The following theorem states some results for specific systems which will clarify the role played by the directedness axiom (TL3). Observe that the formula is not provable in DEK, i.e., it may be the case that both and are true but is not true. Generally, if a valid inference rule has at least 2 premises, and if each of these premises will be known after some course of thought, then it is not necessarily the case that the conclusion will be known. Such situations are precluded in the presence of the directedness axiom.
Utilizing the previous result we can establish an embedding relation between K and DEK. Similar relations obtain between other normal modal systems and their dynamic-epistemic counterparts which contain schema (TL3).