Inquiry Into Inquiry

Animated Logical Graphs • 49

Posted on December 3, 2020 by Jon Awbrey

Re: Richard J. Lipton • The Art Of Math
Re: Animated Logical Graphs • (30) (45) (46) (47) (48)

Dualities are symmetries of order two and symmetries bear on complexity by reducing its measure in proportion to their order. The inverse relationship between symmetry and all those dissymmetries from dispersion and diversity to entropy and uncertainty is governed in cybernetics by the Law of Requisite Variety, the medium of which exchanges C.S. Peirce invested in his formula, Information = Comprehension × Extension.

The duality between entitative and existential interpretations of logical graphs is one example of a mathematical symmetry but it’s not unusual to find symmetries within symmetries and it’s always rewarding to find them where they exist. To that end let’s take up our Table of Venn Diagrams and Logical Graphs on Two Variables and sort the rows to bring together diagrams and graphs having similar shapes. What defines their similarity is the action of a mathematical group whose operations transform the elements of each class among one another but intermingle no dissimilar elements. In the jargon of transformation groups these classes are called orbits. We find the sixteen rows partition into seven orbits, as shown below.

Venn Diagrams and Logical Graphs on Two Variables • Orbit Order
$\text{Boolean Function}$	$\text{Entitative Graph}$	$\text{Existential Graph}$

$f_{0}$	$\text{false}$	$\text{false}$

$f_{1}$	$\lnot (x \lor y)$	$\lnot x \land \lnot y$

$f_{2}$	$\lnot x \land y$	$\lnot x \land y$

$f_{4}$	$x \land \lnot y$	$x \land \lnot y$

$f_{8}$	$x \land y$	$x \land y$

$f_{3}$	$\lnot x$	$\lnot x$

$f_{12}$	$x$	$x$

$f_{6}$	$x \ne y$	$x \ne y$

$f_{9}$	$x = y$	$x = y$

$f_{5}$	$\lnot y$	$\lnot y$

$f_{10}$	$y$	$y$

$f_{7}$	$\lnot x \lor \lnot y$	$\lnot (x \land y)$

$f_{11}$	$x \Rightarrow y$	$x \Rightarrow y$

$f_{13}$	$x \Leftarrow y$	$x \Leftarrow y$

$f_{14}$	$x \lor y$	$x \lor y$

$f_{15}$	$\text{true}$	$\text{true}$

Resources

cc: Cybernetics Communications (1) (2) • FB | Logical Graphs • Ontolog Forum (1) (2)
cc: Peirce (1) (2) (3) (4) (5) (6) (7) (8) (9) • Structural Modeling (1) (2) • Systems (1) (2)

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 • 48

Posted on November 30, 2020 by Jon Awbrey

Re: Richard J. Lipton • The Art Of Math
Re: Animated Logical Graphs • (30) (45) (46) (47)

A more graphic picture of Peirce duality is given by the next Table, which shows how logical graphs map to venn diagrams under entitative and existential interpretations. Column 1 shows the logical graphs for the sixteen boolean functions on two variables. Column 2 shows the venn diagrams associated with the entitative interpretation and Column 3 shows the venn diagrams associated with the existential interpretation.

Logical Graphs • Entitative and Existential Venn Diagrams
$\text{Logical Graph}$	$\text{Entitative Interpretation}$	$\text{Existential Interpretation}$

$\texttt{(} ~ \texttt{)}$	$\text{true}$ $f_{15}$	$\text{false}$ $f_{0}$

$\texttt{(} x \texttt{)(} y \texttt{)}$	$\lnot x \lor \lnot y$ $f_{7}$	$\lnot x \land \lnot y$ $f_{1}$

$\texttt{(} x \texttt{)} y$	$x \Rightarrow y$ $f_{11}$	$x \nLeftarrow y$ $f_{2}$

$\texttt{(} x \texttt{)}$	$\lnot x$ $f_{3}$	$\lnot x$ $f_{3}$

$x \texttt{(} y \texttt{)}$	$x \Leftarrow y$ $f_{13}$	$x \nRightarrow y$ $f_{4}$

$\texttt{(} y \texttt{)}$	$\lnot y$ $f_{5}$	$\lnot y$ $f_{5}$

$\texttt{(} x \texttt{,} y \texttt{)}$	$x = y$ $f_{9}$	$x \ne y$ $f_{6}$

$\texttt{(} x y \texttt{)}$	$\lnot (x \lor y)$ $f_{1}$	$\lnot (x \land y)$ $f_{7}$

