What is a logic? Towards axiomatic emptiness
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07.04.2010
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Abstract
We first recall the original Greek sense of the word logic and how logic was developed on the one hand as an efficient way of reasoning by the use of reduction to the absurd and on the other hand as a useless system of logic by Aristotle. Then we discuss the changes of the modern conception of logic: the rejection of the principle of noncontradiction considered as fundamental by Aristotle and the structuralist move breaking the Aristotelian accident/essence dichotomy. Finally we explain why and how in universal logic — like in universal algebra — axiomatic emptiness prevails: a logical structure is a structure obeying no axioms.
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How to Cite
Beziau J.-Y. What is a logic? Towards axiomatic emptiness // Logicheskie Issledovaniya / Logical Investigations. 2010. VOL. 16. C. 272-279.
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Papers
References
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Beziau, J.-Y. Universal logic // Logica’94, P.Kolar et V.Svoboda (eds.), Acadёmie des Sciences, Prague. 1994. P.73-93.
Beziau, J.-Y. Recherches sur la iogique universelle, PhD Thesis, Universite Denis Diderot (Paris 7). 1995.
Beziau, J.-Y. Prom paraconsistent logic to universal logic // Sorites. Vol. 12. 2001. P. 5-32.
Beziau, J.-Y.(ed). Logica Universalis, Birkhauser, Basel. 2005.
Beziau, J.-Y. 13 questions about universal logic // Bulletin of the Section of Logic. Vol. 25. 2006. P.133-150.
Beziau, J.-Y. Les axiomes de Tarski // R. Pouivet and M. Rebuschi (eds), La philosophie en Pologne 1918-1939, Vrin, Paris. 2006. P.135-149.
Beziau, J.- Y. What is “formal logic”? // Revista Brasileira de Filosofia. Vol. 232. 2009. P.197-208.
Beziau, J.-Y., Camielli, W.A. et Gabbay, D.M. (eds) Handbook of Paraconsistency, King’s College, Londres. 2007.
Birkhoff, G. Universal algebra // Comptes Rendus du Premier CongrSs Canadien de Math&natiques, Presses de PUniversite de Toronto, Toronto. 1946. P.310-326.
Birkhoff, G. Universal algebra // Rota, G.-C. and Oliveira J.S. (eds.), Selected Papers on Algebra and Topology by Garret Birkhoff, Birkhauser, Bale. 1987. P.111- 115.
Boole, G. The mathematical analysis of logic, being an essay toward a calculus of deductive reasoning, Cambridge. 1847.
Bourbaki, N. L’architecture des mathematiques // Les grands courants de la pensde math£matique, F. Le Lionnais (ed). 1948. P.35-48.
Breal, M. Essai de sdmantique, Paris, 1897.
Corry, L. Modern Algebra and the Rise of Mathematical Structures, Birkhauser, Boston. 1996.
De Morgan, A. Formal logic: or, the calculus of inference, necessary and probable, London. 1847.
Dieudonne, J. Pour Phonneur de Pesprit humain, Hachette, Paris. 1987.
Praisse, R. La zerologie: une recherche aux frontihres de la logique et de Part: applications a la logique des relations de base vide // international Logic. Review. Vol. 26. 1982. P.67-29.
Glivenko, V. Theorie generale des structures, Hermann, Paris. 1938.
Granger, G.G. Pensee formelle et science de l’homme, Aubier Montaigne, Paris. 1960.
Granger, G.G. L’irrationnel, Odile Jacob. Paris. 1998.
Hertz, P. Uber Axiomensysteme fur beliebige Satzsysteme // Matematische Annalen Vol. 101. 1929. P.457-514.
Koslow, A. A structuralist theory of logic, Cambridge University Press, New-York. 1992.
Los, J. et Suszko, R. Remarks on sentential logics // Indigationes Mathematicae. Vol. 20. 1958. P.177-183.
Lukasiewicz, J. On the principle of contradiction in Aristotle — A critical study, Krakow. 1910.
