Several important, fundamental aspects of the grinding behaviour of mineral systems are addressed. The Levenberg-Marquardt algorithm for systems of constrained non-linear equations was used to solve the steady-state and dynamic population balance model (PBM) grinding equations to obtain the mill matrices and grinding selection and breakage rate functions, respectively. The fact that the PBM model equation Inverse Problem is degenerate or underspecified is demonstrated. Multiple solutions to the same PBM equations are provided. It is shown that there is no unique solution to the Inverse Problem unless additional constraints are provided or assumptions are made such as the Arbiter-Bhrany normalisation assumption. The severity of the non-uniqueness problem for steady state grinding is demonstrated in several examples using typical feed distributions and mill matrices. In addition, it is also demonstrated that when higher than single powers are used in steady-state mill matrix expressions during simulation or calculation (or iterative procedures are used during numerical simulation) and four or more size intervals are used considerable amount of error is propagated throughout any calculation. Each solution to a PBM, while giving the same prediction during a single mill pass, gives different solutions or predictions for mill composition upon subsequent passes. In addition, it is shown that there is a problem with building up a grinding mill knowledge base with PBMs. A similar analysis was done for the dynamic or kinetic PBM equations. The fact that the dynamic PBM Problem is degenerate or underspecified is also demonstrated. Multiple solutions to the same dynamic PBM equations are provided. Again, it is shown that there is no unique solution to the Inverse Problem unless additional constraints are provided or assumptions are made such as the normalisation assumption. Each solution to a dynamic PBM, while giving the same prediction for a given grinding time interval, gives different solutions or predictions for mill composition for other grinding times. Actual experimental grinding data was assessed to determine the functionality of mill selection and breakage functions. The functionalities obtained through constraints were compared with those obtained with the normalisation assumption of Arbiter-Bhrany which relates breakage functions to particle size distribution. The capability of the population balance model to predict grinding behavior over time in various mineral grinding systems was assessed. The required functionality of selection and breakage functions for effective prediction of grinding behavior in mineral systems is discussed.