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Summary Godelian arguments use Godel's incompleteness theorems to argue against the possibility of human-level computer intelligence.  Godel proved that any number system strong enough to do arithmetic would contain true propositions that were impossible to prove within the system. Let G be such a proposition, and let the relevant system correspond to a computer.  It seems to follow that no computer can prove G (and so know G is true), but humans can know that G is true (by, as it were, moving outside of the number system and seeing that G has to be true to preserve soundness).  So, it appears that humans are more powerful than computers restricted to just implementations of number systems.  This is the essence of Godelian arguments. Many replies to these arguments have been put forward.  An obvious reply is that computers can be programmed to be more than mere number systems and so can step outside number systems just like humans can.  
Key works Probably the central paper using Godelian arguments against AI is Lucas 1961. Another good paper is Benacerraf 1967.  For what is often regarded as the classic reply to Lucas, see Putnam 1960.
Introductions Penrose 1994 and Penrose 1989.
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  1. This Sentence Does Not Contain the Symbol X.Samuel Alexander - 2013 - The Reasoner 7 (9):108.
    A suprise may occur if we use a similar strategy to the Liar's paradox to mathematically formalize "This sentence does not contain the symbol X".
  2. A Machine That Knows Its Own Code.Samuel A. Alexander - 2014 - Studia Logica 102 (3):567-576.
    We construct a machine that knows its own code, at the price of not knowing its own factivity.
  3. Gödel, Penrose, E I Fondamenti Dell'intelligenza Artificiale.Aldo Antonelli - 1997 - Sistemi Intelligenti 9 (3):353-376.
    Il dibattito sul ruolo e le implicazioni del teorema di Gödel per l'intelligenza artificiale ha recentemente ricevuto nuovo impeto grazie a due importanti volumi pubblicati da Roger Penrose, The Emperor's New Mind [1989] e Shadows of the Mind [1994]. Naturalmente, Penrose non è il primo né l'ultimo a usare il teorema di Gödel allo scopo di trarne conseguenze per i fondamenti dell'intelligenza artificiale. Tuttavia il recente dibattito suscitato dai due libri di Penrose è significativo sia per ampiezza sia per profondità. (...)
  4. Discrete Tomography of Penrose Model Sets.M. Baake & C. Huck - 2007 - Philosophical Magazine 87 (18-21):2839-2846.
  5. Sex, Equality and Mr Lucas: Discussion.Andrew Belsey - 1980 - Philosophy 55 (213):386-391.
  6. God, the Devil, and Gödel.Paul Benacerraf - 1967 - The Monist 51 (1):9-32.
  7. Roger Penrose's "the Emperor's New Mind": Implications for Critical Thinking.Ronald Marcel Biron - 1992 - Dissertation, Boston University
    The study examines Roger Penrose's arguments against strong AI and for non-algorithmic thinking as presented in his book, The Emperor's New Mind. The primary question addressed is whether or not to accept the computational hypothesis--that human thought is algorithmic. The study concludes that Penrose offers neither convincing logical proof nor sufficient scientific evidence to reject the hypothesis, but does present a comprehensive position that is very suggestive. It is then argued that this position combined with the criticisms voiced against pure (...)
  8. An Analysis On A Machine Type Gödel.Marcel Bodea - 2001 - Studia Philosophica 1.
    The formal domain of abstract mathematical systems and the propositions provable in them can be illustrated using a ‘machine language’: The Gödel’s machine. This article was written primarily as an investigation and introduction to incompleteness theorems in a way wich is a simplifying factor. We tried to show how a certain computing machine can have very general features in subject to Gödel’s argument. We belive that the proofs we give are unusually simple. We then can turn to some incompletness arguments (...)
  9. Mind Versus Godel.Damjan Bojadziev - 1997 - In Matjaz Gams & M. Wu Paprzycki (eds.), Mind Versus Computer. IOS Press. pp. 202-210.
  10. On “Seeing” the Truth of the Gödel Sentence.George Boolos - 1990 - Behavioral and Brain Sciences 13 (4):655-656.
  11. Samaria in lucas—handelingen.G. Bouwman - 2013 - Bijdragen 34 (1):40-59.
  12. Lucas' Number is Finally Up.G. Lee Bowie - 1982 - Journal of Philosophical Logic 11 (3):279-85.
  13. J. R. Lucas, Kurt Godel, and Fred Astaire.David L. Boyer - 1983 - Philosophical Quarterly 33 (131):147-159.
  14. R. Lucas, Kurt Godel, and Fred Astaire.David L. Boyer - 1983 - Philosophical Quarterly 33 (April):147-59.
  15. A Refutation of Penrose's Godelian Case Against Artificial Intelligence.Selmer Bringsjord - manuscript
    Having, as it is generally agreed, failed to destroy the computational conception of mind with the G\"{o}delian attack he articulated in his {\em The Emperor's New Mind}, Penrose has returned, armed with a more elaborate and more fastidious G\"{o}delian case, expressed in and 3 of his {\em Shadows of the Mind}. The core argument in these chapters is enthymematic, and when formalized, a remarkable number of technical glitches come to light. Over and above these defects, the argument, at best, is (...)
