# Early Russell on Types and Plurals

### Abstract

In 1903, in The Principles of Mathematics (PoM), Russell endorsed an account of classes whereupon a class fundamentally is to be considered many things, and not one, and used this thesis to explicate his first version of a theory of types, adding that it formed the logical justification for the grammatical distinction between singular and plural. The view, however, was shortlived— rejected before PoM even appeared in print. However, aside from mentions of a few misgivings, there is little evidence about why he abandoned this view. In this paper, I attempt to clarify Russell’s early views about plurality, arguing that they did not involve countenancing special kinds of plural things distinct from individuals. I also clarify what his misgivings about these views were, making it clear that while the plural understanding of classes helped solve certain forms of Russell’s paradox, certain other Cantorian paradoxes remained. Finally, I aim to show that Russell’s abandonment of something like plural logic is understandable given his own conception of logic and philosophical aims when compared to the views and approaches taken by contemporary advocates of plural logic.### References

Boolos, George. 1984. To Be is to Be the Value of a Variable

(or to Be Some Values of Some Variables). In Logic, Logic

and Logic, edited by Richard Jeffrey, pages 54–72. Cambridge,

Mass.: Harvard University Press.

Frege, Gottlob. 1980. Philosophical and Mathematical Correspondence. Chicago: University of Chicago Press. Edited by HansKaal.

Frege, Gottlob. forthcoming. Basic Laws of Arithmetic (2 vols). Oxford: Oxford University Press. Translated by Philip Ebert

and Marcus Rossberg; originally published in 1893–1902 as

Grundgesetze der Arithmetik (Jena: H. Pohle).

Grattan-Guinness, Ivor (ed.) . 1977. Dear Russell–Dear Jourdain. New York: Columbia University Press.

Hazen, Allen P. 1997. Relations in Lewis’s Framework Without

Atoms. Analysis 57: 243–248.

Klement, Kevin C. 2003. Russell’s 1903-05 Anticipation of the

Lambda Calculus. History and Philosophy of Logic 24: 15–37.

Klement, Kevin C. 2004. Putting Form Before Function: Logical Grammar in Frege, Russell and Wittgenstein. Philosopher’s Imprint 4: 1–47.

Klement, Kevin C. 2005. The Origins of the Propositional Functions Version of Russell’s Paradox. Russell 24: 101–32.

Klement, Kevin C. 2010a. The Functions of Russell’s No Class Theory. Review of Symbolic Logic 3: 633–664.

Klement, Kevin C. 2010b. Russell, His Paradoxes, and Cantor’s Theorem: Part I. Philosophy Compass 5: 16–28.

Klement, Kevin C. 2013. PM’s Circumflex, Syntax and Philosophy of Types. In The Palgrave Centenary Companion to Principia Mathematica, edited by Nicholas Griffin and Bernard Linsky. New York: Palgrave Macmillan, 218Ð-47.

Landini, Gregory. 1992. Russell to Frege 24 May 1903: ‘I believe I have Discovered that Classes are Entirely Superfluous’. Russell 12: 160–185.

Landini, Gregory. 1998. Russell’s Hidden Substitutional Theory. Oxford: Oxford University Press.

Lewis, David. 1991. Parts of Classes. Oxford: Basil Blackwell.

Linsky, Bernard. 2004. Russell’s Marginalia in His Copies of

Frege’s Works. Russell n.s. 24: 5–36.

Linsky, Bernard. 2005. Russell’s Notes on Frege for Appendix A of The Principles of Mathematics. Russell n.s. 24: 133–72.

McKay, Thomas. 2006. Plural Predication. Oxford: Oxford University Press.

Oliver, Alex and Smiley, Timothy. 2005. Plural Descriptions and Many-valued Functions. Mind 114: 1039–68.

Proops, Ian. 2007. Russell and the Universalist Conception of

Logic. Noûs 41: 1–32.

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

Russell, Bertrand. 1923. Vagueness. In The Collected Papers of Bertrand Russell, edited by John G. Slater, volume 9, pages 147–154. London: Unwin Hyman.

Russell, Bertrand. 1931. The Principles of Mathematics. Cambridge: Cambridge University Press. (First edition 1903).

Russell, Bertrand. 1956a. Our Knowledge of the External World. New York: Mentor Books, 2nd edition. (First edition 1914).

Russell, Bertrand. 1956b. The Philosophy of Logical Atomism. In Logic and Knowledge, pages 175–281. London: Allen and Unwin. (First published 1918).

Russell, Bertrand. 1973a. Essays in Analysis. New York: George Braziller. Edited by Douglas Lackey.

Russell, Bertrand. 1973b. On ‘Insolubilia’ and their Solution by Symbolic Logic. In Russell 1973a, pages 190–214. (First published 1906 in French as “Les paradoxes de la logique”).

Russell, Bertrand. 1973c. The Substitutional Theory of Classes and Relations. In ?, pages 165–89. (Written 1906).

Russell, Bertrand. 1994a. Meinong’s Theory of Complexes and Assumptions. In Urquart 1994, pages 431–74. (First published 1904).

Russell, Bertrand. 1994b. The Nature of Truth. In Urquart 1994, pages 492–93. (Written 1905).

Spencer, Joshua. 2012. All Things Must Pass Away. Oxford

Studies in Metaphysics 7: 67–92.

Stevens, Graham. 2005. The Russellian Origins of Analytical Philosophy.London: Routledge.

Urquhart, Alasdair (ed.) . 1994. The Collected Papers of Bertrand Russell, volume 4. London: Routledge.

