I'm not sure about the answer to this question in higher-order logics, but in first order logic, you can use the Completeness Theorem. In other words, something is provable from the empty theory if and only if it holds in every model of the empty theory. Thus, you can appeal to a semantic argument to show that a sentence is a necessary truth. For example, consider the first-order statement
$((P \to Q) \to P) \to P$
often known as Pierce's law. Proving this directly is unpleasant. However, let us prove it semantically using the above argument.
(1) $P$ is true and $Q$ is true. Then $P \to Q$ is true, and so $(P \to Q) \to P$ is true, and hence $((P \to Q) \to P) \to P)$ is true.
(2) $P$ is true and $Q$ is false. Then $P \to Q$ is false. Then $(P \to Q) \to P$ is true, and hence $((P \to Q) \to P) \to P)$ is true.
(3) $P$ is false. Then $P \to Q$ is true, and so $(P \to Q) \to P$ is false, and hence $((P \to Q) \to P) \to P$ is true.
We have exhausted all possibilities, and in each one, the statement is true. Then the statement holds in every model, and hence every model of the empty theory. By Completeness, it is provable from the empty theory. As my logic professor liked to say, this is the "coward's approach".