1,2 and m M2 is analogous.

Then, h1, h2 satisfies , denoted by h1, h2 , if h1, h2, m, w for

every m M1 M2 and w Wm. Finally, is valid in the fibred semantics

whenever h1, h2 for every pair h1, h2 as above.

For instance, given h1, h2 as above, let W2,R2, V2 M2 and w2 W2 such

that h2(w2) = W1,R1, V1 , w1 . Then:

h1, h2, W2,R2, V2 , w2 12p

iff h1, h2, W1,R1, V1 , w1 12p

iff h1, h2, W1,R1, V1 , w1 2p, for every w1 such that w1R1w1.

Suppose that h1(w1) = W2,R2, V2 , w2 . Then, the latter is valid iff

h1, h2, W2,R2, V2 , w2 2p, for everyw1 such that w1R1w1; i.e., for

every w1 such that w1R1w1 and for every w2 such that w2R2w2, h1, h2,


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W2,R2,V2 , w2 p. This is equivalent to say that, for every w1 such

that w1R1w1 and for every w2 such that w2R2w2, w2V2(p).

With respect to axiomatics, the logics obtained by fibring (or by a variant of

fibring called dovetailing) can, in some cases, be axiomatized by the union of

the (schema) axioms of the given logics. But some logics may require the

addition of some new bridge principles (mixing rules and axioms) in order to

ensure the preservation of completeness. This may explain some discrepancy

between the approaches of fusion and fibring; the completeness of fibring as

exposed in Gabbay 1999 does not work exactly as a substitute of more

technically intricate completeness proofs as in Kracht and Wolter 1991 and

in Fine and Schurz 1996. For more on this discussion, see Kracht 2004.

The technique of fibring by functions is an interesting alternative to fusion and

products, but, as much as its competitors, it cannot be extended to non-modal

logics in any obvious way (see Coniglio and Fernndez 2005 for an adaptation

of the method of fibring by functions to matrix logics). One reason for the

failure of fibring by functions to what concerns generalizations is that it is not

a universal construction (in categorial terms). Moreover, the lack of a

systematic definition of the axiomatization for the logics obtained by fibring is

another negative aspect of this technique. The next subsection describes a

categorial generalization of fibring which solves all the mentioned problems.

4.3 Algebraic Fibring

In order to overcome the limitations of the original method of fibring as

exposed in the last subsection, A. Sernadas and collaborators propose,

in Sernadas et al. 1999, a general definition of fibring using the conceptual

tools of category theory. The central idea of the generalization is simple:

suppose that L1 and L2 are two propositional logics which are to be combined.

Suppose, for simplicity, that no connectives are to be shared, that is, the

language of the logic L to be obtained is the free combination of theconnectives of both logics. In categorial terms, the signature CofL is the

coproduct (disjoint union) of the signatures, C1 ofL1 and C2 ofL2, in the

underlying category of signatures. Then L, which is the leastlogic defined

over Cwhich extends simultaneously L1 and L2, is defined as the coproduct

ofL1 and L2 in the underlying category of logics. The minimality ofL attends

a criterion expressed in Gabbay 1999 (see also Section 5) and also conforms

with the ideal of fusing logics, cf. Kracht and Wolter 1991. This combination

process, called unconstrained fibring, can be generalized, by

allowingC1 and C2 to share some connectives. Thus, the logic obtained by thissecond kind of fibring is defined in a language such that some connectives


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are new, because the formulas 2p and 2(q21r) do not belong to the

language ofL1. Analogous replacements apply, of course, to other inference

rules and axioms ofL1 L2.

Continuing with this example, suppose now that we want to share (or identify)both negations, as well as both implications: this is a reasonable move when,

for instance, these connectives are classic. In such a case (1 2) would

represent the same proposition as (2 1).

In order to do this, the signature C0 just containing and is taken into

consideration, and so 1 is identified with 2 in C1 C2, as well as 1 is

identified with 2. The resulting signature is C, which just contains the

connectives , , 1 and 2. In the resulting logic L, defined over C, there is

now just one rule ofModus Ponens:

1 (12)

2

However, there remains two necessitation rules, since there are still two boxes

in C. The resulting L is thus the fibring ofL1 andL2 constrained by C0. This

procedure precisely coincides with fusion of modal logics. The novelty here is

that this technique applies to a wide class of logics, which are not necessarily

restricted to (normal) modal logics, as in the case of fusion.

