Inquiry Into Inquiry

Re: FB | Foundations of Mathematics • John Corcoran

There was a huge — and of course ultimately futile — discussion of truth theories back in 2005 when the Wikipediot article on Truth was under development.  Pragmatists of one stripe or another from the Peirce List ventured in vain to explain the difference between (1) “classical” correspondence theories, (2) consensus or “social” theories, and (3) Peircean pragmatic — I’m guessing what Tarski meant by “utilitarian” — theories of truth.  I’ll dig up some links and forks when I get a chance.

cc: Cybernetics • Ontolog Forum • Peirce List • Structural Modeling • Systems Science

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 , 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.

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   (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.

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

Re: Ontolog Forum • Mihai Nadin
Re: Peirce List • Helmut Raulien

I first encountered Peirce’s dimensions of generality and vagueness — two measures of determinacy on sign relations describing the extent to which objects are determined by signs and interpretant signs — while exploring the closely related subjects of definition and determination.

Lately I’ve noticed Peirce’s treatment of objectively indeterminate signs has a bearing on my approach to Category Theory through the Logic of Relatives, so it looks worth paying attention to their potential relationships.  To get things rolling, here’s a good entry point:

• Peirce (CP 5.447)

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

Re: Ontolog Forum • Jon Awbrey • John Sowa

We are called to recall Peirce’s deeper meanings
of ideas like Form (think Platonic Ideas and the
way Aristotle compounded Form and Matter).  When
we come to Sentiment, by any other word, Feeling,
that is the medium of Aesthetics, which concerns
Beauty in no merely skin-deep sense but all that
embodies and manifests ‘the admirable in itself’,
thus every form of life worth living.  So Peirce
stands the normative science of Logic on grounds
within the pale of Ethics and fixes the sight of
Ethics on the prize Aesthetics picks to steer by.

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