I think we're probably using some words differently, and that's making you think my claim that deductive reasoning is a special case of Bayes is stronger than I mean it to be.
All I mean, approximately, is:
Bayes theorem: p(B|A) = p(A|B)*p(B) / p(A)
Deduction : Consider a deductive system to be a set of axioms and inference rules. Each inference rule says: "with such and such things proven already, you can then conclude such and such". And deduction in general then consists of recursively turning the crank of the inference rules on the axioms and already generated results over and over to conclude everything you can.
Think of each inference rule "i" as i(A) = B, where A is some set of already established statements and B corresponds to what statements "i" let's you conclude, if you already have A.
Then, by deduction we're just trying to say that if we have generated A, and we have an inference rule i(A) = B, then we can generate or conclude B.
The connection between deduction and Baye's is to take the generated "proofs" of the deductive system as those things to which you assign probability of 1 using Bayes.
So, the inference rule corresponds to the fact that p(B | A) = 1. The fact that A has been already generated corresponds to p(A) = 1. Also, since A has already been generated independently of B, p(A | B) = 1, since A didn't need B to be generated. And we want to know what p(B) is.
Well, plugging into Bayes:
p(B|A) = p(A|B)p(B) / p(A)
i.e. 1 = 1 p(B) / 1
i.e. p(B) = 1.
In other words, B can be generated, which is what we wanted to show.
So basically, I think of deductive reasoning as just reasoning with no uncertainty, and I see that as popping out of bayes in the limiting case. If a certain formal interpretation of this leads me into Godelian problems, then I would just need to weaken my claim somewhat, because some useful analogy is clearly there in how the uncertain reasoning of Bayes reduces to certain conclusions in various limits of the inputs (p=0, p=1, etc.).
I think I would describe what you are talking about as being Bayesian statistics - plus a whole bunch of unspecified rules (the "i" s).
What I was saying is that there isn't a standard set of rules of deductive reasoning axioms that is considered to be part of Bayesian statistics. I would not dispute that you can model deductive reasoning using Bayesian statistics.
In response to the The uniquely awful example of theism, I presented myself as a datapoint of someone in the group who disagrees that theism is uncontroversially irrational.
With a loss of considerable time, several karma points and two bad posts, I now retract my position.
Because I have deconverted? (Sorry, but no.)
I had a working assumption (inferred from here) that rationality meant believing that all beliefs must be rigorously consistent with empirical observation. I now think of this as a weak form of rationalism (see full definition below). A stronger form of rationalism held by (many, most?) rationalists is that there is no other valid source of knowledge. If we define a belief system as religious if and only if it claims knowledge that is independent of empirical experience (i.e., metaphysical) then it is trivially true that all religions are irrational -- using the stronger definition of rational.
A disagreement of definitions is not really a disagreement. Someone suggested on the April open thread that we define "rationality". My idea of a definition would look something like this:
Rationality assumes that:
(1) The only source of knowledge is empirical experience.
(2) The only things that are known are deduced from empirical experience by valid logical reasoning and mathematics.
Weak Rationality assumes that:
(1) The first source of knowledge is empirical experience.
(2) The only things that are known with certainty are deduced from empirical experience by valid logical reasoning and mathematics.
(3) Define a belief system as all knowledge deduced from empirical observation with all metaphysical beliefs, if any. Then the belief system is rational (nearly rational or weakly rational) if the belief system is internally consistent.
Probably these definitions have been outlined somewhere better than they are here. Perhaps I have misplaced emphasis and certainly there are important nuances and variations. Whether this definition works or not, I think it's important to have a working set of definitions that we all agree upon. The wiki has just started out, but I think it's a terrific idea and worth putting time into. Every time you struggle with finding the right definition for something I suggest you add your effort to the group knowledge by adding that definition to the Wiki.
I made the accusation that the consensus about religion was due to "group think". In its pejorative sense, group think means everyone thinks the same thing because dissent is eliminated in some way. However, group think can also be the common set of definitions that we are working with. I think that having a well-defined group think will make posting much more efficient for everyone (with fewer semantic confusions) and will also aid newcomers.
The "group think" defined in the Wiki would certainly need to be dynamic, nuanced and inclusive. A Wiki is already dynamic. To foster nuance and inclusion, the wiki might prompt for alternatives. For example, if I posted the two definitions of rationality above I might also write, "Do you have another working definition of rationalism? Please add it here." so that a newcomer to LW would know they were not excluded from the "group of rationalists" if they have a different definition.
What are some definitions that we could/should add to the Wiki? (I've noticed that "tolerance", as a verb or a noun, is problematic.)