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
