Abstract
We have now examined the philosophical framework surrounding De Morgan’s views on relations, and we have also seen how these views show the need for a logic of relations. In this chapter, we will discuss De Morgan’s central contribution to the logic of relations, which he published in 1860 under the title, “On the Syllogism: IV and on the Logic of Relations.” In this classic memoir, De Morgan moves beyond his relational analysis of the syllogism and the bicopular syllogism to something that may justifiably be called a logic of relations: that is, the specification and systematization of previously unrecognized valid forms of relational inference.
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Notes
S4, 220–27. For discussions of this topic see: I.M. Bochenski, A History of Formal Logic, tr. Ivo Thomas (Notre Dame, Indiana: University of Notre Dame Press, 1961), 375–77;
Jörgen Jorgensen, A Treatise of Formal Logic Vol. I (Copenhagen: Levin & Munksgaard, 1931), I, 92–97;
William Kneale and Martha Kneale, The Development of Logic (Oxford: Clarendon Press, 1962), 427–28;
C.I. Lewis, A Survey of Symbolic Logic (1918; reprint, New York: Dover Publications, 1960), 43–51;
A.N. Prior, Formal Logic (Oxford: Clarendon Press, 1955), 152–54;
Boruch A. Brody, “The Rise of the Algebra of Logic” (diss., Princeton University, 1967), 184–202;
Carl Gottschall, “An Examination and Evaluation of the Historical Importance of the Principal Innovative Contributions to Formal Logic of Augustus De Morgan” (diss., New York University, 1980), 149–70;
Benjamin S. Hawkins, “A Reassessment of Augustus De Morgan’s Logic of Relations: A Documentary Reconstruction,” International Logic Review 10 (1979): 32–61;
R.M. Martin, “De Morgan and the Logic of Relations,” in Peirce’s Logic of Relations and Other Studies (Lisse: Peter de Ridder Press, 1979), 46–53.
As stated by De Morgan, this may carry the presupposition that there is at least one L of Y This is not intended. A better version might be “X is not an L of Y.”
See “On the Foundation of Algebra, No. IV, on Triple Algebra,” Transactions of the Cambridge Philosophical Society 8 (1842–47): 241.
Alfred Tarski, “On the Calculus of Relations,” Journal of Symbolic Logic 6 (1941): 73–89;
for the more recent approach see Alfred Tarski and Steven Givant, A Formalization of Set Theory Without Variables (Providence: American Mathematical Society, 1987), esp. 45–51. Aristotle also uses individual cases to establish the basic general laws of his logic.
De Morgan proposed this as a puzzle in Notes and Queries (14 January 1860): 25, and reported on its solution in Notes and Queries (10 March 1860): 184–85. He also posed the problem to Sir William Rowan Hamilton. See De Morgan to Hamilton, 22 August 1859; Hamilton to De Morgan, 26 August 1859 (misdated 1850); and De Morgan to Hamilton, 7 September 1859 (Sir William Rowan Hamilton Correspondence, Library of Trinity College, Dublin). He also sent the problem to Sir John Herschel (Herschel Correspondence, Royal Society of London). This letter is undated, but De Morgan’s address on it means that it was not written before early summer of 1859.
I owe this point to John Corcoran (private communication), who suggests that De Morgan’s conjecture might be modified to read, that every relation which is convertible and weakly reflexive satisfies condition D. I do not know whether this conjecture is true or false.
Bertrand Russell claims that De Morgan was the first person to use “transitive” in this sense. See Principles of Mathematics, 2nd ed. (New York: W.W. Norton & Co., 1938), 218n.
“Logic Notebook,” Ms. 399, 50r; in the C.S. Peirce Papers, Harvard University.
“Description of a Notation for the Logic of Relatives.”
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© 1990 Kluwer Academic Publishers
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Merrill, D.D. (1990). The Logic of Relations. In: Augustus De Morgan and the Logic of Relations. The New Synthese Historical Library, vol 38. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2047-7_5
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