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Peirce’s philosophy of logical notation
by Zhang Liu-hua
In some measure, logic, or at least modern logic, can be perceived as “notation of reasoning”. It is known that logic treats reasoning (and argumentation) as the central subject. However, logic consists in not only study of reasoning, but in a crucial sense it consists in the study of reasoning by convenient notation, which tell you how to represent reasoning in some suitable notation and make reasoning efficiently by the notation. Many facts in history of logic make it clear that logical notation, making much sense, should be regarded as an important matter.
Logic and notation is not a new topic, but in the past it often behaves as relation between sign and inference. As mathematical notation played a revolutionary role in the development of algebra and geometry in the abstract, the new logical notation mainly illuminated by the algebraic notation brought on the birth of modern logic in a more direct sense. American logician C.S.Peirce, one of the founders of modern logic, shows greater zeal for exploration and research in logical notation than his many contemporaries. He, for the first time in history of logic, made an inviting exposition and discussion on the profound significance of logical notation, and advanced “philosophy of notation” in his terminology. The dissertation titled “Peirce’s philosophy of logical notation”, concentrates on discussing the methodological bearing of Peirce’s notation on his elementary logic, Peirce’s logical thoughts on types of signs, and Peirce’s quest for an ideal logical notation, trying to touch into and reveal Peirce’s comprehensive and original contribution through his investigation on logical notation on one hand and attach more importance than people commonly would to logical notation on the other hand.
Firstly, the article expounded that Peirce’s unique semiotics offered the widest context and frame for investigating logical notation, particularly from points of view of sign revealing the inherent links between notation and logic: semiotics as logic. In general, logical notation is sign or sign system. Logicians all through the history concerned studies of signs. Peirce, a distinguish logic historian, grounding on his irreductive triadic relation between Sign, Object and Iinterpretant, developed his Semieotic which is distinct from Saussure’s Semiology, and then used the semiotic as the comprehensive context for investigating logical notation and all his logic theory. Having synthesized and abstracted types of signs in the logic history, he classified signs into icons, indices and symbols. Subsequently, this notable classification of signs was adopted by many logicians, linguists, philosophers and cognitive scientists. Next, he pointed out that each of icon, index and symbol is essential for a perfect logical notation, as becomes a foundational idea of Peirce’s philosophy of logical notation in some degree. In the whole, Peirce’s investigation on signs is always along with his logical research. He said, “Logic, in its general sense, is, as I believe I have shown, only another name for semiotic”.
Secondly, the article, via a few instances of logical notation in modern logic, analyzed the refinement and profoundness of Peirce’s algebraic notation, mirrored some important contribution in his early logical research, and indicated Peirce’s algebraic notation as a source of standard logical notation today. Inclusive disjunction “+”, limited discourse of universe “1”, material implication“—<” and universal and particular quantifiers (in a sense of modern logic) “∏ ” “ ∑” not only spread out Peirce’s great improvement on notation of Boolean algebra, but also took on severe logical significance in Peirce. In practice, Peirce’s various thoughts of algebraic notation directly speeded up maturing propositional logic and predicate logic in initial stages of modern logic.
Lastly, the article, introducing Peirce’s Existential Graphs on the backgrounds of logical diagrams in history, analyzed and assessed Peirce’s Existential Graphs and its superiority and developmental perspectives, trying to claim that Peirce’s graphic notation not only radically improves on logical diagrams in history but also shows to logicians in right of its tremendous logical expressive power that, besides traditionally algebraic notation, there is an independent diagrammatical logical notation. Existential Graphs, which Peirce dedicated his later life to, is a two-dimensional notation which is different from his early linear algebraic notation, as is called by Peirce “my chef d’oeuvre”. His finished “Alpha” and “Beta” portions of Existential Graphs correspond respectively to propositional logic and predicate logic we commonly use today, and have proved to be consistent and complete by some peircians and logicians. Moreover the “Gamma” portion of Existential Graphs which Peirce himself hadn’t finished in his life covers evidently the thoughts of modal logic and higher-order logic. This notation is a natural deduction system in inferring rules, and its visual features are higher than algebraically natural deduction system, which was created thereafter by Gentzen. Peirce’s work on the graphic notation is continued by Barwise, Sowa and other logicians and computer scientists of the day.
Summing up， Peirce’s philosophy of logical notation contains many illuminating suggestions as well as much original and extensive logical contribution. If examine philosophy of notation with the development nowadays of linguistics, computer science and philosophy, then we would have a conclusion: logical notation is not trivial matter, so how to achieve a optimizing weaving of icons, indexes and symbols, is of essential issues which is referred to the progress of logic, no longer only a question of arbitrary formal arrangement.
C.S.Peirce; logical notation; philosophy of notation; semiotics; algebraic notation; graphic notation; Existential Graphs