Walking the line between phenomenology and mathematics, let us cast our eyes on the prize of defining logic. Peirce defines logic as formal semiotic — and that in turn calls for a definition of sign.
Here is a place where he defines logic and signs in one deft pass —
Peirce defines logic as formal semiotic. We know semiotic is the doctrine or theory of signs, but the current passage leaves us with a promissory note on the meaning of formal. Luckily, though, it is easy enough to find other places where he tells us that formal is pragmatically synonymous with normative, and that puts logic squarely within the normative sciences, as classical tradition always said it ought to be.
Peirce defines a sign in relational terms, as one role out of three, the other two roles being the role of its interpretant sign and the role of its object. This is a very different matter from defining an essence, that is, an inalienable, inherent, intrinsic property of a “thing in itself”.
Peirce is emphatic about the independence of his joint definition from any reference to human cognition, just as normative sciences ought to be orthogonal to descriptive sciences, and yet he insists that his “non-psychological conception of logic” virtually inheres in the general idea of logic, however incognito it may abide there.