Theme One Program • Motivation 3

Sometime around 1970 John B. Eulenberg came from Stanford to direct Michigan State’s Artificial Language Lab, where I would come to spend many interesting hours hanging out all through the 70s and 80s.  Along with its research program the lab did a lot of work on augmentative communication technology for limited mobility users and the observations I made there prompted the first inklings of my Learner program.

Early in that period I visited John’s course in mathematical linguistics, which featured Laws of Form among its readings, along with the more standard fare of Wall, Chomsky, Jackendoff, and the Unified Science volume by Charles Morris which credited Peirce with pioneering the pragmatic theory of signs.  I learned about Zipf’s Law relating the lengths of codes to their usage frequencies and I named the earliest avatar of my Learner program XyPh, partly after Zipf and playing on the xylem and phloem of its tree data structures.

Resources

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Theme One Program • Motivation 2

A side-effect of working on the Theme One program over the course of a decade was the measure of insight it gave me into the reasons why empiricists and rationalists have so much trouble understanding each other, even when those two styles of thinking inhabit the very same soul.

The way it came about was this.  The code from which the program is currently assembled initially came from two distinct programs, ones I developed in alternate years, at first only during the summers.

In the Learner program I sought to implement a Humean empiricist style of learning algorithm for the adaptive uptake of coded sequences of occurrences in the environment, say, as codified in a formal language.  I knew all the theorems from formal language theory telling how limited any such strategy must ultimately be in terms of its generative capacity, but I wanted to explore the boundaries of that capacity in concrete computational terms.

In the Modeler program I aimed to implement a variant of Peirce’s graphical syntax for propositional logic, making use of graph-theoretic extensions I had developed over the previous decade.

As I mentioned, work on those two projects proceeded in a parallel series of fits and starts through interwoven summers for a number of years, until one day it dawned on me how the Learner, one of whose aliases was Index, could be put to work helping with sundry substitution tasks the Modeler needed to carry out.

So I began integrating the functions of the Learner and the Modeler, at first still working on the two component modules in an alternating manner, but devoting a portion of effort to amalgamating their principal data structures, bringing them into convergence with each other, and unifying them over a common basis.

Another round of seasons and many changes of mind and programming style, I arrived at a unified graph-theoretic data structure, strung like a wire through the far‑flung pearls of my programmed wit.  But the pearls I polished in alternate years maintained their shine along axes of polarization whose grains remained skew in regard to each other.  To put it more plainly, the strategies I imagined were the smartest tricks to pull from the standpoint of optimizing the program’s performance on the Learning task I found the next year were the dumbest moves to pull from the standpoint of its performance on the Reasoning task.  I gradually came to appreciate that trade-off as a discovery.

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Theme One Program • Motivation 1

The main idea behind the Theme One program is the efficient use of graph-theoretic data structures for the tasks of “learning” and “reasoning”.

I am thinking of learning in the sense of learning about an environment, in essence, gaining information about the nature of an environment and being able to apply the information acquired to a specific purpose.

Under the heading of reasoning I am simply lumping together all the ordinary sorts of practical activities which would probably occur to most people under that name.

There is a natural relation between the tasks.  Learning the character of an environment leads to the recognition of laws which govern the environment and making full use of that recognition requires the ability to reason logically about those laws in abstract terms.

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Higher Order Sign Relations • 6

Re: Higher Order Sign Relations • 4
Re: Relations, Types, Functions
Re: CyberneticsCliff Joslyn

Cliff Joslyn recommends the following books.

Dear Cliff,

The following Survey page gives a hint of the tack I’ve been taking with category theory since the early days but definitely moving into higher gear during my year at Illinois in the mid 1980s.  John Gray taught a course joint between math and computer science on the Applications of Lambda Calculus and David Plaisted taught a course on Resolution-Unification Theorem Proving, both of which I took and followed up with independent studies.  I spent a heady year making the circuit between math, computer science, and psychology departments and a lot of what I work on today goes back to issues raised in those days.

I know that Survey from a couple years ago still looks a little sketchy but I’ll be working to make it less so as time goes by, especially if I ever get around to unpacking my notes from the basement boxes.

I have been sampling current approaches to categories at sundry sites around the web over the last two decades — John Baez, nCafe, nLab, Zulip Category Chat, Topos Institute, etc.  As great as all that is there’s a reason why it bears but tangentially on the questions I’ve been pursuing.  That has to do with the Peirce Factor and how far a given line of inquiry takes account of it.

As luck would have it, one of the texts John Gray used for his course, Lambek and Scott’s Introduction to Higher Order Categorical Logic, resonated strongly with themes I knew from Peirce and that led me to many adventures of ideas still in progress.  The following set of excerpts I shared with the Standard Upper Ontology Group back in the day may suggest the character of that work.

