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Fibring: completeness preservation

Published online by Cambridge University Press:  12 March 2014

Alberto Zanardo ,
Amilcar Sernadas  and
Cristina Sernadas
Alberto Zanardo
Affiliation:
Dipartimento di Matematica Pura ed Applicata, University of Padova, Italy, E-mail: [email protected]
Amilcar Sernadas
Affiliation:
CMA, Departamento de Matematica, 1st, Portugal, E-mail: [email protected]
Cristina Sernadas
Affiliation:
CMA, Departamento de Matemática, 1st, Portugal, E-mail: [email protected]
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Abstract

A completeness theorem is established for logics with congruence endowed with general semantics (in the style of general frames). As a corollary, completeness is shown to be preserved by fibring logics with congruence provided that congruence is retained in the resulting logic. The class of logics with equivalence is shown to be closed under fibring and to be included in the class of logics with congruence. Thus, completeness is shown to be preserved by fibring logics with equivalence and general semantics. An example is provided showing that completeness is not always preserved by fibring ligics endowed with standard (non general) semantics. A categorial characterization of fibring is provided using coproducts and cocartesian liftings.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 1 , March 2001 , pp. 414 - 439
Copyright
Copyright © Association for Symbolic Logic 2001

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References

REFERENCES

[1990] Barr, M. and Wells, C., Category theory for computing science, Prentice Hall, 1990.Google Scholar
[1978] van Benthem, J., Two simple incomplete logics, Theoria, vol. 44 (1978), pp. 25–37.CrossRefGoogle Scholar
[1983] van Benthem, J., The logic of time, D. Reidel P. C., Dordrecht, 1983.CrossRefGoogle Scholar
[1985] van Benthem, J., Modal logic and classical logic, Bibliopolis, Napoli, 1985.Google Scholar
[1997] Blackburn, P. and de Rijke, M., Why combine logics?, Studia Logica, vol. 59 (1997), no. 1, pp. 5–27.CrossRefGoogle Scholar
[1997] Chagrov, A. and Zakhryaschev, M., Modal logic, Clarendon Press, Oxford, 1997.CrossRefGoogle Scholar
[1996] del Cerro, L. F. and Herzig, A., Combining classical and intuitionistic logic, Frontiers of combining systems (Baader, F. and Schulz, K., editors), Kluwer Academic Publishers, 1996, pp. 93–102.CrossRefGoogle Scholar
[1996a] Gabbay, D., Fibred semantics and the weaving of logics: part 1, this Journal, vol. 61 (1996), no. 4, pp. 1057–1120.Google Scholar
[1996b] Gabbay, D., An overview of fibred semantics and the combination of logics, Frontiers of combining systems (Baader, F. and Schulz, K., editors), Kluwer Academic Publishers, 1996, pp. 1–55.Google Scholar
[1999] Gabbay, D., Fibring logics, Oxford University Press, 1999.Google Scholar
[1950] Henkin, L., Completeness in the theory of types, this Journal, vol. 15 (1950), pp. 81–91.Google Scholar
[1996] Hughes, G. E. and Cresswell, M. J., A new introduction to modal logic, Routledge, London, 1996.CrossRefGoogle Scholar
[1991] Kracht, M. and Wolter, F., Properties of independently axiomatizable bi-modal logics, this Journal, vol. 56 (1991), no. 4, pp. 1469–1485.Google Scholar
[2000] Sambin, G., Battilotti, G., and Faggian, C., Basic logic: Reflection, symmetry, visibility, this Journal, vol. 65 (2000), pp. 979–1013.Google Scholar
[1997a] Sernadas, A., Sernadas, C., and Caleiro, C., Synchronization of logics, Studia Logica, vol. 59 (1997), no. 2, pp. 217–247.CrossRefGoogle Scholar
[1997b] Sernadas, A., Synchronization of logics with mixed rules: Completeness preservation, Algebraic methodology and software technology - amast'97, Springer-Verlag, LNCS 1349, 1997, pp. 465–478.CrossRefGoogle Scholar
[1999] Sernadas, A., Fibring of logics as a categorial construction, Journal of Logic and Computation, vol. 9 (1999), no. 2, pp. 149–179.CrossRefGoogle Scholar
[2000] Sernadas, A., Sernadas, C., Caleiro, C., and Mossakowski, T., Categorial fibring of logics with terms and binding operators, Frontiers of combining systems - FroCoS'98 (de Rijke, M. and Gabbay, D., editors), Research Studies Press, 2000, pp. 295–316.Google Scholar
[1996] Wolter, F., A counterexample in tense logic, Notre Dame Journal of Formal Logic, vol. 37 (1996), no. 2, pp. 167–173.CrossRefGoogle Scholar