This is the only mammal that can fly.
We were taught that an independent clause must have a complete thought/idea. How can "This is the only mammal" be treated as an independent clause?
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BillJ's comment is to the effect that "that can fly", while a dependent clause, is not separate from the main clause in the way that, say, "because it can fly" would be.
"that can fly" is a relative clause, which is embedded in the NP (noun phrase) "the only mammal that can fly". This means that, in your example sentence "This is the only mammal" is not an independent clause. (It can be one, as RichF exhibits, but it isn't in this case). "That can fly" is embedded in, and part of, the main clause.
Edit: To clarify: in this particular sentence, "this is the only mammal" is not a clause. It could be one, as RichF's answer says; but here it is not, it is just a string of words. The clause that it is part of is "This is the only mammal that can fly" - which is the whole sentence. It is true that "that can fly" is also a clause, but it is not separate from the main clause, it is part of it.
it is like asking about the status of "I want to go" in the sentence "I want to go home", or of "Give me a big red juicy" in "Give me a big red juicy apple": those strings of words are not constituents of the sentence, and if you ask about their grammatical status the only sensible answer is "They haven't got a grammatical status, because they are not grammatical units">
In the sentence: "This is the only mammal that can fly," "This is the only mammal" is indeed the main clause. The term "independent" may in fact be misleading because main clauses often contain elements that link them with subordinate or dependent clauses, and if we take the latter ones out, the main clause will appear to be incomplete.
In the sentence at issue, that word within the main clause establishing a liaison with the subordinate one is, obviously, the adjective "only".
In this other sentence:
the main clause is "This bird is so strong," but once again we find an element that links it with the dependent clause "that it can fly for hours." That element is the adverb "so."
The question is, then, how to spot the main clause in a sentence. You'd better forget about calling it independent. The main clause will be the one that is not introduced by a subordinating word, be it a relative pronoun or adverb or a conjunction.
Notice that the relative (dependent) clause "that can fly" is introduced by the relative (subordinating) pronoun "that": then, that is the dependent clause and whatever remains is the main clause.
In the other example I provided, the adverbial (dependent) clause of result "that it can fly for hours" is introduced by the subordinating conjunction "that": then, that is the dependent clause and whatever remains is the main clause.
The main clause may be even shorter. In a sentence like: "Whoever arrives first is the winner," the subject "Whoever arrives first" is a nominal (dependent) clause, so only the predicate "is the winner" is the main clause, with an implicit or tacit subject "he/she". In a sentence like: "Say what you think," the direct object "what you think" is a content (dependent) clause and only the verb "Say" (with the tacit subject "you") is the main clause.
The main clause is actually "independent" in what we call the "deep structure" of the sentence, and sticking to the words that appear "on the surface" will not help you detect it, nor will the concept of a standalone, independent clause always help. The best thing for you to do is discard all the dependent clauses (not phrases), that is, smaller embedded sentences consisting of subject and predicate, and the main clause will be what remains.
I'll answer by example.
Across the stage, you see a goldfish, a snake, and a lizard. Next to me is a dog. This is the only mammal.
(added about an hour after original post)
The incomplete feel of "This is the only mammal" on its own is that our minds expect more information. In other words, it greatly benefits from added context. That context was provided in your original example by appending, "that can fly." I did it above by preceding the sentence with a situation I hoped you could picture.