Yes. Far too advanced for most.

That’s a pretty lame critique.

He gives no examples of what he claims to find when he says “his methods press well beyond the confines of any form of axiomatic probability theory” … okay, where exactly do I do that? And in what way would that even be bad, since not everything is about axiomatic probability theory?

I have no idea what he means by “in general, heuristic arguments are not only invalid but also useless from a purely logical framework.” What is he critiquing with that statement? What heuristic arguments? What in my book is he claiming is a heuristic argument? And what does he mean by saying all heuristic arguments are invalid and useless? (Really? All heuristic arguments, by definition?)

His claim that “most irritating was Carrier’s insistence that proving something to be unlikely is equivalent to proving something false” seems to be ignorant of basic epistemology (if being unlikely is not what we mean by saying something is false, then nothing can ever be claimed to be false, since everything has a nonzero probability of being true: Axiom 4, pp. 23-25).

His reference to the Banach-Tarski Paradox has no discernible relevance to anything I argue in my book. He doesn’t even provide an explanation of what relevance he thinks it has to anything I actually argue in my book.

As far as his claiming “I would consider his entire premise suspect, due to his insistence on applying subjective quantities to an objective theorem and general lack of mathematical rigour,” that suggests he didn’t actually read the book, which addresses that objection extensively and in detail (so if he has no argument against what my book says about that, and it appears he does not, then it appears he did not actually read the book).

And as far as his vague and undefended complaint against my “fast-and-loose treatment of mathematics and logic,” I think my article here (above) addresses that more than adequately.

http://www.amazon.com/review/R392IPXC3QP131/ref=cm_cr_pr_viewpnt#R392IPXC3QP131

Carrier: “The short answer is that Cartesian Demons have vanishingly small priors and therefore can be ruled out ”

Ian: Not in situations where we’re explicitly being asked to determine if a supernatural event occurred.

Untrue. The nature of the claim makes no difference. Either way you are still positing an entity with an extremely low prior. Indeed, a lower one, since a God who acts like a Cartesian Demon is inherently less likely than a God in general (this is thus just another instance of gerrymandering that halves the prior, at best: as I explain on pp. 80-81, and that’s even assuming that there is a straight 50/50 chance that if God exists, he acts like this, which IMO is an absurdly generous hypothesis, especially if one tries to make it compatible with the definition of any God anyone actually believes in).

Its a different kettle of fish to finding correlations in gene sequences and ignoring the possibility that a deceptive God is playing with our instruments. We’re not talking about a Cartesian Demon here, but a teleological purpose for the evidence, that is related to the hypothesis. I think there’s a very good reason to enforce methodological naturalism on the discussion in all cases and to explicitly disregard any supernatural event. Sure, you lose any leverage over true believers, then, but I suspect you didn’t have much of that to start with.

I fully agree with all of this. But that last point has to have a logically valid reason. BT provides that reason. And the first point makes no difference to what I was saying: you do not know the prior probability of a meddling god, yet you know it is low enough to exclude it. Thus “not knowing the prior” is not a valid argument against modeling reasoning with BT, whether in science or history.

Carrier: “It is simply not necessary to know exact numbers in order to know general facts like these. That’s why human reasoning is even possible in the first place. We couldn’t find our way to the bathroom otherwise.”

Ian: Of course this is true. But I simply don’t see how it addresses the point.

That depends on what the point was.

If it was that some cases are undecidable on BT, that point is moot. Because that simply translates what is already true generally: with or without BT, some cases are undecidable. BT just tells us when and why.

But if it was that BT cannot usefully model historical reasoning because the probabilities are always fuzzy, it certainly addresses the point. Because “being fuzzy” does not translate to “are not known to any useful degree.”

Almost all human reasoning, and historical reasoning especially, deals in fuzzy probabilities. That does not make such knowledge impossible–not generally, nor in the hands of BT.

So on neither the first point nor the second do I see any relevant objection being made.