Constrained and unconstrained fibring, being categorial, are universal

constructions, and so enjoy well-defined and theoretically predictable formal

properties. Profiting from universal constructions, in order to handle algebraic

fibring, it is enough to define appropriate categories of signatures and logic

systems. Indeed, the same fibring construction (coproduct or cocartesian

lifting) can be performed in differentcategories of logic systems. This is a

remarkable advantage of the categorial perspective for fibring. There are

several proposals in the literature devoted to combining logics presented indifferent ways by means of algebraic fibring: propositional Hilbert calculi,

first-order modal logics, higher-order modal logics, non-truth-functional

logics, logics semantically presented through ordered algebras (encompassing

generalized Kripke models) etc.

An important question connected to combination of logics (and, in particular,

to algebraic fibring) is the preservation of metaproperties such as

completeness, interpolation etc. For instance, when L1 and L2 are complete

logic systems presented both semantically and syntactically, under whichcondition is their fibring also complete? In this regard, Zanardo et al.


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2001 andSernadas et al. 2002a give a partial solution to this question. On the

other hand, transfer results have been extensively studied in the case of fusion

of modal logics, as already mentioned in Subsection 4.1.

The relationship between fusion and algebraic fibring deserves somecomments. When restricted to modal propositional logics, fusion is a

particular case of algebraic fibring in the category of interpretation systems,

where logics are presented through ordered algebras: it is enough to consider

interpretation systems defined over power set algebras induced by Kripke

models. At the syntactical level, fusion is also a particular case of algebraic

fibring in the category of Hilbert calculi, in the realm of propositional

signatures. As much as first-order modal logics are concerned, the approaches

diverge, mainly because there are different semantical accounts for treating

first-order modalities. For instance, when considering algebraicfibring, Sernadas et al., 2002a offer a different semantical approach to modal

first-order logics than that ofKracht and Kutz 2002 for fusion.

The fact that algebraic fibring generalizes (at least at the propositional level)

the fusion of modal logics makes the former method become very natural and

useful. Moreover, the universality of the construction allows to define

algebraic fibring in very different logical contexts (categories of logics), such

as non-truth-functional logics, higher-order logics, sequent calculi etc. As it

will be shown in Section 5, the different notions of morphisms between logics

affect the strength of the logics obtained by algebraic fibring in the different

categories of logic systems. For general accounts of algebraic fibring see, for

instance, Caleiro et al. 2005and Carnielli et al. 2008.

4.4 Possible-Translations Semantics

The methods for combining logics described above adhere to the splicing

methodology: they are used to combine logics creating new systems which

extend the given logics.

As mentioned in Section 2, there is a converse direction: the splitting

methodology in which a given logic system is decomposed into other systems.

The possible-translations semantics (in short, PTS), introduced in Carnielli

1990, is one of the few supporters of this viewpoint.

The notion of PTS was originally defined as an attempt to endow certain

logics with recursive and palatable semantic interpretations. Concretely,

several paraconsistent logics which are not characterizable by finite matrices

can be characterized by suitable combinations of many-valued logics. Theidea of the decomposition is quite natural: given a logic L, presented as a


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pair L = C, L in which Cis a signature and L is a consequence relation,

a family of translationsfi :L(C) L(Ci) (for i I) is taken into consideration.

Here,L(C) andL(Ci) denote the algebra of formulas defined by the

signature Cand Ci, respectively. Recall that a translation from a logic L into a

logic L is a mappingfbetween the respective sets of formulas which

preserves derivability, that is: L (in the source logic L) implies thatf( )

Lf() (in the target logic L).

A pair P = {Li}i I, {fi}i I as above is called apossible-translations

frame forL. We say that P is apossible-translations semantics forL if, for

every {} L(C),

L ifffi() L

if

i(), for every i I.

