A Simple Solution to the “Preface Paradox”

Many philosophers argue that paradoxes like the so-called “preface paradox” show that it is not a requirement of rationality that the contents of one’s beliefs should all be consistent with each other. (For example, David Christensen argues for this in his recent book Putting Logic in its Place.) I believe that this argument is mistaken.

As a matter of fact, I don’t actually accept that it is a requirement of rationality that the contents of one’s beliefs should all be consistent with each other. If the contradictions are lurking in obscure hidden parts of one’s belief set – so that it would take an extremely long and complicated chain of reasoning to derive any contradiction from the contents of one’s beliefs, and one has not in fact performed any such long and complicated chain of reasoning – then it need not be irrational for one to persist in one’s beliefs (or so I am inclined to think).

However, I am prepared to defend a different connection between consistency and rational belief. If one consciously considers a set of propositions, and this set of propositions is obviously logically inconsistent, then one should revise one’s beliefs in such a way as to avoid fully believing all members of this inconsistent set. (The paradigm case of a set of propositions that is “obviously logically inconsistent” is the set {p, ‘¬ p’}, but other sets of propositions may also count as “obviously logically inconsistent” as well, such as {p, q, ‘¬ (p & q)’}, and so on.)

Still, I don’t think that the preface paradox is a good argument against the stronger consistency requirement. In fact, the flaw that I see in the argument has already been pointed out some years ago by Simon Evnine – albeit in the course of arguing for a very strong claim that I wouldn’t myself accept, that rationality requires one to believe the conjunction of all of one’s beliefs (see his “Believing Conjunctions”, Synthese 1999).

The paradox is that it seems rational to believe the “Preface Proposition”:

At least one of the propositions that I believe (other than this proposition) is false.

But if one believes this proposition, that guarantees that at least one of one’s beliefs is false! So it seems that it is rational to believe this proposition, even though if one does so, one’s beliefs are inconsistent.

In fact, this argument is fallacious: even if one believes the “Preface Proposition”, it does not follow that the contents of one’s beliefs – that is, the set of propositions that one believes – are logically inconsistent. Unlike most of the propositions that one believes, the Preface Proposition is a higher-order proposition. As Evnine rightly insists, the conjunction of the propositions that one believes is not logically equivalent to the proposition that everything that one believes is true; and the proposition that something that one believes is false is not equivalent to the disjunction of the negations of the propositions that one believes. Even if one believes the “Preface Proposition” there is a possible situation in which all of the propositions that one actually believes are true – namely, a situation in which one in addition believes some further proposition (which one does not actually believe) and that further proposition is false.

Christensen objects to Evnine’s point by saying that an ideally self-aware thinker would recognize that a certain long conjunction was equivalent to the conjunction of all the propositions that she believes, and so would take the Preface Proposition to be equivalent to the negation of this long conjunction (and so, presumably, this ideally self-aware thinker would also infer the negation of this long conjunction from the Preface Proposition, in which case the contents of this thinker’s beliefs would be logically inconsistent). It would be “unpersuasive”, Christensen says, to insist that it is only thinkers who are not ideally self-aware in this way who are rationally entitled to believe the Preface Proposition. Our lack of such ideal self-awareness is a “superficial limitation” (Putting Logic in its Place, p. 38).

But it seems to me that our lack of such ideal self-awareness is anything but a “superficial limitation”. This inability to survey such large totalities of propositions seems one of the fundamental limitations of any finite mind. Of course an infinite or omniscient mind (in so far as we can make sense of what such a mind would be like) would not be subject to such limitations, but presumably it would not be rational for such an omniscient mind to believe the Preface Proposition.

Contrast the following two cases. In the first case, I write an extremely short “book” containing just one simple assertion. It seems that it would be irrational for me to both to believe that assertion and to believe that the book contains errors. In the second case, I write an enormous 600-page book containing thousands upon thousands of assertions. Then it seems to me that may well be perfectly rational for me to believe each assertion in the book but also to believe that the book must contain some errors. It seems to me that one crucial difference between these cases is that it is so much harder to survey the totality of propositions that are asserted in the longer book. No rational thinker who has evidence of his own fallibility would believe of any single proposition that it is logically equivalent to the conjunction of all and only the assertions in that book. So I don’t think that Christensen has identified a problem with this aspect of Evnine’s response to the so-called “preface paradox”.


