How to convert sentence that contains “no more than 3” into predicate logic sentence

Question

How to convert sentence that contains “no more than 3” into predicate logic sentence?

For example: "No more than three $x$ satisfy $R(x)$"
using predicate logic.

This is what I have for "exactly one $x$ satisfies $R(x)$":
$\exists x(R(x) \land \forall y(R(y) \rightarrow (x = y)))$

Answer 1

Interpreting 'no more than three' as 'at most three' (i.e. it could be three, two, one, or maybe just none at all), you can do:

$$\exists x \exists y \exists z \forall u (R(u) \rightarrow (u = x \lor u = y \lor u = z))$$