Modulated fibring and the collapsing problem

Sernadas, Cristina; Rasga, João; Carnielli, Walter

doi:10.2178/jsl/1190150298

Cited by 35 publications

(27 citation statements)

References 12 publications

(36 reference statements)

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“…It is also worthwhile to point out that the graph-theoretic setting provides the means to avoid some well known collapses [7] in a very natural way, different from previous works on the topic [21,3]. For instance, in the case of the modulated fibring, the collapse was avoided by restricting instantiation in derivation which is not the case in the graph-theoretic approach.…”

Section: Discussionmentioning

confidence: 97%

“…Namely, the well known collapse when combining classical propositional logic with intuitionistic propositional logics does not appear [7,21,3]. This happens since in our setting the fibred semantics is in a sense an exhaustive interleaving of the semantics of the components, and so the characteristics of the semantics of the components are present in the fibring.…”

Section: Introductionmentioning

confidence: 93%

See 1 more Smart Citation

On Graph-theoretic Fibring of Logics

Sernadas¹,

Sernadas²,

Rasga³

et al. 2009

Journal of Logic and Computation

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A graph-theoretic account of fibring of logics is developed, capitalizing on the interleaving characteristics of fibring at the linguistic, semantic and proof levels. Fibring of two signatures is seen as an m-graph where the nodes and the m-edges include the sorts and the constructors of the signatures at hand. Fibring of two models is an m-graph where the nodes and the m-edges are the values and the operations in the models, respectively. Fibring of two deductive systems is an m-graph whose nodes are language expressions and the m-edges represent the inference rules of the two original systems. The sobriety of the approach is confirmed by proving that all the fibring notions are universal constructions. This graph-theoretic view is general enough to accommodate very different fibrings of propositional based logics encompassing logics with non-deterministic semantics, logics with an algebraic semantics, logics with partial semantics, and substructural logics, among others. Soundness and weak completeness are proved to be preserved under very general conditions. Strong completeness is also shown to be preserved under tighter conditions. In this setting, the collapsing problem appearing in several combinations of logic systems can be avoided.

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Section: Discussionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 93%

On Graph-theoretic Fibring of Logics

Sernadas¹,

Sernadas²,

Rasga³

et al. 2009

Journal of Logic and Computation

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show abstract

“…These kind of preservation results are, despite their importance, usually shown in a case by case basis, although recently there can be seen a widely and growing effort to generalize preservation results so that they can be applied once to a whole class of entities [4,5,26,28].…”

Section: Introductionmentioning

confidence: 99%

Interpolation via translations

Rasga¹,

Carnielli

Sernadas

2009

Mathematical Logic Qtrly

Self Cite

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A new technique is presented for proving that a consequence system enjoys Craig interpolation or Maehara interpolation based on the fact that these properties hold in another consequence system. This technique is based on the existence of a back and forth translation satisfying some properties between the consequence systems. Some examples of translations satisfying those properties are described. Namely a translation between the global/local consequence systems induced by fragments of linear logic, a Kolmogorov-GentzenGödel style translation, and a new translation between the global consequence systems induced by full Lambek calculus and linear logic, mixing features of a Kiriyama-Ono style translation with features of a Kolmogorov-Gentzen-Gödel style translation. These translations establish a strong relationship between the logics involved and are used to obtain new results about whether Craig interpolation and Maehara interpolation hold in that logics.

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“…Instead, it uses mixed axioms and incorporates syntactic restrictions on their instantiation. A first general solution to the collapsing problem, modulated fibring, was proposed in [21] using similar ideas.…”

Section: Introductionmentioning

confidence: 99%

From Fibring to Cryptofibring. A Solution to the Collapsing Problem

Caleiro

Ramos

2007

Log. univers.

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Abstract. The semantic collapse problem is perhaps the main difficulty associated to the very powerful mechanism for combining logics known as fibring. In this paper we propose cryptofibred semantics as a generalization of fibred semantics, and show that it provides a solution to the collapsing problem. In particular, given that the collapsing problem is a special case of failure of conservativeness, we formulate and prove a sufficient condition for cryptofibring to yield a conservative extension of the logics being combined. For illustration, we revisit the example of combining intuitionistic and classical propositional logics.

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Modulated fibring and the collapsing problem

Cited by 35 publications

References 12 publications

On Graph-theoretic Fibring of Logics

On Graph-theoretic Fibring of Logics

Interpolation via translations

From Fibring to Cryptofibring. A Solution to the Collapsing Problem

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