(1) It won't take at least 15 minutes to walk there.
(2) It won't take more than 15 minutes to walk there.
Am I right in thinking (2) sounds natural but (1) doesn't?
If so, what is the reason for this? Two phrases at least X and more than X are similar in that the former implies ">= X" and the latter ">X".
I agree with you that (1) is unnatural. I think it's not just a grammar issue but also a logical one: it's kind of hard to work out what the phrase "it won't take at least" should mean, and I would argue that it doesn't mean "it will take less than", at least not unambiguously so.
We can start by noting that the phrase is a double negative: "at least" means "not less than", so if you say
it won't take at least 15 minutes
it means
it is not the case that the time taken will not be less than 15 minutes
Double negatives are generally considered unnatural in English, so that's one reason this sentence sounds odd.
As T.E.D. and R.M. pointed out in the comments, when a double negatives does appear in English it often serves to amplify the negative rather than cancel it out, as in "I ain't afraid of no ghosts." So if "it won't take at least 15 minutes" did appear as an English idiom it might end up meaning "it will definitely take 15 minutes or more". (I don't think it's likely someone would say it with this intended meaning but it does add to the ambiguity.)
However, let's continue trying to extract meaning from the phrase in a more literal-minded way. At first glance, it seems to say that the time taken will be not(not(less than 15 minutes)), i.e. it will be less than 15 minutes. But if we start to pick it apart a bit more this becomes less clear. If you say
It will take at least 15 minutes to walk there.
That could mean something like
You can't walk there in less than 15 minutes even if you walk really fast.
If you use that meaning for "at least" then "it won't take at least 15 minutes to walk there" means
It is not the case that you can't walk there in less than 15 minutes even if you walk really fast.
Which in turn means that you can get there in less than 15 minutes if you walk fast, but it might take longer otherwise. In other words, we can read "it won't take at least 15 minutes" as meaning not "it will take less than 15 minutes" but "it might take less than 15 minutes."
We can even make sense of this mathematically. Other answers have noted that this kind of phrase refers to a range of possible times, so let's write W ⊆ ℝ for the range of times it might take. Then "it will take at least 15 minutes" corresponds to the statement
∀ w ∈ W : w ≥ 15.
(In other words, no matter how long it takes to walk, that time will be 15 minutes or more.) So then "it won't take at least 15 minutes" corresponds to
¬(∀ w ∈ W : w ≥ 15),
which is logically equivalent to
∃ w ∈ W : w < 15.
(i.e. it is possible that the time taken will be less than 15 minutes.) But "it will take less than 15 minutes" corresponds to
∀ w ∈ W : w < 15,
which is a different statement.
So the sentence has two possible meanings in addition to being a double negative, and I think this is why it sounds so weird.
The meaning from other usage customarily heard, comes across as if the adverbial phrase 'at least' refers to the outer limit (upper limit as user405662 mentions) and is the comparative qualifier of time in this sentence. The walk should not take more than that specified amount of time and probably a little less.
I have heard a number of variations, examples below, but never the given sentence 1 as a substitute for 2. Sentence 2 matches in meaning with the variations here. (Because I use trains, taxis and pedal power a lot so...)
If we try to read given sentence 1 literally so we interpret it as not less than the specified amount of time then we either need another clause or sentence to figure the literal meaning such as, "..there; it's a good 2 miles from here." "...there. How do you figure that? It'll take a lot longer."
And if we read it as I think it's meant to be said, it's (Sentence 1) clumsy through and through, the construction and the hearing of it.
To supplement the more detailed answers with a brief intuitive take: "At least" implies that the value given is a minimum, and the actual amount could be larger. The construct "it won't take" implies that the value is a maximum, and the actual amount could be smaller. It sounds unnatural because it seems to be implying two mutually exclusive conditions.
(A famous example of abused English is "Everything $1 or less and up." This makes the same mistake, but is too inclusive rather than mutually exclusive.)
