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TRACTARIAN FIRST-ORDER LOGIC: IDENTITY AND THE N-OPERATOR

Published online by Cambridge University Press:  02 April 2012

BRIAN ROGERS  and
KAI F. WEHMEIER
BRIAN ROGERS*
Affiliation:
Logic & Philosophy of Science, UC Irvine
KAI F. WEHMEIER*
Affiliation:
Logic & Philosophy of Science, UC Irvine
*
*LOGIC & PHILOSOPHY OF SCIENCE, UC IRVINE, IRVINE, CA 92697 E-mail: brogers@uci.edu (Brian Rogers)
†LOGIC & PHILOSOPHY OF SCIENCE, UC IRVINE, IRVINE, CA 92697 E-mail: wehmeier@uci.edu (Kai Wehmeier)
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Abstract

In the Tractatus, Wittgenstein advocates two major notational innovations in logic. First, identity is to be expressed by identity of the sign only, not by a sign for identity. Secondly, only one logical operator, called “N” by Wittgenstein, should be employed in the construction of compound formulas. We show that, despite claims to the contrary in the literature, both of these proposals can be realized, severally and jointly, in expressively complete systems of first-order logic. Building on early work of Hintikka’s, we identify three ways in which the first notational convention can be implemented, show that two of these are compatible with the text of the Tractatus, and argue on systematic and historical grounds, adducing posthumous work of Ramsey’s, for one of these as Wittgenstein’s envisaged method. With respect to the second Tractarian proposal, we discuss how Wittgenstein distinguished between general and non-general propositions and argue that, claims to the contrary notwithstanding, an expressively adequate N-operator notation is implicit in the Tractatus when taken in its intellectual environment. We finally introduce a variety of sound and complete tableau calculi for first-order logics formulated in a Wittgensteinian notation. The first of these is based on the contemporary notion of logical truth as truth in all structures. The others take into account the Tractarian notion of logical truth as truth in all structures over one fixed universe of objects. Here the appropriate tableau rules depend on whether this universe is infinite or finite in size, and in the latter case on its exact finite cardinality.

As it is obviously easy to express how propositions can be constructed by means of this operation and how propositions are not to be constructed by means of it, this must be capable of exact expression.

5.503

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Type
Research Articles
Information
The Review of Symbolic Logic , Volume 5 , Issue 4 , December 2012 , pp. 538 - 573
Copyright
Copyright © Association for Symbolic Logic 2012

