The choice of postulates is somewhat arbitrary in the sense that given a set of postulates you almost always can find an alternative set. The choice is guided by subjective criteria such as simplicity, closeness to experiment, or theoretical elegance.

However there are situations where some postulates do not make sense. For instance, $[\hat{x},\hat{p}] = i\hbar$ makes no sense in the Wigner & Moyal formulation of quantum mechanics, neither as postulate nor as theorem, because this formulation of quantum mechanics does not use operators:

The chief advantage of the phase space formulation is that it makes quantum mechanics appear as similar to Hamiltonian mechanics as possible by avoiding the operator formalism, thereby "'freeing' the quantization of the 'burden' of the Hilbert space.

In the phase space formulation of quantum mechanics the CCR can be still obtained as a theorem when one makes the transition from the general phase space state to the configuration space wavefunction $\rho_W(p,x;t) \rightarrow \Psi(x;t)$. Precisely, an explicit derivation of the CCR $[\hat{x},\hat{p}] = i\hbar$ is given in my paper Positive definite phase space quantum mechanics

The choice of postulates is somewhat arbitrary in the sense that given a set of postulates you almost always can find an alternative set. The choice is guided by subjective criteria such as simplicity, closeness to experiment, or theoretical elegance.

However there are situations where some postulates/theorems do not make sense. For instance, $[\hat{x},\hat{p}] = i\hbar$ makes no sense in the Wigner & Moyal formulation of quantum mechanics, neither as postulate nor as theorem, because this formulation of quantum mechanics does not use operators:

The chief advantage of the phase space formulation is that it makes quantum mechanics appear as similar to Hamiltonian mechanics as possible by avoiding the operator formalism, thereby "'freeing' the quantization of the 'burden' of the Hilbert space.

Although the phase space formulation of quantum mechanics does not use commutation relations, them can be still obtained as a theorem when one makes the transition from the general phase space state to the configuration space wavefunction: $W(p,x;t) \rightarrow \Psi(x;t)$. Precisely, an explicit derivation of the $[\hat{x},\hat{p}] = i\hbar$ is given in my paper Positive definite phase space quantum mechanics

The choice of postulates is somewhat arbitrary in the sense that given a set of postulates you almost always can found an alternative set. The choice is guided by subjective criteria such as simplicity, closeness to experiment, or theoretical elegance. The CKC formulation of thermodynamics uses postulates close to experimental findings, whereas the MTE formulation uses a set of abstract postulates guided by criteria of conciseness and elegance. Both formulations are equivalent and explain the same experiments.

However there are situations where some postulates do not make sense. For instance, the above CCR postulate $[\hat{x},\hat{p}] = i\hbar$ makes no sense in the Wigner & Moyal formulation of quantum mechanics, because this formulation does not require the use of operators!

In a phase space formulation the CCR can be obtained as a theorem when one makes the transition from the phase space state to the configuration space wavefunction. Precisely, a explicit derivation of both the expression for the momentum and position operators and of the CCR $[\hat{x},\hat{p}] = i\hbar$ is given in my paper Positive definite phase space quantum mechanics

The choice of postulates is somewhat arbitrary in the sense that given a set of postulates you almost always can find an alternative set. The choice is guided by subjective criteria such as simplicity, closeness to experiment, or theoretical elegance.

However there are situations where some postulates do not make sense. For instance, $[\hat{x},\hat{p}] = i\hbar$ makes no sense in the Wigner & Moyal formulation of quantum mechanics, neither as postulate nor as theorem, because this formulation of quantum mechanics does not use operators:

The chief advantage of the phase space formulation is that it makes quantum mechanics appear as similar to Hamiltonian mechanics as possible by avoiding the operator formalism, thereby "'freeing' the quantization of the 'burden' of the Hilbert space.

In the phase space formulation of quantum mechanics the CCR can be still obtained as a theorem when one makes the transition from the general phase space state to the configuration space wavefunction $\rho_W(p,x;t) \rightarrow \Psi(x;t)$. Precisely, an explicit derivation of the CCR $[\hat{x},\hat{p}] = i\hbar$ is given in my paper Positive definite phase space quantum mechanics