Aristotle’s Prototype Rule-Based Underlying Logic

Abstract

This expository paper on Aristotle’s prototype underlying logic is intended for a broad audience that includes non-specialists. It requires as background a discussion of Aristotle’s demonstrative logic. Demonstrative logic or apodictics is the study of demonstration as opposed to persuasion. It is the subject of Aristotle’s two-volume Analytics, as its first sentence says. Many of Aristotle’s examples are geometrical. A typical geometrical demonstration requires a theorem that is to be demonstrated, known premises from which the theorem is to be deduced, and a deductive logic by which the steps of the deduction proceed. Every demonstration produces (or confirms) knowledge of (the truth of) its conclusion for every person who comprehends the demonstration. Aristotle presented a general truth-and-consequence theory of demonstration meant to apply to all demonstrations: a demonstration is an extended argumentation that begins with premises known to be truths and that involves a chain of reasoning showing by deductively evident steps that its conclusion is a consequence of its premises. In short, a demonstration is a deduction whose premises are known to be true. Aristotle’s general theory of demonstration required a prior general theory of deduction presented in the Prior Analytics. His general immediate-deduction-chaining theory of deduction was meant to apply to all deductions: any deduction that is not immediately evident is an extended argumentation that involves a chaining of immediately evident steps that shows its final conclusion to follow logically from its premises. His deductions, both direct and indirect, were rule-based and not tautology-based. The idea of tautology-based deduction, which dominated modern logic in the early years of the 1900s, is nowhere to be found in Analytics. Rule-based (or “natural”) deduction was rediscovered by modern logicians. To illustrate his general theory of deduction, Aristotle presented a prototype: an ingeniously simple and mathematically precise special case traditionally known as the categorical syllogistic. With reference only to propositions of the four so-called categorical forms, he painstakingly worked out exactly what those immediately evident deductive steps are and how they are chained to complete deductions. In his specialized prototype theory, Aristotle explained how to deduce from a given categorical premise set, no matter how large, any categorical conclusion implied by the given set. He did not extend this treatment to non-categorical deductions, thus setting a program for future logicians. The prototype, categorical syllogistic, was seen by Boole as a “first approximation” to a comprehensive logic. Today, however it appears more as the first of the dozens of logics already created and as the first exemplification of a family that continues to expand.

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References

  1. 1.

    Aristotle: Prior analytics. In: Smith, T.R. (eds) Aristotle’s Prior Analytics. Hackett, Indianapolis (1989)

  2. 2.

    Aristotle: Posterior analytics. In: Mure, T.G.R.G (eds) McKeon (1947)

  3. 3.

    Beth, E.: Foundations of Mathematics. North-Holland, Amsterdam (1959)

    Google Scholar 

  4. 4.

    Boger, G.: Aristotle’s underlying logic. In: Gabbay, D., Woods, J. (eds.) Handbook of the History of Logic. Elsevier, Amsterdam (2004)

    Google Scholar 

  5. 5.

    Boole, G.: Laws of Thought. Macmillan, Cambridge (1854/2003) (Reprinted with introduction by J. Corcoran. Buffalo: Prometheus Books)

  6. 6.

    Robert, Audi. (ed.): The Cambridge Dictionary of Philosophy. Cambridge University Press, Cambridge (1999)

  7. 7.

    Corcoran, J.: Completeness of an ancient logic. J. Symb. Log. 37, 696–702 (1972)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Corcoran, J. (ed.): Ancient Logic and Its Modern Interpretations. Kluwer, Dordrecht (1974)

  9. 9.

    Corcoran, J.: Aristotle’s natural deduction system. Corcoran 1974, 85–131 (1974)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Corcoran, J.: Ockham’s syllogistic semantics. J. Symb. Log. 57, 197–8 (1981)

    Google Scholar 

  11. 11.

    Corcoran, J.: Review of “Aristotelian Induction”, Hintikka 1980. Math. Rev. 82m, 00016 (1982)

    Google Scholar 

  12. 12.

    Corcoran, J.: Deduction and reduction: two proof-theoretic processes in prior analytics I. J. Symb. Log. 48(1983), 906 (1983)

    Google Scholar 

  13. 13.

