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Compare these 2 sentences:
I have a few questions:
The capital of Belgium is Brussels.
The subject is the phrase in front of the verb. The entire phrase is a noun phrase. The verb is the present tense, third person singular of be: is. It is intransitive in that it doesn't take a direct object. Here, it is a linking verb; there is no action.
The noun which follows the verb is the subject complement.
Brussels is the capital of Belgium.
The subject here is Brussels. The subject compliment is the noun phrase the capital of Belgium.
The linguist R. M. W. Dixon presents another way of understanding this type of sentence in English in the context of language typology. This is from his ‘Basic Linguistic Theory’, Vol. 2 Ch.14: Copula Clauses and Verbless Clauses:
Each language has intransitive and transitive clause types. There is frequently a further minor – but still important – type: copula clause. This has as predicate a copula verb, taking two core arguments, a copula subject (CS) and a copula complement (CC). The predicate in an intransitive or transitive clause has reference. A copula verb as predicate is different in that it has relational rather than referential meaning. [Italics mine.]
He then lists some of the relations that can be expressed with a copula verb in English. They include:
Identity: This man is a doctor.
Attribution: This man is clever.
Possession: This book is John’s.
Location: The apple tree is in the garden.
The nature of the relationship expressed depends on the copula complement – for identity it is a noun (as in your examples), for attribution an adjective, etc.
Your basic question can be reinterpreted in these terms as ‘when are CS and CC in the Identity relation reversible?’ To answer this, Dixon distinguishes three possible types:
Most combinations of CS and CC in these categories can be reversed, sometimes with a shift of meaning. For example, these two sentences are equivalent:
Mary Smith (1) is the owner (2).
The owner (2) is Mary Smith (1).
The Brussels example is of the above type (specific referent (1) + specific description(2)). However, there is one combination – 1 and 3 – that is not reversible. For example, the second sentence here doesn’t work:
Mary Smith (1) is a teacher (3).
*A teacher (3) is Mary Smith (1).
And how do we identify CS and CC in cases where the Identity relation is reversible? Dixon (p. 172):
“The principle is basically pragmatic, relating to what is topic within a section of discourse, what has a known referent, and what is new, and so on.”
This is just one way of looking at things. The copula verb (or just copula) in fact is not so different from what other answers call a linking verb. Other linguists avoid using either term – you could look up other ELU questions with the “copula” tag if you are curious.
When parsing out a sentence, you normally 'drop out' clauses that complicate the issue to show the essential structure. the word 'of' is an indicator of a phrase that can usually be discarded....so drop out 'of Belgium' whereever you see it.
The capital of Belgium is Brussels. --> The capital is Brussels.
So now you can see the subject, verb, noun structure. Similarly:
Brussels is the capital of Belgium. --> Brussels is the capital.
So you see that this sentence is just flipping the first around the verb 'is'.
