Peircean Algebraic Logic and Peirce's Reduction Thesis
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Abstract
Robert Burch describes Peircean Algebraic Logic (PAL) as a language to express Peirce's “unitary logical vision” (1991: 3), which Peirce tried to formulate using different logical systems. A “correct” formulation of Peirce's vision then should allow a mathematical proof of Peirce's Reduction Thesis, that all relations can be generated from the ensemble of unary, binary, and ternary relations, but that at least some ternary relations cannot be reduced to relations of lower arity.
Based on Burch's algebraization, the authors further simplify the mathematical structure of PAL and remove a restriction imposed by Burch, making the resulting system in its expressiveness more similar to Peirce's system of existential graphs. The drawback, however, is that the proof of the Reduction Thesis from Burch (A Peircean reduction thesis: The foundations of topological logic, Texas Tech University Press, 1991) no longer holds. A new proof was introduced in Hereth Correia, and Pöschel (The teridentity and Peircean algebraic logic: 230–247, Springer, 2006) and was published in full detail in Hereth (Relation graphs and contextual logic: Towards mathematical foundations of concept-oriented databases, Technische Universität Dresden dissertation, 2008).
In this paper, we provide proof of Peirce's Reduction Thesis using a graph notation similar to Peirce's existential graphs.
© 2011 Walter de Gruyter GmbH & Co. KG, Berlin/Boston
Articles in the same Issue
- Introduction: Diagrammatical reasoning and Peircean logic representations
- Images, diagrams, and narratives: Charles S. Peirce's epistemological theory of mental diagrams
- The fine structure of Peircean ligatures and lines of identity
- When is a bunch of marks on paper a diagram? Diagrams as homomorphic representations
- Ligatures in Peirce's existential graphs
- Iconic thought and diagrammatical scripture: Peirce and the Leibnizian tradition
- Linear notation for existential graphs
- Peircean Algebraic Logic and Peirce's Reduction Thesis
- Remarks on the iconicity and interpretation of existential graphs
- Cognitive conditions of diagrammatic reasoning
- External diagrammatization and iconic brain co-evolution
- Computers as medium for mathematical writing
- Peircean diagrams of time
- Space, complementarity, and “diagrammatic reasoning”
- Diagrams, iconicity, and abductive discovery
- Moving pictures of thought II: Graphs, games, and pragmaticism's proof
- Peirce's alpha graphs and propositional languages
- Peirce's tutorial on existential graphs
- On operational and optimal iconicity in Peirce's diagrammatology
- Existential graphs and proofs of pragmaticism
Articles in the same Issue
- Introduction: Diagrammatical reasoning and Peircean logic representations
- Images, diagrams, and narratives: Charles S. Peirce's epistemological theory of mental diagrams
- The fine structure of Peircean ligatures and lines of identity
- When is a bunch of marks on paper a diagram? Diagrams as homomorphic representations
- Ligatures in Peirce's existential graphs
- Iconic thought and diagrammatical scripture: Peirce and the Leibnizian tradition
- Linear notation for existential graphs
- Peircean Algebraic Logic and Peirce's Reduction Thesis
- Remarks on the iconicity and interpretation of existential graphs
- Cognitive conditions of diagrammatic reasoning
- External diagrammatization and iconic brain co-evolution
- Computers as medium for mathematical writing
- Peircean diagrams of time
- Space, complementarity, and “diagrammatic reasoning”
- Diagrams, iconicity, and abductive discovery
- Moving pictures of thought II: Graphs, games, and pragmaticism's proof
- Peirce's alpha graphs and propositional languages
- Peirce's tutorial on existential graphs
- On operational and optimal iconicity in Peirce's diagrammatology
- Existential graphs and proofs of pragmaticism