    Corcoran, J.: Review of G. Saccheri. Euclides Vindicatus (1733), edited and translated by G. B. Halsted, 2nd ed. (1986), In: Mathematical Reviews 88j:01013 (1988)

  14. 14.

    Corcoran, J.: Argumentations and logic. Argumentation 3, 17–43 (1989). (Spanish translation by R. Fernández and J. M. Sagüillo: Corcoran 1994a)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Corcoran, J.: Logical methodology: Aristotle and Tarski. J. Symb. Log. 57, 374 (1992)

    Google Scholar 

  16. 16.

    Corcoran, J.: Argumentaciones y lógica. Ágora 13/1, 27–55 (1994a). (Spanish translation by R. Fernández and J. M. Sagüillo of a revised and expanded version)

    Google Scholar 

  17. 17.

    Corcoran, J.: The founding of logic. Anc. Philos. 14, 9–24 (1994f)

    Article  Google Scholar 

  18. 18.

    Corcoran, J.: Logical form. In: Audi, R. (ed.) Cambridge Dictionary of Philosophy, 2nd edn. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  19. 19.

    Corcoran, J.: Aristotle’s prior analytics and Boole’s laws of thought. Hist. Philos. Log. 24, 261–288 (2003a)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Corcoran, J.: Introduction. In: Prometheus Books, George Boole’s Laws of Thought. Reprint Buffalo (2003b)

  21. 21.

    Corcoran, J.: Schemata: the concept of schema in the history of logic. Bull. Symb. Log. 12, 219–40 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Corcoran, J.: Existential import. Bull. Symb. Log. 13, 143–144 (2007i)

    Google Scholar 

  23. 23.

    Corcoran, J.: Scientific revolutions. In: Lachs, J., Talisse, R. (eds.) Encyclopedia of American Philosophy. Routledge, New York (2008s)

    Google Scholar 

  24. 24.

    Corcoran, J.: Aristotle’s demonstrative logic. Hist. Philos. Log. 30, 1–20 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Corcoran, J.: Hidden consequence and hidden independence. Bull. Symb. Log. 16(2010), 443 (2010)

    Google Scholar 

  26. 26.

    Corcoran, J.: The Aristotle Łukasiewicz omitted. Bull. Symb. Log. 21, 237–238 (2015)

    Article  Google Scholar 

  27. 27.

    Corcoran, J., Hamid, I.S.: Investigating knowledge and opinion. In: Buchsbaum, A., Koslow, A. (eds.) The Road to Universal Logic, vol. I, pp. 95–126. Springer, Berlin (2015)

    Google Scholar 

  28. 28.

    Corcoran, J., Tracy, K.: Review of Joray 2017. Math. Rev. MR3681098 (2018)

  29. 29.

    Davenport, H.: Higher Arithmetic. Harper, New York (1952/1960)

  30. 30.

    Encyclopedia Britannica: Encyclopedia Britannica, Inc.: Chicago (1980)

  31. 31.

    Euclid: Elements. 3 vols. Heath, T (tr). New York: Dover (1956)

  32. 32.

    Feferman, A., Feferman, S.: Alfred Tarski: Life and Logic. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  33. 33.

    Flannery, K.: Ways into the Logic of Alexander of Aphrodisias. Brill, Leiden (1995)

    Google Scholar 

  34. 34.

    Galen: Institutio Logica. Kieffer, J. (ed) Johns Hopkins UP: Baltimore (1964)

  35. 35.

    Gasser, J.: Essai sur la nature et les critères de la preuve. Editions DelVal, Cousset (Switzerland) (1989)

  36. 36.

    Gasser, J.: Aristotle’s logic for the modern reader. HPL 12, 235–240 (1991)

    MathSciNet  Google Scholar 

  37. 37.

    Heath, T.: Tr. Euclid’s Elements, vol. 3. Dover, New York (1908/1925/1956)

  38. 38.

    Hintikka, J.: Aristotelian induction. Rev. Int. Philos. 34(1980), 422–439 (1980)

    MathSciNet  Google Scholar 

  39. 39.

    Hughes, R.: Philosophical Companion to First-Order Logic. Hackett, Indianapolis (1993)

    Google Scholar 

  40. 40.

