Quantum mechanics uses the wave function to calculate probabilities by taking the square modulus of the wave function as requirement by Max Born. Why should this (squaring of the wave function) be so, was there a logical argument for this?

I did a little research and found the four Born conditions. The sites told me exactly what they were but fell short of actually explaining the logic behind them. I can understand the modulus part but why square the wave function, is there an explanation?