Abstract
Those who go in for attacking AI can be indefatigable. John Searle tirelessly targets at least one brand of AI (“Strong” AI) with variations on his Chinese Room. (Searle fired the first shot in (Searle 1980). Two decades later, CR is alive and well, as the arrival of a new book (Bishop & Preston 2002) reveals.). Bringsjord has at last count published 16 formal arguments against “Strong” AI. (For a sampling, recall Table 1.) And Roger Penrose appears to have the same endless energy when it comes to producing Gödelian attacks on AI: Having, as it is generally agreed, failed to improve on Lucas’ at-best-controversial primogenitor (Lucas 1964) with the argument as formulated in his The Emperor’s New Mind (ENM) (Penrose 1989),1 Penrose returned, armed with a new Gödelian case, expressed in his Shadows of the Mind (SOTM) (Penrose 1994). This case, unlike its predecessor, does more than recapitulate Lucas’ argument, but it nonetheless fails, as we shall see. The great irony is that this case is based on Penrose’s near-deification of logico-mathematical reasoning, but such reasoning, as we show herein, can be used to refute Penrose’s case.
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Footnotes
For an explanation of why standard Gödelian attacks fail, see “Chapter VII: Gödel” in (Bringsjord 1992).
Recall our discussion of TT and TTT in Chapter 1. The straight Turing Test, again, tests only for linguistic performance. Harnad (1991)’sTTT, on the other hand, requires that the human and robot (or android) players compete across the full range of behavior. For example, the judge in TTT can ask questions designed to provoke an emotional response, and can then observe the facial expressions of the two players. We discussed all of this, of course, in Chapter 1.
Newell expressed his dream for a unified (production system-based) theory for all of human cognition in (Newell 1973).
Feferman (1995) and Davis (1980) catalogue the inadequacy of Penrose’s scholarship when it comes to mathematical logic. However, we don’t think the quality of this scholarship, however questionable it may be, creates any fundamental problems for Penrose’s core Gödelian arguments against “Strong” AI.
It’s important at this point to note that while many of the mathematical issues relating to Penrose’s Gödelian arguments are not expressible in LI, the core arguments themselves must conform to the basic, inviolable principles of deductive reasoning that form the foundation of technical philosophy. SOTM is an essay in technical philosophy; it’s not a mathematical proof. The present book, of course, is itself technical philosophy. We make reference to logical systems beyond LI, but our core reasoning is intended to meet standards for deductive reasoning circum-scribed in LI and LII. See (Ebbinghaus et al. 1984) for a nice introduction to these logical systems, as well as more advanced ones, such as the infinitary systems related to Yablo’s Paradox. These systems and this paradox are discussed later in the chapter.
We make no comments about what we’ve called ‘Lemma 1.’ For an elegant proof of this lemma (which, if formalized, would require more lines centered around the rules ∃E, ∀E, ∃I, ∀I), see (Boolos & Jeffrey 1989).
For details on how to construct ConsisФ, see Chapter X in (Ebbinghaus et al. 1984).
That is, Zermelo-Fraenkel axiomatic set theory with the axiom of choice. ZFC plays a more important role in Chapter 4 than in the present chapter, and is partially described in Chapter 4 (where five of the nine axioms composing ZFC are presented). Readers wanting a fuller treatment are directed to (Ebbinghaus et al. 1984).
In the articles discussing Yablo’s paradox, writers refer to an infinitary version of Yablo’s paradox. For example, both Priest (f997) and Beall (2001) refer to such a formulation in an unpublished note by Forster (n.d.), but the formulation simply isn’t there. Forster (n.d.) provides only the onc-linc kernel of an Lω1ω-based definition of our function s. Likewise, Priest (1997) says that Hardy (1995) provides an infmitary, ω-rule-based version of Yablo’s paradox, but actually Hardy only proves, indirectly, that Yablo’s paradox entails ω-inconsistency. We specify an infinitary version of Yablo’s paradox, expressed in the “background” logic that allows for meta-proofs regarding infinitary logical systems like Lω1ω (this particular system will turn out to be at the heart of the argument given in the next chapter). This system is presented in encapsulated form in (Ebbinghaus et al. 1984), from which the student interested in infinitary logic can move to (Karp 1964), then to (Keisler 1971), and then to (Dickmann 1975).
In personal conversation Penrose seemed to be inclined to complain that the original version of Yablo’s paradox appears to go beyond what can be derived from axiomatic set theory (e.g., ZFC). This reply is odd, for SOTM is quite literally filled with reasoning that appears to go beyond first-order logic. (E.g., consider the diagrammatic “proof concerning hexagonal numbers we visited above. Such diagrams seem to move beyond first-order logic (Bringsjord & Bringsjord 1996).)
The array Penrose presents can be positioned on three tapes of a five-tape Turing machine, and the operation of rotation can result in three dots being placed contiguously on all five tapes. For a nice introduction to Turing machines with k tapes, see the textbook: (Lewis & Papadimitriou 1981). For a discussion of much more sophisticated image-processing AI systems than this hypothetical Turing machine, in the context of imagistic reasoning seemingly more exotic than what Penrose describes here, see (Bringsjord & Bringsjord 1996).
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Bringsjord, S., Zenzen, M. (2003). A Refutation of Penrose’s Gödelian Case. In: Superminds. Studies in Cognitive Systems, vol 29. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0283-7_2
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