The Logical Significance of Assertion: Frege on the Essence of Logic
DOI:
https://doi.org/10.15173/jhap.v5i8.3073Abstract
Assertion plays a crucial dual role in Frege's conception of logic, a formal and a transcendental one. A recurrent complaint is that Frege's inclusion of the judgement-stroke (the formal counterpart of assertion) in the Begriffsschrift is either in tension with his anti-psychologism or wholly superfluous. Assertion, the objection goes, is at best of merely psychological significance. In this paper, I defend Frege against the objection by giving reasons for recognising the central logical significance of assertion in both its formal and its transcendental role.
References
The following abbreviations are used to refer to Frege’s works. Page numbering is to the German editions first:
BS: Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle: L. Nebert, 1879. Translated as Conceptual Notation and Related Articles, edited by T. W. Bynum. Oxford: Clarendon Press, 1972.
GL: Die Grundlagen der Arithmetik. Breslau: W. Koebner, 1884. Translated as The Foundations of Arithmetic, translated by J. L. Austin. Oxford: Blackwell, 1953.
GG: Grundgesetze der Arithmetik, 2 vols., Jena: H. Pohle, 1893 and 1902. Translated as Frege (1893/2013).
BW: Wissenschaftlicher Briefwechsel Hamburg: Verlag, 1976. Translated as Philosophical and Mathematical Correspondence, translated by H. Kaal. Oxford: Blackwell, 1980.
NS: Nachgelassene Schriften. Hamburg: Verlag, 1983. Translated as Posthumous Writings, translated by Peter Long and Roger White. Oxford: Blackwell, 1979.
KS: Kleine Schriften. Hildesheim: Verlag, 2011. Translated as Collected Papers on Mathematics, Logic, and Philosophy, edited by B. McGuinness. Oxford: Blackwell, 1980.
Other abbreviations:
TLP: Wittgenstein, Ludwig. Tractatus Logicus Philosophicus, translated by C. K. Ogden. London: Routledge and Kegan Paul, 1922.
PI: Wittgenstein, Ludwig. Philosophical Investigations, edited and translated by G. E. M. Anscombe. Oxford: Blackwell, 1953.
NB: Wittgenstein, Ludwig. Notebooks 1914–1916, edited by G. H. von Wright and G. E. M. Anscombe. Oxford: Blackwell, 1961.
Other works cited:
Anscombe, G. E. M., 1959. An Introduction to Wittgenstein’s Tractatus: Themes in the Philosophy of Wittgenstein. South Bend, IN: St. Augustine’s Press.
Baker, G. P. and P. M. S. Hacker, 1984. Frege: Logical Excavations. New York: Oxford University Press.
Bell, David, 1979. Frege’s Theory of Judgement. Oxford: Clarendon Press.
Black, Max, 1964. A Companion to Wittgenstein’s Tractatus. Ithaca, NY: Cornell University Press.
Broome, John, 2013. Rationality Through Reasoning. Oxford: Wiley-Blackwell.
Burge, Tyler, 1998. “Frege On Knowing The Foundation.” Reprinted in Burge (2005), pp. 317–55.
———, 2005. Truth, Thought, Reason. Essays on Frege. Oxford: Clarendon Press, Oxford.
Cohen, L. Jonathan, 1964. “Do Illocutionary Forces Exist?” Philosophical Quarterly 14: 118–37.
Cook, Roy, 2013. “How To Read Grundgesetze.” In Frege (1893/2013), pp. A1–A42.
Currie, Gregory, 1987. “Remarks on Frege’s Conception of Inference.” Notre Dame Journal of Formal Logic 28: 55–68.
Dudman, V. H., 1971. “Peano’s Review of Frege’s Grundgesetze.” Southern Journal of Philosophy 9: 25–37.
Dummett, Michael, 1973. Frege. Philosophy of Language, 2nd ed. London: Duckworth.
Ebert, Philip A. and Marcus Rossberg, 2013. “Translators’ Introduction.” In Frege (1893/2013), pp. xiii–xxvi.
Fitting, Melvin and Richard L. Mendelsohn, 1998. First-Order Modal Logic. Dordrecht: Kluwer.
Frege, Gottlob, 1882/1964a. “Über den Zweck der Begriffsschrift.” In Begriffsschrift und andere Aufsätze, edited by Ignacio Angelellli, pp. 97–106. Hildesheim: Verlag.
