Inquiry Into Inquiry

Animated Logical Graphs • 31

Posted on September 1, 2019 by Jon Awbrey

Re: Systems Science • Aleksandar Malečić
Re: Animated Logical Graphs • 21

AM:: Each step on its own, as far as I can follow them, makes sense. You are, if I understand it correctly, trying to figure out something fundamental, the rock bottom reality. When can we expect that results of such a research to become “applicable to more than one of the traditional departments of knowledge”? What kinds of tragedy, disaster, misunderstanding, mismanagement, or failure would/will be preventable by your approach?

The larger questions asked above — interdisciplinary inquiry, the interest in integration, the synthesis of ideas across isolated silos of specialization, and what it might mean for the future — are issues Susan Awbrey and I addressed from a pragmatic semiotic perspective:

Awbrey, S.M., and Awbrey, J.L. (2001), “Conceptual Barriers to Creating Integrative Universities”, Organization : The Interdisciplinary Journal of Organization, Theory, and Society 8(2), Sage Publications, London, UK, 269–284. Abstract. Online.
Awbrey, S.M., and Awbrey, J.L. (1999), “Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century”, Second International Conference of the Journal ‘Organization’, Re-Organizing Knowledge, Trans-Forming Institutions : Knowing, Knowledge, and the University in the 21st Century, University of Massachusetts, Amherst, MA. Online.

From that vantage point, what I’m about here is just a subgoal of a subgoal, panning what bits of elemental substrate can be found ever nearer that elusive “rock bottom reality”.

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

Posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Constraint Satisfaction Problems, Deduction, Diagrammatic Reasoning, Duality, Equational Inference, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Painted Cacti, Peirce, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Theorem Proving, Visualization | Tagged Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Constraint Satisfaction Problems, Deduction, Diagrammatic Reasoning, Duality, Equational Inference, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Painted Cacti, Peirce, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Theorem Proving, Visualization | 9 Comments

Animated Logical Graphs • 30

Posted on August 25, 2019 by Jon Awbrey

The duality between Entitative and Existential interpretations of logical graphs is a good example of a mathematical symmetry, in this case a symmetry of order two. Symmetries of this and higher orders give us conceptual handles on excess complexity in the manifold of sensuous impressions, making it well worth the effort to seek them out and grasp them where we find them.

In that vein, here’s a Rosetta Stone to give us a grounding in the relationship between boolean functions and our two readings of logical graphs.

$\text{Boolean Functions on Two Variables}$
$\text{Boolean Function}$	$\text{Entitative Graph}$	$\text{Existential Graph}$
$f_{0}$
$\text{false}$	$\text{false}$	$\text{false}$
$f_{1}$
$\text{neither}~ x ~\text{nor}~ y$	$\lnot (x \lor y)$	$\lnot x \land \lnot y$
$f_{2}$
$y ~\text{and not}~ x$	$\lnot x \land y$	$\lnot x \land y$
$f_{3}$
$\text{not}~ x$	$\lnot x$	$\lnot x$
$f_{4}$
$x ~\text{and not}~ y$	$x \land \lnot y$	$x \land \lnot y$
$f_{5}$
$\text{not}~ y$	$\lnot y$	$\lnot y$
$f_{6}$
$x ~\text{not equal to}~ y$	$x \ne y$	$x \ne y$
$f_{7}$
$\text{not both}~ x ~\text{and}~ y$	$\lnot x \lor \lnot y$	$\lnot (x \land y)$
$f_{8}$
$x ~\text{and}~ y$	$x \land y$	$x \land y$
$f_{9}$
$x ~\text{equal to}~ y$	$x = y$	$x = y$
$f_{10}$
$y$	$y$	$y$
$f_{11}$
$\text{if}~ x ~\text{then}~ y$	$x \Rightarrow y$	$x \Rightarrow y$
$f_{12}$
$x$	$x$	$x$
$f_{13}$
$\text{if}~ y ~\text{then}~ x$	$x \Leftarrow y$	$x \Leftarrow y$
$f_{14}$
$x ~\text{or}~ y$	$x \lor y$	$x \lor y$
$f_{15}$
$\text{true}$	$\text{true}$	$\text{true}$

cc: Cybernetics (1) (2) • Laws of Form • FB | Logical Graphs • Ontolog Forum (1) (2)
cc: Peirce List (1) (2) (3) (4) (5) (6) (7) (8) • Structural Modeling (1) (2) • Systems (1) (2)

