Assuming that the lexical entry let in English can represent two distinct entities, can you (a) describe the distinction, using examples like the following: (i) Let us go, shall we. (ii) Let us go, will you and (b) explain the ungrammaticality of the (iv) example: (iii) Let’s go, shall we. (iv) *Let’s go, will you.

Two common usages of the English word let include let as the beginning word of a proposition or suggestion and let meaning "to be allowed." Because the two usages create an entirely different meaning within a sentence, only the former may be used in the form of a contraction.

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The differences between your statements come from the subtle difference between two versions of the English word let. One usage of let is for when one party proposes an action to another. With the sentence “Let us go, shall we,” the speaker is proposing that the two parties go...

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The differences between your statements come from the subtle difference between two versions of the English word let. One usage of let is for when one party proposes an action to another. With the sentence “Let us go, shall we,” the speaker is proposing that the two parties go now without actually going so far as to make an imperative (command) statement. Another example of a proposition is, “Let us eat.” In this sentence, the speaker is not asking permission to eat, but is suggesting or proposing to another party that they eat now.

In contrast, a second usage of the English word let is a request—or often a demand—to be allowed or permitted to do something. In the situation of “Let us go, will you,” the speaker is asking to be allowed to go. If the second usage of let were being used with the previous “Let us eat” example, the speaker would be asking permission for at least two people to eat now. It is not a proposal, but a demand to be allowed to eat.

Only one of these two usages works as a contraction. Since the first usage is a proposition, let and us are commonly combined into the contraction let’s. With the second usage, however, the verb let meaning “to be allowed” cannot be combined with the us that follows it. The word us is functioning as a direct object of the verb let. Direct objects cannot grammatically be included in contractions.

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