 To introduce other quantifiers, one needs to appreciate what kind of expressions \(\forall\) and \(\exists\) are. Syntactically, they are operators binding one variable in one formula. To see how they work semantically it is useful to rewrite  (1)  and  (2)  slightly. First, every formula \(\p(x)\) with one free variable denotes in a model \(\M\) a subset of M; the set of individuals in M satisfying \(\p(x)\). More generally, if \(\p(x,x_1,\ldots,x_n) = \p(x,\xbar)\) has at most the free variables shown and \(\abar = a_1,\ldots,a_n\) are elements of M, let