Logical reason and truth.

Question

I would like to get some direction to the following question. I am having trouble translating this into a working logical expression.

A very special island is inhabited only by knights and knaves. Knights always tell the truth, and knaves always lie. You meet three inhabitants: Alice, Rex and Bob, where

Alice tells you that "Rex is a knave". Rex tells you that "it's false that Bob is a knave". Bob claims, "I am a knight or Alice is a knight." So who is a knight and who is a knave?

Answer 1

Let $a, r, b$ represent the statements Alice is a knight, Rex is a knight, Bob is a knight respectively. The first statement you gave can be written as:

$$(a\rightarrow(\neg r))\wedge((\neg a)\rightarrow r)$$

Can you find similar expressions for the other statements?

Answer 2

Alice claims Rex lies, Rex claims Bob tells the truth, and Bob claims he and Alice don't both lie. If Bob is lying, so is Alice, so Rex is telling the truth, a contradiction. So Bob is telling the truth, as is Rex, and Alice is lying. In algebra we can denote Alice's statement as $a$ etc. so $a=\neg r,\,r=b,\,b=a\lor b$, and the argument becomes$$\neg b\implies\neg a\implies r\implies b\implies\bot,\,\therefore b\therefore r\therefore\neg a.$$

Answer 3

Here's how you can do these:

Use $A$ for the claim that Alice is a knight

Use $R$ for the claim that Rex is a knight

Use $B$ for the claim that Bob is a knight

Now use the fact that someone from this island is a knight if and only if what they say is true. So, given that:

Alice tells you that "Rex is a knave".

we know:

$A \leftrightarrow \neg R$

Likewise, given:

Rex tells you that "it's false that Bob is a knave".

we have: $R \leftrightarrow \neg \neg B$

And finally:

Bob claims, "I am a knight or Alice is a knight."

gives us: $B \leftrightarrow (B \lor A)$

OK, we can simplify $R \leftrightarrow \neg \neg B$ to $R \leftrightarrow B$

Then, we can treat the biconditionals as a kind of substitution. That is, given $R \leftrightarrow B$, and given $A \leftrightarrow \neg R$, we can infer $A \leftrightarrow \neg B$.

And combining that with $B \leftrightarrow (B \lor A)$, we get $B \leftrightarrow (B \lor \neg B)$, which is equivalent to $B \leftrightarrow \top$, and hence to $B$. So, we know Bob is a knight.

But given $R \leftrightarrow B$, that means Rex is a knight as well, and given $A \leftrightarrow \neg R$, that means Alice is a knave.

More formally:

\begin{array}{lll} 1. & A \leftrightarrow \neg R & Given\\ 2. &R \leftrightarrow \neg \neg B & Given\\ 3. &B \leftrightarrow (B \lor A) & Given\\ 4. &R \leftrightarrow B & Double \ Negation \ 2\\ 5. &A \leftrightarrow \neg B & Biconditional \ Substitution \ 1,4\\ 6. &B \leftrightarrow (B \lor \neg B) & Biconditional \ Substitution \ 3,5\\ 7. &B \leftrightarrow \top & Complement \ 6\\ 8. &B & Biconditional \ Identity \ 7\\ 9. &R & Biconditional \ Elimination \ 4,8\\ 10. &\neg A & Biconditional \ Elimination \ 1,10 \end{array}