More generally, logic can be used to demonstrate the limitations of logic:

https://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_th...

You could say that a neural network found this limitation of neural networks, to the extent that neural networks could be defined in terms mathematical logic. However, it's not guaranteed that the neural networks in our brain could be explained in these terms--the physical processes underlying them are not fully understood.