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C.S. Peirce studied ternary (and higher-order, n-ary logic, too) logic in the context of which he called semiosis,

an action, or influence, which is, or involves, a coöperation of three subjects, […], this tri-relative influence not being in any way resolvable into actions between pairs.

Has any logician applied ternary logic to understand the intricacies of human relationships among three (or more) people?

cf.:

"Triang Relationships"
13 permutations

or "Love Polyhedra" for the higher-order analogue:

Geremia
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  • By "ternary" do you mean 3-place relations are 3-valued logic? Peirce studied both. And semiosis is different from both, it is the process of interpreting signs, although the central relation (object-sign-interpretant) is ternary. Blood relations were the paradigmatic example for developing the logic of relations even before Peirce, in de Morgan's 1860 paper. But finite relations are just equivalent to directed graphs, and [social networks](https://en.wikipedia.org/wiki/Social_network_analysis), etc., are studied under that label. Graphs have greater intuitive appeal. – Conifold Feb 07 '22 at 22:17
  • @Conifold Did Peirce study social networks in the context of his "existential graphs"? And what was De Morgan's 1860 paper? – Geremia Feb 07 '22 at 23:17
  • FOL fragment containing equality with any >=2-ary relation(s) is undecidable for first order validities, so seems there's no essential differences between your termed "twin" and "ternary" logics , but it does indicate an essential difference from 1-ary predicates technically. Metaphysically this is expected without surprise since any difference is essentially between two entities, any difference among ternary relations if there's any at all can be reduced to some relational difference between at least a pair of entities... – Double Knot Feb 08 '22 at 04:44
  • How important is the “3 or more”, and how important is the “human relationship”? From a logical perspective the idea of polyadic relations is quite a natural construct - consider instead the “relationship units” and you can map arbitrarily many people to those units using a “member of” relation or similar - but I don’t have much to contribute to the more human aspect of your puzzle! – Paul Ross Feb 08 '22 at 08:38
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    [De Morgan, On the Syllogism, no. IV, and on the Logic of Relations](https://archive.org/details/transactionsofca10camb/page/330/mode/2up). Peirce did not apply it to social relationships, he was interested in semiotics and metaphysics, but derivatives, like [conceptual graphs](https://en.wikipedia.org/wiki/Conceptual_graph), are applied to all sorts of things, those included, see e.g. [Huang, Understanding Large-Scale Social Relationship Data by Combining Conceptual Graphs and Domain Ontologies](https://www.hindawi.com/journals/ddns/2021/2857611/). – Conifold Feb 08 '22 at 12:06

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