The author of the textbook I'm following has emphasised the importance of a class of judgements include the entailment *:□, that is
\[\Delta; \Gamma \vdash *:\Box\]
He's done this by talking about them a lot, and including them in the exercises a lot. But. he hasn't spelled out why they're important - I guess expected us to work out it.
But I haven't! I'd welcome help...
https://cs.stackexchange.com/questions/175981/importance-of-delta-gamma-vdash-box