Abduction, Deduction, Induction, Analogy, Inquiry • 27

Posted on August 20, 2019 by Jon Awbrey

Re: Gil Kalai • Avi Wigderson : Integrating Computational Modeling, Algorithms, and Complexity into Theories of Nature Marks a New Scientific Revolution!

I took a look at Avi’s paper “On the Nature of the Theory of Computation” (OtNotToC). There is naturally a good dose of TOC but little on the type of World-Objective Contact (WOC) it takes to connect with empirical science. Just on that sample it reminds me of projects like Wolfram’s “New Kind Of Science”. They all do a good job of convincing us to use computational media as virtual laboratories for conducting experimental mathematics, but they leave us hanging when it comes to analyzing the relation between what goes on inside the box of computation and the natural world outside the box.

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

Posted in Abduction, Analogy, Aristotle, Artificial Intelligence, C.S. Peirce, Deduction, Induction, Inquiry, Inquiry Driven Systems, Intelligent Systems Engineering, Logic, Mental Models, Peirce, Scientific Method, Semiotics, Systems | Tagged Abduction, Analogy, Aristotle, Artificial Intelligence, C.S. Peirce, Deduction, Induction, Inquiry, Inquiry Driven Systems, Intelligent Systems Engineering, Logic, Mental Models, Peirce, Scientific Method, Semiotics, Systems | 6 Comments

Animated Logical Graphs • 29

Posted on August 11, 2019 by Jon Awbrey

Re: Animated Logical Graphs • 21
Re: Ontolog Forum • Joseph Simpson

JS:: I tend to view equivalence and distinction as relationships as opposed to operations. I do not know if this makes any significant difference in this context.

Dear Joe,

I invoked the general concepts of equivalence and distinction at this point in order to keep the wider backdrop of ideas in mind but since we’ve been focusing on boolean functions to coordinate the semantics of propositional calculi we can get a sense of the links between operations and relations by looking at their relationship in a boolean frame of reference.

Let $\mathbb{B} = \{ 0, 1 \}$ and $k$ a positive integer. Then $\mathbb{B}^k$ is the set of $k$ -tuples of elements of $\mathbb{B}.$

A $k$ -variable boolean function is a mapping $\mathbb{B}^k \to \mathbb{B}.$
A $k$ -place boolean relation is a subset of $\mathbb{B}^k.$

The correspondence between boolean functions and boolean relations may be articulated as follows.

Any $k$ -place relation $L,$ as a subset of $\mathbb{B}^k,$ has a corresponding indicator function (or characteristic function) $f_L : \mathbb{B}^k \to \mathbb{B}$ defined by the rule that $f_L (x) = 1$ if $x$ is in $L$ and $f_L (x) = 0$ if $x$ is not in $L.$
Any $k$ -variable function $f : \mathbb{B}^k \to \mathbb{B}$ is the indicator function of a $k$ -place relation $L_f$ consisting of all the $x$ in $\mathbb{B}^k$ where $f(x) = 1.$ The set $L_f$ is called the fiber of $1$ or the pre-image of $1$ in $\mathbb{B}^k$ and is commonly notated as $f^{-1}(1).$

Resources

Logic Syllabus • Relation Theory
Logical Equality • Logical Inequality
Boolean Function • Boolean-Valued Function

cc: Peirce List (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15)
cc: Structural Modeling (1) (2) • Systems Science (1) (2)
cc: Cybernetics (1) (2) • Ontolog Forum (1) (2)
cc: FB | Logical Graphs • Laws of Form

Abduction, Deduction, Induction, Analogy, Inquiry • 26

Posted on August 6, 2019 by Jon Awbrey

Re: Gil Kalai • Avi Wigderson : Integrating Computational Modeling, Algorithms, and Complexity into Theories of Nature Marks a New Scientific Revolution!

