Jon and Jerry, JA
triadic relations extend across a threshold of complexity, such that relations of all higher adicities can be analyzed in terms of 1-adic, 2-adic, and 3-adic relations.
No. Peirce never said that. Many logicians have correctly observed that you can replace any triadic relation by three dyadic relations plus an additional quantified variable. In a graph, the node that represents the variable will be linked to the three dyadic relations. For example, consider the following sentence and its translation to two different formulas in predicate calculus: x gives y to z. ∃x ∃y ∃z gives(x,y,z). ∃x ∃y ∃z ∃w (give(w) & agent(w,x) & theme(w,y) & recipient(w,z)). The second formula has a new entity named w, which is linked to three dyadic relations. There is still an implicit triad in the formula. In an earlier note, I showed the sentence "Sue gives a child a book" as two different conceptual graphs. In the attached giveEGCG.jpg, I show that sentence translated to the same two conceptual graphs and to their translations as existential graphs. To show the mappings to the algebraic formulas, I also annotated the lines of identity: x, y, and z represent the same lines in both EGs. But w represents a ligature of *four* lines of identity that are connected at a "tetra-identity". What Peirce showed is that any connection of four or more lines may be replaced by connections of just three lines (called teridentities). In the diagram giveEGCG.jpg, you can replace the ligature labeled w with a ligature of 5 lines of identity linked by two teridentities. JA
In mathematics, category theory is largely based on the prevalence of functions in mathematical practice, and functions are dyadic relations.
Not just "largely based", but "completely based". And note that the "functions" of plus, minus, times, and divide map two arguments to a single value. For generality, mathematicians say that functions map elements from one domain to another, but those elements may be pairs, N-tuples, or structures of any kind. I agree with Jerry: JLRC
The mappings may represent a vast range of mathematical structures and be constrained to oriented graphs.
Yes. Graphs are convenient because they can show some logical connections more clearly than a linear notation. But the basic principles are independent of notation. John
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