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Sentence: The two types of beaver are.

I was asked in class if this was a fragment. It clearly sounds like one, but it caused me to wonder if the word are can be used intransitively. I know it is technically a form of be, and the infinitive of be can mean to exist. So could the sentence be saying that two types of beaver exist?

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    You might quibble with the definite article "The" at the beginning of the sentence, given that, although two species of the genus Castor (the North American beaver and the Eurasian beaver) are extant today (according to Wikipedia), there have been other members of the family Castoridae in the past, including two species of giant beavers of Pleistocene North America, which most certainly are not (anymore). Hence it would be truer to say "Two types of beaver are, but at least two others are not."
    – Sven Yargs
    Commented Feb 21, 2019 at 4:36
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    It’s not that they exist, but that they are “whatever was being discussed”: Here we see a group of animals including a fox, an elephant two types of beaver, and a bird. Question: Which of these animals are capable of building a dam? Answer: The two types of beaver are.
    – Jim
    Commented Feb 21, 2019 at 18:23

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Yes, be, in any of its conjugations, can mean exist. To quote the OED:

be (verb) I.1.a: to have place in the objective universe or realm of fact, to exist; (spec. of God, etc.) to exist independently of other beings. Also: to exist in life, to live.

However, your sentence sounds unnatural to me. At least in my experience, the use of be as above often connotes a degree of solemnity. Consider the examples given in the OED:

The great beasts came first, strange forms that were when the world was new. (E. Nesbit)

To be, or not to be, that is the question. (Shakespeare)

God is, nay alone is. (Thomas Carlyle)

These examples all deal with remarkable topics: dragons, human existence, and God! For a more down-to-earth topic, like types of beavers, I would instead use the following.

There are two types of beavers.

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