Abstract. In this paper we give an informally semi-rigorous explanation of the notion of sequentiality. We argue that the classical definition, due to Pudlák, is slightly too narrow. We propose a wider notion m-sequentiality as the notion that precisely captures the intuitions behind sequentiality. The paper provides all relevant separating examples minus one. 1.

The notion of a critical successor [dJV90] has been central to all modal completeness proofs in interpretability logics. In this paper we shall work with an alternative notion, that of an assuring successor. As we shall see, this makes life a lot easier. After a general treatment of assuringness, we shall apply it to obtain a completeness results for ILW, a result first proved by de Jongh and Veltman [dJV99]. In our proof, the gain of assuringness becomes very clear. 1