The Difference That Makes A Difference That Peirce Makes : 23

Re: Structural ModelingJoseph Simpson
Re: Peirce’s 1870 Logic Of RelativesSelection 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 \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.

cc: Systems ScienceStructural ModelingOntolog ForumLaws of FormCybernetics

This entry was posted in Analogy, C.S. Peirce, Communication, Descriptive Science, Fixation of Belief, Formal Systems, Information, Inquiry, Logic, Logic of Relatives, Logic of Science, Logical Graphs, Mathematics, Normative Science, Paradigms, Peirce, Pragmatic Maxim, Pragmatism, Relation Theory, Semiotics, Sign Relations, Triadic Relations, Triadicity and tagged , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

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