One of the parts of the first edition of IFL that I liked rather better, and which seemed to work with students, was the bunch of chapters motivating the quantifier-variable notation and introducing the standard language of quantification theory. Which isn’t to say that the chapters have survived into IFL2 without some radical rewriting. But I hope the heavily revised versions still work! I think, at any rate, they are clearer than many about what is the essential Fregean insight, and what are artefacts of supplementary and optional decisions (e.g. because we decide to keep the quantifiers unary and single-sorted).
So here for your delight and delectation are the draft Chapters 25 to 29. All comments most gratefully received — ideally in the next five weeks or so, as June is the month for getting the final corrections/revisions of the book done before sending it off to CUP. Ever the optimist, here!