Formal polynomials, heuristics and proofs in logic
##plugins.themes.bootstrap3.article.sidebar##
Published:
07.04.2010
##plugins.themes.bootstrap3.article.main##
Abstract
This note surveys some previous results on the role of formal polynomials as a representation method for logical derivation in classical and non-classical logics, emphasizing many-valued logics, paraconsistent logics and modal logics. It also discusses the potentialities of formal polynomials as heuristic devices in logic and for expressing certain meta-logical properties, as well as pointing to some promising generalizations towards algebraic geometry.
##plugins.generic.usageStats.downloads##
##plugins.generic.usageStats.noStats##
##plugins.themes.bootstrap3.article.details##
How to Cite
Carnielli W. Formal polynomials, heuristics and proofs in logic // Logicheskie Issledovaniya / Logical Investigations. 2010. VOL. 16. C. 280-294.
Issue
Section
Papers
References
Agudelo, J.C. and Carnielli, W.A. Polynomial ring calculus for modal logics: a new semantics and proof method for modalities. Pre-print available at CLE e-Prints 9(4), 2009, at http://www.cle.unicamp.br/e-prints/vol_9,n_4,2009.html.
Banaschewski, B. On G. Spencer Brown’s Laws of Form. Notre Dame Journal of Formal Logic 18(3):507-509, 1977.
Bazhanov, V.A. New archival materials concerning P.S. Poretskij. Modem Logic 3(1) pp. 80-81, 1992.
Beziau, J.-Y. Classical negation can be expressed by one of its halves. Logic Journal of IGPL 7(2):145-151, 1999.
Butler, S. Miscellaneous Thoughts, in: The Poems of Samuel Butler, Vol. II. (Chiswick: C. Willingham, 1822), p. 281
Carnielli, W.A. Formal polynomials and the laws of form. In ‘The Multiple Dimensions of Logic”, Colecao CLE volume 54, UNICAMP, Brazil (Eds. Jean-Yves Beziau and Alexandre Costa-Leite), pp. 202-212, 2009.
Carnielli, W.A. A polynomial proof system for Lukasiewicz logics. Second Principia International Symposium. August 6-10, 2001 Florianypolis, SC, Brazil.
Carnielli, W.A. Polynomial ring calculus for many-valued logics. Proceedings of the 35th International Symposium on Multiple-Valued Logic. IEEE Computer Society. Calgary, Canada. IEEE Computer Society, pp. 20-25, 2005. Pre-print available at CLE e-Prints 6(3), 2006, at http://www.cle.unicamp.br/e~prints/vol_6,n_3, 2006.html.
Carnielli, W.A. Polynomizing: Logic inference in polynomial format and the legacy of Boole. In: Model-Based Reasoning in Science, Technology, and Medicine (Editors, L. Magnani and P. Li). Series “Studies in Computational Intelligence”, volume 64, pp. 349-364. Springer Berlin-Heidelberg, 2007. Pre-print available under the title “Polynomial ring calculus for logical inference” at CLE e-Prints 5(3), 2005, at http://www.cle.unicamp.br/e-prints/vol_5,n_3,2005.html.
Caleiro, C., Carnielli, W.A., Coniglio, _.E. and Marcos, J. Two’s company:‘The humbug of many logical mines”. In Logica Universalis, 169—189, editor Beziau, J.-Y., Birkhauser Verlag, Basel, Switzerland, 2005.
Carnielli, W.A., Coniglio, _.E. and Marcos, J. Logics of formal inconsistency. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume 14, pages 15—107. Springer, 2nd edition, 2007. Preprint available from CLE e-Prints 5(1), 2005 at http://www.cle.unicamp.br/e-prints/vol5,nl,2005.html.
Carnielli, W.A. and Coniglio, _.E. Splitting Logics. In "We Will Show Them! Essays in Honour of Dov Gabbay" pages 389-414. College Publications, 2005.
Carolino, P.K. Polinomization of Logics: Problems and Perpscrives. Master Dissertation (in Portuguese). IFCH-UNICAMP, Campinas, SP, Brazil, 2009.
Clegg, M., Edmonds, J., and Impagliazzo, R. Using the Grobner basis algorithm to find proofs of unsatisfiability. In Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, pages 174-183, Philadelphia, PA, May 1996.
De Morgan, A. On the syllogism, no. IV, and on the logic of relations. Trans. Cambridge Philosophical Soc 10:331-358, 1860.
Effinger, G., Hicks, K, and Mullen, G.L. Integers and polynornails: comparing the close cousins Z and Fq[x]. The Mathematical Intelligencer 27(2):26-34, 2005.
Fisch, M. and Turquette, A. Peirce’s Triadic Logic. Transactions of the Charles S. Peirce Society 11:71-85, 1966.
Gottwald, S. A Treatise on Many-Valued Logics, Studies in Logic and Computation, Research Studies Press Ltd. Hertfordshire, England, 2001.
Humberstone, L. Вeziau’s translation paradox. Theoria 71:138-18, 2005.
Kneebone, G.T. Mathematical Logic and the Foundation of Mathematics: An introductory Survey. Courier Dover Publications, 2001.
Polya, G. How to Solve It: A New Aspect of Mathematical Method. Princeton, NJ: Princeton University Press, 1945.
