Abstract
In this chapter I argue that human beings should reason, not in accordance with classical logic, but in accordance with a weaker ‘reticent logic’. I characterize reticent logic, and then show that arguments for the existence of fundamental Gödelian limitations on artificial intelligence are undermined by the idea that we should reason reticently, not classically.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
If the word ‘never’ raises intuitionistic worries about permanently undetermined truth-values, then it can be added as an extra stipulation that there is a time limit of 1 h for accepting sentences. The truth-values of 1 and 3 will be determined once and for all when the hour expires.
- 2.
Is the prisoner’s argument a counterexample to modus ponens? No—or at least, not if by ‘counterexample’ we mean a case where both ϕ and ϕ → ψ are true but ψ is false. The prisoner’s argument is instead a case in which ϕ and ϕ → ψ can both be true only if ψ is not accepted.
- 3.
Lingering suspicions that 3 is liar-like should be put to rest by noticing that Gödel’s (1931) diagonalization procedure for generating self-referential sentences with well defined truth-conditions can be used to manufacture a version of 3. See Sect. 10.5, below, for an explanation of how this procedure can be applied to English.
- 4.
I borrow the ‘□’ notation from provability logic, wherein the intended meaning of ‘□ϕ’ is ‘ϕ is provable in Peano Arithmetic’. In using this notion I don’t mean to suggest that RL is a standard modal logic. (It isn’t.)
- 5.
The perversity check will be straightforward if the language is that of propositional logic or unary predicate logic, since it will then be decidable whether S⊢¬□ϕ is classically valid. For richer languages it will be necessary to make do with an incomplete perversity testing method, that errs by sometimes failing to classify perverse arguments as perverse. For every such method there will be a corresponding version of Basic RL, with its own strengths and weaknesses where its ability to detect perversities is concerned. The question as to which of such methods are ‘best’ is rich and complex, but I say no more about it here.
- 6.
From the fact that the system has accepted □ϕ, it does not follow that ϕ is true. But it does follow that if the system fails to accept ϕ, then its risk of having accepted a falsehood is 100 %.
- 7.
There is some question as to precisely how Penrose’s ‘new argument’ is supposed to go (see Chalmers 1995; Penrose 1996; Lindström 2001, 2006; Shapiro 2003). My Argument B is closely based on Penrose’s (1994) original, informal presentation of the argument, and its essential logic is similar to Lindström’s (2001) formulation. Departing from Penrose, I frame the argument in terms of the consistency of the formal systems in question, instead of the soundness of these systems, with the reason being that the former notion is less demanding and more general than the latter but still adequate for the argument’s purposes.
- 8.
Most of these objections were initially conceived as objections to the original version of the mathematical argument, but apply equally against Argument J.
- 9.
Chalmers uses Löb’s theorem, rather than Gödel’s theorem (but these two theorems are intimately related).
- 10.
This formulation of Gödel’s diagonalization procedure is based on (Rucker 1982, p. 284).
- 11.
When we ask whether the human protagonist in the mathematical argument can prove things a formal system cannot, we should not force her to use Gödel numbers and Peano arithmetic, which play to the strengths of formal systems, instead of names and natural language, which play to the strengths of the human mind. To do so would be to make her fight with one arm tied behind her back.
- 12.
The corresponding premise in Argument H is H5, which is problematic for the same reasons as J8.
References
Barr, M. (1990). Review: The emperor’s new mind. By Roger Penrose. The American Mathematical Monthly, 97(10), 938–942.
Benacerraf, P. (1968). God, the devil and Gödel. The Monist, 51, 9–32.
Boolos, G. (1990). On seeing the truth of the Gödel sentence. Behavioral and Brain Sciences, 13(4), 655–656.
Bowie, G. L. (1982). Lucas’ number is finally up. Journal of Philosophical Logic, 41(3), 279–285.
Chalmers, D. (1995). Minds, machines, and mathematics. A review of shadows of the mind by Roger Penrose. Psyche, 2, 11–20.
Gaifman, H. (2000). What Gödel’s incompleteness result does and does not show. The Journal of Philosophy, 97(8), 462–470.
Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monash Mathematical Physics, 38, 173–198.
Gödel, K. (1951). Some basic theorems on the foundation of mathematics and their implications. In S. Feferman (Ed.) (1995), Kurt Gödel Collected works, vol. III: Unpublished essays and lectures. (pp. 304–323). Oxford: Oxford University Press.
Grush, R., & Churchland, P. (1995). Gaps in Penrose’s toilings. Journal of Consciousness Studies, 2(1), 10–29.
Lindström, P. (2001). Penrose’s new argument. Journal of Philosophical Logic, 30, 241–250.
Lindström, P. (2006). Remarks on Penrose’s new argument. Journal of Philosophical Logic, 35, 231–237.
Lucas, J. R. (1961). Minds, machines and Gödel. Philosophy, 36, 112–127.
Lucas, J. R. (1996). Minds, machine and Gödel: A retrospect. In P. J. R. Millican & A. Clark (Eds.), Machines and thought: The legacy of Alan Turing (pp. 103–124). Oxford: Oxford University Press.
McCullough, D. (1995). Can humans escape Gödel? A review of shadows of the mind by Roger Penrose. Psyche, 2(23), 57–65.
Nagel, E., & Newman, J. (1957). Gödel’s proof. New York: New York University Press.
Penrose, R. (1989). The emperor’s new mind. Oxford: Oxford University Press.
Penrose, R. (1994). Shadows of the mind. Oxford: Oxford University Press.
Penrose, R. (1996). Beyond the doubting of a shadow. Psyche, 2(23), 89–129.
Putnam, H. (1960). Minds and machines. In Sidney Hook (Ed.), Dimensions of mind: A symposium. New York: New York University Press. Reprinted in A. R. Anderson (1964) (Ed.), Minds and machines. Prentice-Hall, 77.
Raatikainen, P. (2014). Gödel’s incompleteness theorems. In Edward N. Zalta (Ed.), The Stanford encyclopedia of philosophy (Spring 2014 Edition). http://plato.stanford.edu/archives/spr2014/entries/goedel-incompleteness/
Robinson, W. (1992). Penrose and mathematical ability. Analysis, 52(2), 80–87.
Rucker, R. (1982). Infinity and the mind: The science and philosophy of the infinite. Princeton: Princeton University Press.
Shapiro, S. (2003). Mechanism, truth and Penrose’s new argument. Journal of Philosophical Logic, 32, 19–42.
Sorensen, R. (1988). Blindspots. Oxford: Clarendon.
Turing, A. M. (1947). Lecture to the London Mathematical Society on 20 February 1947. Reprinted in D. C. Ince (1992) (Ed.), Collected works of A.M. Turing: Mechanical intelligence. Amsterdam: North Holland.
Turing, A. M. (1948). Intelligent machinery. Reprinted in D. C. Ince (1992) (Ed.), Collected works of A.M. Turing: Mechanical intelligence. Amsterdam: North Holland.
Turing, A. M. (1950). Computing machinery and intelligence. Reprinted in D. C. Ince (1992) (Ed.), Collected works of A.M. Turing: Mechanical intelligence. Amsterdam: North Holland.
Acknowledgements
Many thanks to Thomas Forster, Matthew Grice, Doukas Kapantais and Michael-John Turp for comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Campbell, D. (2016). Why We Shouldn’t Reason Classically, and the Implications for Artificial Intelligence. In: Müller, V.C. (eds) Computing and Philosophy. Synthese Library, vol 375. Springer, Cham. https://doi.org/10.1007/978-3-319-23291-1_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-23291-1_10
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-23290-4
Online ISBN: 978-3-319-23291-1
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)