———, 1882/1964b. “On the Scientific Justification of a Concept-script.” Mind 73: 155–60.
———, 1893/2013. Basic Laws of Arithmetic, edited and translated by Philip A. Ebert and Marcus Rossberg. Oxford: Oxford University Press.
Gabriel, Gottfried, 1996. “Frege’s ‘Epistemology in Disguise’.” In Frege: Importance and Legacy, edited by Matthias Schirn, pp. 330–46. Berlin: Walter de Gruyter.
Geach, Peter T., 1965. “Assertion.” In Logic Matters, pp. 254–69. Oxford: Blackwell.
———, 1976. “Saying and Showing in Frege and Wittgenstein.” Acta Philosophical Fennica 28: 54–70.
Gödel, Kurt, 1961. “The Modern Development of the Foundations of Mathematics in the Light of Philosophy.” In Collected Works, vol. III, Unpublished Essays and Lectures, edited by Solomon Feferman, pp. 374–87. Oxford: Oxford University Press.
Greimann, Dirk, 2000. “The Judgement-Stroke as a Truth-Operator: A New Interpretation of the Logical Form of Sentences in Frege’s Scientific Language.” Erkenntnis 52: 213–38.
———, 2014. “Frege on Truth, Assertoric Force and the Essence of Logic.” History and Philosophy of Logic 35: 272–88.
Hanks, Peter, 2015. Propositional Content. Oxford: Oxford University Press.
Heck, Richard G., 1996. “The Consistency of Predicative Fragments of Frege’s Grundgesetze der Arithmetik.” History and Philosophy of Logic 17: 209–20.
———, 2012. Reading Frege’s Grundgesetze. Oxford: Clarendon Press.
Hilbert, David, 1925. “On the Infinite.” Reprinted in From Frege to Gödel, edited by Jean van Heijenoort, pp. 367–92. Cambridge, MA: Harvard University Press, 1967.
Hylton, Peter, 1990. Russell, Idealism, and the Emergence of Analytic Philosophy. Oxford: Oxford University Press.
Kant, Immanuel, 1787. Kritik der reinen Vernunft. 2nd ed. Leipzig: L. Voss.
Kanterian, Edward, 2012. Frege: A Guide for the Perplexed. London: Continuum.
Kenny, Anthony, 1995. Frege. Harmondsworth: Penguin Books.
Kitcher, Philip, 1979. “Frege’s Epistemology.” Philosophical Review 88: 235–62.
Klement, Kevin C., 2002. Frege and the Logic of Sense and Reference. New York: Routledge.
Kremer, Michael, 2000. “Judgment and Truth in Frege.” Journal of the History of Philosophy 38: 549–81.
———, 2010. “Sense and Reference: the Origins and Development of the Distinction.” In The Cambridge Companion to Frege, edited by Michael Potter and Tom Ricketts, pp. 220–92. Cambridge: Cambridge University Press.
Künne, Wolfgang, 1992. “Hybrid Proper Names.” Mind 101: 721–31.
———, 2010. Die Philosophische Logik Gottlob Freges. Ein Kommentar. Frankfurt am Main: Klostermann.
———, 2013. “Merely Entertaining a Thought, Judging and Asserting: Notes on a Passage in Frege’s ‘The Thought’.” In Judgement and Truth in Early Analytic Philosophy and Phenomenology, edited by Mark Textor, pp. 52–73. Basingstoke: Palgrave Macmillan.
Leech, Jessica, 2015. “Logic and the Laws of Thought.” Philosopher’s Imprint 15: 1–27.
Macbeth, Danielle, 2014. Realizing Reason: A Narrative of Truth and Knowing. Oxford: Oxford University Press.
Mendelsohn, Richard, 2001. “Frege and the Grammar of Truth.” In Grammar in Early Twentieth-Century Philosophy, edited by Richard Gaskin, pp. 28–53. London and New York: Routledge.
Pedriali, Walter B., 2012. “Solvitur Ambulando. Meaning-constitutive Principles and the Inscrutability of Inference.” In Insolubles and Consequences: Essays in Honour of Stephen Read, edited by Catarina Dutilh Novaes and Ole Thomassen Hjortland, pp. 163–85. London: College Publications.
———, 2017. “Sense, Incomplete Understanding, and the Problem of Normative Guidance.” Grazer Philosophische Studien 94: 1–37.