That leaves the possibility that your point was something other than those two things. In which case, you’ll need to explain what your point is, in such terms as to make clear how it is not either of those.

MalcolmS: …when one does an a fortiori argument one needs to use ranges for the inputs (P(H), P(E|~H), P(E|H)).

You don’t need to give a range. Because the other end of the margin is irrelevant. Only one margin is relevant: the one that draws the line between whether you are right or wrong, and does so beyond where that line actually is, in favor of your being wrong. Hence producing a conclusion a fortiori (“from the stronger reason”). The other margin does not produce an a fortiori argument but an a tenuiori argument, which is a fallacy (because that line between whether you are right or wrong is the weakest, being much less likely correct, making your argument look the stronger but actually making it weaker). You therefore don’t need to bother with that other bound.

That doesn’t mean you can’t work with different ranges, if there is something useful in doing so, e.g. to show what the conclusion is with different assumptions.

For example, I estimate the prior probability of a passage about Jesus in a non-Christian source before the 4th century being an interpolation is a fortiori better than 1 in 200. We could say the other bound is then 1 in 1 (i.e. that they are all interpolations and no evidence could ever show otherwise), but what use is that? None. Obviously if the most against-interpolation bound (1 in 200) produces a result, in the face of the evidence for a specific passage, that that passage is probably an interpolation, then using the other bound (1 in 1) will make the probability that that passage is an interpolation much higher (in fact, it automatically becomes 100% no matter what evidence you did or didn’t produce). But of what use is knowing that? None. Only the a fortiori bound produces a conclusion that can be at all persuasive. Moreover, only the a fortiori bound produces a conclusion that we can have a high confidence in. I cannot have a high confidence in a conclusion produced from the other bound (of 1 in 1, or even anything close to 1 in 1). So the conclusion that comes from that bound produces no appreciable confidence. It’s therefore useless even to us, much less for the task of persuading others. And this is true even in scenarios like you describe.

But what if the lower bound for P(E|~H) is also zero?

It depends on whether the “lower” bound here is a fortiori or not. If not, it’s moot (since, as I just explained, such a bound cannot produce an a fortiori argument nor generate confidence and is therefore useless). But if it is a fortiori, as are all the other bounds being used, and the scenario is as you describe:

Thus in the case where both P(H) (or P(E|H)) and P(E|~H) are known to be small, but we can’t say which one is larger, we will get a range for P(H|E) of 0 to 1.

If we assume P(E|H) is high, then this translates to: we don’t know. You have just mathematically modeled a class of historical claims whose truth is unknowable on present evidence. That is not a problem. Because we already know that most historical claims are such. Unless you want to claim all historical claims are described by this scenario; but clearly you can’t be saying that. So what use is this as an objection to anything I argue in PH? That some claims are undecidable is already affirmed repeatedly in PH.

But let’s instead imagine a scenario in which all three are low (P(H), P(E|H) and P(E|~H)), that is, the a fortiori bound for each is somewhere uncertainly close to zero.

Note then that the key statement here would be “we can’t say which one is larger.” In the case of the likelihoods, that translates to: we do not know the ratio of likelihoods would be in the Odds Form of BT–in fact, not only do we not know them, but we don’t even know them a fortiori. Which means, for all we know, that ratio is 1 to 1: we have no knowledge establishing it is any higher, nor any knowledge establishing it is any lower–because if we had either, “we can’t say which one is larger” would be false, and the scenario would not apply.

But if for all we know that ratio is 1/1, and the ratio of priors is approaching 0, then we should conclude H is probably false. Until we get information that allows us to argue that the ratio of likelihoods (i.e. the evidence) favors H over ~H.

Indeed, that’s the definition of “having evidence for H” : having a ratio of likelihoods that favors H.

The question then is how much evidence favoring H do we need in order to argue that H is true when H has a very small prior. That is the unusual scenario you are describing. And that gets into a whole slew of other questions.

