On Operator N and Wittgenstein’s Logical Philosophy


  • James R Connelly Trent University Peterborough and Oshawa, ON, Canada




In this paper, I provide a new reading of Wittgenstein’s N operator, and of its significance within his early logical philosophy. I thereby aim to resolve a longstanding scholarly controversy concerning the expressive completeness of N. Within the debate between Fogelin and Geach in particular, an apparent dilemma emerged to the effect that we must either concede Fogelin’s claim that N is expressively incomplete, or reject certain fundamental tenets within Wittgenstein’s logical philosophy. Despite their various points of disagreement, however, Fogelin and Geach nevertheless share several common and problematic assumptions regarding Wittgenstein’s logical philosophy, and it is these mistaken assumptions which are the source of the dilemma. Once we recognize and correct these, and other, associated expository errors, it will become clear how to reconcile the expressive completeness of Wittgenstein’s N operator, with several commonly recognized features of, and fundamental theses within, the Tractarian logical system.

Author Biography

James R Connelly, Trent University Peterborough and Oshawa, ON, Canada

Assistant Professor
Department of Philosophy


Anscombe, G. E. M., 1959. An Introduction to Wittgenstein’s Tractatus. London: Hutchinson University Library.

Fogelin, R. J., 1982. “Wittgenstein’s Operator N.” Analysis 42: 124–27.

———, 1987. Wittgenstein, 2nd ed. London: Routledge and Kegan Paul.

Frascolla, Pasquale, 1994. Wittgenstein’s Philosophy of Mathematics. London and New York: Routledge.

Geach, P. T., 1981. “Wittgenstein’s Operator N.” Analysis 41: 168–70.

———, 1982. “More on Wittgenstein’s Operator N.” Analysis 42: 127–28.

Hintikka, Jaakko, 1956. “Identity, Variables, and Impredicative Definitions.” Journal of Symbolic Logic 21: 225–45.

Jacquette, Dale, 2001. “Analysis of Quantifiers in Wittgenstein’s Tractatus: A Critical Survey.” Logical Analysis and History of Philosophy 4: 191–202.

King, John and Desmond Lee, eds., 1980. Wittgenstein’s Lectures, Cambridge 1930–32. Oxford: Blackwell Publishers.

Landini, Gregory, 2007. Wittgenstein’s Apprenticeship with Russell. New York: Cambridge University Press.

Marion, Mathieu, 1998. Wittgenstein, Finitism, and the Foundations of Mathematics. New York: Oxford University Press.

McGray, James W., 2006. “The Power and the Limits of Wittgenstein’s N Operator.” History and Philosophy of Logic 27: 143–69.

Methven, S. J., 2015. Frank Ramsey and the Realistic Spirit. Basingstoke: Palgrave Macmillan.

Moore, G. E., 1955. “Wittgensten’s Lectures in 1930–33.” Mind 64: 1–27.

Morris, Michael, 2008. Routledge Philosophy Guidebook to Wittgenstein and the Tractatus. London and New York: Routledge.

Proops, Ian, 2001. “The New Wittgenstein: A Critique.” European Journal of Philosophy 9: 375–404.

Ramsey, Frank, 1931. The Foundations of Mathematics and Other Logical Essays. London: Routledge and Kegan Paul.

Rogers, Brian and Kai F. Wehmeier, 2012. “Tractarian First-Order Logic: Identity and the N-Operator.” The Review of Symbolic Logic 5: 538–73.

Russell, Bertrand, 1919. Introduction to Mathematical Philosophy. London: Allen & Unwin.

Soames, Scott, 1983. “Generality, Truth-Functions, and Expressive Capacity in the Tractatus.” The Philosophical Review 92: 573–89.

Stern, David, Brian Rogers and Gabriel Citron, eds., 2016. Wittgenstein: Lectures, Cambridge 1930–1933: From the Notes of G. E. Moore. Cambridge: Cambridge University Press.

Wittgenstein, Ludwig, 1922. Tractatus Logico-Philosophicus, translated by C. K. Ogden. London: Routledge & Kegan Paul. (TLP 1922)

———, 1953/2001. Philosophical Investigations, 3rd ed., translated by G. E. M. Anscombe. Malden: MA: Blackwell Publishing.

———, 1961. Tractatus Logico-Philosophicus, translated by D. Pears and B. F. McGuinness. Malden: MA: Blackwell Publishing. (TLP 1961)

———, 2005. The Big Typescript: TS 213, German–English Scholar’s Edition, edited and translated by C. Grant Luckhardt and Maximilian A. E. Aue.. Malden, MA: Blackwell Publishing.