Lately I’ve seen a flurry of repeated mistakes in reasoning about probability. I realized a primer is needed to correct some people so they can stop making those mistakes (assuming you care about not making mistakes; an alarming number of people don’t, but I can’t help them). I’ve discussed related issues in probability theory before (see any of my articles on Bayesian Reasoning, especially Advice on Probabilistic Reasoning, and likewise Innumeracy: A Fault to Fix and Everything You Need to Know about Coincidences). Today I’ve chosen three of the most common errors I’ve been seeing lately: the Fallacy of Uncompared Probabilities; the Fallacy of Unconditioned Probabilities; and the Fallacy of Neglected Total Probability. And to illustrate I’ll be using, again, the example of the historicity of Jesus (which I have discussed the mathematics of before, from Doing the Math: Historicity of Jesus Edition to Kamil Gregor on the Historicity of Jesus, and others to be mentioned here).

The Fallacy of Uncompared Probabilities

Someone recently produced an argument more or less like this: there’s a 75% chance the Gospels contain some historically true facts about Jesus, and a 75% chance Paul is talking about biology when he mentions Brothers of the Lord, and a 60% chance the Jewish execution of Jesus in 1 Thessalonians 2 has not been interpolated, and a 60% chance Josephus recorded something true about Jesus, and a 50% chance Papias knew actual eyewitnesses to a historical Jesus, therefore Jesus more likely existed. Because even if we assigned a 20% chance to each of these, there is still a better than average chance at least one of them is legit, because there are so many of them. In fact one could calculate, on this reasoning, the probability is 1 – (1 – p)^n or 1 – (1 – 0.2)^5, which is roughly 0.67, for at least a 67% chance Jesus existed. So doubting historicity requires too many things to be true.

That would be correct if we were talking about assumptions and not evidence. For example, adopting unevidenced assumptions to rescue a theory do stack to reduce its prior probability; and assuming repeated excuses across dozens of occasions does reduce the probability of the evidence on your theory (e.g. OHJ, pp. 518-19, n. 13). But centuries ago, Thomas Bayes discovered that that isn’t how evidence works. By the reasoning used above, because every probability is nonzero, and all facts are “evidence” to some trivial degree for any hypothesis, all conclusions would approach 100% certainty—even flatly contradictory conclusions. For example, if there is a 0.1% chance that each person who doubts the historicity of Jesus is right, and there are a thousand such persons, the probability that they are all wrong is vanishingly small, and therefore the probability that Jesus didn’t exist is 1 – (1 – 0.001)^1000, which is roughly 63%. So by the same reasoning we have arrived at a 63% chance he didn’t exist and a 67% chance he did exist, a logical contradiction. The method doesn’t work.

And you can’t escape this by claiming disanalogy. Because we can do the same thing with any chain of analogous evidence, too—in fact, the exact same evidence. If there is a 20% chance each of those same five mentioned items of evidence attests a historical Jesus, there could be, say, a 30% chance each attests a non-historical Jesus…or a 60% chance—or even an 80% chance! So the probability that all five would fail to attest a non-existent Jesus could then be 1 – (1 – 0.8)^5, which is 99.968%. So the historicist requires too many things to be true. In fact, we even win by this reasoning, because it’s over 99% that at least one of those items of evidence proves Jesus didn’t exist, but only 67% that one of them proves he did. But of course this is all nonsense. Neither of those probabilities can be simultaneously true. This is simply an illogical procedure, start to finish.

What we are seeing here is the Fallacy of Uncompared Probabilities, where simply “multiplying probabilities” in neglect of the alternatives gives us contradictory outcomes. In actuality, the probability of a conclusion is not simply any kind of product of all the probabilities that each item of evidence attests a particular conclusion. If you say we need five things to be true for a conclusion to be true, then the probability indeed goes down; but if the alternative also needs five things to be true, its probability also goes down. But how can both probabilities go down? When they sum to all possibilities, they should always sum to 1 (because there is a 100% chance that either one or the other is true). So if one goes down, the other must go up. So how do we ensure it does?

