Abstract
G. Priest's anti-consistency argument (Priest 1979, 1984, 1987) and J. R. Lucas's anti-mechanist argument (Lucas 1961, 1968, 1970, 1984) both appeal to Gödel incompleteness. By way of refuting them, this paper defends the thesis of quartet compatibility, viz., that the logic of the mind can simultaneously be Gödel incomplete, consistent, mechanical, and recursion complete (capable of all means of recursion). A representational approach is pursued, which owes its origin to works by, among others, J. Myhill (1964), P. Benacerraf (1967), J. Webb (1980, 1983) and M. Arbib (1987). It is shown that the fallacy shared by the two arguments under discussion lies in misidentifying two systems, the one for which the Gödel sentence is constructable and to be proved, and the other in which the Gödel sentence in question is indeed provable. It follows that the logic of the mind can surpass its own Gödelian limitation not by being inconsistent or non-mechanistic, but by being capable of representing stronger systems in itself; and so can a proper machine. The concepts of representational provability, representational maximality, formal system capacity, etc., are discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson, Alan Ross (ed.): 1964, Minds and Machines — Contemporary Perspectives in Philosophy Series, Prentice-Hall, Inc., Englewood Cliffs NJ.
Arbib, Michael A.: 1987, Brains, Machines, and Mathematics, Springer-Verlag, New York.
Barwise, J. (ed.): 1978, Handbook of Logic, North-Holland, Amsterdam.
Bell, J. L. and M. Machover: 1977, A Course in Mathematical Logic, North-Holland, Amsterdam.
Benacerraf, Paul: 1967, ‘God, The Devil, and Gödel’, Monist 51, 9–32.
Boyer, David L: 1983, ‘J. R. Lucas, Kurt Gödel, and Fred Astaire’, Philosophical Quarterly 33(131), 147–59.
Butrick, R.: 1965, The Gödel Formula: Some Reservations, Mind IXXIV, 411–14.
Chihara, Charles S.: 1984, ‘Priest, the Liar and Gödel’, Journal of Philosophical Logic 13, 117–24.
Church, A.: 1956, Introduction to Mathematical Logic, Princeton University Press, Princeton.
Dennett, David: 1978, Brainstorms, Bradford Books Publishers, Cambridge MA.
Grim, Patrick: 1988, ‘Logic and Limits of Knowledge and Truth’, Noûs XXII(3), 341–67.
Hilbert, David: 1964, ‘On the Infinite’, in P. Benacerraf and H. Putnam (eds.), Philosophy of Mathematics: Selected Readings, Prentice-Hall, Inc., Englewood Cliffs, pp. 134–51.
Hofstadter, Douglas R.: 1979, Gödel, Escher, Bach: An Eternal Golden Braid, Basic Books, Inc., New York.
Kirk, Robert: 1986, ‘Mental Machinery and Gödel’, Synthese 66, 437–52.
Lucas, J. R.: 1961, ‘Minds, Machines and Gödel’, Philosophy XXXVI, 112–27 (reprinted in Anderson, 1964, pp. 43–59).
Lucas, J. R.: 1968, ‘Satan Stultified: A Rejoinder to Paul Benacerraf’, Monist LII, 145–58.
Lucas, J. R.: 1970, The Freedom of the Will, Clarendon Press, Oxford.
Lucas, J. R.: 1984, ‘Lucas, Gödel, and Astaire: A Rejoinder’, Philosophical Quarterly 34, 507–08.
Mendelson, Elliott: 1990, ‘Second Thoughts about Church's Thesis and Mathematical Proofs’, The Journal of Philosophy LXXXVII, 225–33.
Myhill, John: 1964, ‘The Abstract Theory of Self Reproduction’, in M. D. Mesarovic (ed.), Views on General Systems Theory, John Wiley & Sons, New York, pp. 106–18.
Napoli, Ernesto: 1985, ‘Priest's Paradox’, Logique et Analyse 28, 403–07.
Nelson, R. J.: 1980, ‘Margaret Boden's Artificial Intelligence and Natural Man’, Synthese 43, 433–51.
Nelson, R. J.: 1982, The Logic of Mind, D. Reidel, Dordrecht, Holland.
Penrose, Roger, et al.: 1990, ‘Precis of the Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics’, and many other comments from other authors, and Penrose's reply to them, Behavioral and Brain Sciences 13, 643–705.
Post, Emil. L.: 1967, ‘Absolutely Unsolvable Problems and Relatively Undecidable Propositions: Account of an Anticipation’, in M. Davis (ed.), The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions, Raven Press, New York, pp. 338–433.
Priest, Graham: 1979, ‘The Logic of Paradox’, Journal of Philosophical Logic 8, 219–41.
Priest, Graham: 1984, ‘Logic of Paradox Revisited’, Journal of Philosophical Logic 13, 153–79.
Priest, Graham: 1987, In Contradiction — a Study of the Transconsistency, Martinus Nijhoff Publishers, Dordrecht.
Putnam, Hilary: 1960, ‘Minds and Machines: A Symposium’, in Anderson, 1964, pp. 72–97.
Van Heijenoort, Jean (ed.): 1967, From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, Harvard University Press, Cambridge, MA.
Van Heijenoort, J.: 1967a, ‘Gödel's Theorem’, in Paul Edwards (ed.), Encyclopedia of Philosophy, Vol. 3, Macmillan Publishing, pp. 348–57.
Wang, Hao: 1974, From Mathematics to Philosophy, Humanities Press, New York.
Webb, Judson Chambers: 1980, Mechanism, Mentalism, and Metamathematics — An Essay on Finitism, D. Reidel, Dordrecht, Holland.
Webb, Judson Chambers: 1983, ‘Gödel's Theorem and Church's Thesis — A Prologue to Mechanism’, in R. S. Cohen and M. W. Wartofsky (eds.), Language, Logic, and Method, D. Reidel, Dordrecht, Holland.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Yu, Q. Consistency, mechanicalness, and the logic of the mind. Synthese 90, 145–179 (1992). https://doi.org/10.1007/BF00485195
Issue Date:
DOI: https://doi.org/10.1007/BF00485195