What to cover in the Guide straight after standard classical FOL?
Theories expressed in first-order languages with a first-order logic turn out to have their limitations — that’s a theme that will recur when we look at model theory, theories of arithmetic, and set theory. You will find explicit contrasts being drawn with richer theories expressed in second-order languages with a second-order logic. That’s why — although this is of course a judgement call — I do on balance think it is worth knowing just something early on about second-order logic, in order to be in a position to understand something of the contrasts being drawn. Hence this next short chapter.
There are no very substantive changes from the previous version. But it is a little tidier in some respects. So here is Chapter 4: Second-order logic, quite briefly.