Polynomizing: Logic Inference in Polynomial Format and the Legacy of Boole

  • Walter Carnielli
Part of the Studies in Computational Intelligence book series (SCI, volume 64)

Summary. Polynomizing is a term that intends to describe the uses of polynomiallike representations as a reasoning strategy and as a tool for scientific heuristics. I show how proof-theory and semantics for classical and several non-classical logics can be approached from this perspective, and discuss the assessment of this prospect, in particular to recover certain ideas of George Boole in unifying logic, algebra and the differential calculus.


Boolean Algebra Propositional Logic Logic Inference Algebraic Logic Boolean Ring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Agudelo, J.C., Carnielli, W.A.: Quantum algorithms, paraconsistent computa-tion and Deutsch’s problem. In Bhanu Prasad et al., eds.: Proceedings of the 2nd Indian International Conference on Artificial Intelligence, Pune, India, December 20-22 (2005) IICAI 2005, 1609-1628. Pre-print available from CLE e-Prints vol. 5(10) (2005)
  2. 2.
    Ahmed, T.S.: Algebraic logic, where does it stand today? Bull. Symbolic Logic 11(4) (2005) 465-516zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Beame, P., Impagliazzo, R., Krajicek, J., T. Pitassi, T., Pudlak, P.: Lower bounds on Hilbert’s Nullstellensatz and propositional proofs. Proceedings of the London Mathematical Society 73 (1996) 1-26zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Boole, G.: The Mathematical Analysis of Logic, Being an Essay Towards a Calculus of Deductive Reasoning. Macmillan, Barclay and Macmillan, London (1847) (Reprinted by Basil Blackwell, Oxford, 1965)Google Scholar
  5. 5.
    Boole, G.: The calculus of logic. Cambridge and Dublin Math. Journal 3 (1848) 183-198Google Scholar
  6. 6.
    Boole, G.: An Investigation of the Laws of Thought, on Which are Founded the Mathematical Theories of Logic and Probabilities. Walton and Maberley, London (1854) (Reprinted by Dover Books, New York, 1954)Google Scholar
  7. 7.
    Boole, G.: Calculus of Finite Differences. 5th Edition, Chelsea Publishing (orig-inally published in 1860) (1970)Google Scholar
  8. 8.
    Burris, S.: The laws of Boole’s thought. Unpublished (2002), Preprint at
  9. 9.
    Caleiro, C., Carnielli, W.A., Coniglio, M.E., Marcos, J.: Two’s company: “The humbug of many logical values”. In Béziau J.-Y., Birkhäuser, eds.: Logica Universalis. Verlag, Basel, Switzerland (2005) 169-189 Preprint available at
  10. 10.
    ın, A.: Completud de dos cálculos lógicos de Leibniz. Theoria 16(3)(2001)539-558Google Scholar
  11. 11.
    Carnielli, W.A.: Systematization of the finite many-valued logics through the method of tableaux. The Journal of Symbolic Logic 52(2) (1987) 473-493zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Carnielli, W.A.: A polynomial proof system for Lukasiewicz logics. Second Prin-cipia International Symposium. August 6-10, Florianpolis, SC, Brazil (2001)Google Scholar
  13. 13.
    Carnielli, W.A.: Polynomial ring calculus for many-valued logics. Procee-dings of the 35th International Symposium on Multiple-Valued Logic. IEEE Computer Society. Calgary, Canad. IEEE Computer Society, pp. 20-25, 2005. Available from CLE e-Prints vol. 5 (3) (2005) at http://www.cle. 5,n 3,2005.html
  14. 14.
    Carnielli, W.A., Coniglio, M.E.: Polynomial formulations of non-deterministic semantics for logics of formal inconsistency. Manuscript (2006)Google Scholar