———, forthcoming. “When Logic Gives Out: Frege on Basic Logical Laws.” In Essays on Frege’s Basic Laws of Arithmetic, edited by Marcus Rossberg and Philip A. Ebert. Oxford: Oxford University Press.
Potter, Michael, 2009. Wittgenstein’s Notes on Logic. Oxford: Oxford University Press.
Proops, Ian, 1997. “The Early Wittgenstein on Logical Assertion.” Philosophical Topics, 25: 121–44.
Reck, Erich H., 1997. “Frege’s Influence on Wittgenstein: Reversing Metaphysics via the Context Principle.” In Early Analytic Philosophy: Frege, Russell, Wittgenstein. Essays in Honor of Leonard Linsky, edited by William W. Tait, pp. 123–85. Chicago: Open Court.
———, ed., 2002a. From Frege to Wittgenstein. Oxford: Oxford University Press.
———, 2002b. “Wittgenstein’s ‘Great Debt’ to Frege: Biographical Traces and Philosophical Themes.” In Reck (2002a), pp. 3–38.
———, 2007. “Frege on Truth, Judgment, and Objectivity.” In Essays on Frege’s Conception of Truth, edited by Dirk Greimann, pp. 149–73. Amsterdam: Rodopi.
Ricketts, Thomas, 1986. “Objectivity and Objecthood: Frege’s Metaphysics of Judgement.” In Frege Synthesized, edited by L. Haaparanta and J. Hintikka, pp. 65–95. Dordrecht: Reidel.
———, 1996. “Logic and Truth in Frege.” Proceedings of the Aristotelian Society suppl. vol. 70: 121–40.
Bertrand Russell, 1903. The Principles of Mathematics. Cambridge: Cambridge University Press.
Sher, Gila and Cory D. Wright. “Truth as a Normative Modality of Cognitive Acts.” In Truth and Speech Acts: Studies in the Philosophy of Language, edited by Dirk Greimann and Geo Siegwart, pp. 280–306. Abingdon: Routledge.
Shieh, Sanford, 2002. “On Interpreting Frege on Truth and Logic.” In Reck (2002a), pp. 96–124.
Sluga, Hans, 2002. “Frege on the Indefinability of Truth.” In Reck (2002a), pp. 75–95.
Smith, Nicholas J. J., 2000. “Frege’s Judgement Stroke.” Australasian Journal of Philosophy 78: 153–75.
———, 2009. “Frege’s Judgement Stroke and the Conception of Logic as the Study of Inference not Consequence.” Philosophy Compass 4: 639–65.
Sullivan, Peter, 2004. “Frege’s Logic.” In Handbook of the History of Logic, vol. 3, The Rise of Modern Logic: From Leibniz to Frege, edited by Dov M. Gabbay and John Woods, pp. 659–750. Amsterdam: Elsevier.
———, 2011. “Is Logic Transcendental?” In Transcendental Philosophy and Naturalism, edited by Joel Smith and Peter Sullivan, pp. 157–83. Oxford: Oxford University Press.
Sundholm, Göran, 1998. “Inference, Consequence, Implication: A Constructivist’s Perspective.” Philosophia Mathematica 3: 178–94.
———, 2012. “‘Inference versus Consequence’ Revisited: Inference, Consequence, Conditional, Implication.” Synthèse 187: 943–56.
Taschek, William W., 2008. “Truth, Assertion, and the Horizontal: Frege on ‘The Essence of Logic’.” Mind 117: 375–401.
Textor, Mark, 2007. “Frege’s Theory of Hybrid Proper Names Developed and Defended.” Mind 116: 947–81.
———, 2010. “Frege on Judging as Acknowledging the Truth.” Mind 119: 615–55.
———, 2015. “Frege’s Theory of Hybrid Proper Names Extended.” Mind 124: 823–47.
van der Schaar, Maria, 2017. “Frege on Judgement and the Judging Agent.” Mind, online pre-print, https://doi.org/10.1093/mind/fzw059.
Wehmeier, Kai F., 1999. “Consistent Fragments of Grundgesetze and the Existence of Non-logical Objects.” Synthèse, 121: 309–28.
Whitehead, Alfred North and Bertrand Russell, 1910/1997. Principia Mathematica to *56. Cambridge: Cambridge University Press.
Wiggins, David, 1994. “Putnam’s Doctrine of Natural Kind Words and Frege’s Doctrines of Sense, Reference, and Extension: Can They Cohere?” In Reading Putnam, edited by Peter Clark and Bob Hale, pp. 201–15. Oxford: Blackwell.
