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For brevity, I symbolise synonymity with ≈. So X ≈ Y means X and Y are synonyms.
From http://www.thefreedictionary.com/let+alone: let alone ≈ not to mention
From Merriam Webster: let alone ≈ much less ≈ still less.
Are these all synonyms? What are the similarities and differences between each?
MacMillan and I agree in disagreeing that not to mention and let alone are synonymous:
not to mention: used for adding a comment that emphasizes the main idea of what you have already said: The weather here is gorgeous, not to mention the wonderful food.
let alone: used for saying that something is even less likely to happen than another unlikely thing: I hardly have time to think these days, let alone relax.
still/much/even less: used after a negative statement in order to emphasize that it applies even more to what you say next: I am no one's spokesman, much less his.
You could make the case that let alone and much less are synonymous, since you could substitute one for the other:
I am no one's spokesman, let alone his.
I hardly have time to think these days, much less relax.
However, substituting not to mention and let alone for each other would result in sentences that sound contradictory:
The weather here is gorgeous, let alone the wonderful food.
I hardly have time to think these days, not to mention relax.
I find the distinction that MacMillan makes between not to mention and the supposedly synonymous let alone and still/much/even less useful:
Some grammarians insist that not to mention and let alone are synonymous; while you can find some cases where substituting one for the other might make sense, I generally find their meaning and usage distinct from each other.
There is a fallacy at the heart of this question.
Let's say word W has two meanings W1 and W2.
The meanings may be completely different, slightly different, or merely different nuances.
The point is that W1 and W2 are different meanings and may be used in different contexts.
Word W also has three synonyms (S1, S2, S3) listed in various dictionaries, thesauri, etc..
As it happens:
Therefore, although
it does not follow that
because they relate to different meanings of the original word.
Hence just because:
it does not follow that
[I am specifically not commenting on whether or not the cited words are synonyms - and they could be. I'm merely explaining that, on the basis on the facts given in the question, it does not necessarily follow that they are all synonyms of one another.]
I would like to point out that these are not really synonyms but represent a decreasing scale of likelihood or relevance:
I could not afford a bicycle, let alone a motorbike, much less a car and still less a limousine.
To me, not to mention is not in the same group, it is used slightly differently as explained in @Gnawme's answer. For example,
The food was excellent, not to mention the wine!
These are all Constructions.
They have semantic properties, syntactic peculiarities, pragmatic uses, metaphoric sources,
and plenty of idiomaticity to go around.
Luckily, the let alone construction has been analyzed in great and precise detail in a famous paper by Fillmore, Kay, and O'Connor, "Regularity and Idiomaticity in Grammatical Constructions: The Case of Let Alone", Language, Vol 64, No 3 (1988:501-38).
This doesn't explain the differences with the other ones,
but it does give you an idea how complex the subject can get.