  15. 15.
    Clegg, M., Edmonds, J., Impagliazzo, R.: Using the Gröbner bases algorithm to find proofs of unsatisfiability. Proceedings of the 28th Annual ACM Symposium on Theory of Computing. Philadelphia, Pennsylvania, USA (1996) 174-183Google Scholar
  16. 16.
    Corcoran, J.: Aristotle’s Prior Analytics and Boole’s Laws of Thought. Hist. and Ph. of Logic 24 (2003) 261-288zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Dummett, M.: Review of “Studies in Logic and Probability by George Boole”. Rhees, R., Open Court, 1952, J. of Symb. Log. 24 (1959) 203-209MathSciNetGoogle Scholar
  18. 18.
    Eves, H.: An Introduction to the History of Mathematics. (6th ed.) Saunders, New York (1990)Google Scholar
  19. 19.
    Euler, L.: Variae observations circa series infinitas, Commentarii academiae sci-entiarum Petropolitanae 9 (1737) 160-188. Reprinted in Opera Omnia, Series I volume 14, Birkhuser, 216-244. Available on line at
  20. 20.
    Giusti, E.B.: Cavalieri and the Theory of Indivisibles. Cremonese, Roma (1980)zbMATHGoogle Scholar
  21. 21.
    Gottwald, S.: A Treatise on Many-Valued Logics, Studies in Logic and Compu-tation. Research Studies Press Ltd. Hertfordshire, England (2001)Google Scholar
  22. 22.
    Hailperin, T.: Boole’s Logic and Probability: A Critical Exposititon from the Standpoint of Contemporary Algebra, Logic, and Probability Theory. North-Holland Studies In Logic and the Foundations of Mathematics (1986)Google Scholar
  23. 23.
    Lloyd, G.: Finite and infinite in Greece and China. Chinese Science 13 (1996) 11-34Google Scholar
  24. 24.
    MacHale, D.: George Boole: His Life and Work. Boole Press (1985)Google Scholar
  25. 25.
    Mates, B.: Stoic Logic. University of California Press, Berkeley, CA (1953)Google Scholar
  26. 26.
    Martzloff, J.-C.: A History of Chinese Mathematics. Springer-Verlag, Berlin (1997)zbMATHGoogle Scholar
  27. 27.
    Roy, R.: The discovery of the series formula for π by Leibniz, Gregory and Nilakantha. Mathematics Magazine 63(5) (1990) 291-306zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Schroeder, M.: A brief history of the notation of Boole’s algebra. Nordic Journal of Philosophical Logic 2(1) (1997) 41-62zbMATHMathSciNetGoogle Scholar
  29. 29.
    Schröder, E.: Vorlesungen über die Algebra der Logik (exakte Logik) 2 1, B.G. Teubner, Leipzig, 1891 (reprinted by Chelsea, New York, 1966).Google Scholar
  30. 30.
    Stone, M.H.: The theory of representations for boolean algebras. Trans. of the Amer. Math. Soc 40 (1936) 37-111zbMATHCrossRefGoogle Scholar
  31. 31.
    Styazhkin, M.I.: History of Mathematical Logic from Leibniz to Peano. The M.I.T. Press, Cambridge (1969)zbMATHGoogle Scholar
  32. 32.
    Wu, J.-Z., Tan, H.-Y., Li, Y.: An algebraic method to decide the deduction problem in many-valued logics. Journal of Applied Non-Classical Logics 8(4) (1998) 353-60zbMATHMathSciNetGoogle Scholar
  33. 33.
    Weyl, H.: God and the Universe: The Open World. Yale University Press (1932) Reprinted as “The Open World” by Ox Bow Press, 1989.Google Scholar
  34. 34.
    Zhegalkin, I.I.: O tekhnyke vychyslenyi predlozhenyi v symbolytscheskoi logykye (On a technique of evaluation of propositions in symbolic logic). Matematicheskii Sbornik 34 (1) (1927) 9-28 (in Russian)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Walter Carnielli
    • 1
  1. 1.Centre for Logic, Epistemology and the History of ScienceUNICAMPCampinasBrazil

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