Cactus Language • Overview 3

Posted on March 7, 2025 by Jon Awbrey

In the development of Cactus Language to date the following two species of graphs have been instrumental.

  • Painted And Rooted Cacti (PARCAI).
  • Painted And Rooted Conifers (PARCOI).

It suffices to begin with the first class of data structures, developing their properties and uses in full, leaving discussion of the latter class to a part of the project where their distinctive features are key to developments at that stage.  Partly because the two species are so closely related and partly for the sake of brevity, we’ll always use the genus name “PARC” to denote the corresponding cacti.

To provide a computational middle ground between sentences seen as syntactic strings and propositions seen as indicator functions the language designer must not only supply a medium for the expression of propositions but also link the assertion of sentences to a means for inverting the indicator functions, that is, for computing the fibers or inverse images of the propositions.

Given a body of conceivable propositions we need a way to follow the threads of their indications from their object domain to their values for the mind and a way to follow those same threads back again.  Moreover, we need to implement both ways of proceeding in computational form.  Thus we need programs for tracing the clues sentences provide from the universe of their objects to the signs of their values and, in turn, from signs to objects.  Ultimately, we need to render propositions so functional as indicators of sets and so essential for examining the equality of sets as to give a rule for the practical conceivability of sets.  Tackling that task requires us to introduce a number of new definitions and a collection of additional notational devices, to which we now turn.

Resources

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Posted in Automata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Differential Logic, Equational Inference, Formal Grammars, Formal Languages, Graph Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Propositional Calculus, Visualization | Tagged , , , , , , , , , , , , , , , , | 3 Comments

Cactus Language • Overview 2

Posted on March 6, 2025 by Jon Awbrey

In order to facilitate the use of propositions as indicator functions it helps to acquire a flexible notation for referring to propositions in that light, for interpreting sentences in a corresponding role, and for negotiating the requirements of mutual sense between the two domains.  If none of the formalisms readily available or in common use meet all the design requirements coming to mind then it is necessary to contemplate the design of a new language especially tailored to the purpose.

In the present application, there is a pressing need to devise a general calculus for composing propositions, computing their values on particular arguments, and inverting their indications to arrive at the sets of things in the universe which are indicated by them.

For computational purposes it is convenient to have a middle ground or an intermediate language for negotiating between the koine of sentences regarded as strings of literal characters and the realm of propositions regarded as objects of logical value, even if that makes it necessary to introduce an artificial medium of exchange between the two domains.

If the necessary computations are to be carried out in an organized fashion, and ultimately or partially by familiar classes of machines, then the strings expressing logical propositions are likely to find themselves parsed into tree‑like data structures at some stage of the game.  As far as their abstract structures as graphs are concerned, there are several species of graph‑theoretic data structures fitting the task in a reasonably effective and efficient way.

Resources

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cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Posted in Automata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Differential Logic, Equational Inference, Formal Grammars, Formal Languages, Graph Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Propositional Calculus, Visualization | Tagged , , , , , , , , , , , , , , , , | 4 Comments

Cactus Language • Overview 1

Posted on March 1, 2025 by Jon Awbrey

Thus, what looks to us like a sphere of scientific knowledge more accurately should be represented as the inside of a highly irregular and spiky object, like a pincushion or porcupine, with very sharp extensions in certain directions, and virtually no knowledge in immediately adjacent areas.  If our intellectual gaze could shift slightly, it would alter each quill’s direction, and suddenly our entire reality would change.

Herbert J. Bernstein • “Idols of Modern Science”

The following report describes a calculus for representing propositions as sentences, that is, as syntactically defined sequences of signs, and for working with those sentences in light of their semantically defined contents as logical propositions.  In their computational representation the expressions of the calculus parse into a class of graph‑theoretic data structures whose underlying graphs are called painted cacti.

