Important to learn is this stuff

Question

This song is fun to sing.

This pizza is too hot to eat.

Is the infinitive there considered a complement of the predicate adjective?

Answer 1

1. This song is fun to sing.

2. This pizza is too hot to eat.

Sentence 1

In the first sentence the infinitival clause to sing is a hollow infinitival clause functioning as the Complement of the adjective fun. Notice that the song itself could be depressing but it could be fun to sing. The fact that to sing is a hollow clause here is specifically because of the adjective fun. The adjective fun dictates that our interpretation of the Object of the infinitival clause is determined by the Subject of the matrix clause. We understand the sentence like this:

• This song is fun to sing [it]

The Subject of the infinitival clause is just determined from the context. We could reconstruct the sentence like this:

• This song is fun [for people to sing [it]]

In some grammars these types of adjectives are called TOUGH ADJECTIVES.

Sentence 2

Sentence (2) looks very similar to sentence (1). However, it's very different. Consider the following sentences:

• ^?This pizza is hot to give to a child
• ^?This course is long to finish.

Neither of these sentences is very good. The reason is that neither hot nor long tend to take hollow infinitival clauses as Complements. They are not very good at being tough-adjectives.

However, if we modify these adjectives with the degree adverb too, they become perfectly grammatical and felicitous:

• This pizza is too hot to give to a child.
• This course is too long to finish.

The reason that these sentences are ok is that the hollow infinitival clause is licensed by the deictic degree adverb too. Similarly, the infinitival clause hot to eat in the Original Poster's sentence number (2) is similarly licensed by the adverb too. Notice, however, that the adverb too cannot license the infinitival clause unless it has a following adjective to modify:

• *The pizza is too to eat. (ungrammatical)

Also notice that unlike other words that take Complements, the word too does not form a phrase directly with its Complement. For this reason CaGEL say that the infinitival clause is the Indirect Complement of the adjective hot. Notice that this is very different from being a bona fide Complement of an adjective as in sentence (1).

Conclusion

The infinitival phrase is a Complement of the adjective fun in sentence (1). However, the infinitival clause is not a Complement of the adjective in sentence (2) where it is licensed by the Modifier too not by the adjective itself. Some grammars would refer to the infinitival clause here as an Indirect Complement of the adjective hot.

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^References:

^{CaGEL is an abbreviation for The Cambridge Grammar of the English Language by Huddleston & Pullum, 2002.}