Inquiry Into Inquiry

Theme One Program • Motivation 5

Posted on June 7, 2024 by Jon Awbrey

Since I’m working from decades‑old memories of first inklings I thought I might peruse the web for current information about Zipf’s Law. I see there is now something called the Zipf–Mandelbrot (and sometimes –Pareto) Law and that was interesting because my wife Susan Awbrey made use of Mandelbrot’s ideas about self‑similarity in her dissertation and communicated with him about it. So there’s more to read up on.

Just off‑hand, though, I think my Learner is dealing with a different problem. It has more to do with the savings in effort a learner gets by anticipating future experiences based on its record of past experiences than the savings it gets by minimizing bits of storage as far as mechanically possible. There is still a type of compression involved but it’s more like Korzybski’s “time‑binding” than space‑savings proper. Speaking of old memories …

The other difference I see is that Zipf’s Law applies to an established and preferably large corpus of linguistic material, while my Learner has to start from scratch, accumulating experience over time, making the best of whatever data it has at the outset and every moment thereafter.

Resources

cc: FB | Theme One Program • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Posted in Algorithms, Animata, Artificial Intelligence, Boolean Functions, C.S. Peirce, Cactus Graphs, Computation, Computational Complexity, Cybernetics, Data Structures, Differential Logic, Form, Formal Languages, Graph Theory, Inquiry, Inquiry Driven Systems, Intelligent Systems, Laws of Form, Learning, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Pragmatics, Programming, Propositional Calculus, Propositional Equation Reasoning Systems, Reasoning, Semantics, Semiotics, Sign Relations, Spencer Brown, Syntax, Visualization | Tagged Algorithms, Animata, Artificial Intelligence, Boolean Functions, C.S. Peirce, Cactus Graphs, Computation, Computational Complexity, Cybernetics, Data Structures, Differential Logic, Form, Formal Languages, Graph Theory, Inquiry, Inquiry Driven Systems, Intelligent Systems, Laws of Form, Learning, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Pragmatics, Programming, Propositional Calculus, Propositional Equation Reasoning Systems, Reasoning, Semantics, Semiotics, Sign Relations, Spencer Brown, Syntax, Visualization | 3 Comments

Theme One Program • Motivation 4

Posted on June 6, 2024 by Jon Awbrey

From Zipf’s Law and the category of “things that vary inversely with frequency” I got my first brush with the idea that keeping track of usage frequencies is part and parcel of building efficient codes.

In its first application the environment the Learner has to learn is the usage behavior of its user, as given by finite sequences of characters from a finite alphabet, which sequences of characters might as well be called “words”, together with finite sequences of those words which might as well be called “phrases” or “sentences”. In other words, Job One for the Learner is the job of constructing a “user model”.

In that frame of mind we are not seeking anything so grand as a Universal Induction Algorithm but simply looking for any approach to give us a leg up, complexity wise, in Interactive Real Time.

Resources

cc: FB | Theme One Program • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Theme One Program • Motivation 3

Posted on June 5, 2024 by Jon Awbrey

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

cc: FB | Theme One Program • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Theme One Program • Motivation 2

Posted on June 4, 2024 by Jon Awbrey

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.

Resources

cc: FB | Theme One Program • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Theme One Program • Motivation 1

Posted on June 3, 2024 by Jon Awbrey

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.

Resources

cc: FB | Theme One Program • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Precursors Of Category Theory • 6

Posted on May 30, 2024 by Jon Awbrey

Hilbert and Ackermann • Principles of Mathematical Logic (1928)

For the intuitive interpretation on which we have hitherto based the predicate calculus, it was essential that the sentences and predicates should be sharply differentiated from the individuals, which occur as the argument values of the predicates. Now, however, there is nothing to prevent us from considering the predicates and sentences themselves as individuals which may serve as arguments of predicates.

