Paradox vs. Surprise

A paradox is different from a result that is merely surprising, but what is the difference? This question touches on matters beyond epistemology, but it is applicable to the major epistemic paradoxes, including preface, lottery, surprise quiz, and knowability. It is the latter that prompts my question.

In the knowability paradox, we purportedly demonstrate that if all truths are knowable, then all truths are known. There is no question that the result is surprising, but what makes it paradoxical? Compare it with Godel’s incompleteness theorems, for example, which are also quite surprising, but not paradoxical. Or compare it to the ontological argument, where it is purportedly shown that if a certain description is possibly exemplified, then it is necessarily exemplified. This, too, is quite surprising, but I doubt it is paradoxical. So, what is the difference?

Perhaps the difference is psychological. Logical results are suprising when they go beyond what we presently believe to be true, and when they concern issues of significance to us. We notice the result which, prior to the proof, we doubted; after seeing the proof, we are convinced and thereby surprised. When the result is paradoxical, however, something additional happens. The proof threatens our intellectual commitments in some way, it threatens our firmly held opinions on matters that are significant to us. Admitting the soundness of a proof to the contrary thus engenders a bit of mental apoplexy: we know something has to give, but it’s hard to see what.

Is there a different account of the distinction? I’m not sure; if you have a different account, please share it. But if we suppose that this account is on track, one has to dig a bit to find a paradox in the knowability result. The result is a conditional: if every truth is knowable, then every truth is known. That’s not a denial of any deeply entrenched viewpoint I hold on issues that are significant to me. So why the fuss? I think there is something paradoxical in the neighborhood here, and I think it has important lessons. But since it depends on the psychological account of the difference between surprising and paradoxical derivations, I’ll hold off to see if there might be a better account of the difference.


Paradox vs. Surprise — 8 Comments

  1. Would it be a paradox, or merely surprising, if the ‘Knowability Paradox’ turned out actually to be a surprise, and not really a paradox? 😉

    Slightly more seriously: I doubt that the concept of the paradox would sustain a very high intensity of analysis. It’s too much a term of art, and not enough a folk term. (Consider what gets termed a paradox in Pirates of Penzance, for example — not something that philosophers would typically call one, I would think.) I imagine that there are a number of different interesting ways of defining “paradox”, each of which would capture a large set of what we typically label with that term, and each of which would leave out at least one or two such typically-labeled puzzlers. So everyone should feel free to draw the paradox/non-paradox line wherever it best suits their theoretical interests.

  2. Jonathan, you’re right that there’s not much hope of finding a natural kind here. And your joke about the knowability paradox is exactly on target–Williamson, for example, holds that the result is merely surprising and not paradoxical at all.

    So suppose we restrict the question to things of philosophical interest. In our own context, what’s the difference between a merely surprising logical result and a paradoxical one?

  3. But is there even one unique context of philosophical interest? I would think that different philosophical interests might draw different lines here. If you’re operating within a very formal branch of philosophy, you may be inclined to draw the distinction along the lines of logical form: a paradox involves an argument to a conclusion that is a logical contradiction, whereas a surprise lacks that form but is merely, well, surprising. I suppose that would render the lottery paradox a paradox (something like
    (Ex)Fx & (~Fa & ~Fb & ~Fc & …) & (Ax)(x = a v x = b v …)
    But the result of the knowability ‘paradox’ comes out as just a surprise.

    But working within a context that allows for more substantive appeal to human psychology — and I take it that your most natural working context is one of these — one could indeed give a ‘cognitive apoplexy’ definition of a paradox. (Which may have to be relativized to individual psyches: e.g., general relativity is apoplexy-producing in most folks, but we defer such matter to a community, namely professional physics, that finds it merely surprising.) Maybe on this account knowability is a paradox (for most of us), too.

    And so on — lots of contexts, lots of theoretical interests, and lots of resulting notions of paradox. For example, I can imagine an Austinian or Wittgensteinian context in which anytime an argument compels you to say something that is wildly inconsistent with everyday usage, it would count as a paradox.