Thomas Bayes solved this. What he observed is that we shouldn’t be looking for a mere list of probabilities, like “there is a 75% chance Paul attests to a historical Jesus and a 75% chance the Gospels do as well,” because each of those is actually a conclusion and not a premise. Hence we might just as well say “there is a 20% chance Paul attests a non-historical Jesus and a 20% chance the Gospels do as well,” and so on down the line, getting well above a 67% chance Jesus didn’t exist and yet on the other side get the exact same result for Jesus existing. Contradictory results. Wrong. What we need is the ratio between them. This was Bayes’ first fundamental insight. His second was regarding the crucial and inescapable roll of the prior odds (and hence cumulative odds, as every prior is the posterior of a previous accounting of evidence); but there the same principles apply: that, too, is ultimately a ratio of two probabilities. And you can’t ignore that, either (see Only Those Who Don’t Really Understand Bayesianism Are Against It and The Principle of Indifference). But here I want to focus on the role of evidence.

What we want to know, Bayes found, is not simply what “the probability is” that some item of evidence attests to one state of affairs or its opposite. What we want to know is the probability that that evidence would exist on each competing explanation—and it is their ratio that determines whether that fact is actually evidence or not, as well as whether it is weak or strong evidence, and how weak or strong (see Empirical Logic and Romans 1:3 for a discourse on this). If some fact is equally likely on either theory—if it is just as expected whether some hypothesis h is true or if h is false—then it is not evidence at all. It increases neither theory’s probability of being true. This is because their ratio of expected probabilities will always be 50/50, which is simply 1/1 and hence 1, and anything multiplied by 1 (like, say, the probability of h) remains unchanged by that fact’s existence. If the fact in question is slightly more probable on one theory than another, then it is evidence (albeit weak evidence) for that theory against the other. And if some fact is a great deal more probable on that theory than on any other, then it is quite strong evidence for that theory against the others. This is how evidence works.

So in our current example what we want to know is the following:

• What is the probability that the Gospels we have would contain what they do if Jesus existed?
• What is the probability that the Gospels we have would contain what they do if Jesus did not exist?
• … And hence what is the ratio between those two probabilities?

For examples of how to argue this way, see Historicity Big and Small: How Historians Try to Rescue Jesus. In short, you need some fact in the Gospels that is more likely to be there if Jesus existed, than if he didn’t. And that requires you to be specific: what fact are you talking about? I show there really aren’t any such facts in Proving History (chapter 5), and in added detail in Historicity (chapter 10).

The same follows all the way down the line. We need to know:

• What is the probability that Paul would use the phrase Brothers of the Lord where and how he does if Jesus existed?
• What is the probability that Paul would use the phrase Brothers of the Lord where and how he does if Jesus did not exist?
• … And hence what is the ratio between those two probabilities?

And:

• What is the probability that the Jewish execution of Jesus as described in 1 Thessalonians 2 would be there if Jesus existed?
• What is the probability that the Jewish execution of Jesus as described in 1 Thessalonians 2 would be there if Jesus did not exist?
• … And hence what is the ratio between those two probabilities?

And:

• What is the probability that the content of Josephus would be as we find it if Jesus existed?
• What is the probability that the content of Josephus would be as we find it if Jesus did not exist?
• … And hence what is the ratio between those two probabilities?

And:

• What is the probability that Papias would say the things he does if Jesus existed?
• What is the probability that Papias would say the things he does if Jesus did not exist?
• … And hence what is the ratio between those two probabilities?

At each step, from start to finish across all five facts examined, we then multiply: all these ratios multiply together. That way, we don’t get illogical results where both probabilities go up and down, but they remain in ratio to each other, completing the entire probability space of all possibilities—because there is a 100% chance Jesus either existed or didn’t, so the probability that he did and the probability that he didn’t must always sum to 100%. If in each case the evidence is more likely if he existed than if he didn’t, then the probability he existed will rise, and the probability he didn’t will decline. If the other way around, then the other way around. And if it’s a mixed bag (if some facts are evidence one way, and some another), we simply have to multiply them all together and see where the net effect of all that evidence lands. That’s how it is done.

And I take pains in On the Historicity of Jesus to do all this. So if you want a different result, you have to take the same pains—and address the evidence I present for my probabilities, and justify, and that means justify empirically, the probabilities you want to replace them with (see Doing the Math: Historicity of Jesus Edition). For example, I myself already assign “50/50” (the same 50%) to the evidence in Papias: I conclude it is as likely on historicity as on non-historicity. Neither theory makes the content of Papias more likely. I do not have to conclude anything further. I do not “need” Papias to only evince legendary material; all I need is for us to not know whether he is evincing historical material. Not knowing if something is true is not the same thing as saying it’s false. We simply can’t know if anything pertinent in Papias is true. So we just can’t use him as evidence for anything. Maybe there’s some true stuff in there. But maybe there is not. The probability is equal either way. So we get 50% one way, 50% another way, for a ratio of 50/50, which reduces to 1/1, which is simply 1. And any probability multiplied by 1 remains the same. So nothing in Papias can change the probability Jesus existed—up or down.

