 Modern predicate logic fixes the meaning of \(\forall\) and \(\exists\) with the respective clauses in the truth definition, which specifies inductively the conditions under which a formula \(\f(x_1,\ldots,x_n)\) (with at most \(x_1,\ldots,x_n\) free) is satisfied by corresponding elements \(a_1,\ldots,a_n\) in a model \(\M = (M,I)\) (where M is the universe and I the interpretation function assigning suitable extensions to non-logical symbols): \(\M \models \f(a_1,\ldots,a_n)\). The clauses are (where “iff” as usual stands for “if and only if”)