It's often easier to translate a sentence into idiomatic English if you translate it into some kind of 'loglish' first; the same is true of formalising English sentences. In this case it would be: 'For all x there exists a y such that x is smaller than y.' In idiomatic English, clearly all this says for a domain of discourse containing only positive integers is that for any positive integer there is always a larger positive integer, so you are indeed correct.

Can I run all four by you:

1. Universal quantifier (and upside down A), then an x, then an existential quantification (a backwards E) then an Mxy.

This one we discussed.

2. An existential quantification (a backwards E), then a y, then a universal quantifier (an upside down A) with an x beside it, and then a Mxy.

I interpret this to mean that some integer is greater than every integer, which is going to be false whether we are dealing with only positive integers, or all integers...

3. Universal quantifier (an upside down A), with a y beside it, then an existential quantification (a backwards E) with an x beside it, and then a Mxy.

I interpret this to mean that every integer is greater than some integer, which is false if we're dealing only with positive integers, but true if we're dealing with all integers.

4. Existential quantification (backwards E) with an x, then a universal quantifier (uspide down A) with a y, then a Mxy.

I interpret this to mean that some integers are less than every integer which is true if we're dealing only with positive integers, but false if we're dealing with all integers.

Am I translating correctly, and then determining properly whether each is true or false? Appreciate any help.

There is a problem here. ∃x∀yMxy Does indeed say, 'there exists at least one positive integer that is smaller than all positive integers', but it isn't true. You're probably thinking that 1 satisfies it, but it isn't smaller than all positive integers; it isn't smaller than 1. Remember, it's possible to instantiate two different variables with the same object.

However, that's an easy mistake to make. After the misapplication of inference rules in proof systems, getting confused about what happens when quantifiers start moving about is probably the most common problem in any introductory logic course, but you seem to understand it (almost) perfectly well.

I would say the best way to think about it is that whenever you have two or more different quantifiers at the beginning of a sentence, what type of sentence it is is determined by which quantifier comes first. If you have a universal quantifier first, the sentence is a universal claim about all of the objects in the domain. If this is followed by an existential quantifier, then we are saying that for any given object in the domain, there is at least one object in the domain of discourse that bears a certain relation to it (the nature of the relation doesn't really matter because from a mathematical point of view it will just be a set of ordered pairs). If you have an existential quantifier first, the sentence is an existential claim, and if it is followed by a universal quantifier, we are claiming the existence of one or more specific objects that bear a certain relation to all of the objects in the domain.

Thank you. I am doing a distance ed logic course, and appreciate the community here who have been so helpful...