When we get to the passages in Josephus, even if we set aside the vast evidence confirming they are fake (see Josephus on Jesus? Why You Can’t Cite Opinions Before 2014), we are left with the fact that we cannot establish any of their content does not come from the Gospels or Christians reliant on the Gospels (and in fact the longer passage in Josephus has been proved to derive from Luke). Not being independent, and lacking (as we do) any evidence of further fact-checking, the probability that later authors would repeat the content of the Gospels (directly or through Christian informants) is essentially 100% whether that content was true or false. So if an earthly Jesus didn’t exist, the Gospels made him up, and Josephus is simply repeating what they said. And if Jesus did exist, the Gospels embellished his story, and Josephus is simply repeating what they said. The outcome is the same either way. Thus we are in the same position as with Papias: it is equally likely that that content would be there whether Jesus existed or not, so it’s 50/50, hence 1/1, which is just 1; and anything multiplied by 1 stays unchanged.

To change this you need something about the content of Josephus that is unlikely unless Josephus confirmed it in more reliable sources—in other words, unlikely unless Jesus really existed. But there isn’t anything like that. This again does not “require” me to insist there isn’t. All I need show is that there is no way to know that there is. And that I’ve done. We simply don’t know if anything in Josephus goes to sources beyond the Gospels. And it is what we do not know that we cannot assert. Not the other way around. Even with respect to the shorter passage, about James, even if it’s wholly authentic (and abundant evidence proves it’s not), still all Josephus tells us is that someone known as a Brother “of the one called” Christ was executed; but there is no evidence Josephus was at all aware of the fact that all baptized Christians called each other Brothers of that same Lord (as Paul explains extensively in his letters). Josephus would simply be repeating what he was told, being none the wiser that “brother” here was a cultic designation, not a biological reality. Given the facts we have (that all baptized Christians called each other “brothers” of this Christ; and that Josephus would have no evident way of knowing this), the evidence is equally likely whether Jesus existed or didn’t. This does not require me to say this is not a reference to a biological brother of Jesus; all it requires me to say is that, on the information we have, we cannot know that it is a reference to a biological brother of Jesus. The evidence is 50/50 either way. So this evidence is, again, unusable.

This same fact prevents us saying “it is more likely” that Paul would refer to Brothers of the Lord if they were real biological brothers of Jesus than that he would if they were only cultic (fictive) brothers of Jesus exactly as he himself and every other baptized Christian imagined. We have no evidence that Paul even knew of biological brothers of the Lord, and he makes no such distinction. Whereas we do know he knew of fictive brothers of the Lord. And given that, we’d sooner expect him to have to specify biological and not fictive brothers, if he knew of both kinds; so that it doesn’t occur to him that he has to specify which he means, suggests he had no knowledge of there even being biological brothers. He only says he knew of the fictive ones. So this evidence is not more likely if Jesus existed. It’s actually less. However, to account for a wide margin of uncertainty, in my study I allowed this evidence to be twice as likely if Jesus existed than if he did not (OHJ, pp. 591–92, 594–95), for a ratio of 2/1, which breaks down to an equivalent of saying it is 67% likely that these passages attests a historical Jesus, which is pretty near the 75% our imagined historicist was claiming. And yet still this wasn’t enough to make the overall historicity of Jesus probable. When we add up all the mixed bag of evidence, the balance still tips against Jesus’s historicity, even after counting this as evidence for it (OHJ, Ch. 12).

Meanwhile with the Gospels, we are back to 50/50 at best. There is nothing in the Gospels not as expected on myth as on history. For example, that it includes historical facts and persons was a common enough feature of ancient myth and fiction, as well as of myths and urban legends generally (see Jan Harold Brunvand, The Vanishing Hitchhiker: American Urban Legends and Their Meanings; Alan Cameron, Greek Mythography in the Roman World; and M. David Litwa, How the Gospels Became History: Jesus and Mediterranean Myths), so it is not “unexpected” at all. We see the same thing in the Talmudic version of the Gospel (see OHJ, Ch. 8.1), where Jesus is executed a hundred years earlier, by stoning in Lydda, not crucifixion in Jerusalem: that Jesus co-exists or even interacts with actual historical rabbis and political leaders of that time, and its account of how legal trials operated is actually more historically accurate than any of our Gospels portray. Yet it cannot be true. It is surely mythical. So if such details don’t make that more likely historical than mythical, they cannot make our Gospels so either.

