In the way that I've always heard them used, there tends to be a different nuance to each. Usually who you're commanding. Here's an example that might clarify:
Let us go!
The first usually refers to the group you're with, telling them that we should go. The second usually refers to someone holding us captive and a desperate plea to let me and my friends leave. The second can be used for the first, but it's a bit outdated.
Yes that is correct. Perhaps building on that explanation I would say the contraction Let's go will never get you released if you are a prisoner. It is not sufficiently positive!
It will usually refer to some action you were expecting to do anyway in your group i.e. let's go NOW. If you say - let's go - when a captive your guards would think they should go with you!
Let us means allow us or permit us and is addressed to someone outside your group as you say. Always use the full version to get results! OR the full version to be more positive or thoughtful as in - Let's imagine - as against - let us imagine. -
Let us imagine - means the same but would always be used in a college lecture or in a thesis. It is more formal.
Let us is NEVER actually wrong but may infer there is something or someone who could stop you.
Let's may be insufficient but will do where permission is not needed - Let's find a good restaurant - for example. - Let us find a good restaurant may perhaps mean someone is stopping you.
But if in doubt use - Let us.
True for "Let's go" and "Let us go", but in the case of "Let's continue" and "Let us continue" they mean exactly the same thing. Then again you could say "Let us go away from here" and it would be the same as "Let's go away from here", (the former slightly more dramatic).
In most cases your kind of asking permission but from who it might depend.
I was confused by this sentence too, as I read it as "we say" or "we are saying" and I couldn't figure out how that was an imperative - I guess it is the same word for both and I just need to get used the that!
I am not an expert on grammar, but having read the comments below, I think that there is a difference between let's say and let us say in English. Or let's go and let us go, or let's do and let us do etc. They can mean the same thing, but they are more likely to have different meanings - let's go is more likely to be with a group of people where you are suggesting or urging an action by the group ("what will we do now - let's go to the park", or "hurry up, lets go!). "Let us go" is more likely to be used in a situation where you are demanding that someone else do something - a demand from a group that has been detained by the authorities for some reason. In English "Let's say we do not know it" could be "if we are questioned, let's say we do not know it", or in a situation where you are being asked about something it could be a kind of vague, nudge nudge wink wink kind of answer -ie let's say that we(the group being questioned) do not know it and then you (the person asking the question) can focus on more important issues - it brings the other person into collusion with the group. On the other hand "Let us say that we do not know it" sounds more like a demand to be allowed to speak and to say that we do not know it. This particular sentence is not something that we would be likely to say in English, but I think in general the difference between the two forms would hold for most sentences of that type, and that the difference would be obvious from the context.
In Romance languages first person plural (noi) usually has a single form for several different verb tenses/moods. A similar case in Italian is (informal) second person plural (voi) with two matching tenses/moods.
- (noi) diciamo = we say / we are saying
dite= you (all) say / you (all) are saying
- se (noi) diciamo = if we say
- (different form for voi)
- diciamo (noi) = let's say
dite(voi) = say! (plural)