Despite knowing how the phrase "not to mention …" is often used, it still grates on me to use it because I am in the act of "mentioning" even as I use it. I found it helpful to read the origins of the phrase here, but I would really like to use an alternative word or phrase that means the same thing. Any suggestions?
Learn English – Alternative to the phrase “not to mention”
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In the phrase you quoted, the implication is one of using two different words for the same exact concept, implying a difference where none exists. It is not simply about disagreement, and indeed implies the opposite: you are not in actual disagreement with each other, but both parties want it to appear that they are referring to different things anyway.
In the original song, the examples that were given are all of "upper class" pronunciations vs. "common man" pronunciations. For example, at the time, an upper-class American would be more likely to say "toh-MAH-toh", to distinguish themselves from the average person that would say "toh-MAY-toh". But both are words for the same thing -- a tomato.
In modern usage, the phrase has taken on a broader meaning of selectively using vocabulary to distinguish between otherwise identical concepts. In the Time example, depending on your particular political affiliation, you are more likely to call the government subsidizing of private companies a "bailout" (if you opposed it) or a "loan" (if you support it), but ultimately you are talking about the same exact action. The Time headline is meant to be understood as reading "Some say 'loan', others say 'bailout', but the result is the same"; they are using the reader's presumed familiarity with the more general phrase for effect.
In your example from Japanese politics, you are talking about a case where an actual distinction exists between the two options. There is a real choice between two possible actions, and the words you use to describe each of those options differ because they must. This is not a case where you would likely use the "You say tomato..." phrase.
If someone who opposed a tax increase began referring to it as a "penalizing the rich", for example, then it would be more appropriate to use that particular turn of phrase.
I'd have expected your first option to be far less common than the other two, but a couple of Google Ngrams show that this view is probably unwarranted. Only 'at least' seems to be commonly used, though the other two are not unknown. I'm surprised; I'm quite familiar with 'at the very least' as a pragmatic evaluator.
At least (as a pragmatic marker – mitigating), as you say, has a different meaning, unwanted here.
I'd avoid at the least as sounding unnatural, though there's no grammatical reason why at the very least should be correct and at the least not so.
You could go with at the very least, which sounds a little over the top to me. I'd rephrase to 'It causes at least one problem: ...; it might also cause others.' This usage of 'at least' is as a quantifier rather than a pragmatic marker.
Best Answer
The mentioning while professing not to mention is a figure of speech known as apophasis.
An alternative to not to mention is let alone.