Similarly, the evidence establishing an interpolation in 1 Thessalonians 2 is overwhelming (and this is agreed by most mainstream experts), and yet the probability of finding such a fiction inserted into the letters of Paul—given all the evidence we have of interpolations, doctored manuscripts, forgeries, and manifest fictions from Christians back then—is simply the same whether Jesus existed or not. We fully expect this passage to be here in the form we find it if Jesus did not exist. We also expect it if he did, since that in no way impaired their desire to fabricate (as dozens of fake Gospels, and even a fake letter Jesus wrote to King Abgar, attest). This conclusion is contingent, however, on that key phrase “in the form we find it,” as it is that form that demonstrates Paul did not write it. Had he actually written it, it would look different than this; and had it looked like that, we’d have to come to a different conclusion about it. So we could have had evidence here establishing the historicity of Jesus. We just don’t. And yet that “the material we have” is indeed unlikely if it were authentic is not itself unlikely if Jesus existed—because lots of forged stuff was produced, so we know it was being produced even if he existed. Consequently, that this is fake is not evidence against the existence of Jesus. It’s 100% expected whether “mythicism” were true or not. It’s thus simply unusable.

It is, however, in that one sole case that I do actually need something to be true: mythicism does not require that we know everything in the Gospels is fake, only that we don’t know any of it is true; but mythicism does require that we know that this passage in 1 Thessalonians 2 is fake (at least to a reasonably high certitude). But this is not an assumption. It’s an empirical conclusion. That’s why an independent analysis of its authenticity is fundamental, and also why it matters that this conclusion was already the broadest consensus among experts before mythicists even started noting it in their own peer-reviewed studies. Whereas on the other four points of evidence we have no need of anything, other than a lack of knowing anything. We don’t “need” the Gospels to be fake; we only need to not know they are historical. We don’t “need” Paul to mean cultic brothers; we only need to not know whether he meant biological brothers. And so on.

But that won’t be apparent if you are still hung up on the Fallacy of Uncompared Probabilities and think all you have to do is multiply a bunch of probabilities together. That’s illogical. The correct logical procedure for assessing the effect of evidence on the probability of a conclusion is always comparative: you have to assess the ratios between evidential probabilities, which means you have to take seriously, and actually credibly answer, the question of how likely any given fact in evidence is given that mythicism is true. You have to take seriously, and actually for a moment assume, the mythicist thesis is actually what happened, and deduce from that how expected or unlikely what we have is (and then of course switch modes and do the same for the historicist thesis, preferably the most defensible version thereof). And then you compare those two probabilities. There is no other logically valid way to do it.

The Fallacy of Unconditioned Probabilities

This leads to the second most common error of late: the Fallacy of Unconditioned Probabilities. When I say you “have to take seriously, and actually for a moment assume, the mythicist thesis is actually what happened, and deduce from that how expected or unlikely what we have is” and “then of course switch modes and do the same for the historicist thesis,” that means: “given everything else we know about the world and that period and place.” Because in a Bayesian model (which in Proving History I prove is fundamentally the only logically valid model for inducing empirical probabilities) every single probability is conditional on this information, which is usually designated b or k for “background information.” This is not arguable. Because it is obvious that the probability of a thing is always contingent on the total field of pertinent facts affecting that probability.

This means that all probabilities (priors and likelihoods) are conditional on background information, and therefore when estimating them, you must always consider that information and its effect. Like the previous example, when you are estimating the probability of a fictional text in antiquity containing historical people and facts, you must take into account whether that did or didn’t commonly enough happen then, or in general. You can’t just declare “it didn’t,” without checking; least of all if when you check you find it did. What is likely is determined by what was usual, or at least usual enough to render it indeterminative. If we find historical facts and persons in ancient myth and fiction, then finding historical facts and persons in any story simply does not rule it out as myth or fiction—especially when every historical fact serves some narrative purpose for the author to have included it. When that’s the case, as it is with the Gospels, you simply have to look for other markers—or admit there are none.

