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The Rules Of Logic Are Just Probability Theory Without Uncertainty

05 Jun

bayes theorem

So it looks like for the summer I won’t be having any grad courses. Which means I can go back to blogging a bit and commenting on the multitude of things I find dealing with religion and/or rationality that I come across on the web. Maybe even finish reading some books I’ve bought and blogging about them too!

One thing I read on Quora is an intersection of religion and rationality: Using Bayes Theorem in history. Unfortunately this won’t be a post praising the argument; rather, it’ll be one explaining the author’s fail at rationality:

To begin with, it’s illustrative to note who uses Bayes Theorem to analyse history and who does not. In the first category we have William Lane Craig, the conservative Christian apologist, who uses Bayes Theorem to “prove” that Jesus actually did rise from the dead. And we also have Richard Carrier, the anti-Christian activist, who uses Bayes Theorem to “prove” that Jesus didn’t exist at all. Right away, a curious observer would find themselves wondering how, if this Theorem is the wonderful instrument of historical objectivity both Craig and Carrier claim it to be, two people can apply it and come to two completely contradictory historical conclusions. After all, if Jesus didn’t exist, he didn’t do anything at all, let alone something as remarkable as rise from the dead. So both Carrier and Craig can’t both be right. Yet they both use Bayes Theorem to “prove” historical things. Something does not make sense here.

Yes something doesn’t make sense here, and one can tell what that is by inference from the title of this current blog post.

As I wrote above, logic is just probability without the attendant uncertainty. Which should sorta be uncontroversial since logic and math are highly interconnected, just like math and probability are interconnected. I’m also not the first to point this out; I first read this connection in Jaynes.

But let me offer a couple of demonstrations. How about the basic syllogism with a conjunction as the major premise:


1. P ^ Q (true)
2. P (true)
Therefore Q

If I give a probability value to the major and minor premise, we can find out what conclusion follows:


1. P ^ Q (100%)
2. P (100%)
Therefore Q (100%)

This follows both logically and mathematically / probabilistically. If P * Q is 1, and P is 1, then Q must also be 1. So the answer is the same for both the formal logic formulation and the probabilistic formulation. Another example, using the same format:


1. ~(P ^ Q)
2. P (true)
Therefore ~Q

So if you can’t understand the fancy symbols, this reads that if you have a conjunction P and Q that is false, and you also know that P is true, then it follows necessarily that Q is false. The same conclusion will follow if we substitute probabilities:


1. P ^ Q (0%)
2. P (100%)
Therefore Q (0%)

This reads if the probability of P and Q is 0%, and we know that P is 100% then it must mean that Q is 0%. It’s a straightforward algebraic solve-for-x deal. The conjunction of the major premise of this case can be converted into a disjunction using DeMorgan’s law:


1. ~P v ~Q (true)
2. P (true)
3. Therefore ~Q

Does using probability yield the same conclusion?


1. ~P v ~Q (100%)
2. P (100%)
3. Therefore ~Q (100%)

Since this is a disjunction, we are no longer using multiplication to find the answer.

The point with this is that the underlying mechanisms are the same: conjunctions in propositional logic have the same “mechanism” for finding conclusions that math/probability do. The main difference between logic and probability is that logic is binary (yes/no) whereas probability is comparative. If we know that A is greater than B, and B is greater than C, then A must be greater than C. The shortcut for those sorts of comparisons is using numbers. And more relevantly, if history is about comparing explanations — which is a measure of uncertainty — the only clear way to do so is by using numbers: Probability.

So let’s substitute “Bayes theorem” with “propositional logic” in the original quote and see if this still makes sense:

To begin with, it’s illustrative to note who uses [modus tollens] to analyse history and who does not. In the first category we have William Lane Craig, the conservative Christian apologist, who uses Bayes Theorem to “prove” that Jesus actually did rise from the dead. And we also have Richard Carrier, the anti-Christian activist, who uses [modus tollens] to “prove” that Jesus didn’t exist at all. Right away, a curious observer would find themselves wondering how, if this [propositional logic] is the wonderful instrument of historical objectivity both Craig and Carrier claim it to be, two people can apply it and come to two completely contradictory historical conclusions. After all, if Jesus didn’t exist, he didn’t do anything at all, let alone something as remarkable as rise from the dead. So both Carrier and Craig can’t both be right. Yet they both use [modus tollens] to “prove” historical things. Something does not make sense here

And there we have it. It is indeed true that both Carrier and Craig have attempted to use propositional logic to defend their cases. This must mean that historians need to do away with using formal rules of logical inference because they can lead to different, contradictory conclusions. Clearly, this now means that the whole gamut of logical fallacies is now in play to argue anything one wants in historical analysis!

This reminds me of how Creationists and other anti-science types think that the scientific enterprise is wholly corrupt because sometimes the scientific method produces two contradictory studies.

But yes. Both probability and logic (and science) follow the GIGO rule: Garbage in, garbage out. We can’t argue against a tool just because it follows GIGO.

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