# Predicativity, the Russell-Myhill Paradox, and Church’s Intensional Logic

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## Abstract

This paper sets out a predicative response to the Russell-Myhill paradox of propositions within the framework of Church’s intensional logic. A predicative response places restrictions on the full comprehension schema, which asserts that every formula determines a higher-order entity. In addition to motivating the restriction on the comprehension schema from intuitions about the stability of reference, this paper contains a consistency proof for the predicative response to the Russell-Myhill paradox. The models used to establish this consistency also model other axioms of Church’s intensional logic that have been criticized by Parsons and Klement: this, it turns out, is due to resources which also permit an interpretation of a fragment of Gallin’s intensional logic. Finally, the relation between the predicative response to the Russell-Myhill paradox of propositions and the Russell paradox of sets is discussed, and it is shown that the predicative conception of set induced by this predicative intensional logic allows one to respond to the Wehmeier problem of many non-extensions.

## Keywords

Russell-Myhill Intensional logic Predicativity## Notes

### Acknowledgments

I was lucky enough to be able to present parts of this work at a number of workshops and conferences, and I would like to thank the participants and organizers of these events for these opportunities. I would like to especially thank the following people for the comments and feedback which I received on these and other occasions: Robert Black, Roy Cook, Matthew Davidson, Walter Dean, Marie Duží, Kenny Easwaran, Fernando Ferreira, Martin Fischer, Rohan French, Salvatore Florio, Kentaro Fujimoto, Jeremy Heis, Joel David Hamkins, Volker Halbach, Ole Thomassen Hjortland, Luca Incurvati, Daniel Isaacson, Jönne Kriener, Graham Leach-Krouse, Hannes Leitgeb, Øystein Linnebo, Paolo Mancosu, Richard Mendelsohn, Tony Martin, Yiannis Moschovakis, John Mumma, Pavel Pudlák, Sam Roberts, Marcus Rossberg, Tony Roy, Gil Sagi, Florian Steinberger, Iulian Toader, Gabriel Uzquiano, Albert Visser, Kai Wehmeier, Philip Welch, Trevor Wilson, and Martin Zeman. This paper has likewise been substantially bettered by the feedback and comments of the editors and referees of this journal, to whom I express my gratitude. While composing this paper, I was supported by a Kurt Gödel Society Research Prize Fellowship and by Øystein Linnebo’s European Research Council funded project “Plurals, Predicates, and Paradox.”

## References

- 1.Anthony Anderson, C. (1980). Some new axioms for the logic of sense and denotation: alternative (0).
*Noûs*,*14*(2), 217–234.CrossRefGoogle Scholar - 2.Anthony Anderson, C. (1984). General intensional logic In D. Gabbay, & F. Guenthner (Eds.),
*Handbook of Philosophical Logic. Vol. II: Extensions of Classical Logic, volume 165 of Synthese Library, pp. 355–385*. Dordrecht: Reidel.Google Scholar - 3.Anthony Anderson, C. (1986). Some difficulties concerning Russellian intensional logic.
*Noûs*,*20*(1), 35–43.CrossRefGoogle Scholar - 4.Anthony Anderson, C. (1987). Semantical antinomies in the logic of sense and denotation.
*Notre Dame Journal of Formal Logic*,*28*(1), 99–114.CrossRefGoogle Scholar - 5.Anthony Anderson, C. (1998). Alonzo Church’s contributions to philosophy and intensional logic.
*Bulletin of Symbolic Logic*,*4*(2), 129–171.CrossRefGoogle Scholar - 6.Barwise, J. (1975).
*Admissible sets and structures*. Berlin: Springer.CrossRefGoogle Scholar - 7.Bealer, G. (1982).
*Quality and concept*. Oxford: Oxford University Press.CrossRefGoogle Scholar - 8.Blass, A., Dershowitz, N., & Gurevich, Y. (2009). When are two algorithms the same?