This is what happens to Paul’s remarks about Brothers. When estimating the probability of this if Jesus didn’t exist, we have to include the background fact that Paul says all baptized Christians are Brothers of the Lord, and the background fact that Paul never mentions knowing any other kind of Brother of the Lord. These facts affect our probability estimates. They cannot be ignored; and that means our estimates cannot ignore their effect on what’s expected. It cannot be denied that this substantially changes our expectation that Paul will call someone a Brother of the Lord if Jesus didn’t exist (radically changes, I would say). But note how this doesn’t work the other way around. The historicist has to assume Paul means biological brothers in these cases—because there is no background evidence in Paul indicating that he does. So the historicist is importing unevidenced assumptions—whereas the mythicist is simply reading the texts of Paul as he wrote them.

This would be different if in our background knowledge we had data indicating Paul knew a difference between biological and cultic brethren. We just don’t. In fact there is no evidence of any Christian knowledge of brothers of Jesus being involved in church business until a hundred years later. Even Acts has no knowledge of such persons ever being involved in the public history of Christianity, which it relates from chapter 2 on; the idea that he even had brothers at all appears to be an invention of the Gospels (OHJ, pp. 453–56). But we need evidence in Paul or contemporary to Paul to interpret Paul; and there isn’t any that pertains here. And we must avoid the fallacy of “unconditioned” probabilities. Our probabilities must always be properly and informedly conditioned on total background information. So, for example, on the theory that Jesus didn’t exist, the Gospels did (most likely) invent his brothers, and therefore we cannot assume any such Gospel content was known to Paul when predicting what Paul meant or would say. We simply have to condition on what he would say given that there were no brothers—and then compare that with the probability of what he would say if there were brothers.

Conversely, of course, when estimating the probability of any proposed evidence we are looking at, we have to estimate it for the evidence as we actually have it. For example, we are not asking what the probability is that Paul would mention Brothers of the Lord just anywhere and in any way—we are asking what the probability is that Paul would mention Brothers of the Lord in the particular way we find him doing (when that makes a difference; I’ll discuss below what happens when such differences don’t). Thus, it affects our estimates that the only time he ever uses the full pleonastic phrase brethren “of the Lord” is when he needs to make a distinction between Apostolic Christians (who are already known to also be Brothers of the Lord, just of higher rank) and non-Apostolic Christians (as we see him doing in 1 Corinthians 9 and Galatians 1: see OHJ, Ch. 11.10). That is exactly the kind of way we expect him to use the fictive label, if such he is doing; and one might claim the same of the biological label, if such he is doing instead, but that still leaves us at a wash. Had Paul used the phrase in ways more peculiar to a biological intention we’d have evidence for historicity. But he didn’t. So we don’t.

That is how conditioning on background information is necessary and significant whenever estimating the probability of some fact on any given theory or its competitor(s). You can’t just “assign probabilities” without reference to that information. And that is why I devote so much time to documenting that information in OHJ: it conditions all our probabilities. We can’t estimate odds without it. If we don’t know of it, or ignore it, our estimates will be objectively inaccurate and unreliable. Likewise if we disregard its effects. And indeed, you already know this. Just as when someone claimed it’s unexpected of a myth to contain historical detail: they are assuming something about the conditioning background information—in this case, that all its myth and fiction contains little or no historical detail. It thus is important for you to check if that’s true first. Because often you’ll find, as in this case, it’s not. You can’t actually tell if something is myth or history by its inclusion of such detail. At least not merely. If some particular manner of such inclusion would have that effect, then you need to show this is the case by a survey of the background information: is there a feature that allows you to reliably make such a prediction; and does such a feature exist anywhere in the Gospels?

All probabilities of the evidence are also conditioned on the hypotheses being compared. That’s a point I’ve made a few times now but it bears repeating, because again, you are arguing fallaciously if you disregard this. When estimating the probability that Paul would call certain people Brothers of the Lord if Jesus didn’t exist, you have to estimate that probability not only by taking into account the background information (as just surveyed) but also by taking into account the theory itself. You can’t say “it’s just unlikely Paul would speak this way.” You have to assume as a fact that Jesus didn’t exist and then deduce what to expect. That is how deductive evidential probability works. If Jesus didn’t exist and (as we know) Paul believed all baptized Christians were Brothers of the Lord, then how likely is it that we will occasionally catch him referring to people as Brothers of the Lord? Likewise, if Jesus existed and he had brothers (as the historicist has to presume; because a historical Jesus wouldn’t necessarily have still-living or involved brothers, and the evidence for his having any is scarcer and more problematic than usually thought), then how likely is it that Paul would call people Brothers of the Lord but never specify which kind he meant? What are the odds that he’d forget to say he meant these were biological brothers and not the usual cultic brothers he more usually goes on about? To be honest, it’s not that likely. Certainly not as likely as in the other case.

