{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,17]],"date-time":"2025-09-17T15:43:44Z","timestamp":1758123824399},"reference-count":57,"publisher":"Duke University Press","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Notre Dame J. Formal Logic"],"published-print":{"date-parts":[[2009,7,1]]},"DOI":"10.1215\/00294527-2009-013","type":"journal-article","created":{"date-parts":[[2009,11,10]],"date-time":"2009-11-10T14:07:39Z","timestamp":1257862059000},"source":"Crossref","is-referenced-by-count":6,"title":["George Boole's Deductive System"],"prefix":"10.1215","volume":"50","author":[{"given":"Frank Markham","family":"Brown","sequence":"first","affiliation":[]}],"member":"73","reference":[{"key":"1","unstructured":"[1] Beth, E. W., <i>The Foundations of Mathematics: A Study in the Philosophy of Science<\/i>, Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Company, Amsterdam, 1959."},{"key":"2","unstructured":"[2] Boole, G., <i>The Mathematical Analysis of Logic, Being an Essay Towards a Calculus of Deductive Reasoning<\/i>, Cambridge: Macmillan, Barclay, &amp; Macmillan; London: George Bell, 1847. Reprinted by Philosophical Library, New York, 1948; in <i>Studies in Logic and Probability by George Boole<\/i>, Watts &amp; Co., London, 1952; by Basil Blackwell, Oxford, U.K., 1965; by Thoemmes Press, Bristol, U.K., 1998; and by Kessenger Publishing, Whitefish, MT, 2007."},{"key":"4","doi-asserted-by":"crossref","unstructured":"[4] Boole, G., <i>An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities<\/i>, Walton, London, 1854. Reprinted by Open Court Publishing Co., Chicago &amp; London, 1911; by Dover Books, New York, 1951; by Prometheus Books, Buffalo, 2003; and by Cosimo Books, New York, 2007.).","DOI":"10.5962\/bhl.title.29413"},{"key":"5","doi-asserted-by":"publisher","unstructured":"[5] Boros, E., and P. L. Hammer, \"Pseudo-Boolean optimization\", <i>Discrete Applied Mathematics<\/i>, vol. 123 (2002), pp. 155--225. Workshop on Discrete Optimization, DO'99 (Piscataway, NJ).","DOI":"10.1016\/S0166-218X(01)00336-5"},{"key":"7","doi-asserted-by":"crossref","unstructured":"[7] Brown, F. M., <i>Boolean Reasoning: The Logic of Boolean Equations<\/i>, Kluwer Academic Publishers, Boston, 1990. Second edition, Dover Publications, Mineola, 2003.","DOI":"10.1007\/978-1-4757-2078-5_6"},{"key":"8","doi-asserted-by":"publisher","unstructured":"[8] Brown, F. M., and S. Rudeanu, \"Consequences, consistency, and independence in Boolean algebras\", <i>Notre Dame Journal of Formal Logic<\/i>, vol. 22 (1981), pp. 45--62.","DOI":"10.1305\/ndjfl\/1093883339"},{"key":"9","unstructured":"[9] Brown, F. M., and S. Rudeanu, \"Recurrent covers and Boolean equations\", pp. 55--86 in <i>Contributions to Lattice Theory (Szeged, 1980)<\/i>, vol. 33 of <i>Colloquia Mathematica Societatis J\u00e1nos Bolyai<\/i>, North-Holland, Amsterdam, 1983."