I often hear both non-theists and theists alike say it is impossible to disprove God’s existence because it is impossible to disprove a universal negative. This conception, though popular, is mistaken.
While a universal negative cannot be proven empirically, it can be proven logically. If something is logically contradictory, or incoherent, we can be sure it does not exist. For example, I can prove there are no square circles. I cannot, and need not do so empirically, but I can do so logically. To prove that God does not exist, then, does not require omniscience so long as one can demonstrate that there is something in the very concept of God that is rationally incoherent. Of course, that is difficult for atheists to do because there doesn’t seem to be anything about the idea of a divine, transcendent being that is internally incoherent or self-contradictory. Nevertheless, if they could find one, they could disprove God’s existence.