Teaching Predicate Logic
I'm thinking about how to introduce the semantics of predicate logic to beginning philosophy students. In particular, I'm interested in the interpretation of predicates and quantifiers. Last year in logic class, it seemed that most students were rather unhappy with the formal recursion on truth we were teaching them.
So I've just picked 15 random logic textbooks to see how they are doing it.
Group 1 (functions and sets): Interpretations are introduced as entities that assign to each n-ary predicate symbol a class of n-tuples of elements of the domain. (Machover, Beckermann, Bostock, Newton-Smith, Mendelson, Kutschera, Allen/Hand, Bühler)
Alternatively, interpretations are sometimes specified as assigning to each n-ary predicate symbol a mapping from classes of n-tuples of elements of the domain to truth values. (Boolos/Jeffrey, Restall)
Group 2 (translation tables): Interpretations are not defined. Rather, it is said how they are to be given, by some kind of translation table. There are three variations of how this is might look for predicate symbols:
a) H: Human; L: Loves; (Bucher)
b) H: ... is human; L: ... loves ---; (Tomassi)
c) Hx: x is human; Lxy: x loves y; (Allen/Hand, Newton-Smith)
Group 3 (colloquial explanation): Neither interpretations nor translation tables are introduced. Instead the relevant rules are only stated as an explanation of what is going on in translating entire sentences. Again, there are the three possibilities from group 2: Gensler and Bühler explain that "H" is the translation of "Human"; Hurley that "H..." translates "... is human"; Cauman that "Hx" translates "x is human".
Some authors occur twice in this list because they use one of the simple methods first, and switch to method 1 much later in a more advanced chapter, e.g. on soundness and completeness. I think this way of mixing methods 2 and 1 is a good strategy. Still have to think about whether to choose 2a, 2b or 2c.
Over to quantification.
Group 1 (Recursion on satisfaction): 
xA is satisfied by a
valuation V under an interpretation I iff A is satisfied by every valuation
V' under I that differs from V at most by what it assigns to x. (Machover,
Bostock)
Alternatively, Tarskian sequences are sometimes used instead of valuations. (Newton-Smith, Mendelson)
Group 2 (Recursion on truth): xA is true under an interpretation I iff A(x/a), that is, A
with every free occurrance of x replaced by some new individual constant a,
is true under every interpretation I' that differs from I at most by what
it assigns to a. (Boolos/Jeffrey, Bostock, Beckermann, Kutschera, Bühler)
Group 3 (Substitution): xA is true (under some interpretation I) iff A(x/a) is true
(under I) for all individual constants a. (Something like this is used by
Restall.)
Group 4 (Expansion): xA is true (under some interpretation I) iff the conjunction of all
instances of it are true (under I). (Allen/Hand)
Group 5 (colloquial explanation): No formal rules are given.
Instead, it is explained that "x" means "everything" or "for all x (in the domain)",
followed usually by lots of sample translations of more complex
quantifications. (Bucher, Allen/Hand, Newton-Smith, Hurley, Cauman,
Bühler, Tomassi)
Again, I think a mixed approach might be best for our purposes. This time I favour mixing methods 5 and 2.