},{"key":"13","doi-asserted-by":"publisher","unstructured":"[13] Corcoran, J., \"Aristotle's Prior Analytics and Boole's Laws of Thought\", <i>History and Philosophy of Logic<\/i>, vol. 24 (2003), pp. 261--88. A Festschrift in honor of Professor Ivor Grattan-Guinness.","DOI":"10.1080\/01445340310001604707"},{"key":"14","doi-asserted-by":"publisher","unstructured":"[14] Corcoran, J., and S. Wood, \"Boole's criteria for validity and invalidity\", <i>Notre Dame Journal of Formal Logic<\/i>, vol. 21 (1980), pp. 609--38.","DOI":"10.1305\/ndjfl\/1093883246"},{"key":"15","unstructured":"[15] Couturat, L., <i>L'alg\u00e8bre de la Logique<\/i>, 2d edition, Librairie Scientifique et Technique Albert Blanchard, Paris, 1905. English translation by Lydia G. Robinson: Open Court Publishing Co., Chicago &amp; London, 1914. Reprinted by Dover Publications, Mineola, 2006."},{"key":"16","doi-asserted-by":"publisher","unstructured":"[16] Deschamps, J.-P., \"Parametric solutions of Boolean equations\", <i>Discrete Mathematics<\/i>, vol. 3 (1972), pp. 333--42.","DOI":"10.1016\/0012-365X(72)90090-8"},{"key":"18","unstructured":"[18] Feys, R., \"Boolean methods of development and interpretation\", <i>Proceedings of the Royal Irish Academy. Sect. A.<\/i>, vol. 57 (1955), pp. 107--112. Celebration of the Centenary of ``The Laws of Thought'' by George Boole, 24th May, 1954."},{"key":"21","unstructured":"[21] Hailperin, T., <i>Boole's Logic and Probability. A Critical Exposition from the Standpoint of Contemporary Algebra, Logic and Probability Theory<\/i>, 2d edition, vol. 85 of <i>Studies in Logic and the Foundations of Mathematics<\/i>, North-Holland Publishing Co., Amsterdam, 1986."},{"key":"22","doi-asserted-by":"crossref","unstructured":"[22] Hammer, P. L., and S. Rudeanu, <i>Boolean Methods in Operations Research and Related Areas<\/i>, vol. 7 of <i>Econometrics and Operations Research<\/i>, Springer-Verlag New York, Inc., New York, 1968.","DOI":"10.1007\/978-3-642-85823-9"},{"key":"23","unstructured":"[23] Hammer, P., I. Rosenberg, and S. Rudeanu, \"On the determination of the minima of pseudo-Boolean functions (in Romanian)\", <i>Studii\u0119<\/i>dla si Cercetri Matematice, vol. 14 (1963), pp. 359--64."},{"key":"29","unstructured":"[29] Kneale, W., and M. Kneale, <i>The Development of Logic<\/i>, The Clarendon Press, Oxford, 1962."},{"key":"32","doi-asserted-by":"publisher","unstructured":"[32] Laita, L. M., L. de Ledesma, E. Roanes-Lozano, A. P\u00e9rez, and A. Brunori, \"Boole's logic revisited from computer algebra\", <i>Mathematics and Computers in Simulation<\/i>, vol. 51 (2000), pp. 419--39. Nonstandard applications of computer algebra, II (Wailea, HI, 1997\/Prague, 1998).","DOI":"10.1016\/S0378-4754(99)00134-2"},{"key":"35","doi-asserted-by":"publisher","unstructured":"[35] L\u00f6wenheim, L., \"\u00dcber die Aufl\u00f6sung von Gleichungen im logischen Gebietekalkul\", <i>Mathematische Annalen<\/i>, vol. 68 (1910), pp. 169--207.","DOI":"10.1007\/BF01474159"},{"key":"42","doi-asserted-by":"crossref","unstructured":"[42] Papaioannou, S. G., and W. A. Barrett, \"The real transform of a Boolean function and its applications\", <i>Computers and Electrical Engineering<\/i>, vol. 2 (1975), pp. 215--24.","DOI":"10.1016\/0045-7906(75)90008-7"},{"key":"43","doi-asserted-by":"publisher","unstructured":"[43] Peirce, C. S., \"On the Algebra of Logic\", <i>American Journal of Mathematics<\/i>, vol. 3 (1880), pp. 15--57. \\zhrefhttp:\/\/www.emis.de\/cgi-bin\/MATH-item?12.0041.01, JFM 12.0041.01.","DOI":"10.2307\/2369442"},{"key":"44","unstructured":"[44] Rudeanu, S., \"On the definition of Boolean algebras by means of binary operations (in Russian)\", <i>Revue de Math\u00e9matiques Pures et Appliqu\u00e9es<\/i>, vol. 6 (1961), pp. 171--83."