*Bulletin of Symbolic Logic*,*15*(2), 145–168.CrossRefGoogle Scholar - 9.Boolos, G. (1971). The iterative conception of set.
*The Journal of Philosophy*,*68*, 215–232. Reprinted in [10].CrossRefGoogle Scholar - 10.Boolos, G. (1998).
*Logic, logic, and logic*. Cambridge: Harvard University Press. Edited by Richard Jeffrey.Google Scholar - 11.Chalmers, D.J. (2011). Propositions and attitude ascriptions: a Fregean account.
*Noûs*,*45*(4), 595–639.CrossRefGoogle Scholar - 12.Chihara, C. (1973).
*Ontology and the vicious circle principle*. Ithaca: Cornell University Press.Google Scholar - 13.Church, A. (1943). [review of [72]].
*The Journal of Symbolic Logic*,*8*(1), 45–47.CrossRefGoogle Scholar - 14.Church, A. (1946). A formulation of the logic of sense and denotation [Abstract].
*The Journal of Symbolic Logic*,*11*(1), 31.Google Scholar - 15.Church, A. (1951). A formulation of the logic of sense and reference In P. Henle, H.M. Kallen, & S.K. Langer (Eds.),
*Structure, Method and Meaning: Essays in Honor of Henry M. Sheffer, pp. 2–24*: Liberal Arts Press.Google Scholar - 16.Church, A. (1974). Outline of a revised formulation of the logic of sense and denotation. II.
*Noûs*,*8*(2), 135–156.CrossRefGoogle Scholar - 17.Church, A. (1984). Russell’s theory of identity of propositions.
*Philosophia Naturalis*,*21*, 513–522.Google Scholar - 18.Church, A. (1993). A revised formulation of the logic of sense and denotation. Alternative (1).
*Noûs*,*27*(2), 141–157.CrossRefGoogle Scholar - 19.de Rouilhan, P. (2005). Russell’s logics. In
*Logic Colloquium 2000, volume 19 of Lecture Notes in Logic, pp. 335–349. Association for Symbolic Logic, Urbana*.Google Scholar - 20.Demopoulos, W. (1994). Frege Hilbert and the conceptual structure of model theory.
*History and Philosophy of Logic*,*15*(2), 211–225.CrossRefGoogle Scholar - 21.Demopoulos, W. (Ed.) (1995).
*Frege’s philosophy of mathematics*. Cambridge: Harvard University Press.Google Scholar - 22.Demopoulos, W. (1998). The philosophical basis of our knowledge of number.
*Noûs*,*32*(4), 481–503.CrossRefGoogle Scholar - 23.Demopoulos, W., & Clark, P. (2005). The logicism of Frege, Dedekind, and Russell In S. Shapiro (Ed.),
*The Oxford handbook of philosophy of mathematics and logic, pp. 129–165*. Oxford: Oxford University Press.Google Scholar - 24.Dowty, D.R., Wall, R.E., & Peters, S. (1981).
*Introduction to Montague semantics*. Dordrecht: Reidel.Google Scholar - 25.Dummett, M (1963). The philosophical significance of Gödel’s theorem.
*Ratio*,*5*, 140–155.Google Scholar - 26.Dummett, M. (1978).
*Truth and other enigmas*. Cambridge: Harvard University Press.Google Scholar - 27.Dummett, M. (1981).
*Frege: philosophy of language*, 2nd edn. New York: Harper & Row.Google Scholar - 28.Dummett, M. (1991).
*Frege: philosophy of mathematics*. Cambridge: Harvard University Press.Google Scholar - 29.Dummett, M. (1993).
*The seas of language*. Oxford: Clarendon.Google Scholar - 30.Dummett, M. (1994). What is mathematics about? In
*Mathematics and Mind, Logic and Computation in Philosophy, pp. 11–26*. Reprinted in [29]: Oxford University Press.Google Scholar - 31.Duží, M., Jespersen, B., & Materna, P. (2010).