This sounds like a rebuttal of a previously given statement "It will take at least 15 minutes to walk there". A polite way to say "This is nonsense, the walk will be significantly shorter".
I think it important to clarify the two forms of expression and each of their converses in order to consider why the negation of one but not the other is English usage.
The converse of “more than” is “less than” (not “at least”).
The converse of “at least” is “at most”, and it is used in positive, but not negative, expressions:
“It will take at least 15 min…”
= “It will take 15 min or more.”
“It will take at most…”
= “It will take 15 min or less.”
Why is negating one of these forms but not the other common usage?
The argument of the OP would seem to be that both “It will take more than 15 min.” and “It will take at least 15 min.” are similar (if not mathematically identical) in meaning:
“The minimum time it will take is 15 min.”
However there is a subtle psychological difference that in the former the minimum time is not defined — it is 15 min ‘plus’, so we tend to envisage two time ranges, of which 15 min is the boundary. Hence, whether logical or not, we have become accustomed to the negation of one range implying the other range:
“More than 15 min” = >15–∞ min
“Not more than 15 min” = 0–15 min
However the focus on the precise minimum time of 15 min in “at least 15 min” psychologically has no natural negation. One thinks of precise alternatives — “at least 20 min”, “at least 25 min” — rather than the range 0–15 min.
It may not be justified mathematically, but, whether for historical or psychological reasons, that is English usage.
I think the simplest reason is this. What would this mean?
It won't take 15 minutes to walk there.
This would almost universally be understood as it won't take 15 minutes or more to walk there, i.e. exactly the same meaning as (1). So the "at least" in (1) is redundant and would normally be omitted.
This wouldn't work for (2), where the meaning is that it could take 15 minutes to walk there, but not longer than that.
At least is an example of a positive polarity item, an expression that isn't allowed (or at least is very awkward) in negated contexts.
There are others. A 1970 paper by C. L. Baker, "Double Negatives", gives these examples:
Already:
a. The Sox have already clinched the pennant.
b. *The Sox haven't already clinched the pennant.Would rather:
a. I would rather go to New Orleans.
b. *I wouldn't rather go to New Orleans.Could just as well:
a. Jacques could just as well have taken the train.
b. *Jacques couldn't just as well have taken the train.
A footnote acknowledges that
There is one particular circumstance in which sentences like this are acceptable; namely, when they represent, word by word, an emphatic denial of a preceding speaker's assertion.
There are also negative polarity items, like any in We don't want any. / *We want any. Polarity-sensitive items exist in virtually all languages, according to Anastasia Giannakidou's chapter on the subject in Semantics: An International Handbook of Natural Language Meaning.
In Cambridge Handbook of Germanic Linguistics (2020), Doris Penka writes: "[A]cross languages, expressions that are polarity sensitive tend to have certain properties concerning their lexical semantics." That is, there are common themes in the meanings of PPIs, even across different languages. For example, many have an attenuating meaning, like rather in He's rather frustrated. / *He's not rather frustrated.
But Penka notes that it's hard to justify a purely semantic view—that a PPI is positive-polarity purely because if negated it would be too confusing—since many PPIs can be paired with a corresponding NPI, like some and any, or already and yet.
You asked why the example "it won't take at least 15 minutes" is unnatural. That's because at least is a PPI. But as to why PPIs are like that: I don't know. Linguists are apparently interested in the theory of PPIs right now.
"More than" and "less that" are proper comparatives. "At least" and "at most" are adverbial phrases that are perhaps gradually evolving into true comparatives, but they're not quite there yet.
One sign of this is that, as you've noticed, "at least" and "at most" don't really work in negated comparisons. Another one is that, even though it's nowadays quite common to treat them as if they were integral parts a noun phrase — as in "at least/most 15 minutes" — it's still also possible to pull them away from the phrase they supposedly modify and have them modify the entire sentence instead:
"It takes at most 15 minutes to walk there."
