On the Curious Calculi of Wittgenstein and Spencer Brown
DOI:
https://doi.org/10.15173/jhap.v6i10.3400Abstract
In his Tractatus, Wittgenstein sets out what he calls his N-operator notation which can be used to calculate whether an expression is a tautology. In his Laws of Form, George Spencer Brown offers what he calls a “primary algebra” for such calculation. Both systems are perplexing. But comparing two blurry images can reduce noise, producing a focus. This paper reveals that Spencer Brown independently rediscovered the quantifier-free part of the N-operator calculus. The comparison sheds a flood light on each and from the letters of correspondence we shall find that Russell, as one might have surmised, was a catalyst for both.
References
Anscombe, G. E. M., 1959. An Introduction to Wittgenstein’s Tractatus. London: Hutchinson University Library.
Banaschewski, Bernhard, 1977. “On G. Spencer Brown’s Laws of Form.” Notre Dame Journal of Formal Logic 18: 507–09.
Engstrom, Jack, 1999. “G. Spencer-Brown’s Laws of Form as a Revolutionary, Unifying Notation.” Semiotica 125: 33–45.
Fogelin, Robert, 1976. Wittgenstein. New York: Routledge and Kegan Paul. (2nd ed., 1987.)
———, 1982. “Wittgenstein’s Operator N.” Analysis 42: 124–27.
Gardner, Martin, 1982. Logic Machines and Diagrams, 2nd ed. Chicago: University of Chicago Press.
Geach, P. T., 1981. “Wittgenstein’s Operator N.” Analysis 41: 168–71.
———, 1982. “More on Wittgenstein’s Operator N.” Analysis 42: 127–28.
Grattan-Guinness, Ivor, 2000. The Search for Mathematical Roots 1870–1940. Princeton: Princeton University Press.
Jadacki, Jacek J., 1986. “Leon Chwistek–Bertrand Russell’s Scientific Correspondence.” Dialectics and Humanism: The Polish Philosophical Quarterly 13: 239–63.
Landini, Gregory, 2007. Wittgenstein’s Apprenticeship with Russell. Cambridge: Cambridge University Press.
McGuinness, B. F. and G. H. von Wright, eds., 1990. “Unpublished Correspondence Between Russell and Wittgenstein.” Russell n. s. 10: 101–24.
Mequire, Philip, 2003. “Boundary Algebra: A Simple Notation for Boolean Algebra and the Truth Functors.” Journal of General Systems 32: 25–87.
Nicod, Jean, 1917. “A Reduction in the Number of the Primitive Propositions of Logic.” Proceedings of the Cambridge Philosophical Society 19: 32–41.
Post, Emil, 1921. “Introduction to a General Theory of Elementary Propositions.” American Journal of Mathematics 43: 163–85.
Quine, W. V., 2008. Quine in Dialogue, edited by D. Føllesdal. Cambridge, MA: Harvard University Press.
Rogers, Brian and Kai Wehmeier, 2012. “Tractarian First Order Logic: Identity and the N-Operator.” Review of Symbolic Logic 5: 538–73.
Russell, Bertrand, 1967–1969. Autobiography, 3 vols. London: Allen and Unwin.
Scharle, Thomas, 1965. “Axiomatization of Propositional Calculus with Sheffer Functors.” Notre Dame Journal of Formal Logic 6: 209–17.
Sheffer, H. M., 1913. “A Set of Five Independent Postulates for Boolean Alegebras, with Application to Logical Constants.” Transactions of the American Mathematical Society 14: 481–88.
Shoup, Richard, 2013. Laws of Form website. Previously available at http://www.lawsofform.org/; archived version available at https://web.archive.org/web/20150620012459/http://www.lawsofform.org/, accessed 17 October 2018.
Soames, Scott, 1983. “Generality, Truth Functions, and Expressive Capacity in the Tractatus.” The Philosophical Review 92: 573–89.
Spencer Brown, D. J. and J. C. Miller, 1966. “An Algorithm for Evaluation of Remote Terms in a Linear Recurrence Sequence.” The Computer Journal 9: 188–90.
Spencer Brown, George, 1971. Laws of Form. New York: Julian Press.
———, 2004. Autobiography, vol. 1, Infancy and Childhood. Leipzig: Bohmeier Verlag.
Tang, Fangfang, 2018. “From Wittgenstein’s N-operator to a New Notation for Some Decidable Modal Logics.” History and Philosophy of Logic, preblished online, https://doi.org/10.1080/01445340.2018.1437519.
The Telegraph, 2016. “George Spencer-Brown, Polymath Who Wrote the Landmark Maths Book Laws of Form.” The Telegraph online, https://www.telegraph.co.uk/obituaries /2016/09/13/george-spencer-brown-polymath-who-wrote- the-landmark-maths-book/, accessed 17 October 2018.
Varela, Francesco, 1979. Principles of Biological Autonomy. New York: North Holland.
Waismann, Friedrich, 1979. Wittgenstein and the Vienna Circle, edited by B. McGuinness, translated by J. Schulte and B. McGuinness. Oxford: Blackwell.
Wehmeier, Kai, 2004. “Wittgenstein’s Predicate Logic.” Notre Dame Journal of Formal Logic 45: 1–11.
Wittgenstein, Ludwig, 1922. Tractatus Logico-Philosophicus, translated by D. Pears and B. McGuinness. London: Routledge, 1961.
———, 1971. Prototractatus: An Early Version of the Tractatus Logico-Philosophicus, translated by B. McGuiness and G. H. von Wright. Ithaca: Cornell University Press.