This means that checking derivability in L is equivalent to checking

derivability in every factor logic Li through the translations. In many cases,

the factor logics Li are presented by finite matrices. Since the length of a

formula is finite, it is enough to test a finite number of translations in order to

determine if a formula ofL is valid in L. Thus, checking the validity of a

formula ofL is equivalent to performing a finite number of finitary tests. This

decidability property is of real advantage when the original logic Lis not

characterizable by finite matrices. For instance, the hierarchy {Cn}n of

paraconsistent logics of N. da Costa, formed by logics which cannot be

characterized by finite matrices, can be represented by means of a PTS whose

factors are presented through finite matrices; this grants a decision procedure

for each logic Cn.

In order to exemplify the concept of PTS as a splitting methodology, consider

the paraconsistent logic bC, introduced inCarnielli and Marcos 2002. This

logic is, in particular, a logic of formal inconsistency, in the sense that there

exists a unary connective expressing the consistency of a formula. Thus,

from and does not follow, in general, an arbitrary formula . However,{,, } entails any formula . The signature CofbC consists of a

paraconsistent negation , a consistency operator , and classical connectives

, ,. It has been proved that bC, and many other logics of formal

inconsistency extending it, cannot be characterized by finite matrices.

Nonetheless, bC is decomposed into several copies of a three-valued logic by

means of possible-translations as follows: consider the signature C1 =

{1,2, 1, 2, 3, , , } consisting of two negations, three consistency

operators, a conjunction, a disjunction and an implication. LetMbe the matrix

over C1 with domain {T, t, F} displayed below, where {T, t} is the set ofdesignated values.


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T t F

T t t F

t t t F

F F F F

T t F

T t t t

t t t t

F t t F

T t F

T t t F

t t t F

F t t t

1 2

T F F

t F t

F T T

1 2 3

T T t F

t F t F

F T t F

Let {fi}i Ibe the family of all the mappingsf:L(C) L(C1) satisfying

clauses (tr0), (tr1), (tr2), (tr3) and (tr4) below.

(tr0) f(p) =p, forp a propositional variable;

(tr1) f() {1f(), 2f()}

(tr2) f(#) = (f()#f()), for # { , , }

(tr3) f( ) { 1f(), 2f(), 3f()}

(tr4) iff() = 2f(), thenf( ) = 1f()

The family of mappings {fi}i Ican be shown to define a PTS which

characterizes bC in a decidable way. As an example, it can be easily checked

that is a theorem ofbC: just consider all its finitely many

translations and test all of them to verify that they are three-valued

tautologies. On the other hand, is not a theorem ofbC, which

can be promptly verified by showing that at least one of its translations is not

a tautology using the three-valued tables above. For an alternative PTS

characterization ofbC and related logics see Marcos 2005.

This example shows that a non-truth functional connective, such as the

paraconsistent negation or the consistency operator ofbC, can be

mimicked by interpreting it (via translations) into different truth-functional

connectives. This is the idea behind A. Avron and I. Lev's non-deterministic

semantics introduced in Avron and Lev 2001 (see also Avron and Lev 2005).

This semantics generalizes logical matrices by considering that multifunctions(rather than functions) interpret the connectives.

As a matter of fact, non-deterministic semantics can be simulated by

appropriate possible-translations semantics (cf. Carnielli and Coniglio 2005).

In particular, the familiar matrix semantics are a particular case of possible-

translations semantics, as well as the historical examples of translations

between logics found in the literature. These facts provide evidence that

possible-translations semantics are a widely applicable conceptual tool for

decomposing logics.


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4.5 Temporalization, Parameterization and Institutions

Apart from the logical and philosophical import of combining logics, there

exists a genuine interest in developing applications based on these techniques.

One of the main areas interested in the methods for combining logics issoftware specification. Certain techniques for combining logics were

developed almost exclusively with the aim of applying them to this area. In

this section some of these methods will be briefly

mentioned: temporalization,parameterization and institutions.

Temporalization was introduced in Finger and Gabbay 1992 (see also Finger

and Gabbay 1996), and generalized in Caleiro et al. 1999 towards the method

called parameterization.

Parameterization, in rough terms, consists of replacing the atomic part of a

given logic L by another logic L. Thus, the logic L is theparameterized logic;

the atomic part is theformal parameterand the logic L is theparameter

logic. Formally, a mixed signature based on the signature ofL is considered,

to which the formulas ofL are added as constants. In the particular case of

temporalization, the parameter logic is a temporal logic. In turn, it can be

proven that parameterization is a particular case of constrained fibring (recall

Subsection 4.3).