$x y$	$x \lor y$ $f_{14}$	$x \land y$ $f_{8}$

$\texttt{((} x \texttt{,} y \texttt{))}$	$x \ne y$ $f_{6}$	$x = y$ $f_{9}$

$y$	$y$ $f_{10}$	$y$ $f_{10}$

$\texttt{(} x \texttt{(} y \texttt{))}$	$x \nLeftarrow y$ $f_{2}$	$x \Rightarrow y$ $f_{11}$

$x$	$x$ $f_{12}$	$x$ $f_{12}$

$\texttt{((} x \texttt{)} y \texttt{)}$	$x \nRightarrow y$ $f_{4}$	$x \Leftarrow y$ $f_{13}$

$\texttt{((} x \texttt{)(} y \texttt{))}$	$x \land y$ $f_{8}$	$x \lor y$ $f_{14}$

	$\text{false}$ $f_{0}$	$\text{true}$ $f_{15}$

Resources

cc: Cybernetics Communications (1) (2) • 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)

Animated Logical Graphs • 47

Posted on November 27, 2020 by Jon Awbrey

Re: Richard J. Lipton • The Art Of Math
Re: Animated Logical Graphs • (30) (45) (46)

A logical concept represented by a boolean variable has its extension, the cases it covers in a designated universe of discourse, and its comprehension (or intension), the properties it implies in a designated hierarchy of predicates. The formulas and graphs tabulated in previous posts are well-adapted to articulate the syntactic and intensional aspects of propositional logic. But their very tailoring to those tasks tends to slight the extensional and therefore empirical applications of logic. Venn diagrams, despite their unwieldiness as the number of logical dimensions increases, are indispensable in providing the visual intuition with a solid grounding in the extensions of logical concepts. All that makes it worthwhile to reset our table of boolean functions on two variables to include the corresponding venn diagrams.

Venn Diagrams and Logical Graphs on Two Variables
$\text{Boolean Function}$	$\text{Entitative Graph}$	$\text{Existential Graph}$

$f_{0}$	$\text{false}$	$\text{false}$

$f_{1}$	$\lnot (x \lor y)$	$\lnot x \land \lnot y$

$f_{2}$	$\lnot x \land y$	$\lnot x \land y$

$f_{3}$	$\lnot x$	$\lnot x$

$f_{4}$	$x \land \lnot y$	$x \land \lnot y$

$f_{5}$	$\lnot y$	$\lnot y$

$f_{6}$	$x \ne y$	$x \ne y$

$f_{7}$	$\lnot x \lor \lnot y$	$\lnot (x \land y)$

$f_{8}$	$x \land y$	$x \land y$

$f_{9}$	$x = y$	$x = y$

$f_{10}$	$y$	$y$

$f_{11}$	$x \Rightarrow y$	$x \Rightarrow y$

$f_{12}$	$x$	$x$

$f_{13}$	$x \Leftarrow y$	$x \Leftarrow y$

$f_{14}$	$x \lor y$	$x \lor y$

$f_{15}$	$\text{true}$	$\text{true}$

Resources

cc: Cybernetics Communications (1) (2) • 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)

Animated Logical Graphs • 46

Posted on November 15, 2020 by Jon Awbrey

Re: Richard J. Lipton • The Art Of Math
Re: Animated Logical Graphs • (30) (45)

Another way of looking at Peirce duality is given by the following Table, which shows how logical graphs denote boolean functions under entitative and existential interpretations. Column 1 shows the logical graphs for the sixteen boolean functions on two variables. Column 2 shows the boolean functions denoted under the entitative interpretation and Column 3 shows the boolean functions denoted under the existential interpretation.