*Procedural semantics for hyperintensional logic: foundations and applications of transparent intensional logic*. Berlin: Springer.Google Scholar - 32.Feferman, S. (1964). Systems of predicative analysis.
*The Journal of Symbolic Logic*,*29*, 1–30.CrossRefGoogle Scholar - 33.Feferman, S. (1968). Lectures on Proof Theory. In
*Proceedings of the Summer School in Logic, pp. 1–107*. Berlin: Springer.Google Scholar - 34.Feferman, S. (2005). Predicativity In S. Shapiro (Ed.),
*The Oxford Handbook of Philosophy of Mathematics and Logic, pp. 590–624*. Oxford: Oxford University Press.Google Scholar - 35.Ferreira, F., & Wehmeier, K.F. (2002). On the consistency of the \({\Delta }_{1}^{1}\)-CA fragment of Frege’s Grundgesetze.
*Journal of Philosophical Logic*,*31*(4), 301–311.CrossRefGoogle Scholar - 36.Fox, C., & Lappin, S. (2005).
*Foundations of intensional semantics*. Malden: Blackwell.CrossRefGoogle Scholar - 37.Frege, G..
*Grundgesetze der Arithmetik: begriffsschriftlich abgeleitet*. Jena: Pohle. 1893, 1903. Two volumes. Reprinted in [38].Google Scholar - 38.Frege, G. (1962).
*Grundgesetze der Arithmetik: begriffsschriftlich abgeleitet*. Hildesheim: Olms.Google Scholar - 39.Frege, G. (2013).
*Basic laws of arithmetic*. Oxford: Oxford University Press. Translated by Philip A. Ebert and Marcus Rossberg.Google Scholar - 40.Gallin, D. (1975). Intensional and higher-order modal logic. North-Holland.Google Scholar
- 41.Gamut, L.T.F. (1991).
*Intensional logic and logical grammar, volume 2 of Logic, Language, and Meaning*. Chicago: University of Chicago Press.Google Scholar - 42.Goldfarb, W. (1988). Russell’s reasons for ramification In C. Wade Savage, & C. Anthony Anderson (Eds.),
*Essays of Bertrand Russell’s Metaphysics and Epistemology, volume 11 of Minnesota Studies in the Philosophy of Science, pp. 24–40*. Minneapolis: University of Minnesota Press.Google Scholar - 43.Heck Jr., R.G. (1996). The consistency of predicative fragments of Frege’s Grundgesetze der Arithmetik.
*History and Philosophy of Logic*,*17*(4), 209–220.CrossRefGoogle Scholar - 44.Heim, I., & Kratzer, A. (1998).
*Semantics in generative grammar*. Malden: Blackwell.Google Scholar - 45.Heinzmann, G. (1986). Poincaré, Russell, Zermelo et Peano. Textes de la discussion (1906–1912) sur les fondements des mathématiques: des antinomie à la prédicativié. Blanchard.Google Scholar
- 46.Horty, J. (2007).
*Frege on definitions*. Oxford: Oxford University Press.Google Scholar - 47.Hrbacek, K., & Jech, T. (1999).
*Introduction to set theory, volume 220 of Monographs and Textbooks in Pure and Applied Mathematics*, 3rd edn. New York: Dekker.Google Scholar - 48.Jech, T. (2003).
*Set theory. Springer Monographs in Mathematics*. Berlin: Springer. The Third Millennium Edition.Google Scholar - 49.Jensen, R.B. (1972). The fine structure of the constructible hierarchy.
*Annals of Mathematical Logic*,*4*, 229–308.CrossRefGoogle Scholar - 50.Kaplan, D. (1975). How to Russell a Frege-Church.
*Journal of Philosophy*,*72*(19), 716–729.CrossRefGoogle Scholar - 51.Keith, J. (1984).
*Devlin. Constructibility. Perspectives in Mathematical Logic*. Berlin: Springer.Google Scholar - 52.Klement, K.C. (2002).