The method can be explained by means of a simple example: consider apropositional modal logic L, to be parameterized with first-order logic Lfol in

order to describe the dynamics of data bases. The resulting logic is defined in

a language whose formulas are obtained by replacing propositional constants

in formulas ofL by first-order formulas. That is, modalities can be freely

used, but quantifiers cannot be applied to modal formulas (other propositional

connectives such as negation and implication are shared).

The semantic structures for the new logic are Kripke structures where the

valuation for propositional constants is replaced by a kind of zooming inmapping (in the sense ofBlackburn and de Rijke 1997) associating a first-

order semantic structure together with a fixed assignment for individual

variables to each state.

The deductive system for the new logic is formed by the rules of both logics.

The rules ofL can be instantiated with formulas of the parameterized

language, but the rules of first-order logic can only be applied to pure first-

order formulas.


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One important difference between parameterization (in particular,

temporalization) and constrained fibring is the degree of symmetry: the

parameterized language and inference rules are intrinsically asymmetric,

while this is not the case of constrained fibring.

Institutions were introduced by J. Goguen and R. Burstall (see Goguen and

Burstall 1984 and Goguen and Burstall 1992) as a kind of abstract model

theory for Computer Science, and are adequate for developing concepts of

specification languages such as structuring of specifications and

implementation.

The theory of institutions is mainly applicable to software specification

defined by multiple logical systems (see, for instance,Diaconescu and

Futatsugi 2002). Thus, under an abstract view of software development,

different components of the same program can be specified using different

formalism in an heterogeneous setting. This is formalized by the use of

institutions and morphisms between them (see, for instance, Tarlecki 2000). A

problem concerning institution morphisms is that formulas involving

connectives from different logics being combined are not allowed. A solution

to this problem was proposed in Goguen and Burstall 1986 and Mossakowski

1996 by using the so-calledparchments andparchment morphisms.

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

Up to this point, several techniques for composing logics have been described

and exemplified. Are these processes appropriate for composing, without

surprises, any pair of logics? In other words, given a pair of logics (presented

in an homogeneous way), are they composable in a meaningful way? Does the

composition make philosophical sense? As pointed bySchurz 1991, it is

conceivable that some multimodal logics obtained as combination of modal

logics by adding arbitrarily chosen bridge principles could be meaningless.

From the technical point of view, there is an important problem concerningcomposition of logics known as the collapsing problem, first identified

in Gabbay 1996b, and later formalized in del Cerro and Herzig 1996. In the

latter paper, it is shown that, by freely combining classical propositional logic

and intuitionistic propositional logic at the semantical level (technically: by

computing their unconstrained fibring in the category of interpretation

systems, recall Subsection 4.3), the resulting logic collapses to classical logic.

More precisely, the resulting logic will consist of two twin copies of classical

propositional logic having two negations, two implications and so on. Clearly,

the respective copies of each connectives will be proved to be inter-derivablein the resulting logic: 1 will be equivalent to 2, (1) will be


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equivalent to (2) and so on. The collapse only happens when

considering the algebraic fibring at the semantical level: in Caleiro and Ramos

2007 was shown that the collapse does not occur when computing the

algebraic fibring of the respective Hilbert calculi.

Basically, the phenomenon arises because both implications collapse, and then

intuitionistic implication becomes classic. From the semantical point of view,

it happens that the models of the fibred logic are Heyting algebras which are

simultaneously Boolean algebras: evidently, the algebras collapse to the

Boolean ones. From the point of view of proof theory, the problem appears as

a consequence of the metaproperty calledDeduction Metatheorem (DMT): let

1 and 2 be the intuitionistic and the classical implications, respectively.

Then

, iff(1)

, iff(2).

Thus, the following argument applies (cf. Gabbay 1996b):

(1) (1) Axiom

(1), DMT for 1

(1) (2) DMT for 2

A similar argument shows that (2) (1). That is, classical and

intuitionistic implications collapse in the combined logic.

It is worth noting that the previous arguments depart from a very strong

assumption: that the metaproperty DMT is preserved in the combined logic.

As we shall see below, this is not the case for algebraic fibring, unless a

stronger notion of morphism between logics is adopted.

