I just started following Alex O'Connor on Twitter and saw he recently posted this neat paradox. I had read about this paradox ages ago but didn't know what it was called. Apparently it's a version of the preface paradox. I'll let Alex explain it:

An interesting paradox: most (humble) people will accept that they are not 100% correct about everything. That is, of all my beliefs, at least some of them will be in fact incorrect.

So, we think that at least one of all the propositions we believe is true is, in fact, false. Therefore, we simultaneously believe (i) that all the propositions we think are true are in fact true, and (ii) that at least one of these propositions is false. This is a contradiction.

Unlike the Mutant Ninja Liar paradox I recently wrote about, this one is not brain-breakingly difficult, so I'll make this short. In case you want to think about it first, I'll put a picture here and put my answer below it.

So the supposed contradiction is in believing two conflicting things:

(i) that all the propositions we think are true are in fact true, and

(ii) that at least one of these propositions is false.

**Here's my Twitter-size answer.** It's a little subtle but we shouldn't, and don't actually believe exactly (i). (i) says that the *combined* statement "A and B and C and D and ..." is true, where A, B, C, D, ... are propositions we believe. But there's no rule of logic that dictates you should believe the combined statement! In other words, from the fact that you believe A, and you believe B, it doesn't follow that you should believe "A and B".

This becomes obvious if you think of "believing" as "being more than x% sure". For example, suppose you think there's only a 1% chance you are wrong about A and a 1% chance you are wrong about B. There are two ways "A and B" can fail, if A is wrong or if B is wrong. So the chance of that is not 1% but higher. If your threshold for believing something is being 99% sure then you shouldn't believe "A and B" even though you believe A and believe B.

**Summary.** You believe many propositions individually. But it doesn't follow that you should believe the single statement that all of them are true.

## 1 Comments - Go to bottom

Well put. In other words the mistake is in believing that it follows from x: “I believe A” and y: “I believe B” that “I believe A and B”. But this doesn’t actually follow. Instead, x&y is translated as “I believe A and I believe B”.

ReplyDeleteBecause “I believe A and B” is not analogous to “I believe A and I believe B”, it follows that there is no contradiction.

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