John Corcoran, Second-order logic explained in plain English, in Logic, Meaning and Computation: Essays in Memory of Alonzo Church, ed. Anderson and Zelëny.

There is something a little bit Guy Steele-ish about trying to explain the fundamentals of second-order logic (SOL, the logic that Quine branded as set theory in sheep's clothing) and its model theory while avoiding any formalisation. This paper introduces the ideas of SOL via looking at logics with finite, countable and uncountable models, and then talks about FOL and SOL as being complementary approaches to axiomatisation that are each deficient by themself. He ends with a plea for SOL as being an essential tool at least as a heuristic.