In Sernadas et al. 2002b, other examples of collapse were presented, and a

solution to the problem was proposed by means of a controlled notion of

algebraic fibring called modulated fibring. An apparently simpler solution to

the collapsing problem appeared in Caleiro and Ramos 2007, using a variant

of the algebraic fibring technique called cryptofibring.

Independently, in Bziau 2004, it was observed that by putting together the

sequent rules for classical conjunction and the rules for classical disjunction,the resulting sequent calculus will (unexpectedly) prove the distributivity


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between conjunction and disjunction. The same phenomenon occurs if we join

the (two-valued) valuation clauses for classical conjunction with the valuation

clauses for classical disjunction. However, this is avoided by considering

algebraic fibring in the usual categories (Hilbert calculi or consequence

relations) with translations between logics as morphisms: the logic obtained isthe logic of lattices, which does not satisfy distributivity (see Bziau and

Coniglio 2005).

This situation, in which new interaction rules between the connectives arise, is

arguably undesirable. In fact, it contradicts a basic criterion of fibring (and

also of fusion), as expressed in Gabbay 1999: given logic systems L1 and L2,

the combination ofL1 andL2 should be the smallest logic system in the

combined language which is a conservative extension of both L1 and L2.

Indeed, the distributivity problem and the collapsing problem are two

instances of the same phenomenon of emergence of unexpected interactions

(such as bridge principles) between connectives caused by combination

processes. In the case of combination of conjunction with disjunction, the

distributive law emerges: this interaction is due to the combination process

and appears without any apparent reason. In turn, the collapsing problem is a

limit case of interactions: the 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 in such cases,

because they derive too many propositions in the new combined language.

On the other hand, the opposite (or dual) situation may also be problematic:

suppose, to fix ideas, that the logic of classical negation is combined with the

logic of classical disjunction. These logics can be presented, for instance,

axiomatically (through Hilbert calculi) or semantically, say, through

valuations over {0,1} (that is, by means of classical truth-tables). Thesemantical presentation of the logic of classical negation consists of the set of

all valuations over {0,1} satisfying the following clause:

v() = 0 iffv() = 1.

On the other hand, the logic of classical disjunction can be characterized by

the set of all valuations over {0,1} such that:

v( ) = 0 iffv() = 0 and v() = 0.


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As a consequence, the combined logic of negation and disjunction (which can

be defined as the logic over and characterized by the valuations over

{0,1} satisfying both clauses above) validates ( ), and so classical logic

is recovered. This is the result obtained by the combination method

called direct union of matrices. However, if algebraic fibring is considered incategories such as those of Hilbert calculi or consequence relations, the

combination between the logic of negation and the logic of disjunction results

in a logic defined over and , which is weaker than classical logic: the

interaction law ( ) is no longer valid. That is, an arguably desirable

interaction between the connectives is lost in the combination process, and

classical logic over and cannot be recovered from its fragments, as long

as algebraic fibring in these categories of logics is used.

Another example of the same kind is the following: the algebraic fibringbetween the logic of classical negation and the logic of classical implication

performed in the categories above does not recover classical logic over

and . Indeed, the resulting logic system, defined over and , cannot

validateEx Contradictione Sequitur Quodlibetwhen presented as an axiom:

( ()).

Interestingly enough,Ex Contradictione Sequitur Quodlibet, presented as a

derivation, holds in the fibred logic:

, .

Observe that ( ()) is an interaction rule between the connectives

of the logics being combined which cannot be obtained by algebraic fibring in

the categories under consideration (however, this principle can be recovered,

e.g., by direct union of matrices). If one is interested in recovering a logic

from its fragments, this result is disappointing.

These examples as well as other along the same lines suggest a dual problem

to that of collapsing and distributivity between conjunction and disjunction:

some expected interaction laws fail to be created by some combination

processes.

In such cases, it could be said that the combined logics are too weak, because

they are unable to derive certain intended propositions in the new combined

language.


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What could be expected when combining logics? Strong logics (guaranteeing,

for instance, that a logic can be recovered from its fragments) or weaker ones

(in which undesirable interactions between connectives are blocked)?

The examples above are evidences against and in favor of both situations: inorder to avoid the collapsing problem, a careful splicing process should be

expected (and so the interaction between both implications would disappear).

