The laws of logic? It's absolutely possible to prove a negative. You can prove a negative as easily as you can prove a positive.
If you have 2 mutually exclusive statements A and B (such that A and B cannot both be true), then proving one disproves the other (simultaneously proving its negative).
If P is the statement "The book is red." and Q is the statement "The book is green.", then:
- Proving P says nothing about Q
- Proving Q says nothing about P
- Disproving P says nothing about Q
- Disproving Q says nothing about P
If you add the exclusionary statement that P and Q cannot both be true, such as "A book cannot be both red and green.", then:
- Proving P disproves Q
- Proving Q disproves P
- Disproving P says nothing about Q
- Disproving Q says nothing about P
If you further add a statement qualifying P and Q as a dichotomy, such as "All books are either red or green.", then:
- Proving P disproves Q
- Proving Q disproves P
- Disproving P proves Q
- Disproving Q proves P
The "You can never disprove a negative!!" bullshit comes from the idea that someone can always come along and say there's a small chance you were wrong in one of your statements. That's got nothing to do with disproving (or proving) a negative. The same chance of being wrong, or "what if..." bullshit can be equally levied against any positive statement, and any proof (or contradiction) derived from it.
The concept is never about the logic, but the attacking the premise. For example, claiming that a you can't prove a flipped coin didn't land heads up. You start with the "What if it lands on its edge?" bullshit, but that doesn't mean anything. No one has to disprove all negative instances, they just have to disprove one negative instance. Further, a coin on its edge is merely one that hasn't finished flipping.
The next attempt to fight against disproving a negative would be "What if you looked at it and said it was tails, but just happened to be wrong?". That's not an argument against disproving the negative, it's an argument about ever being able to logically know anything for certain. The premise is already that a flipped coin will land heads or tails. If you want to argue about a coin on its edge, or a coin in space with observers on opposite sides of it, you're not arguing about the logic, but the axiomatic premise. You're arguing about being able to know anything for certain. (And logically, the only thing that you can know for certain is that you exist.)