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.
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References
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) ftp://logica.cle.unicamp.br/pub/e-prints/MTPs-CompQuant%28Ing%29.pdf
Ahmed, T.S.: Algebraic logic, where does it stand today? Bull. Symbolic Logic 11(4) (2005) 465-516
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) 1-26
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)
Boole, G.: The calculus of logic. Cambridge and Dublin Math. Journal 3 (1848) 183-198
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)
Boole, G.: Calculus of Finite Differences. 5th Edition, Chelsea Publishing (orig-inally published in 1860) (1970)
Burris, S.: The laws of Boole’s thought. Unpublished (2002), Preprint at http://www.thoralf.uwaterloo.ca/htdocs/MYWORKS/PREPRINTS/aboole.pdf
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 http://wslc.math.ist.utl.pt/ftp/pub/CaleiroC/05-CCCM-dyadic.pdf.
ın, A.: Completud de dos cálculos lógicos de Leibniz. Theoria 16(3)(2001)539-558
Carnielli, W.A.: Systematization of the finite many-valued logics through the method of tableaux. The Journal of Symbolic Logic 52(2) (1987) 473-493
Carnielli, W.A.: A polynomial proof system for Lukasiewicz logics. Second Prin-cipia International Symposium. August 6-10, Florianpolis, SC, Brazil (2001)
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. unicamp.br/e-prints/vol 5,n 3,2005.html
Carnielli, W.A., Coniglio, M.E.: Polynomial formulations of non-deterministic semantics for logics of formal inconsistency. Manuscript (2006)
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) 174-183
Corcoran, J.: Aristotle’s Prior Analytics and Boole’s Laws of Thought. Hist. and Ph. of Logic 24 (2003) 261-288
Dummett, M.: Review of “Studies in Logic and Probability by George Boole”. Rhees, R., Open Court, 1952, J. of Symb. Log. 24 (1959) 203-209
Eves, H.: An Introduction to the History of Mathematics. (6th ed.) Saunders, New York (1990)
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 http://www.EulerArchive.org.
Giusti, E.B.: Cavalieri and the Theory of Indivisibles. Cremonese, Roma (1980)
Gottwald, S.: A Treatise on Many-Valued Logics, Studies in Logic and Compu-tation. Research Studies Press Ltd. Hertfordshire, England (2001)
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)
Lloyd, G.: Finite and infinite in Greece and China. Chinese Science 13 (1996) 11-34
MacHale, D.: George Boole: His Life and Work. Boole Press (1985)
Mates, B.: Stoic Logic. University of California Press, Berkeley, CA (1953)
Martzloff, J.-C.: A History of Chinese Mathematics. Springer-Verlag, Berlin (1997)
Roy, R.: The discovery of the series formula for π by Leibniz, Gregory and Nilakantha. Mathematics Magazine 63(5) (1990) 291-306
Schroeder, M.: A brief history of the notation of Boole’s algebra. Nordic Journal of Philosophical Logic 2(1) (1997) 41-62
Schröder, E.: Vorlesungen über die Algebra der Logik (exakte Logik) 2 1, B.G. Teubner, Leipzig, 1891 (reprinted by Chelsea, New York, 1966).
Stone, M.H.: The theory of representations for boolean algebras. Trans. of the Amer. Math. Soc 40 (1936) 37-111
Styazhkin, M.I.: History of Mathematical Logic from Leibniz to Peano. The M.I.T. Press, Cambridge (1969)
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) 353-60
Weyl, H.: God and the Universe: The Open World. Yale University Press (1932) Reprinted as “The Open World” by Ox Bow Press, 1989.
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)
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Carnielli, W. (2007). Polynomizing: Logic Inference in Polynomial Format and the Legacy of Boole. In: Magnani, L., Li, P. (eds) Model-Based Reasoning in Science, Technology, and Medicine. Studies in Computational Intelligence, vol 64. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71986-1_20
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