Triadic Relations • 1

Examples from Mathematics

For the sake of topics to be taken up later, it is useful to examine a pair of triadic relations in tandem.  We will construct two triadic relations, L_0 and L_1, each of which is a subset of the same cartesian product X \times Y \times Z.  The structures of L_0 and L_1 can be described in the following way.

Each space X, Y, Z is isomorphic to the boolean domain \mathbb{B} = \{ 0, 1 \} so L_0 and L_1 are subsets of the cartesian power \mathbb{B} \times \mathbb{B} \times \mathbb{B} or the boolean cube \mathbb{B}^3.

The operation of boolean addition, + : \mathbb{B} \times \mathbb{B} \to \mathbb{B}, is equivalent to addition modulo 2, where 0 acts in the usual manner but 1 + 1 = 0.  In its logical interpretation, the plus sign can be used to indicate either the boolean operation of exclusive disjunction or the boolean relation of logical inequality.

The relation L_0 is defined by the following formula.

L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}.

The relation L_0 is the following set of four triples in \mathbb{B}^3.

L_0 ~=~ \{ ~ (0, 0, 0), ~ (0, 1, 1), ~ (1, 0, 1), ~ (1, 1, 0) ~ \}.

The relation L_1 is defined by the following formula.

L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}.

The relation L_1 is the following set of four triples in \mathbb{B}^3.

L_1 ~=~ \{ ~ (0, 0, 1), ~ (0, 1, 0), ~ (1, 0, 0), ~ (1, 1, 1) ~ \}.

The triples in the relations L_0 and L_1 are conveniently arranged in the form of relational data tables, as shown below.

Triadic Relation L0 Bit Sum 0

Triadic Relation L1 Bit Sum 1

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Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.

cc: CyberneticsOntolog ForumPeirce ListStructural ModelingSystems Science

This entry was posted in C.S. Peirce, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Philosophy, Pragmatic Semiotic Information, Pragmatism, Relation Theory, Semiotics, Sign Relations, Thirdness, Triadic Relations, Triadicity and tagged , , , , , , , , , , , , , , . Bookmark the permalink.

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