Peirce clearly set great store by his 1902 definition of logic as formal semiotic, whose principles he proposed to deduce by evident and rigorous mathematical reasoning from his triadic relational definition of a sign.
- It is from this definition, together with a definition of “formal”, that I deduce mathematically the principles of logic. (NEM 4, 21).
- It is from this definition that I deduce the principles of logic by mathematical reasoning, and by mathematical reasoning that, I aver, will support criticism of Weierstrassian severity, and that is perfectly evident. (NEM 4, 54).
By “criticism of Weierstrassian severity”, Peirce refers to the reconstitution of mathematical analysis precipitated by Karl Weierstrass during the latter part of the 19th Century, replacing many intuitive but problematic concepts with exacting enough definitions to support the development of coherent and powerful theories.
That bit of history teaches an important lesson. Not every form of words that might be cast about an intuitive object or used to express an intuitive concept will exhibit the strength of a formal definition, one that supports mathematical reasoning from evident truths and allows the deduction of a coherent and comprehensive theory of its object domain, for example, in our present instance, “the principles of logic”.
The moral of the story so far is to recognize the qualifications of these “industrial strength” definitions, and to realize what it means for Peirce to be advertizing these very virtues for his 1902 brand of sign definition.