———, 1974. Cambridge Letters, 2nd ed., edited by B. McGuinness and G. H. von Wright. Oxford: Blackwell.
———, 1979. Notebooks 1914–1916, edited by G. H. von Wright and G. E. M. Anscombe. Chicago: University of Chicago Press.
Żyliński, Eustachy, 1925. “Some Remarks Concerning the Theory of Deduction.” Fundamenta Mathematicae 7: 203–09.
Banaschewski, Bernhard, 1977. “On G. Spencer Brown’s Laws of Form.” Notre Dame Journal of Formal Logic 18: 507–09.
Engstrom, Jack, 1999. “G. Spencer-Brown’s Laws of Form as a Revolutionary, Unifying Notation.” Semiotica 125: 33–45.
Fogelin, Robert, 1976. Wittgenstein. New York: Routledge and Kegan Paul. (2nd ed., 1987.)
———, 1982. “Wittgenstein’s Operator N.” Analysis 42: 124–27.
Gardner, Martin, 1982. Logic Machines and Diagrams, 2nd ed. Chicago: University of Chicago Press.
Geach, P. T., 1981. “Wittgenstein’s Operator N.” Analysis 41: 168–71.
———, 1982. “More on Wittgenstein’s Operator N.” Analysis 42: 127–28.
Grattan-Guinness, Ivor, 2000. The Search for Mathematical Roots 1870–1940. Princeton: Princeton University Press.
Jadacki, Jacek J., 1986. “Leon Chwistek–Bertrand Russell’s Scientific Correspondence.” Dialectics and Humanism: The Polish Philosophical Quarterly 13: 239–63.
Landini, Gregory, 2007. Wittgenstein’s Apprenticeship with Russell. Cambridge: Cambridge University Press.
McGuinness, B. F. and G. H. von Wright, eds., 1990. “Unpublished Correspondence Between Russell and Wittgenstein.” Russell n. s. 10: 101–24.
Mequire, Philip, 2003. “Boundary Algebra: A Simple Notation for Boolean Algebra and the Truth Functors.” Journal of General Systems 32: 25–87.
Nicod, Jean, 1917. “A Reduction in the Number of the Primitive Propositions of Logic.” Proceedings of the Cambridge Philosophical Society 19: 32–41.
Post, Emil, 1921. “Introduction to a General Theory of Elementary Propositions.” American Journal of Mathematics 43: 163–85.
Quine, W. V., 2008. Quine in Dialogue, edited by D. Føllesdal. Cambridge, MA: Harvard University Press.
Rogers, Brian and Kai Wehmeier, 2012. “Tractarian First Order Logic: Identity and the N-Operator.” Review of Symbolic Logic 5: 538–73.
Russell, Bertrand, 1967–1969. Autobiography, 3 vols. London: Allen and Unwin.
Scharle, Thomas, 1965. “Axiomatization of Propositional Calculus with Sheffer Functors.” Notre Dame Journal of Formal Logic 6: 209–17.
Sheffer, H. M., 1913. “A Set of Five Independent Postulates for Boolean Alegebras, with Application to Logical Constants.” Transactions of the American Mathematical Society 14: 481–88.
Shoup, Richard, 2013. Laws of Form website. Previously available at http://www.lawsofform.org/; archived version available at https://web.archive.org/web/20150620012459/http://www.lawsofform.org/, accessed 17 October 2018.
Soames, Scott, 1983. “Generality, Truth Functions, and Expressive Capacity in the Tractatus.” The Philosophical Review 92: 573–89.
Spencer Brown, D. J. and J. C. Miller, 1966. “An Algorithm for Evaluation of Remote Terms in a Linear Recurrence Sequence.” The Computer Journal 9: 188–90.
Spencer Brown, George, 1971. Laws of Form. New York: Julian Press.
———, 2004. Autobiography, vol. 1, Infancy and Childhood. Leipzig: Bohmeier Verlag.
Tang, Fangfang, 2018. “From Wittgenstein’s N-operator to a New Notation for Some Decidable Modal Logics.” History and Philosophy of Logic, preblished online, https://doi.org/10.1080/01445340.2018.1437519.
The Telegraph, 2016. “George Spencer-Brown, Polymath Who Wrote the Landmark Maths Book Laws of Form.” The Telegraph online, https://www.telegraph.co.uk/obituaries /2016/09/13/george-spencer-brown-polymath-who-wrote- the-landmark-maths-book/, accessed 17 October 2018.
Varela, Francesco, 1979. Principles of Biological Autonomy. New York: North Holland.
Waismann, Friedrich, 1979. Wittgenstein and the Vienna Circle, edited by B. McGuinness, translated by J. Schulte and B. McGuinness. Oxford: Blackwell.
Wehmeier, Kai, 2004. “Wittgenstein’s Predicate Logic.” Notre Dame Journal of Formal Logic 45: 1–11.
Wittgenstein, Ludwig, 1922. Tractatus Logico-Philosophicus, translated by D. Pears and B. McGuinness. London: Routledge, 1961.
———, 1971. Prototractatus: An Early Version of the Tractatus Logico-Philosophicus, translated by B. McGuiness and G. H. von Wright. Ithaca: Cornell University Press.
———, 1974. Cambridge Letters, 2nd ed., edited by B. McGuinness and G. H. von Wright. Oxford: Blackwell.
———, 1979. Notebooks 1914–1916, edited by G. H. von Wright and G. E. M. Anscombe. Chicago: University of Chicago Press.
Żyliński, Eustachy, 1925. “Some Remarks Concerning the Theory of Deduction.” Fundamenta Mathematicae 7: 203–09.
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