"It takes, at most, 15 minutes to walk there."
"At most, it takes 15 minutes to walk there."
"It takes 15 minutes to walk there, at most."
Some of the sentences above might sound a bit awkward or archaic to some speakers, but they all mean the same thing: even if we only consider the maximum possible time, it still only takes 15 minutes to walk there.
(FWIW, the reason I chose to use "at most" in my example sentences above was to avoid confusion with the alternative usage of "at least" as a sentence modifier to mean "in any event" or "anyway". The phrase "at most" lacks this extra bit of ambiguity.)
In at least(!) the last two or three sentences above it's clear that "at most" isn't modifying just the phrase "15 minutes", but actually the entire sentence. And even though it could be parsed in the first sentence as applying to just the phrase "15 minutes", I would argue that it's not necessary to do so.
And indeed I would argue that (since humans don't really parse sentences based on rigid abstract grammar) for most people the connotation of "at most" (or "at least") indicating that the entire sentence refers to a maximum (or minimum) scenario of some kind is there, at least to some extent, in all of the example sentences above, including the first one.
And that's why "it won't take at least X" sounds unnatural: on one hand, "won't" is clearly a negation of "will", so it should mean the opposite of "it will take at least X". But on the other hand, both of these phrases have the "at least" in there, suggesting that they should both be describing a minimum scenario of some kind.
Your question is very simply answered:
That's the only reason "1" sounds awkward.
A simple observation is that there are any number of very common phrases in - say - Indian English, which are common and normal there, but have the exact "jarring - is that a double negative or what?" quality you alludea to in "1".
Even more simply, the oddball formulations (which we're now familiar with and "sound right") simply once weren't, and others will be in the future.
It's entirely, totally plausible that your "won't take at least" formulation will be trendy in 5 years and completely normal in 50 years.
(Because English is the planet's absurdly flexible, loose, ruleless, morphing language, bizarrely your "won't take at least" formulation is starting to sound natural to me, minute by minute, literally as we discuss it here!)
Steve: "It'll take at least 15 minutes to walk there." Bill: "Bullshit, it won't take at least 15 minutes to walk there, man ..."
Your "1" would have been used many times, in that scenario.
Your "1" is no more or less "logical" than formulations which do ("happen to") exist here and now.
It's an interesting observation that: there are many, many questions on this site (interesting questions) which are about the logicality of an issue in English. (These can range from spelling issues, to idiom-related issue like this one, to grammar issues.) The answer is always the same: logic is irrelevant in English.
We've all heard of double-negatives.
Regarding your formulation,
It won't take at least 15 minutes to walk there.
I have named that a rebuttal-negative.
Using the spectacular observation by @rackandboneman, we know that (today, as of writing), rebuttal-negatives are only used in the unusual situation of a rebuttal to a long-group-idiom. They only (today, as of writing) sound sensible as a rebuttal (by adding the testy word "won't") where you're parodying the other speaker's a long-group-idiom. As in
Steve: "It'll take at least 15 minutes to walk there." Bill: "Bullshit, it won't
take at least 15 minutes to walk there, man, you're full of crap ..."
However! Rebuttal-negatives immediately appeal to me as a speaker. I'm going to start using them even before the other person gives the original idiom which I'm gonna parody.
So it's sort of a naked rebuttal-parody-negative.
I can imagine using this! Someone says ...
Man, it's gonna take ages to walk there, we need a taxi, my legs hurt ...
I then parody the abstract phrase take at least 15 minutes to walk there as if some loser had said the abstract phrase ...
You pansy ass, what's wrong with your feet! It won't (seething parody of something not actually said)
take at least 15 minutes to walk there, what are you saying! Let's do it! Get walking!
You see?
You've invented rebuttal-negatives, bravo. They're kind of a naked rebuttal/parody of an idiom expected in the situation.
(Maybe this exists already, perhaps in stand-up comedy - I don't know.)
