## Relation Theory • Discussion 3

Re: Relation Theory • (1)(2)(3)(4)(5)
Re: Laws of FormJames Bowery

JB:
Thanks for that very rigorous definition of “relation theory”.

Its “trick” of including the name of the $k$-relation in a $(k+1)$-relation’s tuples reminds me Etter’s paper “Three-Place Identity” which was the result of some of our work at HP on dealing with identity (starting with the very practical need to identify individuals/corporations, etc. for the purpose of permitting meta-data that attributed assertions of fact to certain identities aka “provenance” of data).

The result of that effort threatens to up-end set theory itself and was to be fully fleshed out in “Membership and Identity” […]

We were able to get a preliminary review of Three-Place Identity by a close associate of Ray Smullyan.  It came back with a positive verdict.  I believe I may still have that letter somewhere in my archives.

Dear James,

The article on Relation Theory represents my attempt to bridge the two cultures of weak typing and strong typing approaches to functions and relations.  Weak typing was taught in those halcyon Halmos days when functions and relations were nothing but subsets of cartesian products.  Strong typing came to the fore with category theory, its arrows from source to target domains, and the need for closely watched domains in computer science.

Peirce recognizes a fundamental triadic relation he calls “teridentity” where three variables $a, b, c$ denote the same object, represented in his logical graphs as a node of degree three, and at first I thought you might be talking about that.

But I see $x(y = z)$ read as ${}^{\backprime\backprime} x ~\text{regards}~ y ~\text{as the same as}~ z {}^{\prime\prime}$ is more like the expressions I use to discuss “equivalence relations from a particular point of view”, following one of Peirce’s more radical innovations from his 1870 “Logic of Relatives”.

• C.S. Peirce • On the Doctrine of Individuals (1) (2)

Using square brackets in the form $[a]_e$ for the equivalence class of an element $a$ in an equivalence relation $e$ we can express the above idea in one of the following forms.

$\begin{matrix} [y = z]_x & \text{or} & [y]_x = [z]_x & \text{or} & y =_x z \end{matrix}$

I wrote this up in general somewhere but there’s a fair enough illustration of the main idea in the following application to “semiotic equivalence relations”.

• Semiotic Equivalence Relations • (1) (2)

The rest of your remarks bring up a wealth of associations for me, as seeing the triadic unity in the multiplicity of dyadic appearances is a lot of what the Peircean perspective is all about.  I’ll have to dig up a few old links to fill that out …

Regards,

Jon

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