Lukasiewicz, J. et Tarski, A. Untersuchungen fiber den Aussagenkalkfil // Comptes Rendus des s6ances de la Societe des Sciences et des lettres de Varsovie XX11I, Classe III. 1930. P.30-50.
Mossakowski, T., Goguen, J., Diaconescu, R. and Tarlecki, A. What is a Logic? // J.-Y. B6ziau (ed). Logica Universalis, Birkhauser, Basel. 2005.
Porte, J. Recherches sur la th4orie g£n0rale des systemes formels et sur les systёmes connectifs, Gauthier-Villars, Paris et Nauwelaerts, Louvain. 1965.
Rasiowa, H. et Sikorski, R. The mathematics of metamathematics, Academie Polonaise des Sciences, Varsovie. 1963.
Riche, J. From universal algebra to universal logic // J.Y. Beziau and A. Costa- Leite (eds). Perspectives on Universal Logic, Polimetrica, Monza. 2007. P.3-39.
Rougier, L. The relativity of logic // Philosophy and Phenomenological Research. Vol. 2. 1941. P. 137-158.
Rougier, L. Le traits de la connaissance, Gauthiers-Villars, Paris. 1955.
Schanuel, S.H. et Lawvere, F.W. Conceptual mathematics — A first introduction to categories, CUP. 1997.
Scholz, H. Abriss der Geschichte der Logik, Karl Alber, Fribourg. 1931.
Sylvester, J.J. Lectures on the principles of universal algebra // American Journal of Mathematic. Vol. 6. 1884. P.270-286.
Tarski, A. Remarques sur les notions fond ament ales de la methodologie des mathematiques // Annales de la Societe Polonaise de Mathematique. 1928. P.270- 271.
Tarski, A. Uber einige fundamenten Begriffe der Metamathematik // Comptes Rendus dcs seances de la Societe des Sciences et des lettres de Varsovie XXIII, Classe III. 1930. P.22-29.
Tarski, A. Fundamental Begriffe der Methodologie der deduktiven Wissenschaften. I // Monatshefte fur Mathematik und Physik. Vol. 37. 1930. P.361-404.
Whitehead, A.N.A treatise on universal logic, CUP. 1898.
Wolenski, J. Logic and Philosophy in the Lvov-Wasaw School, Kluwer, Dordrecht. 1989.
Beziau, J.-Y. Universal logic // Logica’94, P.Kolar et V.Svoboda (eds.), Acadёmie des Sciences, Prague. 1994. P.73-93.
Beziau, J.-Y. Recherches sur la iogique universelle, PhD Thesis, Universite Denis Diderot (Paris 7). 1995.
Beziau, J.-Y. Prom paraconsistent logic to universal logic // Sorites. Vol. 12. 2001. P. 5-32.
Beziau, J.-Y.(ed). Logica Universalis, Birkhauser, Basel. 2005.
Beziau, J.-Y. 13 questions about universal logic // Bulletin of the Section of Logic. Vol. 25. 2006. P.133-150.
Beziau, J.-Y. Les axiomes de Tarski // R. Pouivet and M. Rebuschi (eds), La philosophie en Pologne 1918-1939, Vrin, Paris. 2006. P.135-149.
Beziau, J.- Y. What is “formal logic”? // Revista Brasileira de Filosofia. Vol. 232. 2009. P.197-208.
Beziau, J.-Y., Camielli, W.A. et Gabbay, D.M. (eds) Handbook of Paraconsistency, King’s College, Londres. 2007.
Birkhoff, G. Universal algebra // Comptes Rendus du Premier CongrSs Canadien de Math&natiques, Presses de PUniversite de Toronto, Toronto. 1946. P.310-326.
Birkhoff, G. Universal algebra // Rota, G.-C. and Oliveira J.S. (eds.), Selected Papers on Algebra and Topology by Garret Birkhoff, Birkhauser, Bale. 1987. P.111- 115.
