In A Student's Introduction to English Grammar, Rodney Huddleston and Geoffrey Pullum pose the following problem:
Leaving aside interjections, the eight parts of speech (as defined by H and P) are nouns, verbs, adverbs, adjectives, prepositions, determinatives, coordinators, and subordinators. Is it possible to construct a sentence that contains one and only one of each of the following parts of speech?
As many of my students figured out, the key problem is the "subordinator" (traditionally, a "subordinating conjunction"): a sentence with a subordinator will necessarily have two verbs (one in the main clause, one in the subordinate clause).
I have come up with a sentence that contains two verbs and one of each of the other parts of speech:
But go slowly if the old dog wakes up.
Some people might argue that the coordinator "but" should not be at the start of a sentence like this, and the presence of a coordinator in a sentence does raise difficulties for the problem stated above, precisely because coordinators mostly coordinate two things with the same form (noun and noun, verb and verb, etc.). But a sentence-initial coordinator overcomes this difficulty, and, as Huddleston and Pullum argue in their book, there is no descriptive justification for the "rule" that says you cannot begin a sentence with a coordinator.
In class, though, I discussed a different problem: which parts of speech can appear in one-word sentences? I assumed that the answer is that only verbs can stand by themselves as sentences, and then only in the imperative form: Stop! But students made two other proposals that intrigued me:
This is not the verb "fire" but the noun: it's what you shout when there's a fire in a crowded theater (or what you don't shout when there isn't one).
At first, I wanted to say that "a sentence has to be a main clause with a finite verb in it, or it's not a sentence." But I was immediately suspicious of this: that's a conventional definition, but perhaps there's a more interesting definition that takes the two possibilities above into account.
I had hoped that Huddleston and Pullum's Cambridge Grammar of the English Language might help me on this score, but they make short shrift of the concept of "sentence" and instead focus their attention on the concept of "clause" (as is always the case with their unconventional moments, the discussion of the point is lively and fascinating).
So I turned where we all turn these days, to Wikipedia. The page is called Sentence (linguistics). The first definition there focuses on something that had crossed my mind: a sentence is a grammatical unit of one or more words that bears minimal syntactic relation to what comes before and after it. To me, that implies that "Fire!" in the above sense can be considered a sentence, because it can stand on its own without any other context, and it will still be understood. In contrast, "So?" might not be a sentence, as it cannot really be fully understood unless you know what it calls into question (that is, what preceded it). — But that argument is not very satisfying.
The second idea of the sentence provided on the Wikipedia page is that a sentence is anything that begins with a capital letter and ends with a period/full stop. This pretty much wipes out the whole issue of which parts of speech can be one-word sentences, because now anything can: