Consider the sentence Ed signed up and someone else did, too. To analyze it as a conjunction, we need to fill out the second clause, not only by replacing did by the phrase signed up but also by making explicit an implicit reference to Ed. The full analysis would proceed as follows:
Ed signed up and someone else did, too
Ed signed up ∧ someone other than Ed signed up
Ed signed up ∧ someone other than Ed is such that (he or she signed up)
Se ∧ (∃x: x is a person other than Ed) x signed up
Se ∧ (∃x: x is a person ∧ x is other than Ed) Sx
Se ∧ (∃x: Px ∧ ¬ x is Ed) Sx
Se ∧ (∃x: Px ∧ ¬ x = e) Sx
Se ∧ ∃x ((Px ∧ ¬ x = e) ∧ Sx)
[P: λx (x is a person); S: λx (x signed up); e: Ed]
That is, the function of the word else here is to restrict an existential claim by requiring that the example it claims to exist be different from a previous reference; in short, else serves to indicate a new example. The restriction of existential claims so that they claim the existence of new examples can be found not only with the word else but also, though less obviously, in a variety of quantifier phrases we have not yet attempted to analyze.