I am confused about how "belief" works in this logical reasoner of type 1.
Suppose I am such a reasoner.  I can be thought of as a theorem-proving
machine who uses logic to draw conclusions from premises.  We can imagine
there is a numbered list of everything I believe and have concluded.
It starts with my premises and then I add to it with my conclusions.

In this case my premises might be:

1. Knights always tell the truth
2. Knaves always lie
3. Every native is either a knight or a knave
4. A native said, "you will never believe I am a knight".

Now we can start drawing conclusions.  Let t be the proposition that
the native is a knight (and hence tells the truth).  Then 3 implies:

5. t or ~t

Point 4 leads to two conclusions:

6. t implies ~Bt
7. ~t implies Bt

Here I use ~ for "not", and Bx for "I believe x."  I am ignoring some
complexities involving the future tense of the word "will" but I think
that is OK.

However now I am confused.  How do I work with this letter B?  What kind
of rules does it follow?

I understand that Bx, I believe x, is merely a shorthand for saying that
x is on my list of premises/conclusions.  If I ever write down "x" on
my numbered list, I could also write down "Bx" and "BBx" and "BBBx" as far
as I feel like going.  Is this correct?

But what about the other direction?  From Bx, can I deduce x?  That's
pretty important for this puzzle.  If Bx merely is a shorthand for
saying that x is on my list, then it seems fair to say that if I ever
write down Bx I can also write down x.  But this seems too powerful.

So what are the correct rules that I, as a simple machine, can follow for
dealing with the letter B?

The problem is that the rules I proposed here lead to a contradiction.
If x implies Bx, then I can write down:

8. t implies Bt

Note, this does not mean that if he is a knight I believe it, but rather
that if I ever deduce he is a knight, I believe it, which is simply the
definition of "believe" in this context.

But 6 and 8 together mean that t implies a contradiction, hence I can conclude:

9. ~t

He is a knave.  7 then implies

10. Bt

I believe he is a knight.  And if Bx implies x, then:

11. t

and I have reached a contradiction with 9.

So I don't think I am doing this right.

Hal Finney

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