Wittgenstein, Ludwig, 1939/1975. Wittgenstein’s Lectures on the Foundations of Mathematics, Cambridge 1939, edited by Cora Diamond. Chicago: University of Chicago Press.
BS: Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle: L. Nebert, 1879. Translated as Conceptual Notation and Related Articles, edited by T. W. Bynum. Oxford: Clarendon Press, 1972.
GL: Die Grundlagen der Arithmetik. Breslau: W. Koebner, 1884. Translated as The Foundations of Arithmetic, translated by J. L. Austin. Oxford: Blackwell, 1953.
GG: Grundgesetze der Arithmetik, 2 vols., Jena: H. Pohle, 1893 and 1902. Translated as Frege (1893/2013).
BW: Wissenschaftlicher Briefwechsel Hamburg: Verlag, 1976. Translated as Philosophical and Mathematical Correspondence, translated by H. Kaal. Oxford: Blackwell, 1980.
NS: Nachgelassene Schriften. Hamburg: Verlag, 1983. Translated as Posthumous Writings, translated by Peter Long and Roger White. Oxford: Blackwell, 1979.
KS: Kleine Schriften. Hildesheim: Verlag, 2011. Translated as Collected Papers on Mathematics, Logic, and Philosophy, edited by B. McGuinness. Oxford: Blackwell, 1980.
Other abbreviations:
TLP: Wittgenstein, Ludwig. Tractatus Logicus Philosophicus, translated by C. K. Ogden. London: Routledge and Kegan Paul, 1922.
PI: Wittgenstein, Ludwig. Philosophical Investigations, edited and translated by G. E. M. Anscombe. Oxford: Blackwell, 1953.
NB: Wittgenstein, Ludwig. Notebooks 1914–1916, edited by G. H. von Wright and G. E. M. Anscombe. Oxford: Blackwell, 1961.
Other works cited:
Anscombe, G. E. M., 1959. An Introduction to Wittgenstein’s Tractatus: Themes in the Philosophy of Wittgenstein. South Bend, IN: St. Augustine’s Press.
Baker, G. P. and P. M. S. Hacker, 1984. Frege: Logical Excavations. New York: Oxford University Press.
Bell, David, 1979. Frege’s Theory of Judgement. Oxford: Clarendon Press.
Black, Max, 1964. A Companion to Wittgenstein’s Tractatus. Ithaca, NY: Cornell University Press.
Broome, John, 2013. Rationality Through Reasoning. Oxford: Wiley-Blackwell.
Burge, Tyler, 1998. “Frege On Knowing The Foundation.” Reprinted in Burge (2005), pp. 317–55.
———, 2005. Truth, Thought, Reason. Essays on Frege. Oxford: Clarendon Press, Oxford.
Cohen, L. Jonathan, 1964. “Do Illocutionary Forces Exist?” Philosophical Quarterly 14: 118–37.
Cook, Roy, 2013. “How To Read Grundgesetze.” In Frege (1893/2013), pp. A1–A42.
Currie, Gregory, 1987. “Remarks on Frege’s Conception of Inference.” Notre Dame Journal of Formal Logic 28: 55–68.
Dudman, V. H., 1971. “Peano’s Review of Frege’s Grundgesetze.” Southern Journal of Philosophy 9: 25–37.
Dummett, Michael, 1973. Frege. Philosophy of Language, 2nd ed. London: Duckworth.
Ebert, Philip A. and Marcus Rossberg, 2013. “Translators’ Introduction.” In Frege (1893/2013), pp. xiii–xxvi.
Fitting, Melvin and Richard L. Mendelsohn, 1998. First-Order Modal Logic. Dordrecht: Kluwer.
Frege, Gottlob, 1882/1964a. “Über den Zweck der Begriffsschrift.” In Begriffsschrift und andere Aufsätze, edited by Ignacio Angelellli, pp. 97–106. Hildesheim: Verlag.
———, 1882/1964b. “On the Scientific Justification of a Concept-script.” Mind 73: 155–60.
———, 1893/2013. Basic Laws of Arithmetic, edited and translated by Philip A. Ebert and Marcus Rossberg. Oxford: Oxford University Press.
