John Turri’s Rounding Error
This post is devoted to refuting an impressively lame epistemology paper. John Turri presents a novel line of reasoning which, he claims, allows one to gain a priori knowledge of contingent facts. The relevant section begins on page 19:
Sam considers whether the most unlikely possible event is not presently occurring. By ‘the most unlikely possible event’, Sam intends to designate whatever was, at the immediately preceding instant, t – 1, the possible event most unlikely to occur at the next instant, t, which is the moment at which her deliberation occurs. Sam understands the proposition in question. Solely in virtue of that understanding, it seems to her—i.e. she intuits—that the proposition is true, though not necessarily so. On the basis of this intuition, she believes that the most unlikely possible event is not presently occurring. Her belief is true.
To be a bit more precise, Sam starts out believing “X will not occur” with probability 1-ε. By hypothesis, she has some justification for believing so. After drawing on her powers of intuition, she believes “X will not occur” with probability 1. Turri claims that this tiny increase in certainty is justified.
Turri never explains why he builds this argument using the “most unlikely” event instead of an unlikely event, but he does admit that ε must be small in absolute terms for his argument to work. He arbitrarily proposes an upper bound of 10-18 for ε.
I maintain that Sam’s intuition has committed a rounding error. Under the circumstances Turri sets up, there is abundant reason to mistrust human intuition. One need travel no farther than the nearest casino to observe humans attempting to do probabilistic reasoning by intuition. Human intuition doesn’t just commit tiny rounding errors, it commits big errors — errors of much greater magnitude than 10-18 — and commits them quite frequently. Given this extremely spotty record, an optimist might hope that Turri would proceed to demonstrate that in this particular case intuition guides us true.
If [a philosopher] reproached Sam, “Pardon me, Miss, but you of course meant that [the unlikely event] is almost certainly not occurring,” she could rightly respond, “Oh, come on! It’s not occurring, and we all know it.”
That’s Turri’s argument.
Actually, there’s a little bit more to Turri’s argument, but it’s really nothing more than semantic smoke and mirrors. He worries extensively about the definitions of knowledge and assertion. Personally, I think that one can do epistemology quite well without ever using those words. But for the sake of argument, I’ll accept whatever definitions Turri happens to prefer. My question is this: are these concepts (however you’ve chosen to define them) suitable for addressing delicate probabilistic reasoning and small variations in degrees of certainty?
If the answer is No, then there simply isn’t any difference between gaining an ε of knowledge and gaining zero knowledge. There’s no longer a meaningful distinction between knowing/asserting that X is true with probability 1 vs. probability 1-ε.
If Turri wants to answer Yes then I want to see him explicitly wed those concepts to the axioms of probability and redo his argument in a correspondingly rigorous fashion.
[...] Jacob Wintersmith objects to a novel line of argumentation made by John Turri which aims to establish the possibility [...]
Jacob,
I think the proposition Turri has in mind, when he is arguing for contingent a priori knowledge, is not that x is not now occurring, but instead that the most unlikely possible event is not now occuring. These are two distinct propositions where the event in question is picked out in two distinct ways. A subject may not have any beliefs about x (perhaps it is some quantum event), but could still know the latter. And know it not because of some intuition he has about x’s probability, but instead because he has picked it out as ‘the most unlikely posssible event.’
I do have some doubts about whether the knowledge in question is truly a priori though.
If I failed to keep those two propositions distinct it’s because I really don’t see how this helps Turri’s argument. What is important here is that the event in question (regardless of how we pick it) has a very low probability of occurring. Turri himself admits this. He then tries to claim that we are justified in believing, with absolute certainty, that the event is not occurring.
Jacob,
You wrote:
He then tries to claim that we are justified in believing, with absolute certainty, that the event is not occurring.
I’m not sure that’s quite right. What he’s claiming is that the belief in question could constitute knowledge if true and held on the basis he describes. He doesn’t need the absolute certainty claim unless we were to insist that nothing could constitute knowledge unless it was absolutely certain. I’m not 100% on board with Turri’s argument, mind you, but it seems we could believe and know that we will be at work tomorrow for lunch even though there is some chance/probability that we will have a heart attack before then (and we can build in the assumption that we don’t work in morgues). I take it that Turri’s challenge will be (in part) to explain how we could accept the non-sceptical view that we have knowledge of contingent matters of fact since (a) it seems that we could have such knowledge only if knowledge doesn’t require absolute certainty when (b) it seems that the only reason to regard his case as a case of ignorance is that knowledge requires absolute certainty. I don’t have a good answer to that challenge. I sort of feel that he’s wrong, but that’s not really an argument.
Clayton,
If that is what Turri is attempting to do with this argument then I agree that the conflict between (a) and (b) is the core problem.
To put it a bit more sharply: If you define knowledge in a way that washes away small differences in degrees of certainty then “I know X will not occur.” follows analytically from “The chance of X occuring is less than ε.” — no intuition needed. Or rather, it would, if Turri were to explicitly lay out the connection between “knowledge” and probabilistic degrees of certainty.
This sort of mess nicely illustrates why I think talking about “knowledge” causes a lot more problems in epistemology than it solves.
Yes, the conceptual leap Turri makes is obvious. My understanding of probability tells me that if the likelihood of an event happening has the probability of 1, then it will happen. This is logical regardless of our definitive parameters of knowledge (Clayton’s a or b), though it depends on a common understanding of probability.
Turri essentially puts forward that human intuition catalyzes 1-ε = 1.
Based on my understanding of terms, the probability that 1-ε ≠ 1 is 1.
Human intuition may play a part in what confidence interval we apply to an estimation of a probability, but that’s all I’m personally willing to concede.