  16. A Refutation of Penrose's New Godelian Case Against the Computational Conception of Mind.Selmer Bringsjord & H. Xiao - 2000 - Journal of Experimental and Theoretical Artificial Intelligence 12.
  17. The Simulation of Human Intelligence.Donald E. Broadbent (ed.) - 1993 - Blackwell.
  18. The Emperor is Still Under-Dressed.David J. Buller, Jerry Fodor & Tessa L. Crume - 2005 - Trends in Cognitive Sciences 9 (11):508-510.
    Replies to Letters from Cosmides et al. (regarding cheater detection), Buss and Haselton (regarding sex differences in jealousy), and Daly and Wilson (regarding child abuse).
  19. On the Nature of Mathematical Judgement Reply to Penrose.Alan Bundy & Roger Penrose - 1990 - Edinburgh University.
  20. John Lucas, Godel and Mechanical Intelligence.A. Buti - 2002 - Rivista di Filosofia Neo-Scolastica 94 (4):637-674.
  21. Lucas Revived? An Undefended Flank.Jeremy Butterfield - 1990 - Behavioral and Brain Sciences 13 (4):658.
  22. Penrose and the Renewal of Goedelian Argument.A. Butti - 2004 - Rivista di Filosofia Neo-Scolastica 96 (4):669-697.
  23. Minds, Machines and Godel : A Reply to Mr Lucas.Whiteley C. H. - 1962 - Philosophy 37 (139):61-.
    In Philosophy for April 1961 Mr J. R. Lucas argues that Gödel's theorem proves that Mechanism is false. I wish to dispute this view, not because I maintain that Mechanism is true, but because I do not believe that this issue is to be settled by what looks rather like a kind of logical conjuring-trick. In my discussion I take for granted Lucas's account of Gödel's procedure, which I am not competent to criticise.
  24. Why We Shouldn't Reason Classically, and the Implications for Artificial Intelligence.Douglas Campbell - 2016 - In C. Vincent Müller (ed.), Computing and Philosophy: Selected Papers From Iacap 2014. Springer Verlag. pp. 151--165.
    In this paper 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.
  25. Theories of Theories of Mind.Peter Carruthers & Peter K. Smith (eds.) - 1996 - Cambridge University Press.
    Theories of Theories of Mind brings together contributions by a distinguished international team of philosophers, psychologists, and primatologists, who between them address such questions as: what is it to understand the thoughts, feelings, and intentions of other people? How does such an understanding develop in the normal child? Why, unusually, does it fail to develop? And is any such mentalistic understanding shared by members of other species? The volume's four parts together offer a state of the art survey of the (...)
  26. Minds, Machines, And Mathematics A Review of Shadows of the Mind by Roger Penrose. [REVIEW]David Chalmers - 1995 - Psyche 2.
  27. Minds, Machines, and Mathematics.David J. Chalmers - 1996 - Psyche 2:11-20.
    In his stimulating book SHADOWS OF THE MIND, Roger Penrose presents arguments, based on Gödel's theorem, for the conclusion that human thought is uncomputable. There are actually two separate arguments in Penrose's book. The second has been widely ignored, but seems to me to be much more interesting and novel than the first. I will address both forms of the argument in some detail. Toward the end, I will also comment on Penrose's proposals for a "new science of consciousness".
  28. LUCAS, J. R. Responsibility. [REVIEW]T. S. Champlin - 1994 - Philosophy 69:254.
  29. L S Penrose's Limit Theorem: Tests by Simulation.Pao-Li Chang, Vincent C. H. Chua & Moshé Machover - unknown
    L S Penrose’s Limit Theorem – which is implicit in Penrose [7, p. 72] and for which he gave no rigorous proof – says that, in simple weighted voting games, if the number of voters increases indefinitely and the relative quota is pegged, then – under certain conditions – the ratio between the voting powers of any two voters converges to the ratio between their weights. Lindner and Machover [4] prove some special cases of Penrose’s Limit Theorem. They give a (...)
  30. Further Comments on Minds, Machines and Godel.C. T. K. Chari - 1963 - Philosophy 38 (April):175-8.
  31. A Theorem About Computationalism and “Absolute” Truth.Arthur Charlesworth - 2016 - Minds and Machines 26 (3):205-226.
    This article focuses on issues related to improving an argument about minds and machines given by Kurt Gödel in 1951, in a prominent lecture. Roughly, Gödel’s argument supported the conjecture that either the human mind is not algorithmic, or there is a particular arithmetical truth impossible for the human mind to master, or both. A well-known weakness in his argument is crucial reliance on the assumption that, if the deductive capability of the human mind is equivalent to that of a (...)
  32. The Comprehensibility Theorem and the Foundations of Artificial Intelligence.Arthur Charlesworth - 2014 - Minds and Machines 24 (4):439-476.
