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Ideal Negative Conceivability and the Halting Problem

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Abstract

Our limited a priori-reasoning skills open a gap between our finding a proposition conceivable and its metaphysical possibility. A prominent strategy for closing this gap is the postulation of ideal conceivers, who suffer from no such limitations. In this paper I argue that, under many, maybe all, plausible unpackings of the notion of ideal conceiver, it is false that ideal negative conceivability entails possibility.

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Notes

  1. The only reason I will be concerned with in this paper, but there are other, probably more important ones—cf. Chalmers (2002).

  2. This should not be taken to mean that the primary/secondary distinction (closely related to the distinction between epistemic and metaphysical possibility, although I will not be discussing it any further) is unimportant or innocuous. On the contrary, it does crucial work in Chalmers’s philosophy of mind, where it is used, for example, to derive anti-materialist conclusions with respect to the mind-body problem—cf. Chalmers (2009). There are probably many things to say about the cogency of this distinction, when it is applied to the philosophically interesting cases, but I will steer clear of this important complication by restricting the discussion to logico-mathematical statements, whose primary and secondary modal profiles are supposed to match.

  3. And, again, the notion of better reasoning at play should not depend in any way on truth-tracking reliability. As we go along, I will signal the places at which this caveat is relevant.

  4. I should also say that, although there may be other reasonably clear ways to spell out this notion, I don’t know of any equally attractive ones.

  5. In fact, for all Ideal claims, different ideal conceivers might be necessary in order to conceive in the undefeatable-reasoning way different substitutions of a statement schema. In the Addition(m,n,o) case, though, a single ideal conceiver can handle them all, as we are about to see.

  6. This very brief summary can be supplemented by any of a number of textbooks, such as Davis et al. (1994, chapter 6).

  7. It might be that pressing for a specification of the peculiar propositional attitude of finding a priori contradiction in p is already a way to put the ideal conceivabilist into trouble. Concentrating in an area of discourse in which checking for truth and falsity is enough, therefore, has the bonus feature of making things as easy as possible for her.

  8. Nothing is written, perhaps, because the Final state is never reached. This will not happen in the simple case of Addition(m,n,o), but will happen for semi-decidable problems such as the ones that figure prominently in the sequel.

  9. Many and not all substitutions, because Turing machines are able to calculate the halting problem for a restricted class of programs and inputs. For example, an adding device, given any two numbers, will calculate their sum and halt, always. So, when ADD is the addition program it implements, the restricted halting problem is trivial:

    • Halt(ADD,i): A Turing machine running program ADD, given input i, eventually halts.

    The answer is ’yes’ for every pair i of numbers.

  10. Again here, Ideal is compatible with more than one ideal conceiver (Turing machine) being needed in order to deal with different substitutions of the Halt(P,i) schema. An ideal conceivabilist may try to exploit this fact by arguing as follows:

    All I have claimed is that, for any statement S, there is a conceiver who has achieved her conclusions by way of undefeatable reasoning, and who finds contradiction in S only if S is false (and hence impossible). Now, such a pairing of statement and conceiver can be trivially made to work for substitutions of the Halt(P,i) schema, in the following way:

    1. 1.

      Consider two Turing machines: the yes machine is such that, for any pair (P,i) provided as input, it immediately prints a 1 at the left of its head and halts. The no machine, instead, immediately prints a 0 and halts.

    2. 2.

      Feed all pairs (P,i) that correspond to a program P that halts when supplied with input i to the no machine, and all other pairs to the yes machine.

    It is, indeed, easy to see that this pair of machines can deal with the halting problem faultlessly. It is equally clear that, if this way of allocating conceivers to statements were intended by ideal conceivabilists as one which vindicates Ideal, we should not really care much about this thesis, which would have turned out to be entirely vacuous.

    I should quickly point out that real-life ideal conceivabilists are under no illusion about this; and, in particular, Chalmers advocates for the much more substantial independence requirement of undefeatable reasoning, which in this case rules out the allocation of conceivers to statements based on (antecedent) information about the truth or falsity of the statement in question.

    The fact remains that any allocation of conceivers (Turing machines) to statements, such that it respects the independence requirement, is unable to solve the halting problem.

  11. Tasks with an infinite number of steps, cf. Benacerraf (1962).

  12. I will concede that computing during transfinitely many steps makes sense.

  13. Halting problems, really—see (Hamkins 2002, p. 535), (Hamkins 2004, p. 153) for details.

  14. That is, if they have not been cherry-picked in a way that violates the independence requirement in a manner analogous to the one described in footnote 10. Incidentally, we may now note that an effective cherry-picking must be done by something that is more powerful computationally than an oracle Turing machine.

  15. There is another way of understanding ATMs which makes them Turing machines—but unable to solve the Turing machine halting problem. For discussion, see Copeland and Shagrir (2011).

  16. See (Hamkins 2004, p. 153f) for a discussion of oracles in the hypercomputer context.

  17. The least powerful oracle Turing machine is the one that can solve the halting problem for standard Turing machines; the following in the hierarchy is the one that can solve the halting problem for the least powerful oracle Turing machine, etc.

  18. There are very many aspects of rationality that cannot, or not clearly, be modelled by Turing machines: say, a certain kind of subtlety in weighing pros and cons in everyday life. This is, I believe, the kind of abilities that make Chalmers talk of rationality as open ended and as resisting complete characterisation. On the other hand, these abilities are irrelevant to the issue at hand: the (ideal negative) conceivability of logico-mathematical statements.

  19. Financial support for this work was provided by the DGI, Spanish Government, research project FFI2010-15717, and Consolider-Ingenio project CSD2009-00056. Also by the Generalitat de Catalunya, AGAUR grant SGR 2009-1077.

    I would like to thank Fèlix Bou, David Chalmers, Dan López de Sa, David Pineda, Sònia Roca-Royes, Jonathan Schmidt-Dominé, Giuliano Torrengo, Martin Ziegler and two anonymous reviewers for their comments on different versions of this paper.

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Martínez, M. Ideal Negative Conceivability and the Halting Problem. Erkenn 78, 979–990 (2013). https://doi.org/10.1007/s10670-012-9363-x

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