Is Person A a Knight or a Knave on the Island?

[wololo]

Homework Statement

A person can either be a knight (always tells the truth) or a knave (always tells a lie).

On an island with three persons (A, B and C), A tells "If I am a knight, then at least one of us is a knave".

Homework Equations

Truth tables, logic rules.

The Attempt at a Solution

Using the atoms P=A is a knight, Q=B is a knight, R=C is a knight, and the sentence P⟺P→(¬P∨¬Q∨¬R) I get the following truth table:

How can I find if A is a knight or a knave from this table? My guess would be that he is a knave since it is not a tautology but I am really not sure. Thanks!

 

[phinds]

I haven't really analysed your logic statement but one thing jumps out at me about the TT and that is this:

Consider the statement A => <whatever>

if A is false then it doesn't matter what <whatever> is, the statement A => <whatever> is true. It is what's called a vacuous truth. Your TT does not reflect this.

 

[wololo]

Yeah but the statement the person said P→(¬P∨¬Q∨¬R) will always be TRUE if he is a knight and always FALSE if he is a knave. This is why I added an if and only if in front of it. P⟺P→(¬P∨¬Q∨¬R) then means (if i am a knight, then at least on of us is a knave) is only true when the person speaking is a knight.

 

[phinds]

Yeah but the statement the person said P→(¬P∨¬Q∨¬R) will always be TRUE if he is a knight and always FALSE if he is a knave. This is why I added an if and only if in front of it. P⟺P→(¬P∨¬Q∨¬R) then means (if i am a knight, then at least on of us is a knave) is only true when the person speaking is a knight.

If I understand what you just said, then you are not understanding what I said. Do you dispute what I said? If so why and if not, why are you arguing about <whatever> is since it doesn't matter?

 

[wololo]

I agree that P->Q is only false when P is true and Q is false. If P is false, whatever value Q, P->Q will be true, so we both agree on that. The thing is that my truth table is not for an implication, but for an equivalence between an implication and an atom. Suppose we use a simple statement such as P<=>(P->Q). Then take a look at the row where P is False and Q is True. In that case, the value of P->Q will be true, whereas the value with which we concern ourselves, namely P<=>(P->Q), will be false, because the value of P (false) is not the same value as P->Q (true).

The reason it matters is because my truth table should not reflect P->Q as you say, since it is the truth table for P<=>P->Q (or rather P<=>P->not(P or Q or R) if we use the actual statement).

 

[phinds]

I agree that P->Q is only false when P is true and Q is false.

OK, then we are in agreement. I haven't looked at the rest of what you are doing, I just thought you had that wrong.

 

[wololo]

OK, then we are in agreement. I haven't looked at the rest of what you are doing, I just thought you had that wrong.

The thing is, I was never wrong, since the last column in the truth table is NOT a vacuous truth...

 

[SammyS]

I wasn't familiar with "atoms" as such in the context of logic, so I googled a few relevant phrases, and finally found quite a bit on "atomic propositions". They're statements or assertions that have a definite truth value.

Along the lines of the posts of @phinds , it may be helpful to include a column in your truth table which includes that statement of person A in symbolic form.

¬P ∨ ¬Q ∨ ¬R​

From that, it's clear that A cannot be a knave. If he were, his statement would be true, however vacuously. Therefore, person A must be a knight.

So, clearly you have not hit upon that atomic proposition which you are seeking.You have not used the given information that knights always tell the truth and knaves always lie and each person falls into one of those categories or the other. Incorporating such information may lead you to your goal.

Good fortunes !