If we are dealing with an absurd claim, then we will have a vanishingly small prior and a vanishingly small likelihood ratio. We should conclude against H.

That leaves only one scenario to be concerned with: one where the a fortiori prior for H approaches very near to 0 but the ratio of likelihoods is vastly in favor of H. In that case we can’t use “1” and “0” because now we are dealing with a case where we have to start taking seriously the boundaries of our epistemic certainty. For example, as I said before, though I suspect the prior probability of miracles is 0, my confidence in that bound is not epistemically high. It therefore can never be an a fortiori bound.

Thus, if I were faced with a case where the prior is vanishingly small but the evidence is extraordinarily strong (this has never happened, and is a very bizarre scenario–a red flag for any philosophical argument: if you have to create a completely unrealistic scenario in order to make a point, odds are the point wasn’t worth making, especially if your aim is to discuss how to approach reality), then I have to ask what I think the epistemic prior probability of (let’s say) a miracle really is, in other words how much evidence will finally convince me miracles exist (which translates to: how improbable P(E|~H) must be, relative to P(E|H)).

I know there is some such probability, as I have described scenarios that would persuade me before (in Why I Am Not a Christian, for example), so all I need do is figure what my probability estimates really translated to in those cases, in particular the a fortiori bounds, and then benchmark back to the case at hand. And that can then be open to further debate, if someone wants to insist I’m wrong to set that benchmark. And so progressive debate ensues.

I actually discuss these kinds of bizarre scenarios (and how they differ from actual scenarios of low P(H) and P(E|~H)) in PH, pp. 246-55 (but see again pp. 243-46 for context). The example of Matthias the Galilean industrial mechanic, and what evidence would it take to persuade a historian that such a man existed, is exactly on this issue (being a realistic example): of starting with very low priors, but then getting good evidence (yet notice here the most plausible a fortiori prior, the lowest we can reasonably believe it to be, will not be zero, or anywhere near as low as in the case of successful alchemy or sorcery, the counter-examples I explore).

So, to adapt the Matthias the Galilean industrial mechanic example to your numbers:

Here’s a numerical example. Let’s say we can say with confidence that P(H) < 5% and P(E|~H) < 2%.

This cannot be a fortiori. Because your P(H) < 5% is the wrong bound (the useless one). The bound we want is the lowest we can reasonably believe P(H) to be, which I proposed is P(H) > 0.000001. I also explored the highest reasonable prior and found it to be 0.002, but that is too high, because I know the actual prior is less than that, and so I cannot use that with any confidence. I can accept debate over the other bound of 0.000001, however, since one might make an evidence-based case that that is too low (in fact, I actually do believe it is too low), but even then all they would do is end up making a case that the a fortiori bound is somewhere else (the whole point of that section: these are the kinds of debates historians should be having), although I suspect it will still be closer to 0.000001 than to 0.002.

Your P(E|~H) < 2% however would then be the right bound, since to make an a fortiori argument we want to know the highest this probability could reasonably be. And if that’s what we had, a 1 in 50 chance a source is lying or in error (or whatever), and we were confident the odds were at least that high, then we wouldn’t have sufficient evidence to believe in Matthias the Galilean industrial mechanic; we would believe that the source probably made him up. But of course that assumes that’s what we premised, that the source is that unreliable on claims like this. And that might be very arguable; indeed, it might not be a reasonable belief at all, no confidence being warranted in so high an estimate of the likelihood of fabrication on that kind of point (and so on).

For example, finding a sarcophagus in the Palestinian region for Matthias the Galilean industrial mechanic, much like the one we have found in Turkey (cf. n. 36, p. 331), would not have a 1 in 50 chance of being forged or in error; the odds of that would be many millions to one. It would therefore more than overwhelm even an a fortiori prior of 0.000001. When we turn to the case of a historian referring to him, then the matter may be more complicated, and may end in uncertainty–for example we could conclude that the historian’s reliability on such details must be at least X (X being a frequency of correctness on such points, and thus the prior) in order for us to be confident that Matthias the Galilean industrial mechanic existed (rather than “was made up” etc.).

