What does it mean for a collection of inference rules to form the strongest judgment form that is closed under those rules?

A collection of inference rules is taken to specify a judgment form - assuming it's $-J$ - that is:

• The strongest judgment form:

  • That means $J$ cannot contain elements whose membership cannot be inferred from these inference rules.

    -> $a\in J$ implies that $aJ$ can be proven.

• Closed under the inference rules:

  • That means $J$ must contain all elements whose membership can be inferred from these inference rules.

    -> $aJ$ can be proven implies that $a\in J$.