Consider, for example, a logical expression of the form $(x)(A \rightarrow F(x)).$ This may be interpreted as a predicate $P(A, F)$ whose first argument place is occupied by a sentence $A,$ and whose second argument place is occupied by a monadic predicate $F.$

A false sentence $A$ is related to every $F$ by the relation $P(A, F);$ a true sentence $A$ only to those $F$ for which $(x)F(x)$ holds.

Further examples are given by the properties of reflexivity, symmetry, and transitivity of dyadic predicates. To these correspond three predicates: $\mathrm{Ref}(R),$ $\mathrm{Sym}(R),$ and $\mathrm{Tr}(R),$ whose argument $R$ is a dyadic predicate. These three properties are expressed in symbols as follows:

$\begin{array}{l} \mathrm{Ref}(R) \colon (x)R(x, x), \\[6pt] \mathrm{Sym}(R) \colon (x)(y)(R(x, y) \rightarrow R(y, x)), \\[6pt] \mathrm{Tr}(R) \colon (x)(y)(z)(R(x, y) \And R(y, z) \rightarrow R(x, z)). \end{array}$

All three properties are possessed by the predicate $\equiv(x, y)$ ( $x$ is identical with $y$ ). The predicate $<(x, y),$ on the other hand, possesses only the property of transitivity. Thus the formulas $\mathrm{Ref}(\equiv),$ $\mathrm{Sym}(\equiv),$ $\mathrm{Tr}(\equiv),$ and $\mathrm{Tr}(<)$ are true sentences, whereas $\mathrm{Ref}(<)$ and $\mathrm{Sym}(<)$ are false.

Such predicates of predicates will be called predicates of second level. (p. 135).

We have, first, predicates of individuals, and these are classified into predicates of different categories, or types, according to the number of their argument places. Such predicates are called predicates of first level.

By a predicate of second level, we understand one whose argument places are occupied by names of individuals or by predicates of first level, where a predicate of first level must occur at least once as an argument. The categories, or types, of predicates second level are differentiated according to the number and kind of their argument places. (p. 152).

Reference

Hilbert, D. and Ackermann, W., Principles of Mathematical Logic, Robert E. Luce (trans.), Chelsea Publishing Company, New York, NY, 1950. 1st published, Grundzüge der Theoretischen Logik, 1928. 2nd edition, 1938. English translation with revisions, corrections, and added notes by Robert E. Luce, 1950.

Resources

cc: FB | Peirce Matters • Laws of Form • Ma^thstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Posted in Abstraction, Ackermann, Aristotle, C.S. Peirce, Carnap, Category Theory, Hilbert, Kant, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Relation Theory, Saunders Mac Lane, Sign Relations, Type Theory | Tagged Abstraction, Ackermann, Aristotle, C.S. Peirce, Carnap, Category Theory, Hilbert, Kant, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Relation Theory, Saunders Mac Lane, Sign Relations, Type Theory | 3 Comments

Precursors Of Category Theory • 5

Posted on May 29, 2024 by Jon Awbrey

A demonstration rests in a finite number of steps.

G. Spencer Brown • Laws of Form

David Hilbert • “On the Infinite” (1925)

Finally, let us recall our real subject and, so far as the infinite is concerned, draw the balance of all our reflections. The final result then is: nowhere is the infinite realized; it is neither present in nature nor admissible as a foundation in our rational thinking — a remarkable harmony between being and thought. We gain a conviction that runs counter to the earlier endeavors of Frege and Dedekind, the conviction that, if scientific knowledge is to be possible, certain intuitive conceptions [Vorstellungen] and insights are indispensable; logic alone does not suffice. The right to operate with the infinite can be secured only by means of the finite.

The role that remains to the infinite is, rather, merely that of an idea — if, in accordance with Kant’s words, we understand by an idea a concept of reason that transcends all experience and through which the concrete is completed so as to form a totality — an idea, moreover, in which we may have unhesitating confidence within the framework furnished by the theory that I have sketched and advocated here. (p. 392).