Projects giving a central place to computation in scientific inquiry go back to Hobbes and Leibniz, at least, and then came Babbage and Peirce. One of the first issues determining their subsequent development is the degree to which they identify computation with deduction. The next question concerns how many types of reasoning they count as contributing to the logic of empirical science:

Is deduction alone sufficient?
Are deduction and induction irreducible to each other and sufficient in tandem?
Are there in fact three irreducible types of inference: abduction, deduction, induction?

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

Animated Logical Graphs • 28

Posted on August 3, 2019 by Jon Awbrey

Re: R.J. Lipton and K.W. Regan • Discrepancy Games and Sensitivity
Re: Ontolog Forum • Joseph Simpson
Re: Animated Logical Graphs • 24

I will have to focus on other business for a couple of weeks — so just by way of reminding myself what we were talking about at this juncture where logical graphs and differential logic intersect, here’s my comment on R.J. Lipton and K.W. Regan’s blog post about Discrepancy Games and Sensitivity.

Just by way of a general observation, concepts like discrepancy, influence, sensitivity, and the like are differential in character, so I tend to think the proper grounds for approaching them more systematically will come from developing the logical analogue of differential geometry.

I took a few steps in this direction some years ago in connection with an effort to understand a certain class of intelligent systems as dynamical systems. There’s a Survey of related resources on the following page.

Survey of Differential Logic

Resources

Logical Graphs
Differential Propositional Calculus • Part 1 • Part 2
Differential Logic • Part 1 • Part 2 • Part 3
Differential Logic and Dynamic Systems
• Part 1 • Part 2 • Part 3 • Part 4 • Part 5
Differential Analytic Turing Automata

Animated Logical Graphs • 27

Posted on July 31, 2019 by Jon Awbrey

The rules given in the previous post for evaluating cactus graphs were given in purely formal terms, that is, by referring to the mathematical forms of cacti without mentioning their potential for logical meaning. As it turns out, two ways of mapping cactus graphs to logical meanings are commonly found in practice. These two mappings of mathematical structure to logical meaning are formally dual to each other and known as the Entitative and Existential interpretations respectively. The following Table compares the entitative and existential interpretations of the primary cactus structures, from which the rest of their semantics can be derived.

Animated Logical Graphs • 26

Posted on July 28, 2019 by Jon Awbrey

This post and the next wrap up the Themes and Variations section of my speculation on Futures Of Logical Graphs. I made an effort to “show my work”, reviewing the steps I took to arrive at the present perspective on logical graphs, whistling past the least productive of the blind alleys, cul-de-sacs, detours, and forking paths I explored along the way. It can be useful to tell the story that way, partly because others may find things I missed down those roads, but it does call for a recap of the main ideas I would like readers to take away.

Partly through my reflections on Peirce’s use of operator variables I was led to what I called the reflective extension of logical graphs, or what I now call the “cactus language”, after its principal graph-theoretic data structure. This graphical formal language arises from generalizing the negation operator ${}^{\backprime\backprime} \texttt{(} ~ \texttt{)} {}^{\prime\prime}$ in a particular direction, treating ${}^{\backprime\backprime} \texttt{(} ~ \texttt{)} {}^{\prime\prime}$ as the controlled, moderated, or reflective negation operator of order 1, and adding another operator for each integer parameter greater than 1. This family of operators is symbolized by bracketed argument lists of the forms ${}^{\backprime\backprime} \texttt{(} ~ \texttt{)} {}^{\prime\prime},$ ${}^{\backprime\backprime} \texttt{(} ~ \texttt{,} ~ \texttt{)} {}^{\prime\prime},$ ${}^{\backprime\backprime} \texttt{(} ~ \texttt{,} ~ \texttt{,} ~ \texttt{)} {}^{\prime\prime},$ and so on, where the number of places is the order of the reflective negation operator in question.

Two rules suffice for evaluating cactus graphs.