Thus, the requirements of conditional probability are absolutely essential to take into account if you want to argue about probabilities coherently and competently. There is a reason the full equation (in its Odds Form) is P(h|e.b)/P(¬h|e.b) = P(h|b)/P(¬h|b) × P(e|h.b)/P(e|¬h.b). The term P(h|e.b) means the probability of h conditional on the entire contents of e (evidence) and b (background information); and indeed e and b must together encompass all human knowledge. Otherwise, your results are inapplicable to reality. Likewise, P(¬h|b) means the probability that h is false (and thus some other cause of the evidence e is true) conditional on all known background information. And P(e|¬h.b) means the probability of the evidence, e, conditional on h being false (and thus some other cause of the evidence being true) and conditional, again, on all known background information. And so on. You have to take all these mathematical facts seriously. Otherwise, you won’t be getting any result worth taking seriously.

This also does mean any interdependence of evidence must also be taken into account. The probability of the content of Acts will of course be affected by the content of the Gospels, for example, as the author of Acts is using the Gospels as a source. So if Acts just repeats content from the Gospels, that is 100% expected regardless of whether that information is true or false (unless and until we have evidence the author of Acts was operating differently; and so far, we don’t—in fact rather the contrary: see OHJ, Ch. 9). But conditional probabilities aren’t always 100%. For example, Acts includes two peculiar facts: the family of Jesus (and nearly every other historical agent he interacts with) vanishes completely as soon as Acts begins the public history of Christianity (in Acts 2), and never appears again for all three decades of events covered; and the closing trial speeches of Paul peculiarly fail to ever reference a historical Jesus, despite their being quite lengthy and numerous, and in contexts where such mentions would have been crucial or even unavoidable. These could be causally related in some way, being they are outputs of the same author, but not to 100% certainty. One being the case does not make the other 100% expected, or even expected much at all.

Jesus not existing would cause them both, and so both are effectively 100% expected if Jesus didn’t exist (after accounting for the eliminative filter that destroyed most information about the origins of Christianity whether Jesus existed or not, an example of the contingent effect of background information that I’ll discuss next). But we don’t expect both on historicity. So we do have to independently evaluate the probability of each, and they do compound (the existence of both is even less probable if Jesus existed than the existence of either one alone). But our estimates should account for any evident or likely influence from the one to the other (such as having a common author). Because the causal sequence of evidence and its causal origin can both affect the conditional probability of that evidence. But as long as we haven’t overlooked any such effects, we can proceed; because conditional probabilities also multiply.

The Fallacy of Neglected Total Probability

Finally, we must avoid the Fallacy of Neglected Total Probability. Someone might ask why we can say that Jesus not existing makes the forgery in 1 Thessalonians 2 “effectively 100% expected.” How can it predict that? This confuses specific with general prediction. As I wrote in Proving History (p. 215):

Consider the configuration of the stars in the sky: the probability that the stars would today stand in exactly the pattern they now do is vanishingly small, whereas an intelligent engineer who intended to put them in exactly that pattern would make that pattern 100 percent certain. But surely that does not mean that the stars must be in that pattern because of intelligent design. From the conditions fixed shortly after the Big Bang, that the stars would exist now in some pattern that is similarly complex and comparably arranged is all but 100 percent certain, so their existing in that pattern is no more likely on design than natural causes. This is because the stars would inevitably come out in some such complex and unique pattern. So appealing to the complexity of the pattern is fallacious, since in all probability no matter what pattern it came out to be, it would have been just as complex, yet in the same generic features entirely the same. Thus, the Big Bang Theory does not predict exactly how the stars would be arranged; it predicts only what general pattern they would exhibit

I go on to demonstrate this point with further examples. Hence the difference between predicting general rather than specific outcomes. In Proving History I subsume this under what I call a “coefficient of contingency” (see the index for every discussion of that there). I reference this several times in On the Historicity of Jesus as well (pp. 288–89, n. 18; p. 357, n. 122; and p. 605, n. 10).

The short of it is this: the probability of a specific thing (like the exact text of the Gospel of Mark down to every word and its position) and the probability of a general thing (some text more or less like Mark’s) are different; the latter will be the sum of the probabilities of every possible specific text that would satisfy the condition. Just as all the ways the Big Bang could likely have arranged the stars: each one has a small probability, but you have to add them all up, to produce the combined total probability, the general prediction of the Big Bang that one of those configurations is what we will see, not which one. Hence we can expect to near 100% certainty that whether Jesus existed or not some biographies would have been written of him. Because our background information tells us that that always happened back then: nearly every mythical demigod akin to Jesus (so far as we know) received a historicizing biography about their life and times (and teachings) on Earth. We are not thereby saying we can predict the exact contents of Mark, down to every single word and its sequence. We are rather saying that something like Mark is totally expected; all the possible “Gospels of Mark” that could have been written sum to a total probability that is near to 100%, and thus any one of them would have satisfied our prediction that there would be one.

