Yep. Thanks. =)

I was misunderstanding "equivalency" as "identical in all respects to," rather than seeing equivalency as "exactly as true as."

Yep. Thanks. =)

I was misunderstanding "equivalency" as "identical in all respects to," rather than seeing equivalency as "exactly as true as."

There's confusion here between logical implication and reason for belief.

Duncan, I believe, was expressing belief causality -- not logical implication -- when he wrote "If B, then A." This was confusing because "if, then" is the traditional language for logical implication.

With logical implication, it might make sense to translate "A because B" as "B implies A". However, with belief causality, "I believe A because I believe B" is very different from "B implies A".

For example:

A: Uniforms are good.

B: Uniforms reduce bullying.

C: Uniforms cause death.

Let's assume that you believe A because you believe B, and also that you would absolutely not believe A if it turned out that C were true. (That is, ~C is another crux of your belief in A.)

Now look what happens if B and C are both true. (Uniforms reduce bullying to zero because anyone who wears a uniform dies and therefore cannot bully or be bullied.)

C is true, therefore A is false even though B is true. So B can't imply A.

B is only one reason for your belief in A, but other factors could override B and make A false for you in spite of B being true. That's why you can have multiple independent cruxes. If any one of your cruxes for A turns out to be false, then you would have to conclude that A is false. But any one crux being true doesn't *by itself* imply that A is true, because some other crux could be false, which would make A false.

So with belief causality, "A because B" does not mean that B implies A. What it actually means is that ~B implies ~A, or equivalently, that A implies B -- which in that form sounds counter-intuitive even though it's right.

So for B to be a crux of A means only (in formal logical implication) that A -> B, and definitely not that B -> A. In fact, for a crux to be interesting/useful, you *don't* want a logical implication of B -> A, because then you've effectively made no progress toward identifying the source of disagreement. To make progress, you want each crux to be "saying less than" A.

In response to
Double Crux — A Strategy for Resolving Disagreement

Correct me if I'm wrong. You are searching for a sentence B such that:

1) if B then A

2) if not B, then not A. Which implies if A then B.

Which implies that you are searching for an equivalent argument. How can an equivalent argument have an explanatory power?

The requirement that B is crucial for A is equivalent to "If A then B", not "if B then A".

For example:

A = self-driving cars would be safer

B = the chances that a bug would cause all self-driving cars to crash spectacularly on the same day are small.

If A is true, then B must be true, because if B is false, A is clearly false. But B does not imply A: even if B is true (there could be zero chance of such a bug), A could still be false (self-driving cars might be bad at avoiding crashes due to, say, object recognition being too slow to react in time). So B being crucial for A means that A implies B.

*1 point [-]