  • Lambek, J. and Scott, P.J. (1986), Introduction to Higher Order Categorical Logic

There’s a lot more to say, but that’s all I have time for today …

Regards,

Jon

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Survey of Inquiry Driven Systems • 4

This is a Survey of blog and wiki resources on Inquiry Driven Systems, material I plan to refine toward a more compact and systematic treatment of the subject.

An inquiry driven system is a system having among its state variables some representing its state of information with respect to various topics of interest, for example, its own state and the states of any potential object systems.  Thus it has a component of state tracing a trajectory though an information state space.

Elements

Background

Blog Dialogs

Developments

Applications

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Inquiry Into Inquiry • Discussion 5

Re: Inquiry Into Inquiry • In Medias Res
Re: Inquiry Into Inquiry • Flashback
Re: Inquiry Into Inquiry • Discussion 4

A quick review of the highlights so far, and then I’ll continue from the standpoint I indicated last time.  As you recall, Dan Everett opened with the following problem.

DE:
I am trying to represent two readings of the three juxtaposed sentences in English.  The first reading is that the judge and the jury both know that Malcolm is guilty.  The second is that the judge knows that the jury thinks that Malcolm is guilty.

Daniel Everett • Judge, Jury, Malcolm, Guilty • Graph 1

Daniel Everett • Judge, Jury, Malcolm, Guilty • Graph 2

Do these purported EGs of mine seem correct to you?

Dan’s initial question about logical graphs sent me further down memory lane than I usually go, to my first encounters with extensions vs. intensions in logic, intentional contexts, propositional attitudes, referential opacity, truth-functionality, and triadicity, puzzles about which my first logic prof sent me off to read Quine’s Ways of Paradox and a host of others.

I had been studying Peirce on my own through all my undergrad years and was fortunate at long last to find an advisor who was a fund of knowledge about Peirce and Pragmatism, not to mention the Ancients and philosophy in general.  In several of our discussions from those days I can remember expressing my hunch the problems of intentionality were not due to a distinct modality or quality of propositions but a different quantity or dimension of relations.  I did not get to Russell’s monographs of 1918 and 1913 until much later but when I did I was struck immediately by his use of graphs to represent relations, so like Peirce’s graphs for the logic of relatives.

Othello Believes Desdemona Loves Cassio

To be continued …

Reference

  • Bertrand Russell, “The Philosophy of Logical Atomism”, pp. 35–155 in The Philosophy of Logical Atomism, edited with an introduction by David Pears, Open Court, La Salle, IL, 1985.  First published 1918.

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Higher Order Sign Relations • 5

Re: Higher Order Sign Relations • 4
Re: Conceptual GraphsGary Zhu

GZ:
Is there any good contemporary reading of Peirce & James that you recommend?  Their original works have been quite challenging for me.

Dear Gary,

As fortune would have it, I haven’t found much to recommend in the secondary literature on Peirce over the last couple of decades.  Most of it looks bent on assimilating Peirce to the conventional wits of analytic and continental philosophy.  As a result, I hew pretty close to Peirce himself in my current reading.  You could try the two volumes of the Essential Peirce for general orientation, if a trifle light on the math side of Peirce.

The last contemporary work I read with anything like the spirit of Peirce about it would probably be Sowa’s Conceptual Structures, so try that if you haven’t already.  Still worth reading are Pragmatism by William James and How We Think by John Dewey.  James and Dewey lacked the mathematical perspective needed to take in Peirce’s full scope and Dewey was a little slow getting up to speed with Peirce’s message but he kept at it and had the benefit of living long enough to become an able expositor of pragmatic and scientific ways.  Plus he understood people and society far better than Peirce ever did.

There are a few references at the end of the following paper.

  • Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), 40–52.  ArchiveJournal.  Online (doc) (pdf).

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Higher Order Sign Relations • 4

Re: Higher Order Sign Relations • 3
Re: Relations, Types, Functions
Re: CyberneticsCliff Joslyn

CJ:
Categorical approaches to systems theory have been very attractive to me for a long time.  My current work is categorically adjacent, and I’m funding some efforts in this direction.  The category of binary relations is central to our immediate work in hypergraphs and high-order networks, but is also to any general systems theoretical approach.  I’ve approached topoi and closed Cartesian categories a few times, but admit it’s challenging.  I need something at the level that David Spivak and crew have been developing to become more fluent, if you’re aware of his work.  Any worked examples you could provide would be very useful and welcome.