    Jeffrey, R.: Formal Logic. McGraw-Hill, New York (1967/1991)

  41. 41.

    Joray, Pierre: A completed system for Robin Smith’s incomplete ecthetic syllogistic. Notre Dame J. Form. Log. 58(3), 329–342 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  42. 42.

    Kieffer, J.: Tr. Galen’s Institutio Logica. Johns Hopkins University Press, Baltimore

  43. 43.

    Kuhn, T.: The Copernican Revolution. Harvard University Press, Cambridge (1957)

    Google Scholar 

  44. 44.

    Kuhn, T.: The Structure of Scientific Revolutions. University of Chicago Press, Chicago (1962)

    Google Scholar 

  45. 45.

    Lejewski, C.: History of logic. In: Encyclopedia Britannica, vol. 11. Encyclopedia Britannica Inc., Chicago (1980)

  46. 46.

    Łukasiewicz, J.: Aristotle’s Syllogistic. Oxford University Press, Oxford (1951/1957)

  47. 47.

    McKeon, R.: Introduction to Aristotle. Modern Library, New York (1947)

    Google Scholar 

  48. 48.

    Mueller, I.: An introduction to Stoic logic. In: Rist, J.M. (ed.) The Stoics, pp. 1–26. University of California Press, Berkeley (1978)

    Google Scholar 

  49. 49.

    Newman, J. (ed.): The World of Mathematics, vol. 4. Simon and Schuster, New York (1956)

  50. 50.

    Peirce, C.S.: Boole’s calculus of logic. Peirce 1982 (1865/1982)

  51. 51.

    Peirce, C.S.: Writings of Charles S. Peirce: A Chronological Edition, Vol. I. Indiana University Press, Bloomington (1982)

  52. 52.

    Peirce, C.S.: In: Houser N., Kloesel C. (eds) The Essential Peirce: Selected Philosophical Writings (1867–1893), Vol. I. Indiana University Press, Bloomington (1992)

  53. 53.

    Peirce, C.S.: In: Houser N. et al. (eds) The Essential Peirce: Selected Philosophical Writings (1893–1913). Vol. II. Indiana University Press, Bloomington (1998)

  54. 54.

    Ross, W.D.: Aristotle. Meridian Books, New York (1923/1959)

  55. 55.

    Ross, W.D.: Aristotle’s Prior and Posterior Analytics. Oxford University Press, Oxford (1949)

    Google Scholar 

  56. 56.

    Sgarbi, M., Cosci, M.: The aftermath of syllogism. In: Aristotelian Argument from Avicenna to Hegel. Bloomsbury, London (2018)

  57. 57.

    Smiley, T.: What is a syllogism? JPL 2, 136–154 (1973)

    Article  MATH  Google Scholar 

  58. 58.

    Smiley, T.: Aristotle’s completeness proof. Anc. Philos. 14, 24–38 (1994)

    Google Scholar 

  59. 59.

    Smith, R.: Introduction. In: Aristotle’s Prior Analytics. Hackett, Indianapolis (1989)

  60. 60.

    Tarski, A.: Introduction to Logic and to the Methodology of Deductive Sciences. Trans. O. Helmer. Dover, New York (1941/1946/1995)

  61. 61.

    Tarski, A.: Truth and proof. Sci. Am. June 1969. Reprinted in Hughes 1993 (1969/1993)

  62. 62.

    von Plato, J.: The Great Formal Machinery Works: Theories of Deduction and Computation at the Origins of the Digital Age. Princeton University Press, Princeton (2017)

    Google Scholar 

  63. 63.

    Whately, R.: Elements of Logic. James Monroe and Co, Boston and Cambridge (1855)

    Google Scholar 

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Correspondence to John Corcoran.

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Dedicated to William Corcoran, Ph.D., PE. With love, gratitude, and admiration.

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Corcoran, J. Aristotle’s Prototype Rule-Based Underlying Logic. Log. Univers. 12, 9–35 (2018). https://doi.org/10.1007/s11787-018-0189-4

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Mathematics Subject Classification

  • Primary 01A20
  • Secondary 00A25

Keywords

  • Prototype
  • underlying logic
  • demonstration
  • deduction
  • direct
  • indirect
  • syllogistic