As was requested over email, let me provide a worked out example of something like the sentences from Part I, and a couple of notes about parts II−III.

Part I: Translating quantified arguments into FOPL

Suppose that you were asked to translate (12).

(12) Pat ate every sandwich.

What you need to do is provide a key, and a translation. Probably simplest to make a single key for everything (since at least there are a couple in there that refer to the same individuals, or use the same predicates). But here, since there’s just the one sentence, the key would include:

• EAT(x, y) : x ate y
• SANDWICH(x) : x is a sandwich
• p : Pat

Then the translation of this would use the quantifier ∀ (because the English sentence in (12) contains “every”), and what it says is basically for every x that is a sandwich, Pat ate x. Or, in FOPL:

∀x₁( SANDWICH(x) → EAT(p, x₁) )

That’s it for the sentences in Part I, just do that for each of them.

For part II, keep in mind that there’s a difference between “everything likes Bob” and “everyone likes Bob”—the latter one is only generalizing about people, so you’ll want to use the predicate PERSON (at least, in general, if there are any individuals in the domain D that are not people).

For Part III, the models M₁, M₂, etc. are setting out possible actual states of the world. That is, who the professors are and who the vain people are, etc. You won’t actually need to do anything with the assignment function g. The function g is used in the semantic interpretation of a formula with a quantifier in it by taking g and modifying it, but that’s implicit when you write ∀x₁(…). What that means “behind the scenes” is “go through all of the individuals in the domain D, one by one, and for each of them change g so that g(x₁) is the individual you’re considering. Then see if the (…) part is true (where usually the (…) part will contain at least one x₁ that will be replaced by the individual you are considering when you evaluate it). I reiterate that this is not something you need to be writing down on (this) assignment, but that’s how ∀ works. So, step by step, working out something like “everyone is a professor”, we’d translate it as ∀x₁(PROFESSOR(x)) (no need to worry about people vs. non-people, because D as defined has only people in it). To see if it is true, the procedure below is followed:

• Take the first individual in D. It’s Carol.
• Take the assignment function g and change it so that g(x₁)=Carol.
• Check to see if PROFESSOR(x₁) is true (under M₁ and g).
• This is going to be true if g(x₁) ∈ Val(PROFESSOR).
• g(x₁) is Carol according to the change we made a couple of steps ago.
• Val(PROFESSOR) is {Carol, Paul, Pete} according to the definition of model M₁.
• So Carol ∈ {Carol, Paul, Pete} is true.
• Move on to the next individual in D. It’s Paul.
• Take the assignment function g and change it so that g(x₁)=Paul.
• Check to see if PROFESSOR(x₁) is true (under M₁ and g).
• …
• Having no more individuals left, the whole formula ∀x₁(PROFESSOR(x)) is true if we got only “true”s above when we checked whether a formula was true.

So that’s “behind the scenes.” But you can kind of do this in your head. To re-reiterate, you don’t have to write down anything about g or this semantic evaluation procedure on homework 7, but what I wrote above is explaining how it works in FOPL, based on the semantics we have for FOPL. In particular, that PREDICATE(x₁) is true if g(x₁) ∈ Val(PREDICATE).