    So my point is merely: you don’t need to wait & see whether anyone else has a preference for a nonpsychological account of paradoxes. You can just plump for whatever way of drawing a line here as seems interesting to you, and see where it takes us.

    (Unrelatedly, and this may be a nitpicky kinda point, but nonetheless: I find it a little odd that you appealed to the notion of a natural kind here, as an instance of something we might’ve hoped paradoxes were. My understanding is that natural kinds — with their hidden essences and all — are paradigmatic of things that you just don’t at all expect to do much successful conceptual analysis on. We’re much better off trying to analyze concepts of social kinds, like “senator”, say, or “grandmother” (both of which are fairly easy to give analyses of). This isn’t an absolute rule — some social kinds may be hard to analyze, and some nonsocial kinds (such as mathematical ones) may be easily analyzable — but it’s a general trend.)

  4. Historically ‘paradoxical’ sometimes carried overtones of being contrary to common sense or general belief: so when, for instance, people in the 18th century called Hume a ‘paradoxical author’ they meant not merely that he said surprising things but that he said things contrary to what everyone believed. If we used this, then the difference between a surprise and a paradox would be that a paradox conflicts with some common intuition or near-universal assumption. So this would be something like a generalized version of your suggestion: a sociological account, we might say, rather than psychological one.

    I’m not sure, though, that this historical meaning has entirely survived to our day.

  5. Brandon, that’s a good suggestion. To go sociological helps explain why the Godel results are not paradoxical even though they threaten Hilbert and would contravene his deeply held beliefs.

  6. Jonathan, I’m really not looking for an analysis here. I’m just wanting to know how we tell the difference between surprising results and paradoxical ones. I need to be able to tell the difference if I want to argue that a particular results demands explanation, since if the result is merely surprising but not paradoxical, a legitimate response to the demand for explanation would be to go through the proof again and become convinced that it is sound.

    By the way, here’s why I think the knowability paradox is really paradoxical. It obviously follows from all truths being known that all truths are knowable, so if we grant Fitch’s proof, we have a logical equivalence between universally known truth and universally knowable truth. That involves a collapse of the distinction between actuality and possibility, but without the kind of semantic explanation we get in the similar collapse in S5. So what’s paradoxical is the lost logical distinction between actual universally known truth and possible universally known truth, and that contravenes our pre-theoretical assumptions about modality–not just mine, and not just those of philosophers, but of those who reflect on the matter even for the briefest time.

  7. How about this proposal:

    Paradoxes and surprises are similar in that they can be put in the form of a set of mutually inconsistent but individually plausible claims. For the paradoxes, we find it incredibly hard to decide which member of the set is false. The members of the set are roughly equally intuitive, even after tons of reflection; they are also highly intuitive.

    For the surprises, on the other hand, we do not find it all that hard to decide which member is false AND the one we think is false once was accepted as true. So the surprise is that we used to think it was true but now we see it is false. But it is no paradox because we think we know where the falsehood lies.

    However, I made this up thinking of paradoxes in metaphysics, not epistemology.

  8. Bryan, this is along the lines that I’m thinking, too. I don’t think we should require that a surprise occur regarding something that we used to think was true,though. I think Godel’s incompleteness results are surprising, but I never believed the formalism it threatens. I saw the temptation of formalism, but never thought it could be correct. That didn’t stop the Godel results from surprising me though.

    The important point, though, is the first one. Paradoxes involve claims that are intuitive, even after reflection. Faced with such results, we can’t just jettison one claim to solve the problem. We need a sound motivation to favor one way of resolving the conflict over others.

    In the context of knowability, this is very important since most approaches to the paradox fail to take into account what is paradoxical about the Fitch result. As a result, “solutions” are offered that don’t address the fundamental issue at all. For example, a standard anti-realist line is to find a way of refusing to endorse the knowability of all truth. But, if I’m right, the paradox involves a logical equivalence between possible and actual universally known truth, and what is paradoxical is the loss of this logical distinction. Refusing to endorse the universal knowability claim may preserve one’s theory of truth, but it won’t be relevant to the paradoxicality here.

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