I am teaching logic, for my first time. It's a tools course for grad students, covering the basic logical tools that are of general applicability (plus three lectures on meta-theory): First Order Logic (FOL), basic (ZFC) set theory, basic (Kolmogorovian) probability theory and modal logic, using Barwise and Etchemendy. I've really nervous about the FOL portion, because I've never taken, TA'ed or taught a class on FOL. (There is a joke about how two Jesuits, who taught in a high school, teachers were talking. One says: "Do you know any chemistry?" The other says: "No, I haven't even taught it.") As a mathematics/physics undergraduate, I took a model theory class (with John Bell and William Demopoulos) and did an independent study on topos theory (with John Bell). I then did independent study in category theory and topos theory as a math grad student, and as a philosophy grad student I took an oral exam in logic (covering through the Goedel theorems). It comes by nature to me to approach logic always as yet another branch of mathematics, with languages being an algebraic structure like groups and fields, which can be given a set-theoretic account: "A first-order language is a 12-tuple <L,.,S,C,P,V,a,e,m,o,c,n> such that..." This, however, wouldn't be a pedagogically very good approach for teaching students, and it isn't the approach of Barwise and Etchemendy. (And now that I am no longer a mathematician, I worry that the lucidity of the above approach is illusory, due to an illusion of thinking one knows what one is talking about when one is talking about sets.)
I've found Barwise and Etchemendy extremely difficult to understand. They are apt to say something like that john (I am using bold to render their sans-serif) is an individual constant, Sits is a predicate, and Sits" followed by a parenthesis followed by john followed by a parenthesis is an atomic sentence. And then I get really puzzled. Are the individual constants particular token inscriptions? It seems they are, since they can stand in spatiotemporal relations ("followed by") while types can't. On the other hand, they talk of the same individual constant reappearing in multiple sentences, and that doesn't work if it is a token (I believe multilocation is possible, but, except perhaps for elementary particles, it takes a miracle). So it's a type, but a type that stands in spatiotemporal relations. Weird. And what does an inscription like Sits(john) in the book refer to? Does it refer to the token? No. It refers to some type. But it's made up of several types: Sits, (, john, ). This is really confusing.
I think I finally have two consistent interpretations of the text worked out. But it was hard to get there.