 A theory in second (or first) order logic is said to be categorical if any two of its models are isomorphic. A single sentence (which has a model) is categorical if and only if it characterizes some model up to isomorphism (see  §7.1).  A nice property of a categorical theory \(\Gamma\) is (semantic) completeness: Any sentence \(\phi\) in the language in which the theory is written is decided by \(\Gamma\) in the following sense. Either every model of \(\Gamma\) satisfies \(\phi\) or every model of \(\Gamma\) satisfies \(\neg\phi\). In other words, either \(\phi\) or \(\neg\phi\) is a (semantic) logical consequence of \(\Gamma\). The reason is very simple: \(\Gamma\) has, up to isomorphism, only one model M. Since isomorphism preserves truth in second-order logic, \(\phi\) or \(\neg\phi\) is a (semantic) logical consequence of \(\Gamma\) according to whether \(\phi\) is true in M or false in M.