A Simple Solution to the “Preface Paradox” — 14 Comments

  1. Suppose that Acme manufactures flag poles and that their customers are picky about specifications. So the Acme Quality Department measures the length of each new flag pole several times. They assume that the distribution of error for measurements of length is normal, so they correct their set of measurements for each pole accordingly. Thus, each pole records a corrected measurement of length: Acme model A flag pole, serial number 453-01-120, is 6.003 (+/- 0.001) meters long. Acme advertises that Model A flag poles are 6 meters, plus or minus 0.01 meters. So, pole 453-01-120 goes into the Model A stock.

    Quality Inspector 14 has conscientiously accepted 10,000 poles as Model A poles, and in each case applied his “Inspected by 14” sticker to a pole if and only if he believed that pole was 6.00 (+/- 0.01) meters long.

    Yet, as he consciously considers his life’s work at Acme, which is this set of 10,000 flag poles, it is obvious to him that at least one of the poles that he accepted is nevertheless out of specification, i.e., that it fails to be between 5.99 and 6.01 meters long. It is not simply that he believes that one of the poles is a bad Model A, and that the addition of this believe guarantees that at least one of his beliefs is false, but that the grounds for this belief about a bad Model A among the 10,000 poles he accepted are as overwhelming as the painstaking grounds he gathered when evaluating each flag pole and accepting it into the Model A lot. (One might need to be careful about bookkeeping to make that point salient.)

    So, his life’s work and his bad Model A belief are propositions he consciously considers, and the set is obviously inconsistent. But it is not the case that he should revise his beliefs to avoid this inconsistency. It isn’t clear what revision he should make. The evidence he has for each belief is (or can be made to be) unassailable, even if uncertain. And suspending belief outright invites a form a skepticism.

    Let’s even suppose that Inspector A is aided by his computer records of his work which he may exploit as he wishes to “survey” not only the contents of his 10,000 Model A pole beliefs, but his evidence for each of these beliefs as well. We may even stipulate that the record is perfect, that there are no mistakes in his log, that each classification is according to the evidence and follows best practices. Still, this consideration won’t change his epistemic position with respect to the bad Model A belief.

  2. Thanks, Gregory! I’m afraid that I’m just going to insist that what you’ve given is a different paradox — in effect, a version of the lottery paradox. I.e., it seems, at least prima facie, that for some series of n items, it is reasonable to believe, of every m from 1 to n, that the mth item in the series has a certain property, but it is also reasonable to believe that at least one item in that series does not have that property. In the traditional lottery paradox, the series consists of the lottery tickets, and the property is the property of not being a winning ticket; in your example, the series consists of the flag poles, and the property is the property of being 6 +/- 0.01 meters long.

    I agree that these propositions, which it seems prima facie reasonable to believe, are obviously inconsistent. So I am going to say that when you contemplate this set of propositions, you must revise your beliefs so that you don’t have a full belief in all those propositions. You must retreat to a partial credence in at least some of these propositions.

    My point here was just that the preface paradox, as it is usually stated, doesn’t actually involve any logical inconsistency among the contents of one’s beliefs at all.

  3. I’m a proponent of the view that there is no fundamental difference between the preface and the lottery, although I recognize this not a mainstream view.

    I’m not sure that credal belief (degrees of belief) works very well at extreme points, i.e., very near 1, and very near 0, particularly for measurement cases like this one. One thought is that we need to accept come things before we can get our probabilistic machinery off of the ground, and it seems to me that we don’t build our probabilistic models atop partial credences.

    There is a tendency (in the literature) to say “Ah ha! There is a probability distribution under this particular example. And we know the properties of Kolmogorov probability structures, so [enormous leap] of course one should adjust his attitude from full belief (interpreted to be probability 1) to some degree of belief about the events in question for the sake of consistency.”

    The ironic twist to all this is that the lottery paradox was conceived to be short-hand bumper-sticker for why the Carnap-Jeffrey approach credences opens more problems than it solves; it was not viewed by itself to be an argument against conjunctive closure.

