 The improper modalities are \(p\) and \(\neg p\) (1939: 144). Parry proves that S3 has 42 distinct modalities, and that S4 has 14 distinct modalities. It was already known that system S5 has only 6 distinct modalities since it reduces all modalities to modalities of degree zero or one. Parry introduces system S4.5 by adding to S4 the following axiom: 