Dear Cliff,

There are a few sources I recall most vividly for the way they capture the attractions of categories.  The following references come from a bibliography I collected in the early 90s plus a number I added over the course of that decade.

The following sources may also be of interest.

  • Mili • Program construction and semantics from a relational point of view, using Tarski’s approach to binary relations (Fatma Mili taught a course on this at OU).
  • Freyd and ScedrovCategories, Allegories, a category-theoretic take on binary relations.

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Inquiry Into Inquiry • Discussion 4

Re: Inquiry Into Inquiry • In Medias Res
Re: Inquiry Into Inquiry • Flashback
Re: FB Comment • Daniel Everett

Othello Believes Desdemona Loves Cassio

Dan Everett commented on my post about Russell’s question, “How shall we describe the logical form of a belief?”, giving his take on Russell’s analysis of the example, “Othello believes Desdemona loves Cassio”.

DE:
The most interesting aspect of such constructions from my perspective is that embedding is unnecessary for the reading.  In Piraha you can get independent clauses expressing the same thing.  Or even in English.  Othello believes something.  That something is that Desdemona loves Cassio.  So the advantage of Peircean graphs (and later of Discourse Representation Theory) is that the syntactic feature of embedding is not crucial.  Just as in larger discourse of multiple independent sentences.

I added the following observations.

Russell asks, “How shall we describe the logical form of a belief?”  The question is a good one, maybe too good, loaded with a surplus of meanings for “logical form”.  Read in the spectrum of interpretive lights traditional schools of thought have brought to bear on it, “logical form” hovers between the poles of objective form and syntactic form without ever settling down.  A more stable fix on its practical sense can be gained from the standpoint staked out by Peirce on the basis of the pragmatic maxim, aiming at objective structure and seeing syntactic structure as accessory to that aim.

To be continued …

Reference

  • Bertrand Russell, “The Philosophy of Logical Atomism”, pp. 35–155 in The Philosophy of Logical Atomism, edited with an introduction by David Pears, Open Court, La Salle, IL, 1985.  First published 1918.

Resources

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Inquiry Into Inquiry • Understanding 2

In the passage quoted in the previous post Bertrand Russell addresses the question, “What is the logical structure of the fact which consists in a given subject understanding a given proposition?” and he selects a proposition of the form ``A ~\text{and}~ B ~\text{are similar}" to demonstrate his way of analyzing the fact.  Russell wraps up his discussion of the example in the passage quoted below.

Excerpt from Bertrand Russell • “Theory of Knowledge” (1913)

Part 2. Atomic Propositional Thought

Chapter 1. The Understanding of Propositions

(4). [cont.]  It follows that, when a subject S understands ``A ~\text{and}~ B ~\text{are similar}", “understanding” is the relating relation, and the terms are S and A and B and similarity and R(x, y), where R(x, y) stands for the form “something and something have some relation”.  Thus a first symbol for the complex will be

U \{S, A, B, \mathrm{similarity}, R(x, y) \}~.

This symbol, however, by no means exhausts the analysis of the form of the understanding-complex.  There are many kinds of five-term complexes, and we have to decide what the kind is.

It is obvious, in the first place, that S is related to the four other terms in a way different from that in which any of the four other terms are related to each other.

(It is to be observed that we can derive from our five-term complex a complex having any smaller number of terms by replacing any one or more of the terms by “something”.  If S is replaced by “something”, the resulting complex is of a different form from that which results from replacing any other term by “something”.  This explains what is meant by saying that S enters in a different way from the other constituents.)

It is obvious, in the second place, that R(x, y) enters in a different way from the other three objects, and that “similarity” has a different relation to R(x, y) from that which A and B have, while A and B have the same relation to R(x, y).  Also, because we are dealing with a proposition asserting a symmetrical relation between A and B, A and B have each the same relation to “similarity”, whereas, if we had been dealing with an asymmetrical relation, they would have had different relations to it.  Thus we are led to the following map of our five-term complex.

Russell • Understanding (S, A, B, Similarity, Rxy)

In this figure, one relation goes from S to the four objects;  one relation goes from R(x, y) to similarity, and another to A and B, while one relation goes from similarity to A and B.

This figure, I hope, will help to make clearer the map of our five-term complex.  But to explain in detail the exact abstract meaning of the various items in the figure would demand a lengthy formal logical discussion.  Meanwhile the above attempt must suffice, for the present, as an analysis of what is meant by “understanding a proposition”.  (Russell, TOK, 117–118).

Reference

  • Bertrand Russell, Theory of Knowledge : The 1913 Manuscript, edited by Elizabeth Ramsden Eames in collaboration with Kenneth Blackwell, Routledge, London, UK, 1992.  First published, George Allen and Unwin, 1984.

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