What is the predicate in "Is he happy?"

The equivalent of

Is he happy?

for grammatical analysis purposes is

He is happy.

The subject is "He".

The predicate, or full verb phrase, is "is happy".

(Or is this discussion beyond that level of analysis?)

Is he happy?

Here is a tree diagram showing how the elements are diagrammed. Note the function label of 'prenucleus' for the verb which is co-indexed to the predicator function represented by 'gap':

Thus the prenucleus "is" functions as part of predicate consisting of "is happy".

The, by now, classic, old-fashioned Chomskyan method of analysis here would be that 'is (x) happy' is definitely the VP and predicate because it is a simple question-transformation from 'he is happy'. Since that doesn't seem to be convincing to you from the beginning, I attempt to give a non-transformational justification.

The idea of subject and predicate is all about how the elements of a statement map to logical expression. The predicate is the logical form with a variable and the subject is the instantiation of that variable.

Things like NP and VP are purely syntactic (Or as purely syntactic as they can get since the whole point of syntax is to determine what order or construction of elements imply about the semantic relations of those elements as compositionally as possible.

Must a logical statement implied by a particular utterance correspond directly to a contiguous sequence of words or phrases? Obviously not. Must a syntactic constituent, a node in a parse tree, correspond to a contiguous sequence? This depends on how you define constituent, but I think it is better to allow non-contiguous segments to be constituents, as that allows easier rules for formation.

These are words to describe parts of statements. A question is not a statement. It surely shares a lot with a statement, and in the situation you're describing is a straightforward syntactic transformation of a statement.

All this is to say that for "Is he happy?":

• questions don't have predicates.
• But that is too definitional. Whatever one might say about a question that corresponds to predicate, 'is' and 'happy' somehow together make up that thing corresponding to predicate.
• syntactically, the VP, which happens in this case to correspond to the thing that corresponds to predicate, is 'is (x) happy'.

Whatever all this is, what do you get out of determining for this sentence what the predicate is? I think you get more out of determining what the VP is.

The question simply employs a subject-verb inversion. Without the inversion it would be the declarative statement, "he is happy".

In that case, the predicate, stated as an infinitive, is "to be happy".