1 + 1 = 2 is based on conservation of particles. You put a marble in a bowl, then another marble in the bowl, you now have two marbles in the bowl. If you remove conservation then there is no reason why 1 + 1 should equal 2. 1 + 1 being equal to anything could just be nonsense in that universe, such a construct wouldn't exist and there would be no way to reason about quantities. That isn't a logical inconsistency, so such a universe could exist.
In modulo arithmetic, 1+1 != 2 can work just fine. We're not talking about the literal universe, but the space of possibilities that opens up when you change the rules. E.g. all the amazing power and complexity that comes from imagining the existence of a number i with the property that i^2 = -1. This was initially thought to be a logical impossibility.
True == False does seem pretty broken though. Not sure that can go anywhere.