# Three statements, who is saying the truth?

Bob: Steve is lying

Steve: Bob is lying

Fred: Steve is lying or Bob is lying

"or" is inclusive

How would I approach this question? I honestly have no idea. I was thinking of making a truth table and assuming that each of them is saying the truth once and then going through the statements, but i'm not sure how I would handle the "or" bit within a truth table.

• If we assume there are only three people, then Fred is telling the truth because they both can't be truthful or lying. – Duck Oct 17 '18 at 0:42

## 2 Answers

Just consider each assertion one by one, and its implications.

1. Bob tells the truth.

Therefore Steve is lying and Fred tells the truth => this is possible

1. Steve tells the truth.

Therefore Bob is lying and Fred tells the truth => this is possible

1. Fred says the truth.

Either Steve or Bob is lying (those possibilities were already evaluated as possible) or both are lying. The latter case is impossible since we would then get that both are not lying, according to their respective assertions.

1. Fred is lying.

Then neither Steve nor Bob are lying which is impossible.

Conclusion:

Fred tells the truth, for sure, and either Bob or Steve is lying, but not both.

This can be written: Fred AND (Bob XOR Steve)
Note: XOR here is an exclusive OR, contrary to Fred's inclusive OR

Not really sure about this one, but I think

Fred AND (Bob OR Steve) are telling the truth

Because

One can assume that either Bob or Steve is telling the truth. Let's say Bob is telling the truth.
So if we assume Bob's statement "Steve is lying" is true, then Steve's statement "Bob is lying" is indeed false, because as we assumed, Bob is telling the truth.
They however cannot both be telling the truth because by their statements, if one of them is telling the truth then the other one is lying.
And they cannot both be lying because if one of them is lying, then that means the other is telling the truth.
So I think we can assume that only one of them is lying.