Likewise, as with the configuration of the stars, there are some outcomes (general patterns of arrangement) that would have challenged the Big Bang theory, because they’d be highly unlikely on that theory (like, for example, “they will not likely form a perfect cross in the center of the sky as viewed from Jerusalem at midnight every Yom Kippur”). But there are a vast number of outcomes that totally confirm the Big Bang theory. Expected randomization (some coefficient of contingency) can be deducted, leaving a near 100% certainty that we’d see what we see. Not exactly what we see. But something relevantly enough like it that what we do see is one entirely expected outcome of all those that random factors could have produced and still be in line with the predictions of the Big Bang. So, too, with Mark. There are ways Mark could have been written that would challenge the Jesus myth theory. He could have undertaken a critical, historiographically conscious inquiry, complete with identifying sources, how he interacted with them, and explaining what he did to vet what they said. Instead, he composed a patently mythographic novel. Which is exactly what mythicism predicts—again, not exactly what Mark produced; but something relevantly enough like it. This is what we mean when we say the contents of the Gospels are 100% expected on Jesus not existing: not their specific exact contents; but their general content.

Even the historicist faces this. The probability of any specific thing even on their theory is vanishingly small as well. You can’t predict from “Jesus existed” the exact number, choice, and position of words forming the Gospel of Mark. So what historicists resort to, and correctly, is only predicting the general thing, that “some” text like Mark’s would exist. And since any such outcome will have “some” specific collection and sequence and count of words, the improbability of a specific one is not relevant to whether historicity predicts the general fact of it. Thus we can honestly say that “historicity predicts that some biographical narratives in some form would come to exist with near 100% certainty.” Really, there is only one feature of the Gospels that is unexpected on historicity but totally expected on non-historicity: their extensive mythographic nature. Persons that extensively mythologized tend not to exist. Which is not to say they never do. But rather, usually they turn out to be mythical (OHJ, Ch. 6). If Jesus existed we’d more expect first to see more mundane and straightforward letters, memoirs, and accounts that then get exaggerated and mythologized over time. Instead we see the story is thoroughly mythologized right out of the gate, and in fact becomes more historicizing over time, exactly the reverse order to what we expect if Jesus existed (see How Did Christianity Switch to a Historical Jesus? which is reproduced and expanded in Jesus from Outer Space).

This holds even if we allow for the possibility that even historicists would neglect ever composing biographies, because that contingency is the same on mythicism, so it cancels out mathematically. For example, if we said there is a 50% chance that only these kinds of biographies would be written on historicity, but a 2% chance they’d just never have gotten around to it at all, we would have to say the probability of the evidence being as we observe it is 50% × 98%, the latter accounting for the slight chance they’d not have done it at all and thus we should not see any Gospels in that case. But we have no reason to believe (unless you can present some evidence of one) that this is any differently the case on non-historicity. There could be only a 98% chance such biographies would be written in that case, too; and hence, also, a 2% chance they’d just never have gotten around to it. Whatever causes would lead people to neglect that step on historicity, would also lead people to neglect it on non-historicity. Yet if we assume that isn’t the case, then we’d have a 100% chance of seeing what we do. The Gospels are indeed exactly what we expect—in the required general sense. So if we accounted for the coefficient of contingency (all the random things that could have prevented the production or preservation of any Gospels at all) we’d have 50% × 98% / 100% x 98%; but 98/98 cancels to 1/1 and hence 1, and anything multiplied by 1 stays itself, so we are left with 50% / 100%, or 2/1 against historicity.