On the other hand, if one wants to recover, say, classical logics from its

fragments, a more liberal splicing process would be more adequate, as some

intended interactions between connectives of both logics would be recovered.

With respect to the distributivity problem when combining conjunction and

disjunction, the choice of method is also not determined: distributivity could

be a desired feature if we adopted the viewpoint of recovering a logic from its

fragments. In this case, a combination method defining a stronger logic (such

as direct union of matrices) would be more appropriate than, for instance,

algebraic fibring of Hilbert calculi. If, as argued in Bziau 2004, distributivity

is regarded as an intruder, then a more careful process would be

recommended: algebraic fibring would be more appropriate in this case. To

sum up: the choice of the more adequate combining process depends upon

what one wants to obtain.

At this point, it is convenient to notice that the question about whether or not

interactive principles are generated when combining modal logics is

intrinsically related to Hume's is-ought problem discussed in Section 1.

Indeed, as proven in Schurz 1991, it is possible to obtain nontrivial is-ought

deductions in the combination of alethic and deontic logics provided that

some bridge principles are allowed. Bridge principles as are

nothing else than interaction rules between connectives of the logics being

combined. Such principles enjoy a similar conceptual status as the

distributivity laws between conjunction and disjunction, or as the collapsing

example mentioned above. Thus, in order to satisfy Hume's thesis, a

combination process generating logics without interactions should be

preferred. On the other hand, a combination process allowing the creation of

interactions between the connectives could grant bridge principles violating

Hume's thesis.

Finally, it is noteworthy to observe that algebraic fibring does not intrinsically

forbid the emergence of interactions between connectives of the logics being

combined. In fact, the notion of morphism in the category of logic systems

being employed is the key to create or to block interactions. In order to

exemplify this assertion, consider the case of the failure to recover classical


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logic from its {}-fragment and { }-fragment by algebraic fibring. The key

reason for this failure is that the rule

, ,

,

(*)

of the logic of classical negation is not preserved by algebraic fibring in the

categories of logic systems having translations between logics as morphisms

(recall Subsection 4.4), such as the category of Hilbert calculi or consequence

relations.

When considering algebraic fibring of classical implication with classical

negation in those categories, the missing rule is theDeduction Metatheorem:

,

()

(**)

Categories of logic systems having logic translations as morphisms are suchthat the canonical injections of the coproduct are just inclusion mappings.

Then, given two logics L1 and L2, the only rules of these logics which are

preserved by their algebraic fibring are those of the form:

On the other hand, suppose a category of logic systems in which the

preservation of rules such as (*) or (**) above is required by the very notion

of morphism. Thus, if a logic L is obtained as a combination of two othersystems L1 and L2 then the rules ofL1 and L2 would be faithfully transferred

to L. This is the proposal ofConiglio 2007, in which algebraic fibring in

categories of sequent calculi is investigated, taking into account a notion of

morphism which preserves logical rules of the form

If11 and and nn, then .

In such categories, when a logic system is embedded into a larger one by

algebraic fibring, any rule as above, which can be considered as a meta-theorem of the logic, is preserved by the canonical injections. This is why this


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process is called meta-fibring. From the categorial point of view, the process

is the same as for algebraic fibring, the only difference being that the notion of

morphism is stronger. This illustrates the advantages of using category theory

and its tools for defining combination procedures as universal constructions:

the same construction (in this case, algebraic fibring) can be performed incategories of logic systems with different features obtaining, as a consequence

of this, stronger or weaker logic systems.

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Other Internet Resources

van Benthem, J., 2005, Epistemic logic and epistemology: the state oftheir affairs, Technical report, Institute for Logic, Language and

Computation (ILLC).

FroCoS: Frontiers of Combining Systems FLIRTS: Formalism, Logic, Institution Relating, Translating and

Structuring

Combination Methods in Automated ReasoningRelated Entries

category theory | God, arguments for the existence of: moral

arguments | Hume, David: moral philosophy | logic: classical | logic:

deontic | logic: intuitionistic | logic: many-valued | logic: modal | logic:paraconsistent | logic: temporal | logical consequence |logical consequence:

propositional consequence relations and algebraic logic

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Combining Logics