Studia Logica

, Volume 100, Issue 3, pp 545–581 | Cite as

Importing Logics

  • João Rasga
  • Amílcar Sernadas
  • Cristina Sernadas


The novel notion of importing logics is introduced, subsuming as special cases several kinds of asymmetric combination mechanisms, like temporalization [8, 9], modalization [7] and exogenous enrichment [13, 5, 12, 4, 1]. The graph-theoretic approach proposed in [15] is used, but formulas are identified with irreducible paths in the signature multi-graph instead of equivalence classes of such paths, facilitating proofs involving inductions on formulas. Importing is proved to be strongly conservative. Conservative results follow as corollaries for temporalization, modalization and exogenous enrichment.


Combined logics Importing logics Temporalization Modalization Exogenous enrichment 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baltazar P., Mateus P., Nagarajan R., Papanikolaou N.: Exogenous probabilistic computation tree logic. Electronic Notes in Theoretical Computer Science 190(3), 95–110 (2007)CrossRefGoogle Scholar
  2. 2.
    Blackburn, P., M. de Rijke, and Y. Venema, Modal Logic. Vol. 53 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 2001.Google Scholar
  3. 3.
    Caleiro, C., C. Sernadas, and A. Sernadas, Parameterisation of logics. In J. Fiadeiro (ed.), Recent Trends in Algebraic Development Techniques - Selected Papers, vol. 1589 of Lecture Notes in Computer Science, Springer, 1999, pp. 48–62.Google Scholar
  4. 4.
    Chadha R., Cruz-Filipe L., Mateus P., Sernadas A.: Reasoning about probabilistic sequential programs. Theoretical Computer Science 379(1-2), 142–165 (2007)CrossRefGoogle Scholar
  5. 5.
    Chadha, R., P. Mateus, A. Sernadas, and C. Sernadas, Extending classical logic for reasoning about quantum systems. In D. Gabbay, K. Engesser, and D. Lehmann (eds.), Handbook of Quantum Logic and Quantum Structures: Quantum Logic, Elsevier, 2009, pp. 325–372.Google Scholar
  6. 6.
    del Cerro, L. Fariñas, and A. Herzig, Combining classical and intuitionistic logic. In F. Baader, and K. Schulz (eds.), Frontiers of Combining Systems, Kluwer Academic Publishers, 1996, pp. 93–102.Google Scholar
  7. 7.
    Fajardo, R. A. S., and M. Finger, Non-normal modalisation. In Advances in Modal Logic, vol. 4, King’s College Publications, 2003, pp. 83–95.Google Scholar
  8. 8.
    Finger M., Gabbay D.M.: Adding a temporal dimension to a logic system. Journal of Logic, Language and Information 1(3), 203–233 (1992)CrossRefGoogle Scholar
  9. 9.
    Finger M., Weiss M.A.: The unrestricted combination of temporal logic systems. Logic Journal of the IGPL 10(2), 165–189 (2002)CrossRefGoogle Scholar
  10. 10.
    Goldblatt, R., Mathematical modal logic: A view of its evolution. In D. M. Gabbay, and J.Woods (eds.), Handbook of the History of Logic, vol. 7, Elsevier, 2006, pp. 1–98.Google Scholar
  11. 11.
    Huet G.: Confluent reductions: abstract properties and applications to term rewriting systems. Journal of the Association for Computing Machinery 27(4), 797–821 (1980)CrossRefGoogle Scholar
  12. 12.
    Mateus P., Sernadas A.: Weakly complete axiomatization of exogenous quantum propositional logic. Information and Computation 204(5), 771–794 (2006) ArXivmath.LO/0503453CrossRefGoogle Scholar
  13. 13.
    Mateus, P., A. Sernadas, and C. Sernadas, Exogenous semantics approach to enriching logics. In G. Sica (ed.), Essays on the Foundations of Mathematics and Logic, vol. 1, Polimetrica, 2005, pp. 165–194.Google Scholar
  14. 14.
    Newman M.H.A.: On theories with a combinatorial definition of “equivalence”. Annals of Mathematics. Second Series 43, 223–243 (1942)CrossRefGoogle Scholar
  15. 15.
    Sernadas A., Sernadas C., Rasga J., Coniglio M.: A graph-theoretic account of logics. Journal of Logic and Computation 19(6), 1281–1320 (2009)CrossRefGoogle Scholar
  16. 16.
    van Dalen, D., Intuitionistic logic. In D. Gabbay, and F. Guenthner (eds.), Handbook of Philosophical Logic, vol. III, D. Reidel Publishing Company, 1986, pp. 225–339.Google Scholar
  17. 17.
    von Karger B.: Temporal algebra. Mathematical Structures in Computer Science 8(3), 277–320 (1998)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • João Rasga
    • 1
  • Amílcar Sernadas
    • 1
  • Cristina Sernadas
    • 1
  1. 1.SQIG, Instituto de Telecomunicações and Dep. Matemática, Instituto Superior TécnicoUniversidade Técnica de LisboaLisbonPortugal

Personalised recommendations