    But the literature on the lottery, sometime in the 70s, seems to have become unhitched from the underlying motivations behind the the lottery, which are issues within the philosophy of statistics, and epistemologists instead started talking about lottery tickets and books, whether or not a probability distribution was under the example, whether there was or not a logical contradiction. Pollock is probably responsible for installing the idea that there is a fundamental difference between the Preface and the Lottery. Then people started talking about the lottery (at least) as a mistake in reasoning rather than a genuine paradox.

    This is a long way of saying that, in my view, the discussion of the preface and lottery starts with considering whether one can model a theory of measurement in terms of credal beliefs.

    I don’t think this is nearly as pretty nor as pat as mainstream epistemologists imagine it to be. (Which is to say that it is terribly interesting!)

  4. Thanks again, Gregory! Of course I agree that these epistemological issues are neither pat nor pretty, but are terribly interesting.

    Still, I don’t think that I have to commit myself to any very specific views about the nature of partial credences in order to defend the claims that I make about the lottery paradox. In particular, I don’t have to commit myself to probabilism — i.e., to the idea that the degrees of belief that it is rational for a thinker to have can always be represented by means of a probability function. All that I have to commit myself to is the idea that there is some distinction to be drawn between a partial credence and a full belief. A partial credence is a state in which one attaches some credence to the relevant proposition, but also “hedges one’s bets,” by attaching some credence to other incompatible propositions as well — whereas a full belief is a state in which one simply believes the proposition, without hedging one’s bets in that way.

    My point was that there is a fairly straightforward difference between the lottery paradox and the “preface paradox:” the beliefs in the lottery paradox concern a definite sequence of items, and so it is easy for the thinker to “survey” the relevant set of propositions — whereas in the “preface paradox” there is no way of surveying this set of propositions without resorting to some higher-order quantification over propositions or the like. But then there may be absolutely no logical contradiction between the content of the higher-order belief At least one of my beliefs is false and the various other propositions that one believes.

  5.  I haven’t read Evnine (1999), so I might be missing something, but I see two distinct problems raised for the thesis that believing the “Preface Proposition” necessarily involves having inconsistent beliefs, and only one seems to me a genuine problem.
     Suppose I have good evidence for the following pair of beliefs:
    1. The tallest human being in the world is a male
    2. Jane Smith is a female
    Suppose also that in fact Jane Smith is the tallest human being in the world. The two beliefs are still not such that there is no possible world in which they are both true. But now surely the evidence that I have for 1. is also evidence for
    1A. The actual tallest human being in the world is a male
    1A and 2, supposing gender is an essential property, are such that in the situation imagined there is no possible world in which they are both true. But still, it seems that they are not logically inconsistent (of course, it would be hard work to make clear what “logically” means, but that is a general problem; I am just assuming necessary truth not to be sufficient for logical truth). They only become inconsistent when I add to my beliefs
    3. Jane Smith is the tallest human being in the world
     But 3. is also inconsistent with the original pair; so I think the modal point is not relevant, mutatis mutandis, in the preface case as well (if you believe the Preface Proposition then you should also believe that not all your actual beliefs are true); the real problem is the impossibility, in cases such as the generalized version of the preface, of having a complete list of one’s beliefs.
     To make the parallel explicit; we might have the following beliefs
    1. At least one of my actual beliefs [insert a restriction if you like] is false
    2. B1

    N+1. Bn
    And we still don’t have a contradiction unless we also believe
    N+2 All my actual beliefs [relevant restriction] are included between B1 and Bn.

    In the generalized version of the Preface, we are unable to get a justification for N+2
    There is a further issue; I take it that the problem with the generalized version is not that B1…Bn is a list too long to be kept in mind. A mathematician can understand Fermat’s theorem’s proof, and believe it, even if she is not able to keep it all in mind; and if she were to still believe the negation of the theorem, she could not avoid the charge of inconsistency just because she cannot remember all the passages of the proof by heart. So I think you do get an inconsistency in cases such as the 600 hundred pages book. And those cases are indeed akin to the lottery, as far as I can see.

