## The Difference That Makes A Difference That Peirce Makes : 31

One of the more disconcerting developments, I might say “devolutions”, I’ve observed over the last 20 years has been the general slippage back to absolutist and dyadic ways of thinking, all of it due to the stubborn pull of unchecked reductionism and a failure to comprehend the relational paradigm, especially the basic facts about triadic relations, their irreducibility, and the consequences thereof.

With all that in mind, I’ll return to a point in our earlier discussions, add a bit more on the concept of closure, and continue from there to its bearing on the pragmatic maxim.

### The Difference That Makes A Difference That Peirce Makes : 23

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 $\mathrm{A}$, the result of applying $\mathrm{A}$ to an operand $x$ is $\mathrm{A}x,$ the result of applying $\mathrm{A}$ again is $\mathrm{AA}x,$ the result of applying $\mathrm{A}$ again is $\mathrm{AAA}x,$ and so on.  In general, it is perfectly possible each application yields a novel result, distinct from all previous results.

But a closure operator $\mathrm{C}$ is defined by the property $\mathrm{CC} = \mathrm{C},$ so nothing new results beyond the first application.

### The Difference That Makes A Difference That Peirce Makes : 24

The concepts of closure and idempotence are closely related.

We usually speak of a closure operator in contexts where the objects acted on are the primary interest, as in topology, where the objects of interest are open sets, boundaries, closed sets, etc.  In contexts where we abstract away from the operand space, as in algebra, we tend to say idempotence for the detached application $\mathrm{CC} = \mathrm{C}.$  (If I recall right, it was actually Charles Peirce’s father Benjamin who coined the term idempotence.)

At any rate, I’ll have to mutate the principle a bit to cover the uses Peirce makes of it.

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