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References

BIBLIOGRAPHY

Anscombe, G. E. M. (1959). An Introduction to Wittgenstein’s Tractatus. Philadelphia, PA: University of Pennsylvania Press.Google Scholar
Black, M. (1964). A Companion to Wittgenstein’s ‘Tractatus’. Ithaca, NY: Cornell University Press.Google Scholar
Bogen, J. (1981). Recent Wittgensteiniana. Teaching Philosophy, 4(1), 67–74.CrossRefGoogle Scholar
Boolos, G. S., Burgess, J. P., & Jeffrey, R. C. (2007). Computability and Logic (fifth edition). New York, NY: Cambridge University Press.CrossRefGoogle Scholar
Carnap, R. (1937). The Logical Syntax of Language. Smeaton, A., translator. London: Routledge and Kegan Paul.Google Scholar
Cheung, L. (2000). The Tractarian operation N and expressive completeness. Synthese, 123(2), 241–255.CrossRefGoogle Scholar
Floyd, J. (2002). Number and ascriptions of number in the Tractatus. In Reck, E., editor. From Frege to Wittgenstein: Perspectives on Early Analytic Philosophy. New York, NY: Oxford University Press, pp. 308–352.CrossRefGoogle Scholar
Floyd, J. (2007). Wittgenstein and the inexpressible. In Crary, A., editor. Wittgenstein and the Moral Life: Essays in Honor of Cora Diamond. Cambridge, MA: MIT Press, pp. 177–234.CrossRefGoogle Scholar
Fogelin, R. (1976). Wittgenstein. London: Routledge.Google Scholar
Fogelin, R. (1982). Wittgenstein’s operator N. Analysis, 42(3), 124–127.CrossRefGoogle Scholar
Fogelin, R. (1983). Wittgenstein on identity. Synthese, 56(2), 141–154.Google Scholar
Fogelin, R. (1987). Wittgenstein (second edition). London: Routledge.Google Scholar
Frascolla, P. (2007). Understanding Wittgenstein’s Tractatus. London: Routledge.Google Scholar
Frege, G. (1969). Logik. In Hermes, H., Kambartel, F., and Kaulbach, F., editors. Gottlob Frege—Nachgelassene Schriften. Hamburg, Germany: Felix Meiner, pp. 137–163.Google Scholar
Geach, P. (1981). Wittgenstein’s operator N. Analysis, 41(4), 168–171.Google Scholar
Geach, P. (1982). More on Wittgenstein’s operator N. Analysis, 42(3), 127–128.CrossRefGoogle Scholar
Glock, H. (1996). A Wittgenstein Dictionary. Oxford, UK: Blackwell.CrossRefGoogle Scholar
Hintikka, J. (1956). Identity, variables, and impredicative definitions. Journal of Symbolic Logic, 21, 225–245.Google Scholar
Jacquette, D. (2001). Analysis of quantifiers in Wittgenstein’s Tractatus: A critical survey. Logical Analysis and History of Philosophy, 4, 191–202.CrossRefGoogle Scholar
Kremer, M. (2007). The cardinal problem of philosophy. In Crary, A., editor. Wittgenstein and the Moral Life: Essays in Honor of Cora Diamond. Cambridge, MA: MIT Press, pp. 143–176.Google Scholar
Landini, G. (2007). Wittgenstein’s Apprenticeship with Russell. New York, NY: Cambridge University Press.Google Scholar
McGray, J. (2006). The power and the limits of Wittgenstein’s N operator. History and Philosophy of Logic, 27(2), 143–169.CrossRefGoogle Scholar
McGuinness, B., editor. (2008). Wittgenstein in Cambridge: Letters and Documents 1911–1951. Oxford, UK: Blackwell.CrossRefGoogle Scholar
Miller, H. (1995). Tractarian semantics for predicate logic. History and Philosophy of Logic, 16(2), 197–215.Google Scholar
Morris, M. (2008). Wittgenstein and the Tractatus. London: Routledge.Google Scholar
Ramsey, F. P. (1923). Critical notice of the Tractatus. Mind, 32, 465–478.Google Scholar
Ramsey, F. P. (1991). Identity. In Galavotti, M., editor. Notes on Philosophy, Probability and Mathematics. Naples, Italy: Bibliopolis, pp. 155–169.Google Scholar
Schroeder, S. (2006). Wittgenstein. Cambridge, UK: Polity.Google Scholar
Smullyan, R. M. (1968). First-Order Logic. New York, NY: Springer-Verlag.Google Scholar
Soames, S. (1983). Generality, truth functions, and expressive capacity in the Tractatus. The Philosophical Review, 92(3), 573–589.Google Scholar
Sundholm, G. (1992). The general form of the operation in Wittgenstein’s Tractatus. Grazer Philosophische Studien, 42, 57–76.Google Scholar
Varga von Kibéd, M. (1993). Variablen im Tractatus. Erkenntnis, 39, 79–100.CrossRefGoogle Scholar
Wehmeier, K. F. (2004). Wittgensteinian predicate logic. Notre Dame Journal of Formal Logic, 45, 1–11.Google Scholar
Wehmeier, K. F. (2008). Wittgensteinian tableaux, identity, and co-denotation. Erkenntnis, 69, 363–376.Google Scholar
Wehmeier, K. F. (2009). On Ramsey’s ‘Silly Delusion’ regarding Tractatus 5.53. In Primiero, G., and Rahman, S., editors. Acts of Knowledge: History, Philosophy and Logic. London: College Publications, pp. 353–368.Google Scholar
White, R. (2006). Wittgenstein’s Tractatus Logico-Philosophicus. London: Continuum.Google Scholar
Whitehead, A. N., & Russell, B. (1910). Principia Mathematica, Vol. 1. Cambridge, UK: Cambridge University Press.Google Scholar
Wittgenstein, L. (1922). Tractatus Logico-Philosophicus, translator Ogden, C. K.. London: Routledge and Kegan Paul.Google Scholar
Wittgenstein, L. (1961a). Notebooks 1914–1916. von Wright, G. H., and Anscombe, G. E. M., editors. Oxford, UK: Blackwell.Google Scholar
Wittgenstein, L. (1961b). Tractatus Logico-Philosophicus, translators Pears, D. F. and McGuinness, B. F.. London: Routledge.Google Scholar
Wittgenstein, L. (1971). Prototractatus. McGuinness, B. F., Nyberg, T., and von Wright, G. H., editors. Ithaca, NY: Cornell University Press.Google Scholar