Logical Conjunctions and Probability

I get the feeling when reading writings by theologians on the history of the early Church (and also on Judaism), that a great many of them do not quite understand what happens when you combine probability and logic. Let us consider the following situation:

A ∩ B ∩ C ∩ D ∩ E → F

That is, if A, B, C, D and E hold true, then F holds true as well. In maths we can come to near complete certainty that A, B, C ... hold, and thus also conclude that the conclusion holds (with ever so slightly less certainty).

Now, in fields like history and such, it's quite clear that we never have complete certainty – and here there's an interesting consequence of maths.

If P(A) = 0.9, we can be quite certain that A holds ,right? What if the same probability of 0.9 holds for each of the five antecedents there? The probability of A and B both holding true, if they are independent of one another is P(A ∩ B) = P(A) * P(B). We can now recursively do this as P((A ∩ B) ∩ (C)) = P(A ∩ B) * P(C), and keep reiterating, and this obviously ends up giving us P(A ∩..∩E)  = P(A)*...*P(E).

The probability of F holding true given P(A)=...=P(E) is no longer as impressive: 0.9⁵ ≃ 0.59. Still greater than 0.5, though.

For a fair share of studies regarding, say, ancient Judaism or early Christianity, it seems a lot of the assumptions that do form part of the reasoning is way under 0.9, though - somewhere in the range of 0.6 to 0.8 would be a fair assessment. Even at the top range of that - 0.8 - we go below 0.5 at our fourth assumption.

Certainly, some of the probabilities might not be independent – some of the assumptions cannot hold without the previous one holding, for instance. But in that case, we still end up with probabilities of the form P(A)*P(B|A) – the likelihood of A and B, given that B's probability is conditioned by A happening, so we're still dealing with multiplication - which generally does produce quickly diminishing numbers when all our values are within the range [0,1].


What further bothers me in this situation is that oftentimes, a lot of assumptions remain unstated, and thus the list of antecedents is left incomplete, giving an impression of greater likelihood for the conclusion than is warranted. Certainly some of the assumptions might clock in at 0.99, but due to how multiplication works, ...

This is a thing that started bothering me while reading Margaret Barker's The Great High Priest – the various arguments in favour of her hypothesis all seem somewhat plausible taken in isolation. However, they are not arguments that support one another - the conclusion relies on at least a very great number of them all being independently correct. So, we end up in a situation where the only argument you can present at the level of the individual pieces of evidence is sure, these seem plausible, if not necessarily established facts. But when you look at the whole structure of it, it seems you end up with something that is fairly unlikely. It is a very clever rhetorical trick, that makes any criticism of it seem vague and unclear.

A good example of this is the following: Barker assumes that the esoteric teachings of the priesthood of the first temple were transmitted to the essenes. Further, Jesus was a member of this order. As a member, he acquired these teachings, and taught them to his disciples. The disciples propagated these teachings all the way to Origen more than 150 years later

It might seem somewhat possible that the Essenes had in fact inherited traditions from the original priesthood of the First Temple; let us arbitrarily overestimate this probability at 0.8 - in reality, you have circumstances where it would seem less likely - wars, widespread illiteracy, . I think everyone would agree that this is a fairly kind estimate. As for Jesus and the Essenes, maybe we'll even grant 0.9. We don't know much about the Essenes - did they teach all their doctrine to all their members - if not, was Jesus among those members that got access to the more restricted teachings? If he had access, did he understand it correctly? Let's be kind again, and put the probability at 0.8. Further, did he get around to teach his pupils these teachings? Did they understand them? I'd say 0.9 would be reasonable kind there.

Did these teachings get reliably passed down to Origen? Let's again go for 0.8. Maybe he had sources that did not learn from people who had learned from Jesus, or maybe his sources had misunderstood what they had learned from Jesus, etc.

At this point, we're way down: 0.46. Keep in mind that I find the probabilities that I have assigned to be rather exaggeratedly kind: I genuinely feel like values somewhere in the range [0.2, 0.7] would be closer to the actual probabilities here, although these are guesses. Of course, some less probable things do contribute to a slight likelihood of her being correct about the main conclusion: maybe Origen's sources didn't get it from Jesus, and Jesus didn't get it from the Essenes who didn't get it from the First Temple clergy, but Origen got it by some other route that did go back to the First Temple? Such alternative ways of salvaging her thesis exist, but seem highly unlikely, and putting a number like "0.01 at best" on these seems to be excessively kind as well.

If we were to look into greater detail with regards to the claims, we'd end up finding that the probability of the presented thesis falls far under 10%.


The same problem with regards to compound probabilities seems to beset a lot of work in the same field – I am compiling examples for a bigger post on this issue.

This kind of "fallacy of implication of the intersection of many independently probable propositions" is a thing I've seldom seen discussed as a fallacy, and I think it's an important one. The best way to avoid it is to either support your propositions with very much in ways of evidence, or to argue for things that can be supported by disjunctions of facts instead - or adds up probabilities, whereas and multiplies them together.