},{"key":"45","unstructured":"[45] Rudeanu, S., <i>Boolean Functions and Equations<\/i>, North-Holland Publishing Co., Amsterdam, 1974."},{"key":"46","unstructured":"[46] Schneeweiss, W. G., <i>Boolean Functions with Engineering Applications and Computer Programs<\/i>, Springer-Verlag, Berlin, 1989."},{"key":"48","doi-asserted-by":"publisher","unstructured":"[48] Schroeder-Heister, P., \"Frege and the resolution calculus\", <i>History and Philosophy of Logic<\/i>, vol. 18 (1997), pp. 95--108.","DOI":"10.1080\/01445349708837278"},{"key":"50","doi-asserted-by":"publisher","unstructured":"[50] Smith, G. C., \"Boole's annotations on <i>The Mathematical Analysis of Logic<\/i>\", <i>History and Philosophy of Logic<\/i>, vol. 4 (1983), pp. 27--39.","DOI":"10.1080\/01445348308837043"},{"key":"51","doi-asserted-by":"publisher","unstructured":"[51] Stone, M. H., \"The theory of representations for Boolean algebras\", <i>Transactions of the American Mathematical Society<\/i>, vol. 40 (1936), pp. 37--111.","DOI":"10.2307\/1989664"},{"key":"53","doi-asserted-by":"publisher","unstructured":"[53] van Evra, J. W., \"A reassessment of George Boole's theory of logic\", <i>Notre Dame Journal of Formal Logic<\/i>, vol. 18 (1977), pp. 363--77.","DOI":"10.1305\/ndjfl\/1093888009"},{"key":"54","unstructured":"[54] Venn, J., <i>Symbolic Logic<\/i>, 2d edition, Macmillan, London, 1894. Reprinted, revised and rewritten. Bronx: Chelsea Publishing Co., 1971."},{"key":"55","doi-asserted-by":"publisher","unstructured":"[55] Whitney, H., \"Characteristic functions and the algebra of logic\", <i>Annals of Mathematics. Second Series<\/i>, vol. 34 (1933), pp. 405--14.","DOI":"10.2307\/1968168"},{"key":"3","unstructured":"[3] Boole, G., \"The calculus of logic\", <i>Cambridge and Dublin Mathematical Journal<\/i>, vol. 3 (1848), pp. 183--98."},{"key":"6","unstructured":"[6] Broad, C., ``Review of <i>Collected Logical Works<\/i>. Vol. II. <i>Laws of Thought<\/i>. George Boole,'' <i>Mind<\/i>, vol. 26 (1917), pp. 81--99."},{"key":"10","unstructured":"[10] Burris, S. N., \"A Fragment of Boole's Algebraic Logic Suitable for Traditional Syllogistic Logic\", Preprint, 2003. Available at www.thoralf.uwaterloo.ca."},{"key":"11","unstructured":"[11] Burris, S. N., \"Contributions of the Logicians. Part I: From Richard Whately to William Stanley Jevons\", Preprint, March 2001. Available at www.thoralf.uwaterloo.ca."},{"key":"12","doi-asserted-by":"crossref","unstructured":"[12] Carnielli, W., \"Polynomizing: Logic inference in polynomial format and the legacy of Boole\", <i>Studies in Computational Intelligence<\/i>, vol. 64 (2007), pp. 349--64.","DOI":"10.1007\/978-3-540-71986-1_20"},{"key":"17","unstructured":"[17] Dummett, M., \"Review of <i>Studies in Logic and Probability by George Boole<\/i>: Watts &amp; Co., London, 1952, edited by R. Rhees\", <i>The Journal of Symbolic Logic<\/i>, vol. 24 (1959), pp. 203--209."