Commutation relations in QFT [duplicate]

So I have just started learning QFT. So you take a classical field and turn the degrees of freedom into operators. All fine, just like normal quantum.

However I am confused about the commutation relations.

For the Klein-gordon/spin 0 scalar field we say $$[\psi_\alpha(x),\psi_\beta^\dagger(y)]=\delta_{\alpha\beta}\delta(y-x)$$ however for the dirac/spin half field we say $$\{\psi_\alpha(x),\psi_\beta^\dagger(y)\}=\delta_{\alpha\beta}\delta(y-x).$$

In Tong's lecture notes he seems to justify this by appealing to the fact that it works. However I don't find this very satisfying. Is there a mathematical reason for seeing what the commutation relations are a priori or do you just have to see what works.