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Peirce radically modified, extended, and transformed Boolean algebra, making it applicable to propositions, relations, probability, and arithmetic. Following De Morgan, Peirce laid the foundations of the logic of relations, the key technique for the logical analysis of mathematics. He invented the copula symbol of inclusion, the most important symbol in the logic of classes. Peirce invented two new logical algebras, two new systems of logical graphs, discovered the link between the logic of classes and the logic of propositions, was the first to give the fundamental principle for the logical development of mathematics, and made fundamental contributions to probability theory, induction, statistics, and the logic of scientific methodology. Peirce wrote an elaborate book on advanced logic but could not get it published.
 
Peirce radically modified, extended, and transformed Boolean algebra, making it applicable to propositions, relations, probability, and arithmetic. Following De Morgan, Peirce laid the foundations of the logic of relations, the key technique for the logical analysis of mathematics. He invented the copula symbol of inclusion, the most important symbol in the logic of classes. Peirce invented two new logical algebras, two new systems of logical graphs, discovered the link between the logic of classes and the logic of propositions, was the first to give the fundamental principle for the logical development of mathematics, and made fundamental contributions to probability theory, induction, statistics, and the logic of scientific methodology. Peirce wrote an elaborate book on advanced logic but could not get it published.
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==Pragmatism==
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Peirce was the creator if "Pragmatism;" it emerged from his many discussions in Cambridge at a "metaphysical club" founded in the 1870s. Most influential on peirce were [[Chauncey Wright]], a philosopher; [[William James]], Peirce's lifelong friend and benefactor; and Nicholas St. John Green, a lawyer and follower of utilitarian philosopher [[Jeremy Bentham]] who interpreted doctrines in terms of their effect upon social life. Peirce began reading philosophy and by 1871 he was converted to the version of realism originally propounded by medieval philosopher [[Duns Scotus]]. Peirce remained a realist in philosophy all his life, eventually embracing realism not only about generals or universals but also about modalities such as possibility and necessity. His first major statement of Pragmatism appeared in ''Popular Science Monthly'' in January 1878, under the title "How to Make Our Ideas Clear."<ref> see [http://www.peirce.org/writings/p119.html "How to Make Our Ideas Clear," by Charles S. Peirce; ''Popular Science Monthly'' 12 (January 1878), 286-302]</ref>
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In 1898 William James first publicly used the term "pragmatism" and acknowledged Peirce's priority. Peirce's pragmatism, however, is not the same as James's; it has more in common with the somewhat independently developed idealism of [[Josiah Royce]] and the later views of John Dewey. Peirce shared many of the views characteristic of the pragmatic school, developing them in his own fashion. He believed in the dependence of logic on ethics, argued as early as 1868 against individualism and egoism, and developed social theories of reality and logic. At the same time he believed that the development of business methods should be left to business without law's interference.<ref>See the 1910 quote from Peirce in the third section of "Pragmatism and American Democracy: An Elective Affinity Analysis" by Norbert Wiley, presented at the American Sociological Association Meetings, 2003, [http://www.unlv.edu/centers/cdclv/pragmatism/wiley_pragma_demo.html online edition].</ref> He believed and argued that logic precedes metaphysics, and that metaphysicians failing to appreciate that precedence have always barred the way to inquiry.<ref>See for example "F.R.L." (First Rule of Logic), 1899 manuscript, not to be confused with another piece given that title. Published in ''Collected Papers of Charles Sanders Peirce'' vol. 1, paragraphs 135-140, [http://www.princeton.edu/~batke/peirce/frl_99.htm online edition].</ref>
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Peirce held that a conception's meaning is general and does not boil down to its actual consequences (or "cash value") but consists in the conceived object's ''conceivable'' practical consequences. He disagreed with other Pragmatists over what he called their "angry hatred of strict logic" and their views that, among other things, "truth is mutable" and "infinity is unreal".<ref>Peirce, C.S. (1908), "A Neglected Argument for the Reality of God", ''Hibbert Journal'' vol. 7, pp. 90-112. Reprinted in ''Collected Papers of Charles Sanders Peirce'' vol. 6, paragraphs 452-485 and ''The Essential Peirce'' vol. 2, pp.434-450.</ref>.
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Pragmatism allows ideas to have any origin; what matters is not where we got an idea from but where we can go with it. In the case of science the usefulness of an idea includes reliability and fruitfulness. "Fruitfulness" does not mean practical utility but, rather, the opening up of original lines of successful research, what mathematicians nowadays call fecundity. As reliability is to truth, so fruitfulness is to depth and breadth of understanding. The test of truth and fruitfulness lies in the future. Truth, for the pragmatist, depends on our developing the implications of ideas so that we can find ways of applying them experimentally.<ref> See Short (2001)</ref>
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==Semiotics==
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Peirce's work in logic established the foundation for yet another branch of study, semiotics, which would not flourish until 40 years after his death. The discipline of semiotics studies how information, in the form of language or any other type of communication (music, painting, etc.), is transmitted and received. In Peirce's formulation, signs are presented not only by culture but also by nature (reactions, natural semblances, etc.) and logic (abstract ideas are signs), and there is a triangular relationship between sign, object, and interpretant. A sign represents a particular thing or idea which is being communicated (the object). A sign's object is either immediate to the sign, that is, the object as represented by the sign, or is a dynamic object, the object as it really is, even if fictional such as Rip Van Winkle. An interpretant is a sign's effect that amounts to a further sign, for instance a quality of impression (an immediate interpretant, immediate to the sign) which the sign is suited to produce, or an actually resulting interpretation (a dynamic interpretant) such as a translation, state of agitation, or forming of an idea in an interpreter's mind; or a kind of ideal or norm (a final or normal interpretant), the conduct that <em>would</em> be produced if the sign were to be sufficiently interpreted.
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The sign itself can take on three forms, an icon, an index, or a symbol. An icon directly resembles the object, regardless of factual connection to it (the object need not even exist) and regardless of interpretation (the resemblance is objective though relative to sight, intellect, etc.). An index represents through factual connection to the object; for example an ongoing effect such as smoke is an index of fire; the smoke need not resemble the fire, nor is its connection with fire a matter of interpretation. A symbol does depend on interpretation, and so it requires a dependable rule or habit of interpretation; that rule is essentially the symbol. A symbol represents the object by an interpretive norm or rule that may be logical, cultural, or even inborn but in any case without depending on resemblance or factual connection to the object. Written and spoken language therefore rely heavily on symbolic signs, in that a word's meaning must be understood by both the speaker and the listener in order for communication to take place. Designations such as proper names and words such as "this" Peirce usually considered to be indices. Peirce considered language as involving a mind's ability to supply mental indices and icons to accompany it. Peirce regarded diagrams as icons, and "All___is___" as a linguistic diagram, subject to logical transformations.
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Peirce's ideas were highly influential on later linguistic scholars such as [[Noam Chomsky]], as well as on [[deconstruction|deconstructionists]] Jacques Derrida and Paul le Man. His ideas have also entered [[music theory]], in a separate branch known as musical semiotics. Leading exponents of this field include Wilson Coker, Robert Hatten, David Lidov, and V. Kofi Agawu.
      
==Further reading==
 
==Further reading==
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