To me, man is the subject and it has two verbs — was and known —, making there a complement.
My teacher argued that the verb is "was known".
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Leaving out the Toe Suck Fairy, an independently serious Healthy Hyphenation™ problem,
the sentence to be accounted for is
This is a very simple sentence, suitable for beginning stories, displaying two syntactic processes:
inverts the verb and the original subject, and inserts a dummy there as the new subject.
[A man who was known as
X] was. ==> There was [a man who was known as
(This is, btw, the "existential" sense of be, which mostly occurs with There-Insertion.)
deletes a subject Wh-word, and an auxiliary form of be, from a relative clause
There was a man [who was known as
X]. ==> There was a man [known as
The there that's inserted is not a complement, though it is the subject of was.
The derived participial phrase known as
X is of course a reduced relative clause.
One could go into more detail about Passive Participles here, but let's not, OK?
There was a man known as the Toe Suck Fairy
The structure of this sentence, where Predicator is the function carried out by the verb, is as follows:
That complement is an internal complement of the verb BE. It is sometimes referred to as 'a displaced subject', but this is just doffing ones cap at the semantics, it has nothing to do with the syntax! It's not a grammatical subject in any way!
Notice that the Noun Phrase a man known as the TSF can't be considered a Direct Object here, because a Direct Object is a very specific kind of complement of the verb. In particular, Direct Objects have the thematic role of patient in relation to the verb. This means they are the thing the action's being done to. So in Bob punched me, Bob is the agent and I am the patient (the person who received the punch). So in that particular sentence me is the direct object. In the original question, nothing is being done to the TSF, so it can't be a Direct Object.
The structure of the complement Noun Phrase is a bit ambiguous. This Noun Phrase might involve a noun modified by a relative clause which has undergone Whiz-deletion. This is when the relative pronoun and some form of the verb BE have been deleted. The full underlying Noun Phrase is:
There are a few possible structures for the relative clause here.
Here known is an adjective which takes a dependent preposition as, and therefore a Preposition Phrase headed by as as its complement. In turn, the complement of the preposition as is the Noun Phrase the "TSF".
Here was known is the passive construction, where was is the passive auxiliary BE, and known is the past participle of the lexical verb KNOW. Again the verb know also takes a Preposition Phrase headed by as. The complement of as is, of course, the Noun Phrase the TSF.
Alternatively the Noun Phrase might be considered as simply a noun modified by a postpositive adjective phrase. Attributive Adjective Phrases usually can't occur before the noun with complements or other dependents, and instead the adjective phrase must occur postpositively after the noun in question. In this structure the head of the phrase is the Noun man. This is modified by the Adjective Phrase known as the TSF. The head of the phrase is the adjective known, taking as its complement the Preposition Phrase *as the TSF.
As the word order for the different structures is identical, it's not possible to tell which one is being used. I would probably tend towards, number 2, but who knows!!
Hope this helps!
[Note: I've labelled as the TSF a predicative complement here. Because it's only really the TSF that describes the man and not as the TSF, some people may wish to use a different term here. Huddleston & Pullum 2002 (The Cambridge Grammar of the English Language) use the term marked predicative complement.]
There’s a problem.
There is a well-known man.
There is a man known as the “Toe Suck Fairy.”
These are “existential there” sentences.
“Existential there” sentences have the exceptional feature of having two subjects:
In English, the subject typically immediately precedes the finite verb in declarative clauses. In questions, order inversion occurs, and the subject immediately follows the auxiliary: “Is there a problem?” Clearly, the positional subject fits this criterion.
Semantically, the subject in a sentence in the active voice is an agent or theme performing the action expressed by the verb; or when the verb is a theme, the subject receives a property assigned to it by the predicate. The notional subject fits this criterion.
Another feature of subjects is that in a sentence, the subject generally agrees with the finite verb in person and number. Let’s make the first sentence plural:
(a) There are problems.
(b) There’s problems.
In (a), the verb agrees with the notional subject; in (b) with the positional subject.
The form (a) is correct and most commonly found; although (b) is found too (informal, maybe regional?).
Reference: I haven’t consulted the book, but the existential there section in the Wikipedia article on subject refers for a discussion of the subject status of existential there to: Douglas Biber et al., Longman grammar of spoken and written English, Essex: Pearson Education, 1999, page 944.
Note for completeness: Some grammaticians disagree with the notion that the notional subject (or displaced subject or semantic subject or true subject) is a subject too, and call it something else. Apparently, the Cambridge grammar of the English language by Rodney D. Huddleston and Geoffrey K. Pullum, 2002, has a section “Evidence that subject function is uniquely filled by dummy it and there”, on pages 241‒243. Haven’t found or read it myself.
The notional subject “a man known as the ‘Toe Suck Fairy’”:
in this noun phrase, “known as the ‘Toe Suck Fairy’” is a reduced relative clause. The phrase is a shortened form of “a man who was known as the ‘Toe Suck Fairy’”. In the relative clause, the omitted word “who” is the subject.
As Araucaria explains in another answer, there are 2 ways to analyse the relative clause. In both ways, the omitted word “who” is the subject.
Also, should you be putting “Toe Suck Fairy” in quotations, or should you italicize them?
In this case, I would prefer putting those words in quotations. But style guides’ opinions may vary.
There was once a king who had three daughters.
If some analyse "there" as subject or something like that, e.g. pseudo-subject they have a very queer conception of sentence structures. The subject is, of course, "a king". It is an inverted sentence structure and the adverb "there" only indicates that the subject is in post-position after the verb "was".
We might say in normal word order "A king was there once/A king lived once", but such a sentence is not the optimal thing. The main point of the story at the beginning is the king and by placing "the king" after the verb - a special position - a special weight is put on the word king and that is wanted.
Linguists try to explain such a special word order by using the complicated terms "thema and rhema/rema" (I'm not quite sure whether these are the proper terms in English.) I don't use these terms as they are more confusing and difficult to explain.
It is easier to say that the local adverb "there" can be used as an indicator of inverted word order in order to lay special stress and weight on the subject in post-position.
This special word order, typical for the beginning of fairy-tales, is used in the same way in French, Italian and German:
Il était une fois un roi qui ... C'era una volta un re che ... Es war einmal ein König, der ...