},{"key":"19","unstructured":"[19] Frege, G., \"Boole's logical calculus and the concept-script\", Posthumous, English translation in [frege1979?], pp. 9-46, 1880\/81."},{"key":"20","unstructured":"[20] Frege, G., <i>Gottlob Frege: Posthumous Writings<\/i>, Basil Blackwell, Oxford, 1979. English translation of <i>Nachgelassene Schriften<\/i>, vol. 1, edited by H. Hermes, F. Kambartel, and F. Kaulbach, Felix Meiner, Hamburg, 1969."},{"key":"24","unstructured":"[24] Heath, P. L., \"History of logic\", pp. 541--45 in <i>The Encyclopedia of Philosophy<\/i>, vol. 4, Macmillan, New York, 1967. Collier-Macmillan, London."},{"key":"25","unstructured":"[25] Hoff-Hansen, E., \"En matematisk tolkning av den klassiske utsagnsregning (a mathematical interpretation of the classical propositional calculus)\", <i>Norsk Matematisk Tidsskrift<\/i>, vol. 25 (1943), pp. 6--12."},{"key":"26","unstructured":"[26] Jevons, W. S., <i>Pure Logic, or the Logic of Quality Apart from Quantity<\/i>, Stanford, London, 1864. Also in <i>Pure Logic and Other Minor Works<\/i>, London and New York: Macmillan, 1890. Reprinted by Lincoln-Rembrandt Publishing, Charlottesville, VA (no date)."},{"key":"27","unstructured":"[27] J\u00f8rgensen, J., <i>A Treatise of Formal Logic, Vols. 1, 2, 3<\/i>, Levin &amp; Munksgaard, Copenhagen, 1931. Reprint: New York, Russell &amp; Russell, 1962. \\zhrefhttp:\/\/www.emis.de\/cgi-bin\/MATH-item?57.0050.01, JFM 57.0050.01."},{"key":"28","doi-asserted-by":"crossref","unstructured":"[28] Ketonen, O., \"Review of [hoffhansen?] and [skolem43?]\", <i>The Journal of Symbolic Logic<\/i>, vol. 13 (1948), p. 169.","DOI":"10.2307\/2267869"},{"key":"30","doi-asserted-by":"crossref","unstructured":"[30] Ladd, C., ``On the Algebra of Logic,'' pp. 17--71 of <i>Studies in Logic. By Members of the Johns Hopkins University<\/i>, edited by C. S. Peirce, Little, Brown &amp; Co., Boston, 1883.","DOI":"10.1037\/12811-004"},{"key":"31","doi-asserted-by":"crossref","unstructured":"[31] Ladd Franklin, C., \"On some characteristics of symbolic logic\", <i>American Journal of Psychology<\/i>, vol. 2 (1889), pp. 543--67.","DOI":"10.2307\/1411857"},{"key":"33","unstructured":"[33] Lewis, C. I., <i>A Survey of Symbolic Logic<\/i>, University of California Press, Berkeley, 1918. Reprinted by Dover Publications, Inc., New York, 1960. Chap. II, ``The Classic, or Boole-Schr\u00f6der Algebra of Logic.''"},{"key":"34","unstructured":"[34] Lotze, H., \"Note on the logical calculus\", Book II, Chap. III in <i>Logic<\/i>, edited by B. Bosanquet, vol. 1, Clarendon Press, Oxford, 2d edition, 1888. English translation of <i>Logik,<\/i> 2d edition, S. Hirzel, 1880."},{"key":"36","doi-asserted-by":"crossref","unstructured":"[36] Macfarlane, A., \"The fundamental principles of algebra\", <i>Science, N.S.