References

Hilbert, D. (1925), “On the Infinite”, pp. 369–392 in Jean van Heijenoort (1967/1977).
van Heijenoort, J. (1967/1977), From Frege to Gödel : A Source Book in Mathematical Logic, 1879–1931, Harvard University Press, Cambridge, MA, 1967. 2nd printing, 1972. 3rd printing, 1977.
Spencer Brown, G. (1969), Laws of Form, George Allen and Unwin, London, p. 54.

Resources

cc: FB | Peirce Matters • Laws of Form • Ma^thstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Precursors Of Category Theory • 4

Posted on May 28, 2024 by Jon Awbrey

C.S. Peirce • “Prolegomena to an Apology for Pragmaticism” (1906)

I will now say a few words about what you have called Categories, but for which I prefer the designation Predicaments, and which you have explained as predicates of predicates.

That wonderful operation of hypostatic abstraction by which we seem to create entia rationis that are, nevertheless, sometimes real, furnishes us the means of turning predicates from being signs that we think or think through, into being subjects thought of. We thus think of the thought‑sign itself, making it the object of another thought‑sign.

Thereupon, we can repeat the operation of hypostatic abstraction, and from these second intentions derive third intentions. Does this series proceed endlessly? I think not. What then are the characters of its different members?

My thoughts on this subject are not yet harvested. I will only say that the subject concerns Logic, but that the divisions so obtained must not be confounded with the different Modes of Being: Actuality, Possibility, Destiny (or Freedom from Destiny).

On the contrary, the succession of Predicates of Predicates is different in the different Modes of Being. Meantime, it will be proper that in our system of diagrammatization we should provide for the division, whenever needed, of each of our three Universes of modes of reality into Realms for the different Predicaments. (CP 4.549).

The first thing to extract from the above passage is that Peirce’s Categories, for which he uses the technical term “Predicaments”, are predicates of predicates. Considerations of the order Peirce undertakes tend to generate hierarchies of subject matters, extending through what is traditionally called the logic of second intentions, or what is handled very roughly by second order logic in contemporary parlance, and continuing onward through higher intentions, or higher order logic and type theory.

Peirce arrived at his own system of three categories after a thoroughgoing study of his predecessors, with special reference to the categories of Aristotle, Kant, and Hegel. The names he used for his own categories varied with context and occasion, but ranged from moderately intuitive terms like quality, reaction, and symbolization to maximally abstract terms like firstness, secondness, and thirdness. Taken in full generality, k‑ness may be understood as referring to those properties all k‑adic relations have in common. Peirce’s distinctive claim is that a type hierarchy of three levels is generative of all we need in logic.

Part of the justification for Peirce’s claim that three categories are necessary and sufficient appears to arise from mathematical facts about the reducibility of k‑adic relations. With regard to necessity, triadic relations cannot be completely analyzed in terms or monadic and dyadic predicates. With regard to sufficiency, all higher arity k‑adic relations can be analyzed in terms of triadic and lower arity relations.

Reference

Peirce, C.S. (1906), “Prolegomena to an Apology for Pragmaticism”, The Monist 16, 492–546, CP 4.530–572.

Resources

cc: FB | Peirce Matters • Laws of Form • Ma^thstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Precursors Of Category Theory • 3

Posted on May 27, 2024 by Jon Awbrey

Act only according to that maxim by which you can at the same time will that it should become a universal law.

Immanuel Kant (1785)

C.S. Peirce • “On a New List of Categories” (1867)

§1. This paper is based upon the theory already established, that the function of conceptions is to reduce the manifold of sensuous impressions to unity, and that the validity of a conception consists in the impossibility of reducing the content of consciousness to unity without the introduction of it. (CP 1.545).

§2. This theory gives rise to a conception of gradation among those conceptions which are universal. For one such conception may unite the manifold of sense and yet another may be required to unite the conception and the manifold to which it is applied; and so on. (CP 1.546).

