> Your argument boils down to saying I believe something therefore I am correct.

No it doesn't. It boils down to something appearing to be true to me (or "us") and there being actually no good reasons to think that it is not true.

> Also, that’s really not why scientific facts are considered true. Science is based on positive evidence as in a prediction is made and it turns out the prediction was accurate.

You're wrong, and here is why: first I would ask you what counts as positive evidence that a prediction made was verified. You would give me some theoretical explanation (depending upon the context), and then I would keep asking you more detailed questions about your explanation. Eventually I would ask a question that you wouldn't really be able to answer, other than "look, it just appears to me that such-and-such counts as predictive evidence!" (or something like that).

So for example if we had a scientific theory that all red cars cannot exceed 100mph, and you said "that's not true, for look at my red car going 105mph!", I could ask you "well, yeah, your speedometer reads 105mph, but how do you know you're actually going faster than 100mph?" You would give me an explanation about how speedometers work, and I would say "well yeah, but how do you know that theory can be generalized to this particular case <blah blah blah>". Eventually after this chain of questioning you would be left relying on some sort of mere appearance being true. And of course that is rational. The problem is that this chain of reasoning can also be applied to moral theories as well.

> Simply presupposing something is true isn’t considered evidence that it is true.

I'm not merely presupposing something is true. I'm observing that something first appears to be true, asserting there are no good reasons to think it is not true, and then supposing it is true.

Contrast these two cases:

1. It appears to me that 2+2 = 4, and I can think of no good reasons at this moment to doubt otherwise. I guess, therefore, that 2+2 really is 4.

2. I shall pressuppose that 2+2=5!

You're acting as if I'm taking the strategy of (2), but I am actually taking the strategy of (1). And in fact you are as well. Here's a really simple way to see why.

Suppose you thought of some argument that showed that the Principle of Phenomenal Conservativism was false. But why should you, after pondering that argument, believe that argument? Because it appears to you to be true, and you can give no plausible retort for why it's false. Thus in "refuting" the Principle of Phenomenal Conservatism, you must use it! This shows that it is self-defeating to argue against the Principle.

The Principle of Phenomenal Conservatism has to be true for other reasons as well. For example, the following are all statements that I'm sure you agree are true:

1. 1 = 1.

2. The law of non-contradiction is true.

3. Empirical theories which make testable predictions are more likely to be true than empirical theories which can't.

4. The keyboard in front of you actually exists.

But why do you believe these statements? You won't be able to insert some other empirical theory to justify them. They are instead just sort of raw appearances that we suppose are true (because we can't think of any good reasons to doubt them). We make use of our sensory instruments to do this (e.g, our eyes, our non-moral intuitions, etc). But this is exactly like me asserting that "it's wrong to torture children for fun", which relies on a moral intuition that it is wrong (plus the fact that we can't think of any good reason to think otherwise).

All of this is to say: you can't be a scientific realist without also being a moral realist at the same time, or at least if you can, it's because of some argument I haven't yet considered :]