Re: FB | Peirce Society • John Corcoran
To address Was ist und was soll Information sein? in a Peircean context we need to grasp or at least try to grapple with Peirce’s inklings about information. As it happens, I’ve been engaged in that quest for a number of years. Pilgrims moved by a similar spirit may find my travel logs of use on their way —
Re: Ontolog Forum • John Bottoms
Re: History View Blog • A Yukaghir Girl Writes A Love Letter
Re: Peirce’s 1870 “Logic Of Relatives” • Proto-Graphical Syntax
John Bottoms, writing in the Ontolog Forum, compared the graphic I drew for one of Peirce’s relational formulas to a pictographic image he had been studying.
Peirce introduced the compound term “giver of a horse to a lover of a woman” in Selection 7 to illustrate his use of marks of reference to identify the corresponding correlates of component terms. The symbolic form of this compound term is shown below.
In my comment on Proto-Graphical Syntax I drew lines of identity to connect the corresponding marks of reference, as shown in the following Figure.
John tells us what he sees in this Figure in the following words:
In the image a girl complains to her ex-lover that his new relationship is not useful, and he is missing an opportunity to have children with her. This image was created by a Yukaghir girl for a local game similar to Pictionary.
I started a draft in the middle of the night to draw out the analogies and disanalogies of these two Figures but the clear light of day showed me I would need to deal with a host of preliminary issues before moving on. So I will turn to that task next.
cc: Systems Science • Structural Modeling • Ontolog Forum • Laws of Form • Cybernetics
My long ago encounter with Peirce’s 1870 paper, “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic”, was one of the events precipitating my return from the hazier heights of philosophy to the solid plains of mathematics below. Over the years I copied out various drafts of my study notes to the web, consisting of selections from Peirce’s paper along with my running commentary. A few years back I serialized what progress I had made so far to this blog and this Overview consists of links to those installments.
Re: Laws of Form • James Bowery
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 
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: 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 
But a closure operator 
cc: Systems Science • Structural Modeling • Ontolog Forum • Laws of Form • Cybernetics
Peirce Society Facebook Page • JC • JA • JA • JA • JC
A discussion — well, more like a series of posts and counterposts — arose last week on the Facebook Page of the Charles S. Peirce Society, and I’ve been going back over it this week because it seemed to invite a useful re-examination of some old but important issues. There appears to be some sort of disagreement, or maybe just failure to communicate, but I’m still having trouble putting my finger on what the source of the issue might be.
One factor seems to be different understandings about the relationship between Peirce’s brand of semiotics and standard first order logic. One thing I’ve noticed before is that people who view Peirce’s work through the filter of first order logic are not likely to see what many of us appreciate in his semiotic approach to logic. There are commentators on Peirce’s logical systems who treat them as nothing more than first order logics in other syntaxes, but I am not one of those. There is something more general and powerful going on with Peirce’s conception of “logic as formal semiotic”, in other words, a normative science of signs.
I still see that factor playing a role in the background of the animadversion but I’m beginning to think there’s probably a much simpler explanation.
cc: Systems Science • Structural Modeling • Ontolog Forum • Laws of Form • Cybernetics
Re: Ontolog Forum • John Bottoms
Re: The Difference That Makes A Difference That Peirce Makes : 20
The reflections in my previous blog post developed over several weeks observing various discussions around the web where people seemed to be spending most of their effort talking past each other and hardly ever getting any ideas or information out of one skull and into another. It’s not the first time I’ve noticed belief systems, comfort zones, conceptual silos, paradigms, whatever we call them, acting like immune systems, insulating our mental metabolisms from intellectual antigens.
Anyhow, it’s a working hypothesis to prime future inquiry …
As far as Peirce references go, the choices are legion, so I’ll just link to one place I have in mind at the moment where Peirce sets out a number of truly radical ideas, ones I view as missed opportunities — so far as I know he never fully followed up on them.
cc: Systems Science • Structural Modeling • Ontolog Forum • Laws of Form • Cybernetics
Cross-paradigm communication, like cross-disciplinary and cross-cultural communication, can be difficult. Sometimes people do not even recognize the existence of other paradigms, disciplines, cultures, long before it comes to the question of their value. Readers of Peirce know he often uses important words in more primordial senses than later came into fashion. Other times his usage embodies a distinct analysis of the concept in question. More than once I’ve found myself remarking how Peirce “anticipates” some strikingly “modern” idea in logic, mathematics, or science, only to find its roots lay deep in the history of thought. Whether he anticipates a future sense or preserves an ancient sense is not always easy to answer.
cc: Systems Science • Structural Modeling • Ontolog Forum • Laws of Form • Cybernetics
Prompted by a recent discussion of prime numbers and complex dynamics on one of the Santa Fe Institute’s FaceBook pages, I posted a link to an old project of mine, going back to a time when I was first learning programming in college and working as an orderly in a hospital x-ray department. Something about the collision of those influences in the medium of my gray matter led me to see curious connections among self-documenting programs, self-indexing data structures, and molecular tagging. Shortly afterwards a couple of Mathematical Games columns by Martin Gardner started me thinking about Gödel numbers and links among graph theory, logic, and number theory.
At any rate, here’s a report on what came of that —
In modeling intelligent systems, whether we are trying to understand a natural system or engineer an artificial system, there has long been a tension or trade‑off between dynamic paradigms and symbolic paradigms. Dynamic models take their cue from physics, using quantitative measures and differential equations to model the evolution of a system’s state through time. Symbolic models use logical methods to describe systems and their agents in qualitative terms, deriving logical consequences of a system’s description or an agent’s state of information. Logic‑based systems have tended to be static in character, largely because we have lacked a proper logical analogue of differential calculus. The work laid out in this report is intended to address that lack.
This article develops a differential extension of propositional calculus and applies it to the analysis of dynamic systems whose states are described in qualitative logical terms. The work pursued here is coordinated with a parallel application focusing on neural network systems but the dependencies are arranged to make the present article the main and the more self‑contained work, to serve as a conceptual frame and a technical background for the network project.
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science
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