Polya, G. Mathematical discovery: on understanding, learning, and teaching problem solving. New York, NY: John Wiley and Sons, Inc., 1981
Paturi, R., Pudlak, P., Saks, M. and Zane, F. An improved exponential time algorithm for k-sat. Annual IEEE Symposium on Foundations of Computer Sience, 1998.
Spencer-Brown, G. The Laws of Form. Allen & Unwin, London, 1969.
van der Waerden, B.L. Modern Algebra, Julius Springer, Berlin, 1931.
Zhegalkin, I.I. On the Technique of Calculating Propositions in Symbolic Logic. Matematicheskii Sbornik 43: 9-28, 1927.
Banaschewski, B. On G. Spencer Brown’s Laws of Form. Notre Dame Journal of Formal Logic 18(3):507-509, 1977.
Bazhanov, V.A. New archival materials concerning P.S. Poretskij. Modem Logic 3(1) pp. 80-81, 1992.
Beziau, J.-Y. Classical negation can be expressed by one of its halves. Logic Journal of IGPL 7(2):145-151, 1999.
Butler, S. Miscellaneous Thoughts, in: The Poems of Samuel Butler, Vol. II. (Chiswick: C. Willingham, 1822), p. 281
Carnielli, W.A. Formal polynomials and the laws of form. In ‘The Multiple Dimensions of Logic”, Colecao CLE volume 54, UNICAMP, Brazil (Eds. Jean-Yves Beziau and Alexandre Costa-Leite), pp. 202-212, 2009.
Carnielli, W.A. A polynomial proof system for Lukasiewicz logics. Second Principia International Symposium. August 6-10, 2001 Florianypolis, SC, Brazil.
Carnielli, W.A. Polynomial ring calculus for many-valued logics. Proceedings of the 35th International Symposium on Multiple-Valued Logic. IEEE Computer Society. Calgary, Canada. IEEE Computer Society, pp. 20-25, 2005. Pre-print available at CLE e-Prints 6(3), 2006, at http://www.cle.unicamp.br/e~prints/vol_6,n_3, 2006.html.
Carnielli, W.A. Polynomizing: Logic inference in polynomial format and the legacy of Boole. In: Model-Based Reasoning in Science, Technology, and Medicine (Editors, L. Magnani and P. Li). Series “Studies in Computational Intelligence”, volume 64, pp. 349-364. Springer Berlin-Heidelberg, 2007. Pre-print available under the title “Polynomial ring calculus for logical inference” at CLE e-Prints 5(3), 2005, at http://www.cle.unicamp.br/e-prints/vol_5,n_3,2005.html.
Caleiro, C., Carnielli, W.A., Coniglio, _.E. and Marcos, J. Two’s company:‘The humbug of many logical mines”. In Logica Universalis, 169—189, editor Beziau, J.-Y., Birkhauser Verlag, Basel, Switzerland, 2005.
Carnielli, W.A., Coniglio, _.E. and Marcos, J. Logics of formal inconsistency. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume 14, pages 15—107. Springer, 2nd edition, 2007. Preprint available from CLE e-Prints 5(1), 2005 at http://www.cle.unicamp.br/e-prints/vol5,nl,2005.html.
Carnielli, W.A. and Coniglio, _.E. Splitting Logics. In "We Will Show Them! Essays in Honour of Dov Gabbay" pages 389-414. College Publications, 2005.
Carolino, P.K. Polinomization of Logics: Problems and Perpscrives. Master Dissertation (in Portuguese). IFCH-UNICAMP, Campinas, SP, Brazil, 2009.
Clegg, M., Edmonds, J., and Impagliazzo, R. Using the Grobner basis algorithm to find proofs of unsatisfiability. In Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, pages 174-183, Philadelphia, PA, May 1996.
De Morgan, A. On the syllogism, no. IV, and on the logic of relations. Trans. Cambridge Philosophical Soc 10:331-358, 1860.
Effinger, G., Hicks, K, and Mullen, G.L. Integers and polynornails: comparing the close cousins Z and Fq[x]. The Mathematical Intelligencer 27(2):26-34, 2005.
Fisch, M. and Turquette, A. Peirce’s Triadic Logic. Transactions of the Charles S. Peirce Society 11:71-85, 1966.
Gottwald, S. A Treatise on Many-Valued Logics, Studies in Logic and Computation, Research Studies Press Ltd. Hertfordshire, England, 2001.
Humberstone, L. Вeziau’s translation paradox. Theoria 71:138-18, 2005.
Kneebone, G.T. Mathematical Logic and the Foundation of Mathematics: An introductory Survey. Courier Dover Publications, 2001.
Polya, G. How to Solve It: A New Aspect of Mathematical Method. Princeton, NJ: Princeton University Press, 1945.
Polya, G. Mathematical discovery: on understanding, learning, and teaching problem solving. New York, NY: John Wiley and Sons, Inc., 1981
Paturi, R., Pudlak, P., Saks, M. and Zane, F. An improved exponential time algorithm for k-sat. Annual IEEE Symposium on Foundations of Computer Sience, 1998.
Spencer-Brown, G. The Laws of Form. Allen & Unwin, London, 1969.
van der Waerden, B.L. Modern Algebra, Julius Springer, Berlin, 1931.
Zhegalkin, I.I. On the Technique of Calculating Propositions in Symbolic Logic. Matematicheskii Sbornik 43: 9-28, 1927.