On the Structure of Natural Deduction Derivations for “Generally”

  • Leonardo B. VanaEmail author
  • Paulo A. S. Veloso
  • Sheila R. M. Veloso
Part of the Trends in Logic book series (TREN, volume 39)


Logics for ‘generally’ were introduced as extensions of First-Order Logic (FOL) for handling assertions with vague notions (e.g., ‘generally,’ ‘most,’ and ‘several’) expressed by formulas with generalized quantifiers. Deductive systems have been developed for such logics. Here, we characterize the structure of derivations in natural deduction style for filter logic. This characterization extends the familiar one for FOL.


Logics for ‘generally’ Generalized quantifiers Vague notions Natural deduction Derivation structure Normalization Minimum formula Minimum segment 


  1. 1.
    Carnielli, W. A., & Veloso, P. A. S. (1997). Ultrafilter logic and generic reasoning. In G. Gottlob, A. Leitsch, & D. Mundici (Eds.), Computational Logic and Proof Theory (LNCS) (vol. 1289, pp. 34–53). Berlin: Springer.Google Scholar
  2. 2.
    Veloso, S. R. M., & Veloso, P. A. S. (2001). On a logical framework for ‘generally’. In J. M. Abe & J. I. Silva Filho (Eds.), Logic, Artificial Intelligence and Robotics: Proceedings of LAPTEC (pp. 279–286). Amsterdam: IOS Press.Google Scholar
  3. 3.
    Veloso, S. R. M., & Veloso, P. A. S. (2002). On special functions and theorem proving in logics for ‘generally’. In G. Bittencourt & G. L. Ramalho (Eds.), Advances in Artificial Intelligence: 16th Brazilian Symposium in Artificial Intelligence: SBIA 2002 (LNAI) (vol. 2507, pp. 1–10). Berlin: Springer.Google Scholar
  4. 4.
    Rentería, C. J., Haeusler, E. H., & Veloso, P. A. S. (2003). NUL: Natural deduction for ultrafilter logic. Bulletin of Section of Logic, 32(4), 191–199.Google Scholar
  5. 5.
    Veloso, S. R. M., & Veloso, P. A. S. (2004). On modulated logics for ‘generally’: some metamathematical issues. In J.-Y., Béziau, A. Costa Leite & A. Facchini (Eds.), Aspects of universal logic (pp. 146–168). Neuchatel: University of Neuchâtel.Google Scholar
  6. 6.
    Veloso, P. A. S., & Carnielli, W. A. (2004). Logics for qualitative reasoning. In D. Gabbay, S. Rahman, J. Symons, & J. P. van Bendegem (Eds.), Logic, epistemology and the unity of science (pp. 487–526). Dordretch: Kluwer Press.Google Scholar
  7. 7.
    Veloso, P. A. S., & Veloso, S. R. M. (2004). On ultrafilter logic and special functions. Studia Logica, 78, 459–477.CrossRefGoogle Scholar
  8. 8.
    Veloso, S. R. M., & Veloso, P. A. S. (2005). On logics for ‘generally’ and their relational interpretations. In First World Congress on Universal Logic Handbook (pp. 101–102). Montreux.Google Scholar
  9. 9.
    Veloso, P. A. S., & Veloso, S. R. M. (2005). Functional interpretation of logics for ‘generally’. Logic Journal of the IGPL, 13(1), 127–140.CrossRefGoogle Scholar
  10. 10.
    Veloso, P. A. S., & Veloso, S. R. M. (2005a). On ‘most’ and ‘representative’: Filter logic and special predicates. Logic Journal of the IGPL, 13(6), 717–728.CrossRefGoogle Scholar
  11. 11.
    Vana, L. B., Veloso, P. A. S., & Veloso, S. R. M. (1997). Natural deduction for ‘generally’. Logic Journal of the IGPL, 15, 775–800.CrossRefGoogle Scholar
  12. 12.
    Veloso, P. A. S., Vana, L. B., & Veloso, S. R. M. (2005). Natural deduction strategies for ‘generally’. In Actas de la XI Conferencia de la Asociación Española para la Inteligencia Artificial (pp. 173–182). Santiago de Compostela.Google Scholar
  13. 13.
    Vana, L. B., Veloso, P. A. S., & Veloso, S. R. M. (2005). Sobre lógicas para ‘geralmente’ em ambiente de dedução matural. In XXV Congresso da SBC (ENIA V) (pp. 622–630).Google Scholar
  14. 14.
    Vana, L. B., Veloso, P. A. S., & Veloso, S. R. M. (2007). Sequent calculi for ‘generally’. In Electronic Notes on Theoretical Computer Science Proceedings of the II Workshop on Logical and Semantic Frameworks with Applications (vol. 205, pp. 49–65).Google Scholar
  15. 15.
    Prawitz, D. (1957). Natural deduction: A proof-theoretical study. Stockholm: Almquist & Wiksdell.Google Scholar
  16. 16.
    Mostowski, A. (1957). On a generalization of quantifiers. Fundamenta Mathematicae, 44, 12–36.Google Scholar
  17. 17.
    Keisler, J. H. (1970). Logic with the quantifier there exist uncountably many. Annals of Mathematical Logic, 1, 1–93.CrossRefGoogle Scholar
  18. 18.
    Carnielli, W. A., & Sette, A. M. (1994). Default operators. In Abstracts of Workshop on Logic, Language, Information and Computation. Recife.Google Scholar
  19. 19.
    Schlechta, K. (1995). Defaults as generalized quantifiers. Journal of Logic and Computation, 5, 473–494.CrossRefGoogle Scholar
  20. 20.
    Barwise, J., & Cooper, R. (1981). Generalized quantifiers and natural language. Linguistics and Philosophy, 4, 159–219.CrossRefGoogle Scholar
  21. 21.
    Enderton, H. B. (1972). A mathematical introductionto logic. New York: Academic Press.Google Scholar
  22. 22.
    Vana, L. B. (2008). Dedução natural e cálculo de seqüentes para ‘geralmente’. D. Sc. diss., COPPE-UFRJ, Rio de Janeiro, RJ, Brazil.Google Scholar
  23. 23.
    Gentzen, G. (1969). Investigations into logical deduction. In M. E. Szabo (Ed.), The collected papers of Gerhard Gentzen. Amsterdan: North-Holland.Google Scholar
  24. 24.
    van Dalen, D. (1989). Logic and structure (2nd edn., 3rd prt.). Berlin: Springer.Google Scholar
  25. 25.
    Seldin, J. (1986). On the proof-theory of the intermediate logic MH. Jornal of Symbolic Logic, 5, 626–647.CrossRefGoogle Scholar
  26. 26.
    Medeiros, M. P. N. (2001). Traduções via Teoria da Prova: aplicações à lógica linear. D. Sc. diss., PUC-Rio, Rio de Janeiro, RJ, Brazil.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Leonardo B. Vana
    • 1
    Email author
  • Paulo A. S. Veloso
    • 2
  • Sheila R. M. Veloso
    • 3
  1. 1.Institute of MathematicsUniversidade Federal Fluminense (UFF)NiteróiBrazil
  2. 2.Systems and Computer Engineering Program, COPPEUniversidade Federal do Rio de Janeiro (UFRJ)Rio de JaneiroBrazil
  3. 3.Department of Systems and Computer EngineeringUniversidade Estadual do Rio de Janeiro (UERJ)Rio de JaneiroBrazil

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