Gabriel, Gottfried, 1996. “Frege’s ‘Epistemology in Disguise’.” In Frege: Importance and Legacy, edited by Matthias Schirn, pp. 330–46. Berlin: Walter de Gruyter.
Geach, Peter T., 1965. “Assertion.” In Logic Matters, pp. 254–69. Oxford: Blackwell.
———, 1976. “Saying and Showing in Frege and Wittgenstein.” Acta Philosophical Fennica 28: 54–70.
Gödel, Kurt, 1961. “The Modern Development of the Foundations of Mathematics in the Light of Philosophy.” In Collected Works, vol. III, Unpublished Essays and Lectures, edited by Solomon Feferman, pp. 374–87. Oxford: Oxford University Press.
Greimann, Dirk, 2000. “The Judgement-Stroke as a Truth-Operator: A New Interpretation of the Logical Form of Sentences in Frege’s Scientific Language.” Erkenntnis 52: 213–38.
———, 2014. “Frege on Truth, Assertoric Force and the Essence of Logic.” History and Philosophy of Logic 35: 272–88.
Hanks, Peter, 2015. Propositional Content. Oxford: Oxford University Press.
Heck, Richard G., 1996. “The Consistency of Predicative Fragments of Frege’s Grundgesetze der Arithmetik.” History and Philosophy of Logic 17: 209–20.
———, 2012. Reading Frege’s Grundgesetze. Oxford: Clarendon Press.
Hilbert, David, 1925. “On the Infinite.” Reprinted in From Frege to Gödel, edited by Jean van Heijenoort, pp. 367–92. Cambridge, MA: Harvard University Press, 1967.
Hylton, Peter, 1990. Russell, Idealism, and the Emergence of Analytic Philosophy. Oxford: Oxford University Press.
Kant, Immanuel, 1787. Kritik der reinen Vernunft. 2nd ed. Leipzig: L. Voss.
Kanterian, Edward, 2012. Frege: A Guide for the Perplexed. London: Continuum.
Kenny, Anthony, 1995. Frege. Harmondsworth: Penguin Books.
Kitcher, Philip, 1979. “Frege’s Epistemology.” Philosophical Review 88: 235–62.
Klement, Kevin C., 2002. Frege and the Logic of Sense and Reference. New York: Routledge.
Kremer, Michael, 2000. “Judgment and Truth in Frege.” Journal of the History of Philosophy 38: 549–81.
———, 2010. “Sense and Reference: the Origins and Development of the Distinction.” In The Cambridge Companion to Frege, edited by Michael Potter and Tom Ricketts, pp. 220–92. Cambridge: Cambridge University Press.
Künne, Wolfgang, 1992. “Hybrid Proper Names.” Mind 101: 721–31.
———, 2010. Die Philosophische Logik Gottlob Freges. Ein Kommentar. Frankfurt am Main: Klostermann.
———, 2013. “Merely Entertaining a Thought, Judging and Asserting: Notes on a Passage in Frege’s ‘The Thought’.” In Judgement and Truth in Early Analytic Philosophy and Phenomenology, edited by Mark Textor, pp. 52–73. Basingstoke: Palgrave Macmillan.
Leech, Jessica, 2015. “Logic and the Laws of Thought.” Philosopher’s Imprint 15: 1–27.
Macbeth, Danielle, 2014. Realizing Reason: A Narrative of Truth and Knowing. Oxford: Oxford University Press.
Mendelsohn, Richard, 2001. “Frege and the Grammar of Truth.” In Grammar in Early Twentieth-Century Philosophy, edited by Richard Gaskin, pp. 28–53. London and New York: Routledge.
Pedriali, Walter B., 2012. “Solvitur Ambulando. Meaning-constitutive Principles and the Inscrutability of Inference.” In Insolubles and Consequences: Essays in Honour of Stephen Read, edited by Catarina Dutilh Novaes and Ole Thomassen Hjortland, pp. 163–85. London: College Publications.
———, 2017. “Sense, Incomplete Understanding, and the Problem of Normative Guidance.” Grazer Philosophische Studien 94: 1–37.
———, forthcoming. “When Logic Gives Out: Frege on Basic Logical Laws.” In Essays on Frege’s Basic Laws of Arithmetic, edited by Marcus Rossberg and Philip A. Ebert. Oxford: Oxford University Press.
Potter, Michael, 2009. Wittgenstein’s Notes on Logic. Oxford: Oxford University Press.