Painted cacti are a specialization of what graph‑theorists refer to as cacti, which are in turn a generalization of what they call trees.  The data structures corresponding to painted cacti have especially nice properties, not only useful in computational terms but interesting from a theoretical standpoint.  The remainder of the present Overview is devoted to motivating the development of the indicated family of formal languages, going under the generic name of Cactus Language.

Resource

For readers interested and intrepid enough to read ahead, here’s an outline of my work in progress on the OEIS Wiki, which I’ll be revising and serializing to the present blog.

Part 1

Cactus Language • Syntax

Part 2

Generalities About Formal Grammars

Part 3

References

Document History

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Posted in Automata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Differential Logic, Equational Inference, Formal Grammars, Formal Languages, Graph Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Propositional Calculus, Visualization | Tagged , , , , , , , , , , , , , , , , | 4 Comments

Ouch❢

Posted on February 27, 2025 by Jon Awbrey

A child hears it said that the stove is hot.  But it is not, he says; and, indeed, that central body is not touching it, and only what that touches is hot or cold.  But he touches it, and finds the testimony confirmed in a striking way.  Thus, he becomes aware of ignorance, and it is necessary to suppose a self in which this ignorance can inhere.  …

In short, error appears, and it can be explained only by supposing a self which is fallible.

Ignorance and error are all that distinguish our private selves from the absolute ego of pure apperception.

🙞 C.S. Peirce • “Questions Concerning Certain Faculties Claimed For Man”

Resource

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Posted in C.S. Peirce, Ego, Error, Ignorance, Inquiry, References, Selfhood, Semiotics, Sources | Tagged , , , , , , , , | Leave a comment

Theory and Therapy of Representations • 5

Posted on February 25, 2025 by Jon Awbrey

Re: R.J. Lipton and K.W. Regan • Legal Complexity

I do not pretend to understand the moral universe;
the arc is a long one, my eye reaches but little ways;
I cannot calculate the curve and complete the figure by
the experience of sight;  I can divine it by conscience.
And from what I see I am sure it bends towards justice.

🙞 Theodore Parker

The arc of the moral universe may bend toward justice — there’s hope it will.
For the logic of laws to converge on justice may take some doing on our part.

Resources

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Theory and Therapy of Representations • 4

Posted on February 24, 2025 by Jon Awbrey

Re: Theory and Therapy of Representations • 3
Re: Ontolog Forum • Paola Di Maio

JA:
What are the forces distorting our representations of what’s observed, what’s expected, and what’s intended?
PDM:
The short answer is — the force behind all distortions is our own unenlightened mind, and all the shortfalls this comes with.

I think that’s true, we have to keep reflecting on the state of our personal enlightenments.  If we can do that without losing our heads and our systems thinking caps, there will be much we can do to promote the general Enlightenment of the State.

On both personal and general grounds we have a stake in the projects of self‑governing systems — whether it is possible for them to exist and what it takes for them to thrive in given environments.  Systems on that order have of course been studied from many points of view and at many levels of organization.  Whether we address them under the names of adaptive, cybernetic, error-correcting, intelligent, or optimal control systems they all must be capable to some degree of learning, reasoning, and self‑guidance.

Resources

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Theory and Therapy of Representations • 3

Posted on February 23, 2025 by Jon Awbrey

Representation is a concept we find at the intersection of cybernetics, epistemology, logic, mathematics, psychology, and sociology.  In my studies it led me from math to psych and back again, with sidelong glances at the history of democratic governance.  Its time come round again, I find myself returning to the scenes of two recurring questions.

Scene 1.  Pragmatic Truth • Discussion 18

We do not live in axiom systems.  We do not live encased in languages, formal or natural.  There is no reason to think we will ever have exact and exhaustive theories of what’s out there, and the truth, as we know, is “out there”.  Peirce understood there are more truths in mathematics than are dreamt of in logic — and Gödel’s realism should have put the last nail in the coffin of logicism — but some ways of thinking just never get a clue.

