 Matrices are typically used to show the independence of the axioms of a system as well as their consistency. The consistency of two formulas A and B is established by a matrix that, under an assignment \(\sigma\), assigns designated values to both formulas. The independence of formula B from formula A is established by a matrix that (i) preserves the validity of the rules of the system and that (ii) under an interpretation \(\sigma\) assigns a designated value to A but not to B. Parry (1939) uses the matrix method to show that the number of modalities of Lewis’s systems S3 and S4 is finite. A modality is a modal function of one variable that contains only the operators \(\neg\) and \(\Diamond\). The degree of a modality is given by the number of \(\Diamond\) operators contained. A proper modality is of degree higher than zero. Proper modalities can be of four different forms: 