This X might be inside the range of uncertainty and thus not capable of making an argument a fortiori. In which case we would state as much: in colloquial terms, we would say he might have existed, that it’s plausible but we’re nor sure; in exact terms, we’d say that we can be highly confident he existed only if we adopt assumptions (about the prior probability and/or the likelihood of fabrication or error) in which we are not highly confident, which means we can be confident neither that he did exist, nor that he didn’t (that latter distinguishing a case like this, from a more absurd case like alchemy: see again the distinction drawn between plausible unknowns and effective impossibilities in Axiom 5, pp. 26-29).

Now what is our possible range for P(H|E)? Still 0 to 1. So tightening our range didn’t help at all.

Here you are making a moot point. That the “possible range” includes many other values is of no use knowing. Because that “possible range” will include things like “the frequency of interpolated passages about Jesus in non-Christian literature before the 4th century is 100%” and no a fortiori argument can proceed from a premise like that. So that our “possible range” includes it is irrelevant. We don’t care what the “possible range” is. We only care what the a fortiori result is. And that only uses one bound for each value. It therefore does not produce a range, other than “X or less” or “X or more” (depending on whether we are arguing a fortiori for or against H).

When I made my previous post I had claimed that your example of a claim of worldwide darkness in 1983 falls into this type of situation, but I had overlooked the fact that in your book you explicitly stated that P(H) < P(E|~H)/1000. In that kind of case, where one has a bound of the ratio P(H)/P(E|~H), then this problem is avoided. But having such an estimate for the ratio is crucial; without it one couldn't say anything about P(H|E).

Indeed. This is why I also discuss the Odds Form in the book (someone having rightly convinced me of its importance) and why I discuss the tactic of employing artificial ratios even when using the straight form (as in my discussion of neighbors with criminal records on pp. 74-76).

So that can’t be the problem.

That leaves only this…

For that specific example in your book I also would dispute the estimate of P(E|H) ~ 1, since one must take into account that one had never heard about this claim until recently, and also the fact that one had not witnessed the darkness (or a report of it) even having been alive at the time. For example, if one said that P(E|H) = 0.1%, then one gets about 50% for P(H|E), and this is true no matter how small you choose P(H) and P(E|~H) so long as their ratio is 1:1000. Thus whether H is almost certain, fifty-fifty, or even improbable depends critically on the exact ratio of P(H)/P(E|~H) to P(E|H), which would require a more elaborate argument one than you present in your book.

You are introducing elements not stipulated in the analogy. That makes this a straw man argument. I never said anything about “one had not witnessed the darkness despite being alive at the time” nor does “one had never heard about this claim until recently” make a difference if, for example, you are in school and hearing all sorts of things for the first time. In other words, if “hearing about this claim for the first time” is not unexpected, it makes no difference to the consequent probability; you have to stipulate that it is unexpected, which changes the scenario.

Obviously, if you change the scenario that has been stipulated, then you change how it gets modeled in BT. That is not an argument against a fortiori reasoning.

If we stipulated the scenario you do, that we have no personal memory of the event even though we should have, and only just now are hearing about it and that this would be strange, then indeed we may be in a state of indecision, given all the other evidence there is. That simply isn’t the scenario I posited. But we could posit it, as an example of a bizarre scenario where we might not be able to know what is true. That just wouldn’t be relevant to the point I was making with the analogy there (which was to illustrate what it would take to convince us, not what it would take to produce uncertainty). Nor would that scenario be analogous to any actual situation we are ever in (since I cannot think of a single “comparably incredible claim” for which I have such a comparably vast scale of evidence contradicting my own memory, outside of a Philip K. Dick novel).