$\text{Logical Graphs} \stackrel{_\bullet}{} \text{Entitative and Existential Interpretations}$

Resources

cc: Cybernetics Communications (1) (2) • 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)

Animated Logical Graphs • 45

Posted on November 12, 2020 by Jon Awbrey

Re: Richard J. Lipton • The Art Of Math
Re: Animated Logical Graphs • (30)

There’s a nice interplay between geometric and logical dualities in C.S. Peirce’s graphical systems of logic, rooted in his discovery of the amphecks $\textsc{nand}$ and $\textsc{nnor}$ and flowering in his logical graphs for propositional and predicate calculus. Peirce’s logical graphs bear the dual interpretations he dubbed entitative and existential graphs.

Here’s a Table of Boolean Functions on Two Variables, using an extension of Peirce’s graphs from trees to cacti, illustrating the duality so far as it affects propositional calculus.

$\text{Boolean Functions on Two Variables}$

Resources

cc: Cybernetics Communications (1) (2) • 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)

Problems In Philosophy • 12

Posted on November 10, 2020 by Jon Awbrey

Re: R.J. Lipton and K.W. Regan • The Night Of The Ethical Algorithm
Re: K.W. Regan • The Election Night Time Warp
Re: Ontolog Forum • John Sowa

JFS:: C.S. Peirce made a very clear and sharp distinction between formal or mathematical logic and logic as semiotic.
$\cdots$
Short summary: When Peirce uses the word ‘logic’ by itself, it’s important to check the context to see whether he’s talking about formal logic or logic as semiotic.

Dear John,

The first post of this series was prompted by a post 4 years ago on the Gödel’s Lost Letter and P=NP blog which jumped from the frying pan of problems in programming to the fire of problems in philosophy. Then last week two more posts, linked above, made the leap to two of the most flagrant problems in politics, namely, (1) the passage from effective and efficient algorithms to ethical algorithms and (2) the perils of navigating turbulent seas in a ship of state guided by elective representation, where the people pick their pilots from among themselves to represent their collective will and whatever wits they can muster.

Bearing all that in mind, I would like to keep exploring the ancient issues of aesthetics, ethics, and logic from our contemporary algorithmic perspective. There the descriptive and normative orientations to knowledge parallel the systems-theoretic dimensions of information and control. And there we find normative sciences appearing under the banner of “design sciences”. In that frame the art of crafting a ship of state becomes a question of optimal design for a human society.

When it comes to logic, then, a generic conception will do for now, leaving Peirce’s definition of logic as formal semiotic and fine points of the difference between mathematical logic and mathematics of logic to another day.

Resources

Inquiry
Descriptive Science
Normative Science
Theory and Therapy of Representations
Prospects for Inquiry Driven Systems • Logic, Ethics, Aesthetics
Scott, David K., and Awbrey, Susan M. (1993), “Transforming Scholarship”, Change : The Magazine of Higher Learning, 25(4), 38–43. Online (1) (2) (3).

cc: Cybernetics • Ontolog Forum • Peirce List • Structural Modeling • Systems Science

Posted in Aesthetics, Algorithms, Animata, Automata, Beauty, C.S. Peirce, Ethics, Inquiry, Justice, Logic, Model Theory, Normative Science, Peirce, Philosophy, Pragmatism, Problem Solving, Proof Theory, Summum Bonum, Truth, Virtue | Tagged Aesthetics, Algorithms, Animata, Automata, Beauty, C.S. Peirce, Ethics, Inquiry, Justice, Logic, Model Theory, Normative Science, Peirce, Philosophy, Pragmatism, Problem Solving, Proof Theory, Summum Bonum, Truth, Virtue | 1 Comment

Problems In Philosophy • 11

Posted on November 9, 2020 by Jon Awbrey

Re: Problems In Philosophy 9 • Richard Saunders

RS:: BTW I’m not sure I really see a distinction between descriptive and normative (prescriptive?) science except in the set of aims, goals, etc. that are entertained. It might be useful to try to characterize some distinctions in the goals of each.

Re: Problems In Philosophy 10 • Richard Saunders

RS:: Jon, the philosophy of science is all about the aims of science $\textsc{and}$ good ways of achieving them. I’m still not seeing a clear distinction, traditions notwithstanding, between descriptive and normative science. I do see the recursive entanglement though, and I’m still wondering if we can find common axioms that underlie both.

