Let’s is the english cohortative word, meaning “let us” in an exhortation of the group including the speaker to do something. Lets is the third person singular present tense form of the verb let. Aug 29, 2013i find the distinction that macmillan makes between not to mention and the supposedly synonymous let alone and still/much/even less useful:
The phrases let alone and. Jun 14, 2011i notice that "let alone" is used in sentences that have a comma. Otherwise, know that a basic search will turn up let us in innumerable journal articles, official proclamations, formal.
I think that "let a be a set, let b be a group, and let c be a number." is the most formal phrasing. Feb 23, 2016in "let's get started", the starting point is in view and "let's get going", you are on the starting point already. Oct 3, 2011"let us go/let us pray" is a special verb construction of the type let, imperative+us,accusative+bare infinitive.
Apr 13, 2016the let alone construction has been analyzed in great and precise detail in a famous paper by fillmore, kay, and o'connor:
