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.

S.KYou never reviewed

Possibilities and Paradox: An Introduction to Modal and Many-Valued Logic 1st Edition

by J. C. Beall, Bas C. van Fraassen.

Does that mean that you don’t find it helpful?

Peter SmithIt does get recommended, in fact, at p. 15 of the second part of the Guide, in particular for its treatment of relevant logics.

And it is on my list of books to revisit when updating the chapters on modal and other logics.

Wolfgang SchwarzMight it be worth mentioning the possibility of interpreting higher-order quantifiers as not ranging over sets, but over intensionally understood properties (or propositions, etc.)? This doesn’t come up much in mathematical logic, but it plays a big role in recent metaphysics, as surveyed e.g. in https://onlinelibrary.wiley.com/doi/10.1111/phc3.12756?af=R.

Peter SmithA good thought, yes. And thanks for the reference to Lukas Skiba’s useful survey piece!