'A / One / At least one student entered the room.' Are these the same? (truth-conditionally)

Solution 1:

Truth conditionally, each of the three sentences is the same. What is different is the implications of the different sentences when said out of context.

Generally speaking, we expect the maximum amount of relevance from our speakers. This means that we expect that - if this in keeping with what speakers actually want to and are able to do - they will give us the most informative information that we require, for the least amount of processing effort.

Sentence 1

In sentence (1) there is a relatively weak implication that only one student entered the room. The reason is that it takes just about the same time and effort to say some students entered the room as it does to say a student entered the room. The first sentence would enable the listener to make more inferences and add more items to their stock of beliefs (at the very minimum they would know that not only was it necessarily true that a student entered the room, but it was also true that more than one student entered the room).

This implication may be cancelled out by the context. So for example, imagine there was rule that if even a single student entered a particular laboratory that room must be swept for bugging devices the same day. Now, if you're the person who's meant to do the bug sweeps, and there's been a student-staff meeting in that lab, then if your boss comes up to you and says "why haven't you done a bug sweep?" you might reply:

• Why? Oh, yes, sorry. That's right, a student's entered the lab. I should have done a sweep.

Now because everyone knows that more than one student was in there and because the issue is not how many were in there, but that even one was in there, that sentence no longer carries an implication that less than two student entered. Notice that we still consider this sentence to be true. The fact that more than one student entered doesn't make it false.

Sentence 2

The second sentence implies that a minimum of one and maybe more students entered the room. The phrase at least takes more effort to process than the determiner a. It's a lexical phrase, it's two words and therefore longer and it's fairly likely to carry stress, thereby drawing our attention to it. The implication that at least communicates is that the speaker guarantees a minimum of one student entering, but has reason to believe that there were perhaps more.

This information is worth the extra processing effort on the part of the listener, because it is more informative than a student would be. By using this information together with our other background knowledge, we might be able to make more inferences from it than we would from a student entered the room. However, notice that there is no guarantee on the part of the speaker of more than one student having entered, but there is a guarantee of one student having entered. So, if one or more students entered then the sentence is true, exactly in the same way that sentence (1) would be.

Sentence 3

Sentence (3) carries a relatively strong implication that exactly one student entered the room and not more. The word one is usually stressed. It is often argued, in fact, that the word one is the stressed form of the indefinite article a (historically the two words derive from the same root). The fact that the speaker is using a stressed form here gives us reason to believe that they want to draw our attention to the actual quantity itself. Now, if there was more than one student, why would they emphasise the singular numeral? This would be increasing the processing effort with no benefit in terms of extra informativity.

However, there are many contexts in which we might want to do this. For example, if we know that student Bob was in the room, because he told us so, then if somebody else claimed that no students entered the room, we might say:

• One student entered the room!

Here we would be very likely to stress the word one. In this case, the numeral one is contrastive with the zero implied by the word nobody - which is why we use this form, and why it is stressed. However, in this context we are not implying that no other students entered the room. Our sentence would not be false if it turned out that Maria had also entered the room, for example. The reason that we only used one is that we cannot truthfully attest to more than one, not that there weren't more than one. So again, we have a guarantee that one student entered, but no guarantee that it was only one.

Conclusion

Many of the words that we use in speech are there to guide the reader to the best interpretation of the sentence, but do not actually affect the truth functional meaning of the sentence itself. They also help the reader make useful inferences about the speaker's communicative intentions. The three sentences in the Original Poster's examples use different quantifiers and discourse markers which may carry different implications, but which don't differ in terms of their truth values.

Solution 2:

As an exercise in formal logic, all three are equivalent. In the case of #3, if N students entered the room, so did any subset of those students.

As English sentences, though, they differ. The first and third are close, the former emphasising the student and the latter, the number of students. Although other students might have entered the room as well, both sentences focus on the individual student referenced as the subject of the sentence. The second sentence makes it explicit that there could have been more than one student who entered the room, so is a little different from the other two.