*Frege and the logic of sense and reference*. New York and London: Routledge.Google Scholar - 53.
- 54.Klement, K.C. (2010). The senses of functions in the logic of sense and denotation.
*Bulletin of Symbolic Logic*,*16*(2), 153–188.CrossRefGoogle Scholar - 55.Klement, K.C. (2014). Russell-Myhill Paradox. In
*The Internet Encyclopedia of Philosophy. ISSN 2161-0002*. http://www.iep.utm.edu/. - 56.Kripke, S. (1964). Transfinite recursion on admissible ordinals I, II.
*The Journal of Symbolic Logic*,*29*(3), 161–162.Google Scholar - 57.Kunen, K. (1980). Set theory, volume 102 of Studies in Logic and the Foundations of Mathematics. North-Holland.Google Scholar
- 58.Kunen, K. (2011).
*Set theory*. London: College Publications.Google Scholar - 59.Manzano, M. (1996).
*Extensions of First Order Logic, volume 19 of Cambridge Tracts in Theoretical Computer Science*. Cambridge: Cambridge University Press.Google Scholar - 60.Martin, D.A. (1970). Review of [75].
*The Journal of Philosophy*,*67*(4), 111–114.CrossRefGoogle Scholar - 61.Moschovakis, Y.N. (1993). Sense and denotation as algorithm and value. In
*Logic Colloquium ’90, volume 2 of Lecture Notes Logic, pp. 210–249*. Berlin: Springer.Google Scholar - 62.Myhill, J. (1952). Two ways of ontology in modern logic.
*Review of Metaphysics*,*5*(4), 639–655.Google Scholar - 63.Myhill, J. (1958). Problems arising in the formalization of intensional logic.
*Logique et Analyse*,*1*, 78–83.Google Scholar - 64.Parsons, C. (1982). Intensional logic in extensional language.
*The Journal of Symbolic Logic*,*47*(2), 289–328.CrossRefGoogle Scholar - 65.Parsons, C. (1983).
*Mathematics in philosophy: selected essays*. Ithaca: Cornell University Press.Google Scholar - 66.Parsons, C. (2002). Realism and the debate on impredicativity, 1917–1944. In
*Reflections on the foundations of mathematics (Stanford, CA, 1998), volume 15 of Lecture Notes in Logic, pp. 372–389*. Urbana: Associaton of Symbolic Logic.Google Scholar - 67.Parsons, T. (1987). On the consistency of the first-order portion of Frege’s logical system.
*Notre Dame Journal of Formal Logic*,*28*(1), 161–168. Reprinted in [21].CrossRefGoogle Scholar - 68.Parsons, T. (2001). The logic of sense and denotation: extensions and applications. In
*Logic, Meaning and Computation: Essays in Memory of Alonzo Church, volume 305 of Synthese Library, pp. 507–543*. Dordrecht: Kluwer.Google Scholar - 69.Platek, R.A. (1966). Foundations of recursion theory. Unpublished. Dissertation, Stanford University.Google Scholar
- 70.Poincaré, H. (1910). Über transfinite Zahlen, Teubner, Leipzig/Berlin.Google Scholar
- 71.Priest, G. (2013). Indefinite extensibility—dialetheic style.
*Studia Logica*,*101*(6), 1263–1275.CrossRefGoogle Scholar - 72.Quine, W.V.O. (1943). Notes on existence and necessity.
*The Journal of Philosophy*,*40*(5), 113–127.CrossRefGoogle Scholar - 73.Quine, W.V. (1960). Carnap and logical truth.
*Synthese*,*12*(4), 350–374. Reprinted in [74].CrossRefGoogle Scholar - 74.Quine, W.V. (1966).
*Ways of paradox and other essays*. New York: Random House.Google Scholar - 75.Quine, W.V. (1969).
*Set theory and its logic. Revised edition*. Cambridge: Harvard University Press.Google Scholar - 76.