The rule for evaluating a $k$ -node operator, corresponding to an expression of the form ${}^{\backprime\backprime} x_1 x_2 \ldots x_{k-1} x_k {}^{\prime\prime},$ is as follows.

The rule for evaluating a $k$ -lobe operator, corresponding to an expression of the form ${}^{\backprime\backprime} \texttt{(} x_1 \texttt{,} x_2 \texttt{,} \ldots \texttt{,} x_{k-1} \texttt{,} x_k \texttt{)} {}^{\prime\prime},$ is as follows.

Animated Logical Graphs • 25

Posted on July 26, 2019 by Jon Awbrey

Re: Animated Logical Graphs • 23

Let’s examine the Formal Operation Table for the third in our series of reflective forms to see if we can elicit the general pattern.

Or, thinking in terms of the corresponding cactus graphs, writing ${}^{\backprime\backprime} \texttt{o} {}^{\prime\prime}$ for a blank node and ${}^{\backprime\backprime} \texttt{|} {}^{\prime\prime}$ for a terminal edge, we get the following Table.

Evidently, the rule is that ${}^{\backprime\backprime} \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)} {}^{\prime\prime}$ denotes the value denoted by ${}^{\backprime\backprime} \texttt{o} {}^{\prime\prime}$ if and only if exactly one of the variables $a, b, c$ has the value denoted by ${}^{\backprime\backprime} \texttt{|} {}^{\prime\prime},$ otherwise ${}^{\backprime\backprime} \texttt{(} a \texttt{,} b \texttt{,} c \texttt{)} {}^{\prime\prime}$ denotes the value denoted by ${}^{\backprime\backprime} \texttt{|} {}^{\prime\prime}.$ Examining the whole series of reflective forms shows this is the general rule.

In the Entitative Interpretation $(\mathrm{En}),$ where $\texttt{o}$ = false and $\texttt{|}$ = true,
${}^{\backprime\backprime} \texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)} {}^{\prime\prime}$ translates as “not just one of the $x_j$ is true”.
In the Existential Interpretation $(\mathrm{Ex}),$ where $\texttt{o}$ = true and $\texttt{|}$ = false,
${}^{\backprime\backprime} \texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)} {}^{\prime\prime}$ translates as “just one of the $x_j$ is not true”.

Animated Logical Graphs • 24

Posted on July 26, 2019 by Jon Awbrey

Re: Ontolog Forum • Joseph Simpson

JS:

Today I found an interesting publication that might relate to the current discussion of Animated Logical Graphs (ALG). Please see:

Mathematician Solves Computer Science Conjecture in Two Pages

The “sensitivity” conjecture may be a topic that could be explored using ALG. There appear to be many interesting connections between ALG and the sensitivity conjecture. I am looking for an area where ALG application examples may be created and discussed. My first attempt at an example, using the Augmented Model-Exchange Isomorphism (AMEI), raised a number of conceptual and structural application issues. Which can be addressed as we move forward. My plan is to continue the search for specific application areas. I believe that finding domain specific applications will help me better understand the ALG material.

Take care, be good to yourself and have fun,

Joe

Dear Joe,

Boolean functions $f : \mathbb{B}^k \to \mathbb{B}$ and different ways of contemplating their complexity are definitely the right ballpark, or at least the right planet, for field-testing logical graphs.

I don’t know much about the Boolean Sensitivity Conjecture but I happened to run across an enlightening article about it just yesterday and I did a while ago begin an exploration into what appears to be a related question, Péter Frankl’s “Union-Closed Sets Conjecture”. See the resource pages linked below.

At any rate, now that we’ve entered the ballpark, or standard orbit, of boolean functions, I can skip a bit of dancing around and jump to the next blog post I have on deck.

Resources

R.J. Lipton ∧∨ K.W. Regan

Scott Aaronson • Sensitivity Conjecture Resolved
Frankl Conjecture • (1) • (2) • (3) • (4) • (5) • (6) • (7) • (8) • (9) • (10) • (11) • (12)

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Animated Logical Graphs • 31

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Abduction, Deduction, Induction, Analogy, Inquiry • 27

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