realistcat wrote:
I did see a fragment of Leibniz's notes for a universal characteristica. the terms denote properties, so it was kind of a symbolic Aristotelian syllogistic.

I don't think Peirce had much of anything to do with the development of symbolic logic. There was a "logical algebra" developed in early 1800s...by De Morgan I think...that was a predecessor to Frege. But Frege was the genius whose accomplishment was far above anyone before him. That's because his logic covers the territory of both Boolean propostional calculus and Aristotelian syllogistic, but at a higher level of comperehensiveness. Aristotle's syllogistic has certain informal assumptions which Frege relaxed. Aristotle for example assumes that there can be no empty terms. Now, i think that is probably true. But Frege's logic makes this explicit.

wiki wrote:
Peirce made a number of striking discoveries in foundational mathematics, nearly all of which came to be appreciated only long after his death. He:

* Discovered in 1880[20] how that which is now called Boolean algebra could be expressed by means of a single binary operation, either NAND or its dual, NOR. (See also De Morgan's Laws). This discovery anticipated Sheffer by 33 years.

* In Peirce (1881)[21] set out the now-classic axiomatization of natural number arithmetic, a few years before Dedekind and Peano did so. In the same paper Peirce introduced the first purely cardinal definition of a finite set[22], by the property now called that of being "Dedekind-finite".

* Discovered, independently of Dedekind, an important formal definition of an infinite set, namely, as a set that can be put into a one-to-one correspondence with one of its proper subsets.

* In Peirce (1885), set out what can be read as the first (primitive) axiomatic set theory, anticipating Zermelo by about two decades.

In 1918, the logician C. I. Lewis wrote, "The contributions of C.S. Peirce to symbolic logic are more numerous and varied than those of any other writer — at least in the nineteenth century."[23] Beginning with his first paper on the "Logic of Relatives" (1870), Peirce extended the theory of relations that Augustus De Morgan had just recently awakened from its Cinderella slumbers. Much of the actual mathematics of relations now taken for granted was "borrowed" from Peirce, not always with all due credit (Anellis 1995[24]). Beginning in 1940, Alfred Tarski and his students rediscovered aspects of Peirce's larger vision of relational logic, developing the perspective of relational algebra. These theoretical resources gradually worked their way into applications, in large part instigated by the work of Edgar F. Codd, who happened to be a doctoral student of the Peirce editor and scholar Arthur W. Burks, on the relational model or the relational paradigm for implementing and using databases.

In the four-volume work The New Elements of Mathematics by Charles S. Peirce (1976), mathematician and Peirce scholar Carolyn Eisele published a large number of Peirce's previously unpublished manuscripts on mathematical subjects, including the drafts for an introductory textbook, allusively titled The New Elements of Mathematics, that presented mathematics from a decidedly novel, if not revolutionary, standpoint.

In 1902 Peirce applied to the newly established Carnegie Institution for aid "in accomplishing certain scientific work", presenting an "explanation of what work is proposed" plus an "appendix containing a fuller statement". These parts of the letter, along with excerpts from earlier drafts, can be found in NEM 4 (Eisele 1976). The appendix is organized as a "List of Proposed Memoirs on Logic", and No. 12 among the 36 proposals is titled "On the Definition of Logic", the earlier draft of which is quoted in full below.

On Peirce and his contemporaries Ernst Schröder and Gottlob Frege, Hilary Putnam (1982)[25] wrote that he found through research that, though Frege had priority by four years, it was Peirce and his student Oscar Howard Mitchell who effectively discovered the quantifier for the mathematical world. The main evidence for Putnam's claims is "On the Algebra of Logic: A Contribution to the Philosophy of Notation"[26] (1885), published in the premier American mathematical journal of the day. Peano and Ernst Schröder, among others, cited this article and used or adapted Peirce's notations, which are a typographical variant of those currently used. Peirce apparently was ignorant of Frege's work, despite their rival achievements in logic, philosophy of language, and the foundations of mathematics.

Peirce's other major discoveries in formal logic include:

* Distinguishing (Peirce, 1885) between first-order and second-order quantification.

* Seeing that Boolean calculations could be carried out by means of electrical switches,[27] anticipating Claude Shannon by more than 50 years.

* Devising the existential graphs, a diagrammatic notation for the predicate calculus. These graphs form the basis of John F. Sowa's conceptual graphs and of Sun-Joo Shin's diagrammatic reasoning.

A philosophy of logic, grounded in his categories and semiotic, can be extracted from Peirce's writings. This philosophy, as well as Peirce's logical work more generally, is exposited and defended in Hilary Putnam (1982); the Introduction in Houser et al. (1997)[28]; and Dipert's chapter in Misak (2004)[29]. Jean Van Heijenoort (1967)[30], Jaakko Hintikka in his chapter in Brunning and Forster (1997), and Geraldine Brady (2000)[31] divide those who study formal (and natural) languages into two camps: the model-theorists / semanticists, and the proof theorists / universalists. Hintikka and Brady view Peirce as a pioneer model theorist. On how the young Bertrand Russell, especially his Principles of Mathematics and Principia Mathematica, did not do Peirce justice, see Anellis (1995)[24].