Here are the following two sentences.
At least one person speaks English.
Exactly one person speaks English.
Instead of ∃𝑥E(x), what do I write?
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You sometimes find the notation
as an abbreviation for "exactly one x".
With the standard symbol inventory, "exactly one" can be defined in terms of "at least one and not more than one" as follows:
∃x(E(x) ^ ¬∃y(E(y) ^ ¬(x = y)))
("There exists at least one person who speaks English, and there is noone who also speaks English but is different from that first person")
or more compactly
∃x∀y(E(y) ↔ y = x)
("There exists a person such that the people who speak English are exactly that person").