What is the answer to this logic puzzle: Is the light on or off? I just saw this puzzle at the end of this Veritasium video.
The problem is as follows:

Three people give three statements.
Watson: "Exactly 2 of the 3 of us are lying."
Hudson: "Exactly 2 of the 3 of us are lying."
Sherlock: "The light is off."
Is the light on or off?

My thinking for solving this puzzle was to take every permutation of true/false in order to consider each one.
TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFF
I ruled out TTT and TTF because if both Watson and Hudson tell the truth, then their statements are false; thus they would be lying.
Next I ruled out TFT, TFF, FTT, FTF simply because Watson and Hudson cannot give the same statement and it be simultaneously true and false.
This left FFT and FFF.
If it were FFT, exactly 2 of the 3 would be lying and so we reach a paradox.
Therefore, I concluded the only valid option is if all three were lying. Watson and Hudson are lying because exactly 2 of the 3 are lying is false, instead all three are. And Sherlock is lying and so the light must be on.
Am I correct? Is my logic correct? It seems like I could have made a mistake somewhere. I keep thinking over it get a little confused again.
Is there a nice way to solve this using a tidier method? I would love to know.
 A: Your answer seems valid. Imagine the light was off. Sherlock's statement "The light is off." would be true as the light is indeed off. This, however, presents a problem.  As Hudson and Watson made the same statement, they should have the same truth value. But if they spoke the truth, then no one lied. This means that Hudson and Watson told a lie. But if they lied, then exactly 2 of the 3 people lied. This means that Hudson and Watson told the truth. But if they spoke the truth, th...
Due to this logical paradox, the situation cannot exist by proof of contradiction. Therefore, the light is on.
However, this assumes a very important fact, which @Henry brought up in a comment. You must assume that the statements are jointly meaningful. If you were to gather three people in a room with a lit lightbulb, then say the statements from the problem, the world would not end due to logical paradox. The reason the paradox forms is because you assume all the statements must be either the truth or lies. The statements made by Hudson and Watson are neither and both at the same time if the light is on. As the problem never clarifies that the statements must be true or false, you can't necessarily prove that the light is on. Most logical riddles imply it, though, and if you follow that rule set, then the light must be on.