- 77.Quine, W.V. (1986). Reply to Parsons In L.E. Hahn, & P.A. Schilpp (Eds.),
*The philosophy of W.V. Quine, volume 18 of Library of the Living Philosophers, pp. 398–403. Open Court, La Salle*.Google Scholar - 78.Russell, B. (1903).
*The principles of mathematics*. Cambridge: Cambridge University Press.Google Scholar - 79.Russell, B. (1907). On some difficulities in the theory of transfinite numbers and order types.
*Proceedings of the London Mathematical Society*,*s2-4*(1), 29–53. Reprinted in [80].CrossRefGoogle Scholar - 80.Russell, B. (1973).
*Essays in analysis*. London: George Allen.Google Scholar - 81.Sacks, G.E. (1990).
*Higher recursion theory. Perspectives in Mathematical Logic*. Berlin: Springer.CrossRefGoogle Scholar - 82.Schindler, R., & Zeman, M. (2010). Fine structure In M. Foreman, & A. Kanamori (Eds.),
*Handbook of set theory, vol. 1, pp. 605–656*. Berlin: Springer.Google Scholar - 83.Shapiro, S., & Wright, C. (2006). All things indefinitely extensible In A. Rayo, & G. Uzquiano (Eds.),
*Absolute Generality, pp. 255–304*. Oxford: Clarendon Press.Google Scholar - 84.Shoenfield, J.R. (1961). The problem of predicativity. In
*Essays on the Foundations of Mathematics, pp. 132–139*. Jerusalem: Magnes Press.Google Scholar - 85.Shoenfield, J.R. (1967). Chapter 9: set theory. In
*Mathematical Logic, pp. 238–315*. Reading: Addison-Wesley.Google Scholar - 86.Shoenfield, J.R. (1977). Axioms of set theory In J. Barwise (Ed.),
*Handbook of Mathematical Logic, volume 90 of Studies in Logic and the Foundations of Mathematics. North-Holland*.Google Scholar - 87.Soare, R.I. (1987).
*Recursively enumerable sets and degrees. Perspectives in Mathematical Logic*. Berlin: Springer.CrossRefGoogle Scholar - 88.Taschek, W.W. (2010). On sense and reference: a criticial reception In M. Potter, & T. Ricketts (Eds.),
*The Cambridge companion to Frege, pp. 293–341*. Cambridge.Google Scholar - 89.Thomason, R.H. (1980). A model theory for propositional attitudes.
*Linguistics and Philosophy*,*4*(1), 47–70.CrossRefGoogle Scholar - 90.Tichý, P. (1988).
*The foundations of Frege’s logic*. Berlin: De Gruyter.CrossRefGoogle Scholar - 91.Tichý, P. (2004).
*Pavel Tichý’s collected papers in logic and philosophy*. Dunedin: University of Otago Press.Google Scholar - 92.Walsh, S. (2012). Comparing Hume’s Principle, Basic Law V and Peano Arithmetic.
*Annals of Pure and Applied Logic*,*163*, 1679–1709.CrossRefGoogle Scholar - 93.Walsh, S. (2014). The strength of predicative abstraction. Unpublished. arXiv:1407.3860.
- 94.Walsh, S. Fragments of Frege’s
*Grundgesetze*and Gödel’s constructible universe.*The Journal of Symbolic Logic*. forthcoming. arXiv:1407.3861. - 95.Wehmeier, K.F. (1999). Consistent fragments of Grundgesetze and the existence of non-logical objects.
*Synthese*,*121*(3), 309–328.CrossRefGoogle Scholar - 96.Wehmeier, K.F. (2004). Russell’s paradox in consistent fragments of Frege’s
*Grundgesetze*der Arithmetik. In*One hundred years of Russell’s paradox, volume 6 of de Gruyter Series in Logic and its Applications, pp. 247–257*. Berlin: de Gruyter.Google Scholar - 97.Weyl, H. (1918). Das Kontinuum. Kritische Untersuchungen über die Grundlagen der Analysis. Veit, Leipzig.Google Scholar