Whitehead, A. N. and Russell, Bertrand. 1925–1927. Principia

Mathematica, 3 vols. Cambridge: Cambridge University Press.

(First edition 1910–1913).

Yi, Byeong-uk. forthcoming. The Logic of Classes of the No-

Class Theory. In The Palgrave Centenary Companion to Principia

Mathematica, edited by Nicholas Griffin and Bernard Linsky.

New York: Palgrave Macmillan: 96–129.

(or to Be Some Values of Some Variables). In Logic, Logic

and Logic, edited by Richard Jeffrey, pages 54–72. Cambridge,

Mass.: Harvard University Press.

Frege, Gottlob. 1980. Philosophical and Mathematical Correspondence. Chicago: University of Chicago Press. Edited by HansKaal.

Frege, Gottlob. forthcoming. Basic Laws of Arithmetic (2 vols). Oxford: Oxford University Press. Translated by Philip Ebert

and Marcus Rossberg; originally published in 1893–1902 as

Grundgesetze der Arithmetik (Jena: H. Pohle).

Grattan-Guinness, Ivor (ed.) . 1977. Dear Russell–Dear Jourdain. New York: Columbia University Press.

Hazen, Allen P. 1997. Relations in Lewis’s Framework Without

Atoms. Analysis 57: 243–248.

Klement, Kevin C. 2003. Russell’s 1903-05 Anticipation of the

Lambda Calculus. History and Philosophy of Logic 24: 15–37.

Klement, Kevin C. 2004. Putting Form Before Function: Logical Grammar in Frege, Russell and Wittgenstein. Philosopher’s Imprint 4: 1–47.

Klement, Kevin C. 2005. The Origins of the Propositional Functions Version of Russell’s Paradox. Russell 24: 101–32.

Klement, Kevin C. 2010a. The Functions of Russell’s No Class Theory. Review of Symbolic Logic 3: 633–664.

Klement, Kevin C. 2010b. Russell, His Paradoxes, and Cantor’s Theorem: Part I. Philosophy Compass 5: 16–28.

Klement, Kevin C. 2013. PM’s Circumflex, Syntax and Philosophy of Types. In The Palgrave Centenary Companion to Principia Mathematica, edited by Nicholas Griffin and Bernard Linsky. New York: Palgrave Macmillan, 218Ð-47.

Landini, Gregory. 1992. Russell to Frege 24 May 1903: ‘I believe I have Discovered that Classes are Entirely Superfluous’. Russell 12: 160–185.

Landini, Gregory. 1998. Russell’s Hidden Substitutional Theory. Oxford: Oxford University Press.

Lewis, David. 1991. Parts of Classes. Oxford: Basil Blackwell.

Linsky, Bernard. 2004. Russell’s Marginalia in His Copies of

Frege’s Works. Russell n.s. 24: 5–36.

Linsky, Bernard. 2005. Russell’s Notes on Frege for Appendix A of The Principles of Mathematics. Russell n.s. 24: 133–72.

McKay, Thomas. 2006. Plural Predication. Oxford: Oxford University Press.

Oliver, Alex and Smiley, Timothy. 2005. Plural Descriptions and Many-valued Functions. Mind 114: 1039–68.

Proops, Ian. 2007. Russell and the Universalist Conception of

Logic. Noûs 41: 1–32.

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

Russell, Bertrand. 1923. Vagueness. In The Collected Papers of Bertrand Russell, edited by John G. Slater, volume 9, pages 147–154. London: Unwin Hyman.

Russell, Bertrand. 1931. The Principles of Mathematics. Cambridge: Cambridge University Press. (First edition 1903).

Russell, Bertrand. 1956a. Our Knowledge of the External World. New York: Mentor Books, 2nd edition. (First edition 1914).

Russell, Bertrand. 1956b. The Philosophy of Logical Atomism. In Logic and Knowledge, pages 175–281. London: Allen and Unwin. (First published 1918).

Russell, Bertrand. 1973a. Essays in Analysis. New York: George Braziller. Edited by Douglas Lackey.

Russell, Bertrand. 1973b. On ‘Insolubilia’ and their Solution by Symbolic Logic. In Russell 1973a, pages 190–214. (First published 1906 in French as “Les paradoxes de la logique”).

Russell, Bertrand. 1973c. The Substitutional Theory of Classes and Relations. In ?, pages 165–89. (Written 1906).

Russell, Bertrand. 1994a. Meinong’s Theory of Complexes and Assumptions. In Urquart 1994, pages 431–74. (First published 1904).

Russell, Bertrand. 1994b. The Nature of Truth. In Urquart 1994, pages 492–93. (Written 1905).

Spencer, Joshua. 2012. All Things Must Pass Away. Oxford

Studies in Metaphysics 7: 67–92.

Stevens, Graham. 2005. The Russellian Origins of Analytical Philosophy.London: Routledge.

Urquhart, Alasdair (ed.) . 1994. The Collected Papers of Bertrand Russell, volume 4. London: Routledge.

Whitehead, A. N. and Russell, Bertrand. 1925–1927. Principia

Mathematica, 3 vols. Cambridge: Cambridge University Press.

(First edition 1910–1913).

Yi, Byeong-uk. forthcoming. The Logic of Classes of the No-

Class Theory. In The Palgrave Centenary Companion to Principia

Mathematica, edited by Nicholas Griffin and Bernard Linsky.

New York: Palgrave Macmillan: 96–129.

Published

2014-04-03

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