Cued by Kant’s idea regarding the function of concepts in general, Peirce locates his categories on the highest levels of abstraction able to provide a meaningful measure of traction in practice. Whether successive grades of conceptions converge to an absolute unity or not is a question to be pursued as inquiry progresses and need not be answered in order to begin.

Resources

cc: FB | Peirce Matters • Laws of Form • Ma^thstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Precursors Of Category Theory • 2

Posted on May 26, 2024 by Jon Awbrey

Thanks to art, instead of seeing one world only, our own, we see that world multiply itself and we have at our disposal as many worlds as there are original artists …

☙ Marcel Proust

When it comes to looking for the continuities of the category concept across different systems and systematizers, we don’t expect to find their kinship in the names or numbers of categories, since those are legion and their divisions deployed on widely different planes of abstraction, but in their common function.

Aristotle

Things are equivocally named, when they have the name only in common, the definition (or statement of essence) corresponding with the name being different. For instance, while a man and a portrait can properly both be called animals (ζωον), these are equivocally named. For they have the name only in common, the definitions (or statements of essence) corresponding with the name being different. For if you are asked to define what the being an animal means in the case of the man and the portrait, you give in either case a definition appropriate to that case alone.

Things are univocally named, when not only they bear the same name but the name means the same in each case — has the same definition corresponding. Thus a man and an ox are called animals. The name is the same in both cases; so also the statement of essence. For if you are asked what is meant by their both of them being called animals, you give that particular name in both cases the same definition. (Aristotle, Categories, 1.1^a1–12).

Translator’s Note. “Ζωον in Greek had two meanings, that is to say, living creature, and, secondly, a figure or image in painting, embroidery, sculpture. We have no ambiguous noun. However, we use the word ‘living’ of portraits to mean ‘true to life’.”

In the logic of Aristotle categories are adjuncts to reasoning whose function is to resolve ambiguities and thus to prepare equivocal signs, otherwise recalcitrant to being ruled by logic, for the application of logical laws. The example of ζωον illustrates the fact that we don’t need categories to make generalizations so much as to control generalizations, to reign in abstractions and analogies which have been stretched too far.

References

Aristotle, “The Categories”, Harold P. Cooke (trans.), pp. 1–109 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.
Karpeles, Eric (2008), Paintings in Proust, Thames and Hudson, London, UK.

Resources

cc: FB | Peirce Matters • Laws of Form • Ma^thstodon • Ontolog • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

	Survey of Semiotics,… on Sign Relations • Graphical…
	Sign Relations • Gra… on Survey of Semiotics, Semiosis,…
	Survey of Semiotics,… on Sign Relations • Graphical…
	Sign Relations • Gra… on Survey of Semiotics, Semiosis,…
	Jon Awbrey on Sign Relations • Definiti…
	Jon Awbrey on Sign Relations • Definiti…
	Jon Awbrey on Sign Relations • Definiti…
	Survey of Semiotics,… on Sign Relations • Semiotic Equi…
	Survey of Semiotics,… on Sign Relations • Semiotic Equi…
	Sign Relations • Sem… on Survey of Semiotics, Semiosis,…

Resources

Share this:

Resources

Share this:

Resources

Share this:

Resources

Share this:

Resources

Share this:

Hilbert and Ackermann • Principles of Mathematical Logic (1928)

Reference

Resources

Share this:

David Hilbert • “On the Infinite” (1925)

References

Resources

Share this:

C.S. Peirce • “Prolegomena to an Apology for Pragmaticism” (1906)

Reference

Resources

Share this:

C.S. Peirce • “On a New List of Categories” (1867)

Resources

Share this:

Aristotle

References

Resources

Share this:

Elsewhere

Blogroll

Follow Blog via Email

Recent Posts

Recent Comments

Archives

Blog Stats

Categories

Logic, Mathematics, Philosophy

Current Events

Meta