Proops, Ian, 1997. “The Early Wittgenstein on Logical Assertion.” Philosophical Topics, 25: 121–44.
Reck, Erich H., 1997. “Frege’s Influence on Wittgenstein: Reversing Metaphysics via the Context Principle.” In Early Analytic Philosophy: Frege, Russell, Wittgenstein. Essays in Honor of Leonard Linsky, edited by William W. Tait, pp. 123–85. Chicago: Open Court.
———, ed., 2002a. From Frege to Wittgenstein. Oxford: Oxford University Press.
———, 2002b. “Wittgenstein’s ‘Great Debt’ to Frege: Biographical Traces and Philosophical Themes.” In Reck (2002a), pp. 3–38.
———, 2007. “Frege on Truth, Judgment, and Objectivity.” In Essays on Frege’s Conception of Truth, edited by Dirk Greimann, pp. 149–73. Amsterdam: Rodopi.
Ricketts, Thomas, 1986. “Objectivity and Objecthood: Frege’s Metaphysics of Judgement.” In Frege Synthesized, edited by L. Haaparanta and J. Hintikka, pp. 65–95. Dordrecht: Reidel.
———, 1996. “Logic and Truth in Frege.” Proceedings of the Aristotelian Society suppl. vol. 70: 121–40.
Bertrand Russell, 1903. The Principles of Mathematics. Cambridge: Cambridge University Press.
Sher, Gila and Cory D. Wright. “Truth as a Normative Modality of Cognitive Acts.” In Truth and Speech Acts: Studies in the Philosophy of Language, edited by Dirk Greimann and Geo Siegwart, pp. 280–306. Abingdon: Routledge.
Shieh, Sanford, 2002. “On Interpreting Frege on Truth and Logic.” In Reck (2002a), pp. 96–124.
Sluga, Hans, 2002. “Frege on the Indefinability of Truth.” In Reck (2002a), pp. 75–95.
Smith, Nicholas J. J., 2000. “Frege’s Judgement Stroke.” Australasian Journal of Philosophy 78: 153–75.
———, 2009. “Frege’s Judgement Stroke and the Conception of Logic as the Study of Inference not Consequence.” Philosophy Compass 4: 639–65.
Sullivan, Peter, 2004. “Frege’s Logic.” In Handbook of the History of Logic, vol. 3, The Rise of Modern Logic: From Leibniz to Frege, edited by Dov M. Gabbay and John Woods, pp. 659–750. Amsterdam: Elsevier.
———, 2011. “Is Logic Transcendental?” In Transcendental Philosophy and Naturalism, edited by Joel Smith and Peter Sullivan, pp. 157–83. Oxford: Oxford University Press.
Sundholm, Göran, 1998. “Inference, Consequence, Implication: A Constructivist’s Perspective.” Philosophia Mathematica 3: 178–94.
———, 2012. “‘Inference versus Consequence’ Revisited: Inference, Consequence, Conditional, Implication.” Synthèse 187: 943–56.
Taschek, William W., 2008. “Truth, Assertion, and the Horizontal: Frege on ‘The Essence of Logic’.” Mind 117: 375–401.
Textor, Mark, 2007. “Frege’s Theory of Hybrid Proper Names Developed and Defended.” Mind 116: 947–81.
———, 2010. “Frege on Judging as Acknowledging the Truth.” Mind 119: 615–55.
———, 2015. “Frege’s Theory of Hybrid Proper Names Extended.” Mind 124: 823–47.
van der Schaar, Maria, 2017. “Frege on Judgement and the Judging Agent.” Mind, online pre-print, https://doi.org/10.1093/mind/fzw059.
Wehmeier, Kai F., 1999. “Consistent Fragments of Grundgesetze and the Existence of Non-logical Objects.” Synthèse, 121: 309–28.
Whitehead, Alfred North and Bertrand Russell, 1910/1997. Principia Mathematica to *56. Cambridge: Cambridge University Press.
Wiggins, David, 1994. “Putnam’s Doctrine of Natural Kind Words and Frege’s Doctrines of Sense, Reference, and Extension: Can They Cohere?” In Reading Putnam, edited by Peter Clark and Bob Hale, pp. 201–15. Oxford: Blackwell.
Wittgenstein, Ludwig, 1939/1975. Wittgenstein’s Lectures on the Foundations of Mathematics, Cambridge 1939, edited by Cora Diamond. Chicago: University of Chicago Press.
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