A caveat: I'm not at all sure how much I'm projecting on Peirce as far as this point goes.

Well I saw some interesting ideas about science in Piaget, which is at least as tenuous.

if you want a condensed version of his semiotic look over this. You might actually need to read some of the rest of that article (which, I admit, is a bit long) to put it in more context. Also, this wikipedia page looks pretty comprehensive. I'm pretty confident that they're leaving a bit out that might be clearer if you read Peirce, but I'm not sure of how much instrumental value that would be to you.

Okay, I read most of this, but not in too much detail. I'm guessing that the things like

A rheme (also called sumisign and seme) is a sign that represents its object in respect of quality and so, in its signified interpretant, is represented as a character or mark, though it actually may be icon, index, or symbol. The rheme (seme) stands as its object for some purpose. A proposition with the subject places left blank is a rheme; but subject terms by themselves are also rhemes. A proposition, said Peirce, can be considered a zero-place rheme, a zero-place predicate.

are not the crucial parts.

I'm seeing the idea that one has a partially correct theory explaining one's observations and that it is continuously refined. Is that the main idea or am I missing something? It's valid, but I don't know how it compares to other ideas at the time. Also, the emphasis seems to be on refining one's ideas by continuing to contemplate the same evidence, which isn't very empirical, but I could be misunderstanding.

Peirce's three valued logic predating Post.

That's interesting. There are a lot of people using three valued logic today as if it is a huge insight that we can have a system that classifies statements as known to be true, known to be false, and unknown, or with three other, slightly different, categories, but in Pierce's day it was an important insight (well, there were similar ideas before, but they weren't formalized).