The point of this is to illustrate the role of coefficients of contingency and how they wash out in the math. Complaining about this is the Fallacy of Neglected Total Probability: the 100% expectancy on myth, like the 50% expectancy on historicity, is the total probability, after summing all specific outcomes that would match expectation, and after subtracting contingencies that do not differ from one theory to the other. Just as with the stars or the exact words in Mark. The random chance that Mark would choose one word over another at one particular point is simply the same whether Jesus existed or not—unless it isn’t, in which case we have evidence to attend to. But the rest doesn’t evince anything. So suppose, for example, that the probability that Mark would be exactly as we find it has a probability of 0.000002 either way (most of Mark’s contingent choices of vocabulary, idiom, style, order all being the same whether Jesus existed or didn’t). Then we could say the same as I wrote on OHJ regarding the fact that Jesus was placed in two completely different historical eras a hundred years apart (p. 289):

[If] n = 0.000002, then P(e|historical) = 0.5n and P(e|¬historical) = 1 n; n then cancels out in any full [Bayesian] equation, leaving P(e|historical) = 0.5 and P(e|¬historical) = 1. This does not mean the probability that e on h is literally 50% (as if every other historical person were placed in different periods of history), or that the probability that e on ¬h is literally 100% (as if every mythical person were placed in different periods of history), rather this is simply their relative probabilities, after a mathematical reduction. No matter what the consequent probabilities actually are (i.e. no matter how many historical or mythical persons this actually happens to), they will always reduce to 0.5 and 1, if it happens twice as often to mythical persons as historical. Therefore, if that ratio is a warranted belief, then we don’t need to know the actual frequencies. Because whatever they are, they will always reduce to that same ratio. Another way to think of it is that being placed in different periods of history is the sort of thing we expect of a mythical person more than of a historical one (see Carrier, Proving History, pp. 77-81, 214-28), such that if we cancel out the common contingency between them, it is then 100% expected on ¬h but only 50% expected on h …

In other words, it may be that the actual probabilities are 0.5 x 0.000002 = 0.000001 (or one in ten thousand historical people get placed in different centuries, and thus a one in ten thousandth chance Jesus would be) and 1.0 x 0.000002 = 0.000002 (or one in five thousand mythical people get placed in different centuries, and thus a one in five thousandth chance Jesus would be). But the 0.000002 / 0.000002 cancels to 1/1 which is just 1, leaving us with 50% / 100%, or simply, two to one.

This is why we will say there being an interpolation in 1 Thessalonians 2 is 100% expected on mythicism (and 100% expected on historicity). We do not mean either theory predicted exactly that would happen; rather, we mean both theories predicted something like that would happen, and therefore there is nothing surprising about finding an instance of it. Just as there is nothing surprising about finding the stars in exactly the precise arrangement we now do, complete with an Orion Constellation as viewed from much of the northern hemisphere. We can say that’s 100% expected on standard godless cosmological theories; so we don’t need intelligent design to explain it. Because that there would be some convenient constellations of the kind is 100% expected.

This follows from a proper accounting of the Law of Total Probability. A prediction can be of an entire set of possibilities, when the contingent forces selecting one of those possibilities to be realized is the same across both competing explanations. Only when there is a differential prediction—one theory predicts a member of that set while the other does not, or at least makes this more probable than the other—do we have to tease that out and account for it as evidence for one theory over another. Otherwise, all these contingencies are irrelevant to which hypothesis is true. Which things would happen to get interpolated in which documents is virtually infinite in possibility and thus impossible to specifically predict, while at the same time that something would be interpolated somewhere is entirely expected. This is why finding interpolations (provided we can independently prove them to be almost certainly interpolations) does not support mythicism or historicity, because both theories 100% predict we’d find such things, once we account for all common contingencies that cancel out and thus make no difference.

Conclusion

The Fallacy of Uncompared Probabilities is the mistake of thinking all you have to do is come up with a string of probabilities and multiply them, when in fact you need to estimate the probability of each piece of evidence independently on both competing theories, and it is the ratios between those probabilities that is then multiplied to produce “a probability” for each competing theory being true. The Fallacy of Unconditioned Probabilities is the mistake of thinking you can just assign a probability to something out of the blue, when in fact you must condition those assignments on the total background information—everything pertinent—and the competing hypotheses—taking into account what they predict; and that means what they predict in light of all we know about that place and time, its cultures, its literary trends, as well as human nature, scientific facts of the world, and so on; and what they predict in respect to what they entail (e.g. if Jesus didn’t exist, it will then be a fact that he had no brothers, and all probabilities on that theory must then be conditioned on that being the case). And that further means not what we assume all those things to be, but what we can adduce genuine evidence is actually the case. And finally, the Fallacy of Neglected Total Probability is failing to account for all the contingencies that cancel out because they are the same for all competing theories, and remembering that predictions are usually of general sets of possibilities (all of whose probabilities sum into one total probability) and not specific exact outcomes (which always have vanishingly small probabilities by themselves).