Re: Structural Modeling • Joseph Simpson
Re: Peirce’s 1870 Logic Of Relatives • Selection 1

A critical question in mathematical logic and its applications concerns the threshold of complexity between dyadic (binary) and triadic (ternary) relations, in essence, whether 2-place relations are universally adequate or whether 3-place relations are irreducible, minimally adequate, and even sufficient as a basis for all higher dimensions.

One of Peirce’s earliest arguments for the sufficiency of triadic relative terms occurs at the top of his 1870 “Logic of Relatives”.

The conjugative term involves the conception of third, the relative that of second or other, the absolute term simply considers an object.  No fourth class of terms exists involving the conception of fourth, because when that of third is introduced, since it involves the conception of bringing objects into relation, all higher numbers are given at once, inasmuch as the conception of bringing objects into relation is independent of the number of members of the relationship.  Whether this reason for the fact that there is no fourth class of terms fundamentally different from the third is satisfactory or not, the fact itself is made perfectly evident by the study of the logic of relatives.  (Peirce, CP 3.63).

Peirce’s argument invokes what is known as a closure principle, as I remarked in the following comment:

What strikes me about the initial installment this time around is its use of a certain pattern of argument I can recognize as invoking a closure principle, and this is a figure of reasoning Peirce uses in three other places:  his discussion of continuous predicates, his definition of sign relations, and in the formulation of the pragmatic maxim itself.

In mathematics, a closure operator is one whose repeated application yields the same result as its first application.

If we consider an arbitrary operator , the result of applying to an operand is the result of applying again is the result of applying again is and so on.  In general, it is perfectly possible each application yields a novel result, distinct from all previous results.

But a closure operator is defined by the property so nothing new results beyond the first application.

cc: Systems Science • Structural Modeling • Ontolog Forum • Laws of Form • Cybernetics