Boole, G. The mathematical analysis of logic, being an essay toward a calculus of deductive reasoning, Cambridge. 1847.
Bourbaki, N. L’architecture des mathematiques // Les grands courants de la pensde math£matique, F. Le Lionnais (ed). 1948. P.35-48.
Breal, M. Essai de sdmantique, Paris, 1897.
Corry, L. Modern Algebra and the Rise of Mathematical Structures, Birkhauser, Boston. 1996.
De Morgan, A. Formal logic: or, the calculus of inference, necessary and probable, London. 1847.
Dieudonne, J. Pour Phonneur de Pesprit humain, Hachette, Paris. 1987.
Praisse, R. La zerologie: une recherche aux frontihres de la logique et de Part: applications a la logique des relations de base vide // international Logic. Review. Vol. 26. 1982. P.67-29.
Glivenko, V. Theorie generale des structures, Hermann, Paris. 1938.
Granger, G.G. Pensee formelle et science de l’homme, Aubier Montaigne, Paris. 1960.
Granger, G.G. L’irrationnel, Odile Jacob. Paris. 1998.
Hertz, P. Uber Axiomensysteme fur beliebige Satzsysteme // Matematische Annalen Vol. 101. 1929. P.457-514.
Koslow, A. A structuralist theory of logic, Cambridge University Press, New-York. 1992.
Los, J. et Suszko, R. Remarks on sentential logics // Indigationes Mathematicae. Vol. 20. 1958. P.177-183.
Lukasiewicz, J. On the principle of contradiction in Aristotle — A critical study, Krakow. 1910.
Lukasiewicz, J. et Tarski, A. Untersuchungen fiber den Aussagenkalkfil // Comptes Rendus des s6ances de la Societe des Sciences et des lettres de Varsovie XX11I, Classe III. 1930. P.30-50.
Mossakowski, T., Goguen, J., Diaconescu, R. and Tarlecki, A. What is a Logic? // J.-Y. B6ziau (ed). Logica Universalis, Birkhauser, Basel. 2005.
Porte, J. Recherches sur la th4orie g£n0rale des systemes formels et sur les systёmes connectifs, Gauthier-Villars, Paris et Nauwelaerts, Louvain. 1965.
Rasiowa, H. et Sikorski, R. The mathematics of metamathematics, Academie Polonaise des Sciences, Varsovie. 1963.
Riche, J. From universal algebra to universal logic // J.Y. Beziau and A. Costa- Leite (eds). Perspectives on Universal Logic, Polimetrica, Monza. 2007. P.3-39.
Rougier, L. The relativity of logic // Philosophy and Phenomenological Research. Vol. 2. 1941. P. 137-158.
Rougier, L. Le traits de la connaissance, Gauthiers-Villars, Paris. 1955.
Schanuel, S.H. et Lawvere, F.W. Conceptual mathematics — A first introduction to categories, CUP. 1997.
Scholz, H. Abriss der Geschichte der Logik, Karl Alber, Fribourg. 1931.
Sylvester, J.J. Lectures on the principles of universal algebra // American Journal of Mathematic. Vol. 6. 1884. P.270-286.
Tarski, A. Remarques sur les notions fond ament ales de la methodologie des mathematiques // Annales de la Societe Polonaise de Mathematique. 1928. P.270- 271.
Tarski, A. Uber einige fundamenten Begriffe der Metamathematik // Comptes Rendus dcs seances de la Societe des Sciences et des lettres de Varsovie XXIII, Classe III. 1930. P.22-29.
Tarski, A. Fundamental Begriffe der Methodologie der deduktiven Wissenschaften. I // Monatshefte fur Mathematik und Physik. Vol. 37. 1930. P.361-404.
Whitehead, A.N.A treatise on universal logic, CUP. 1898.
Wolenski, J. Logic and Philosophy in the Lvov-Wasaw School, Kluwer, Dordrecht. 1989.