Full Idea
Extensionality is built into the semantics of ordinary logic. When a name-letter is interpreted as denoting something, we just provide the object denoted. All that we provide for a one-place predicate-letter is the set of objects that it is true of..
Gist of Idea
Extensionality is built into ordinary logic semantics; names have objects, predicates have sets of objects
Source
David Bostock (Intermediate Logic [1997])
Book Reference
Bostock,David: 'Intermediate Logic' [OUP 1997], p.356
A Reaction
Could we keep the syntax of ordinary logic, and provide a wildly different semantics, much closer to real life? We could give up these dreadful 'objects' that Frege lumbered us with. Logic for processes, etc.