<\/i>, vol. 10 (15 Sept. 1899), pp. 345--64.","DOI":"10.1126\/science.10.246.345"},{"key":"37","unstructured":"[37] Macfarlane, A., \"Application of the method of the logical spectrum to Boole's problem\", <i>Proceedings of the American Association for the Advancement of Science<\/i>, vol. 39 (1890), pp. 57--60."},{"key":"38","unstructured":"[38] McColl, H., \"The calculus of equivalent statements (second paper)\", <i>Proceedings of the London Mathematical Society<\/i>, vol. 9 (June 13, 1878), pp. 177--86. \\zhrefhttp:\/\/www.emis.de\/cgi-bin\/MATH-item?10.0035.01, JFM 10.0035.01."},{"key":"39","doi-asserted-by":"crossref","unstructured":"[39] McColl, H., \"The calculus of equivalent statements (third paper)\", <i>Proceedings of the London Mathematical Society<\/i>, vol. 10 (Nov. 14, 1878), pp. 16--28.","DOI":"10.1112\/plms\/s1-10.1.16"},{"key":"40","unstructured":"[40] M\u00fcller, E., \"Das Eliminationsproblem und die Syllogistik\", (1901). Programmabh. des Gymnasiums in Tauberbischofsheim."},{"key":"41","unstructured":"[41] Nambiar, S., <i>The Origin of Boole's Philosophy of Logic: The Assimilation of Traditional Logic into Mathematical Analysis (George Boole)<\/i>, Ph.D. thesis, State University of New York at Buffalo, 2000. Dissertation."},{"key":"47","unstructured":"[47] Schr\u00f6der, E., <i>Vorlesungen \u00fcber die Algebra der Logik. Vol. 1, 1890; Vol. 2, 1891; Vol. 3, 1895; Vol. 2, Part 2, 1905<\/i>, Teubner, Leipzig. Reprints: Chelsea Publishing Co., Bronx, 1966; Thoemmes Press, Bristol, 2000."},{"key":"49","unstructured":"[49] Skolem, T., \"Noen bemerkninger til foranstaende artikkel av E Hoff-Hansen (some remarks on the preceding article of E. Hoff-Hansen)\", <i>Norsk Matematisk Tidsskrift<\/i>, vol. 25 (1943), pp. 13--16."},{"key":"52","unstructured":"[52] Styazhkin, N. I., <i>Concise History of Mathematical Logic from Leibniz to Peano<\/i>, The MIT Press, Cambridge, 1969."},{"key":"56","unstructured":"[56] Wood, S., <i>George Boole's Theory of Propositional Forms<\/i>, Ph.D. thesis, University of Buffalo, Buffalo, 1976."},{"key":"57","unstructured":"[57] Wundt, W., <i>Logik. Eine Untersuchung der Principien der Erkenntniss und der Methoden wissenschaftlicher Forschung<\/i>, F. Enke Verlag, Stuttgart, 1880. \\zhrefhttp:\/\/www.emis.de\/cgi-bin\/MATH-item?38.0094.03, JFM 38.0094.03."}],"container-title":["Notre Dame Journal of Formal Logic"],"original-title":[],"link":[{"URL":"https:\/\/projecteuclid.org\/journalArticle\/Download?urlid=10.1215\/00294527-2009-013","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,1,30]],"date-time":"2024-01-30T20:57:19Z","timestamp":1706648239000},"score":1,"resource":{"primary":{"URL":"https:\/\/projecteuclid.org\/journals\/notre-dame-journal-of-formal-logic\/volume-50\/issue-3\/George-Booles-Deductive-System\/10.1215\/00294527-2009-013.full"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,7,1]]},"references-count":57,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2009,7,1]]}},"URL":"https:\/\/doi.org\/10.1215\/00294527-2009-013","relation":{},"ISSN":["0029-4527"],"issn-type":[{"value":"0029-4527","type":"print"}],"subject":[],"published":{"date-parts":[[2009,7,1]]}}}