  6. For some strange formatting problem, there are two squares and a quotation mark in my post every time there should be closing quotation marks

  7. Ciao, Daniele! Let me clarify: only one of the “two distinct problems” that you mention was a point that I intended to make. I completely agree with you that the modal point (that there is a metaphysically possible world at which all those propositions is true) is not relevant. What is relevant is the logical point (i.e., in model-theoretic terms, that there is a model of the appropriate kind in which all those propositions are true, or in proof-theoretic terms, that there is no way of deriving a contradiction from those propositions).

    The problem also isn’t that this set of propositions can’t be “kept in mind.” The problem is that I have no reliable way of identifying a single proposition that is logically equivalent to the conjunction of all and only those propositions. So I can’t express my sense of my epistemic fallibility by simply denying that proposition; instead, I have to go higher-order, and utter a sentence like “At least one of the propositions that I believe is false.”

    So at least one difference between the “preface paradox” and the lottery paradox seems to be that there is a straightforward way of listing the lottery tickets (or Gregory’s flag poles) so that in those cases I can identify a proposition that is equivalent to the conjunction of all and only the relevant propositions (“None of the lottery tickets from 1 to n will win”) — whereas I have no reliable way of identifying a single proposition that is equivalent to the conjunction of all the claims in the 600-page book.

  8. This is fun!

    I think that this identification property is an artifact of particular thought experiments, and not a durable feature that is helpful to draw a distinction, i.e., not a feature that we can hang a solution on. So, I don’t want to deny that there are differences between examples; but I have grown skeptical of getting very far with this paradox (these paradoxes, if you prefer) from looking too closely at the particular features of the versions of the thought experiments.

    Perhaps this difference in view is already clear. So, let me follow up on an earlier suggestion of Ralph’s, about hedging one’s bets. It will start out a bit tedious, I’m afraid, but I shall try to wheel the discussion around to make a broader point about this less than clear fit between logical rules and reasonable attitudes.

    In fact, why don’t I put this in a different post. I am a bit worried about steering this thread off its natural course.

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  10. I still think I might be missing something; suppose after writing my 600 pages tome I decide to read it in front of an audience of faithful disciples; before starting, I modestly remark that “at least one of the propositions contained in the book is false”. When I reach the end, to clarify I am finished (or perhaps in order to wake some people up), I add “and those where all the propositions contained in the book”. Haven’t I made a logically inconsistent set of assertions?

  11. Daniele, I still don’t see how that is a logical inconsistency. Suppose that I point to someone — say, in fact I point to you — and say “That man is not Italian,” and then simultaneously I say “Daniele Sgaravatti is Italian.” Is this a logical contradiction? I don’t think so!

    Similarly, suppose that I say, “That proposition is true” (pointing to some proposition) and at the same time say “The proposition that p is not true”. Even if the demonstrative “that proposition” in fact refers to the proposition that p, I don’t think that my statements are logically inconsistent with each other (although, to be sure, it is impossible for them both to be true).

  12. I agree with all you say; but the statements become inconsistent, I think, if you add to them respectively “and that man is daniele sgaravatti”, or “and that proposition is the proposition that p”; it’s not the fact that those statements are true that makes my beliefs inconsistent, but rather my adding them to my beliefs.

  13. Well I certainly agree with that point. But if I assert thousands upon thousands of propositions p1, …, pn, and then say “One of those propositions is false”, that is quite different from then saying “Either p1 is false or … or pn is false.” It’s also quite different from saying “One of those propositions is false” and then adding “Those propositions are p1, …, and pn”. This last sentence is too long and complicated for me to grasp the proposition that it expresses, and that, it seems to me, makes a big difference. (So the problem isn’t that I can’t keep all these different propositions in mind at the same time. The problem is that to get a logical inconsistency here, we need a single proposition that is too long and complicated to be grasped at all.)

  14. What if I individuate the propositions p1 to pn through (instead of the demonstrative) a property they have , such as being named in the book (suppose it’s a book Tractatus-like) as p1 to pn; and I then believe each of the propositions in the set made by p1 to pn, the Preface-statement, and “propositions in the list from p1 to pn are all the propositions contained in the book”. I seem to grasp each of the propositions in the set, including the last one.

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