Jon A, I was trying to emphasize the distinction between syntax and semantics. Literally, 'formal' means "according to form"; 'syntactic' means "according to the arrangement (taxis)". For diagrams (in one or more dimensions) the form is the arrangement. Therefore, the word 'formal', when applied to notations in any number of dimensions, is a synonym for 'syntactic'.
Not all formal or syntactic transformations are necessary or even quasi-necessary. But those syntactic transformations that are determined by *sound* rules of inference preserve truth. But the truth of a conclusion depends on both the starting point (axioms) and the method of reasoning. To ensure true conclusions, we need true axioms, sound rules, and formally precise application of those rules. Formality alone is not sufficient. With the above prelude, we can analyze what Peirce said in CP 2.227:
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.
Note that Peirce used the phrase "quasi-necessary, or formal," as adjectives that qualify the "doctrine of signs". In that context, quasi-necessary is determined by formal rules of inference that were proved to be sound (AKA truth preserving). In other contexts, formal transformations serve purposes other than truth preservation. Therefore, formal and quasi-necessary are not, in general, synonyms. For determining truth, Peirce introduced endoporeutic (outside-in evaluation), which is a version of model-theoretic semantics over 30 years before Tarski's. But modern logicians (such as Quine) couldn't understand what Peirce was saying because his terminology was foreign to them. Since I had studied Tarski, Quine, Hintikka, and others before I discovered Peirce's EGs, I could see what Peirce was doing and how it paralleled (and often surpassed) what modern logicians have been doing and saying. For those of us who are trying to convince modern students to study Peirce, we need to become bilingual. We need to show how his terminology and notations map to and from current systems -- more importantly, how they point the way to new discoveries and innovations that are *obscured* by modern methods. John
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