That brings us to Question 1 —

  • What are formalisms and all their embodiments in brains and computers good for?

Scene 2.  Theory and Therapy of Representations • 1

Statistics were originally the data a ship of state needed for stationkeeping and staying on course.  The Founders of the United States, like the Cybernauts of the Enlightenment they were, engineered a ship of state with checks and balances and error‑controlled feedbacks for the sake of representing both reality and the will of the people.  In that connection Max Weber saw how a state’s accounting systems are intended as representations of realities its crew and passengers must observe or perish.

That brings us to Question 2 —

  • What are the forces distorting our representations of what’s observed, what’s expected, and what’s intended?

Resources

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Posted in Accountability, Adaptive Systems, C.S. Peirce, Cybernetics, Democracy, Economics, Education, Expectation, Governance, Information, Inquiry, Intention, Justice, Law, Logic, Max Weber, Observation, Plato, Pragmata, Representation, Science, Semiotics, Society, Statistics, Systems Theory | Tagged , , , , , , , , , , , , , , , , , , , , , , , , | 4 Comments

Theory and Therapy of Representations • 2

Posted on February 22, 2025 by Jon Awbrey

December 19, 2011

In a complex society, people making decisions and taking actions at places remote from you have the power to affect your life in significant ways.  Those people govern your life, they are your government, no matter what spheres of influence they inhabit, private or public.  The only way you get a choice in that governance is if there are paths of feedback permitting you to affect the life of those decision makers and action takers in significant ways.  That is what accountability, response-ability, and representative government are all about.

Naturally, some people are against that.

In the United States there has been a concerted campaign for as long as I can remember — but even more concerted since the Reagan Regime — to get the People to abdicate their hold on The Powers That Be and just let some anonymous corporate entity send us the bill after the fact.  They keep trying to con the People into thinking they can starve the beast, to limit government, when what they are really doing is feeding the beast of corporate control, weakening their own power over the forces that govern their lives.

That is the road to perdition as far as responsible government goes.  There is not much of anything one leader or one administration can do unsupported if the People do not constantly demand a government of, by, and for the People.

Resource

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Posted in Accountability, Adaptive Systems, C.S. Peirce, Cybernetics, Democracy, Economics, Education, Expectation, Governance, Information, Inquiry, Intention, Justice, Law, Logic, Max Weber, Observation, Plato, Pragmata, Representation, Science, Semiotics, Society, Statistics, Systems Theory | Tagged , , , , , , , , , , , , , , , , , , , , , , , , | 5 Comments

Theory and Therapy of Representations • 1

Posted on February 21, 2025 by Jon Awbrey
Again, in a ship, if a man were at liberty to do what he chose, but were devoid of mind and excellence in navigation (αρετης κυβερνητικης), do you perceive what must happen to him and his fellow sailors?

Plato • Alcibiades • 135 A

Statistics were originally the data a ship of state needed for stationkeeping and staying on course.  The Founders of the United States, like the Cybernauts of the Enlightenment they were, engineered a ship of state with checks and balances and error‑controlled feedbacks for the sake of representing both reality and the will of the people.  In that connection Max Weber saw how a state’s accounting systems are intended as representations of realities its crew and passengers must observe or perish.

The question for our time is —

  • What are the forces distorting our representations of what’s observed, what’s expected, and what’s intended?

Repercussions

Resource

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Basal Ingredients Of Society • ℞

Posted on February 20, 2025 by Jon Awbrey

THE SOCIAL COMPACT

If then we discard from the social compact what is not of its essence, we shall find that it reduces itself to the following terms:

“Each of us puts his person and all his power in common under the supreme direction of the general will, and, in our corporate capacity, we receive each member as an indivisible part of the whole.”

Reference

  • Jean Jacques Rousseau, “The Social Contract”, G.D.H. Cole (trans.),
    Great Books of The Western World, Volume 38.

Resource

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