Peirce uses the word “formal” in a sense that gives it normative force. It is this sense in which he defines logic as formal semiotic.
But taking “formal” in a normative sense creates difficulties for John Sowa’s thesis of a “sharp distinction between ‘formal logic’, which is part of mathematics, from logic as a normative science”.
I don’t think it means that “formal logic” is “formal formal semiotic”, much less a part of mathematics. It simply means that logic is inherently formal (= normative) and adding the adjective “formal” is redundant.
Selections from C.S. Peirce, “Carnegie Application” (1902)
No. 12. On the Definition of Logic
Logic will here be defined as formal semiotic. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. Namely, a sign is something, A, which brings something, B, its interpretant sign determined or created by it, into the same sort of correspondence with something, C, its object, as that in which itself stands to C. It is from this definition, together with a definition of “formal”, that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has virtually been quite generally held, though not generally recognized. (NEM 4, 20–21).
Selection from C.S. Peirce, “Ground, Object, and Interpretant” (c. 1897)
Logic, in its general sense, is, as I believe I have shown, only another name for semiotic (σημειωτική), the quasi-necessary, or formal, doctrine of signs. By describing the doctrine as “quasi-necessary”, or formal, I mean that we observe the characters of such signs as we know, and from such an observation, by a process which I will not object to naming Abstraction, we are led to statements, eminently fallible, and therefore in one sense by no means necessary, as to what must be the characters of all signs used by a “scientific” intelligence, that is to say, by an intelligence capable of learning by experience. As to that process of abstraction, it is itself a sort of observation. (CP 2.227).