    Problem-solving software that is not-necessarily infallible is central to AI. Such software whose correctness and incorrectness properties are deducible by agents is an issue at the foundations of AI. The Comprehensibility Theorem, which appeared in a journal for specialists in formal mathematical logic, might provide a limitation concerning this issue and might be applicable to any agents, regardless of whether the agents are artificial or natural. The present article, aimed at researchers interested in the foundations of AI, addresses many questions (...)
  33. On Alleged Refutations of Mechanism Using Godel's Incompleteness Results.C. Chihara - 1972 - Journal of Philosophy 69 (September):507-26.
  34. Gödel's Theorem and Mechanism.David Coder - 2003 - Etica E Politica 5 (1):1.
  35. Godel's Theorem and Mechanism.David Coder - 1969 - Philosophy 44 (September):234-7.
    In “Minds, Machines, and Gödel”, J. R. Lucas claims that Goedel's incompleteness theorem constitutes a proof “that Mechanism is false, that is, that minds cannot be explained as machines”. He claims further that “if the proof of the falsity of mechanism is valid, it is of the greatest consequence for the whole of philosophy”. It seems to me that both of these claims are exaggerated. It is true that no minds can be explained as machines. But it is not true (...)
  36. Are Turing Machines Platonists? Inferentialism and the Computational Theory of Mind.Jon Cogburn & Jason Megill - 2010 - Minds and Machines 20 (3):423-439.
    We first discuss Michael Dummett’s philosophy of mathematics and Robert Brandom’s philosophy of language to demonstrate that inferentialism entails the falsity of Church’s Thesis and, as a consequence, the Computational Theory of Mind. This amounts to an entirely novel critique of mechanism in the philosophy of mind, one we show to have tremendous advantages over the traditional Lucas-Penrose argument.
  37. LUCAS, J. R. - "The Concept of Probability". [REVIEW]L. J. Cohen - 1972 - Mind 81:625.
  38. Artificial Intelligence and Its Applications.A. G. Cohn & J. R. Thomas (eds.) - 1986 - John Wiley and Sons.
  39. Turing's o-Machines, Searle, Penrose, and the Brain.Jack Copeland - 1998 - Analysis 58 (2):128-138.
    In his PhD thesis (1938) Turing introduced what he described as 'a new kind of machine'. He called these 'O-machines'. The present paper employs Turing's concept against a number of currently fashionable positions in the philosophy of mind.
  40. Symmetry Properties of Penrose Type Tilings.N. Cotfas - 2008 - Philosophical Magazine 88 (13-15):2017-2023.
  41. Kurt Gödel: Philosopher-Scientist.Gabriella Crocco & Eva-Maria Engelen (eds.) - 2016 - Marseille: Presses universitaires de Provence.
    This volume represents the beginning of a new stage of research in interpreting Kurt Gödel’s philosophy in relation to his scientific work. It is more than a collection of essays on Gödel. It is in fact the product of a long enduring international collaboration on Kurt Gödel’s Philosophical Notebooks (Max Phil). New and significant material has been made accessible to a group of experts, on which they rely for their articles. In addition to this, Gödel’s Nachlass is presented anew by (...)
  42. How Subtle is Gödel's Theorem? More on Roger Penrose.Martin Davis - 1993 - Behavioral and Brain Sciences 16 (3):611.
  43. Machine Man and Other Writings.Julien Offray de La Mettrie & Ann Thomson - 1996
  44. Od Platona do Penrose'a.Jacek Dębiec - 1996 - Zagadnienia Filozoficzne W Nauce 19.
  45. Murmurs in the Cathedral: Review of R. Penrose, The Emperor's New Mind[REVIEW]Daniel C. Dennett - 1989 - Times Literary Supplement (September) 29.
    The idea that a computer could be conscious--or equivalently, that human consciousness is the effect of some complex computation mechanically performed by our brains--strikes some scientists and philosophers as a beautiful idea. They find it initially surprising and unsettling, as all beautiful ideas are, but the inevitable culmination of the scientific advances that have gradually demystified and unified the material world. The ideologues of Artificial Intelligence (AI) have been its most articulate supporters. To others, this idea is deeply repellent: philistine, (...)
  46. Brainstorms.Daniel C. Dennett - 1978 - MIT Press.
    This collection of 17 essays by the author offers a comprehensive theory of mind, encompassing traditional issues of consciousness and free will.
  47. The Abilities of Men and Machines.Daniel C. Dennett - 1978 - In Brainstorms. MIT Press.
  48. Löb's Theorem as a Limitation on Mechanism.Michael Detlefsen - 2002 - Minds and Machines 12 (3):353-381.
    We argue that Löb's Theorem implies a limitation on mechanism. Specifically, we argue, via an application of a generalized version of Löb's Theorem, that any particular device known by an observer to be mechanical cannot be used as an epistemic authority (of a particular type) by that observer: either the belief-set of such an authority is not mechanizable or, if it is, there is no identifiable formal system of which the observer can know (or truly believe) it to be the (...)
  49. Mind in the Shadows.Michael Detlefsen - 1998 - Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 29 (1):123-136.
  50. Mind in the Shadows.Michael Detlefsen - 1998 - Studies in History and Philosophy of Science Part B 29 (1):123-136.
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