In short, I see no objection here to a fortiori reasoning. All I see is a recognition that some claims are unknowable (both in practice, and in extremely bizarre fiction). Which is already argued in Proving History.

BT accomplishes this not so much based on mathematical rigor, but from explicitly stating premises and arguments.

Spot on.

Dr. Carrier,

You still don’t seem to grasp the full extent of this a fortiori reasoning problem. I’ll try to make my point again, hopefully more clearly.

First of all, even though probabilities are not 0 (unless something really is logically impossible, in which it probably would not be an interesting historical question), when one does an a fortiori argument one needs to use ranges for the inputs (P(H), P(E|~H), P(E|H)). In the case where one cannot give a minimum estimate for a probability the lower bound would be 0. For example, if we say that we know P(H) is very small, say, lower than 1%, then our range for P(H) would be 0<P(H)<0.01, so one still has to deal with a zero bound in calculations, even though the probability itself would never be zero.

So if the lower bound for P(H) or P(E|H) is zero, then the lower bound for P(H|E) is zero. So far no problem. But what if the lower bound for P(E|~H) is also zero? In that case, this gives an upper bound for P(H|E) of 1 (because a lower bound for P(E|~H) gives an upper bound for P(H|E)). Thus in the case where both P(H) (or P(E|H)) and P(E|~H) are known to be small, but we can't say which one is larger, we will get a range for P(H|E) of 0 to 1.

Here's a numerical example. Let's say we can say with confidence that P(H) < 5% and P(E|~H) < 2%. (P(E|H) doesn't matter so much in this example, so let's say it is exactly 1.) What is our possible range for P(H|E)? 0 to 1, i.e., we know nothing. Now let's improve our bounds on P(H) and P(E|~H) to, say, 1% for both. Now what is our possible range for P(H|E)? Still 0 to 1. So tightening our range didn't help at all.

When I made my previous post I had claimed that your example of a claim of worldwide darkness in 1983 falls into this type of situation, but I had overlooked the fact that in your book you explicitly stated that P(H) < P(E|~H)/1000. In that kind of case, where one has a bound of the ratio P(H)/P(E|~H), then this problem is avoided. But having such an estimate for the ratio is crucial; without it one couldn't say anything about P(H|E).

For that specific example in your book I also would dispute the estimate of P(E|H) ~ 1, since one must take into account that one had never heard about this claim until recently, and also the fact that one had not witnessed the darkness (or a report of it) even having been alive at the time. For example, if one said that P(E|H) = 0.1%, then one gets about 50% for P(H|E), and this is true no matter how small you choose P(H) and P(E|~H) so long as their ratio is 1:1000. Thus whether H is almost certain, fifty-fifty, or even improbable depends critically on the exact ratio of P(H)/P(E|~H) to P(E|H), which would require a more elaborate argument one than you present in your book.

Therefore I assert that this case is actually one of them where BT can't help us too much in getting a good handle on whether the claim is true or not. Of course it's not just BT that has this problem – any attempt to reason this out logically would founder on the same problems as well. In fact, BT has an advantage here in that it allows one to see precisely how and why this claim is difficult to evaluate, whereas if one were just to respond intuitively one might not grasp how sensitive one's answer is to small changes in assumptions.

More importantly, this thread is great stuff, really digging in and mucking around in the weeds, but, it’s still in the weeds. I’d like to ask Carrier to rise up out of the weeds and give a big picture summary of this debate. It seems clear to me from both the book and lots on this thread that Carrier’s main point is to improve the results of historical debate, by exposing fallacious reasoning and providing a framework to moderate disputes. BT accomplishes this not so much based on mathematical rigor, but from explicitly stating premises and arguments. Put another way, BT for history is more logical and less statistical. Ian seems stuck in the statistical weeds. He may have very good points, but viewed from afar, it is just a small russle of the grass. Is that right? I also suggest including this in one of your upcoming posts, as very few will have gotten through all the weeds above. Thanks.