$\textsc{Saturday, November 7}$

Dear Richard,

Sue and I will be downing some bubbly and sleeping it off till the dawn’s early light, but Sue was into this Policy-Theory Reunion stuff well before I clued into it, so here’s one of her earlier papers you might find of interest in the interim.

Scott, David K., and Awbrey, Susan M. (1993), “Transforming Scholarship”, Change : The Magazine of Higher Learning, 25(4), 38–43. Online (1) (2) (3).

$\textsc{Monday, November 9}$

I am still trying to unscramble my brains after the week’s events but I’m surprised to see so much difficulty over the difference between descriptive sciences, the special sciences as Peirce called them, and normative sciences like aesthetics, ethics, and logic. I deferred to common idiom and conventional wisdom regarding the irreducibility of “Ought” to “Is” but roughly the same dimension and tension is recognized under a legion of names — policy vs. theory, procedural vs. declarative, deontic vs. ontic, and many others.

A pragmatic semiotician’s ears will naturally perk up at reading the word irreducibility above and lead to wondering whether the irreducibility of normative to descriptive has anything to do with the irreducibility of triadic relations to dyadic relations.

To my way of thinking, yes, it does.

Resources

cc: Cybernetics • Ontolog Forum • Peirce List • Structural Modeling • Systems Science

Problems In Philosophy • 10

Posted on November 7, 2020 by Jon Awbrey

Re: Ontolog Forum • David Whitten

DW:: Why does classical tradition or any tradition consider logic to be a normative science?

Dear David,

A science is called that because it deals in knowledge (Latin scientia). Knowing what is the case in a given domain of experience may be distinguished from knowing what ought to be in a given set of circumstances, and people who think in threes, like Kant and Peirce and me, add knowing what may be hoped to the mix.

In the quest to understand how science works a praxis/pragmatist like myself gives the process, inquiry, equal billing with the product, knowledge. People have gotten used to seeing sciences as bodies of ostensible knowledge (BOOKs) and taking their analysis as a matter of assigning them distinctive catalogue numbers and sorting them to the indicated library shelves. That is all well and good but it leaves an all too static impression of science if we settle for that.

Here are capsule summaries on the Sciences of Is and the Sciences of Ought from the Wikiversity articles on Descriptive Science and Normative Science.

Descriptive Science: A descriptive science, or a special science, is a form of inquiry, typically involving a community of inquiry and its accumulated body of provisional knowledge, which seeks to discover what is true about a recognized domain of phenomena.
Normative Science: A normative science is a form of inquiry, typically involving a community of inquiry and its accumulated body of provisional knowledge, which seeks to discover good ways of achieving recognized aims, ends, goals, objectives, or purposes.; The three normative sciences, according to traditional conceptions in philosophy, are aesthetics, ethics, and logic.

Resources

cc: Cybernetics • Ontolog Forum • Peirce List • Structural Modeling • Systems Science

Problems In Philosophy • 9

Posted on November 5, 2020 by Jon Awbrey

Re: FB | Ecology Of Systems Thinking • Richard Saunders

RS:: Hume’s is/ought dichotomy: are these as Gould said “non-overlapping magisteria” or are they concentric domains? Is a science of aesthetics at the core? If memory serves it seems like that was what Wittgenstein suggested at the end of Tractatus. In The Moral Landscape, Harris narrows the aesthetic focus to a distinction between the minimum and maximum suffering of all sentient beings. Maximum suffering is bad or ugly and minimum suffering is good or beautiful. The relationship of conduct to result is the subject of consequentialism, isn’t it? Isn’t that also the subject of science?

I know a lot of people see a cut and dried dichotomy here and conventional wit says you can’t derive Ought from Is. My tracings of the boundaries though tend to find them recursively entangled.

RS:: Recursively entangled is a nice phrase, like the the chicken and the egg. But I’m still wondering about the catch-22. On what general axiom is aesthetics/ethics/logic based? Harris suggests it’s minimizing net suffering. (That doesn’t imply the elimination of suffering, because some suffering has a net positive result.)

I got no absolutes here. I have my personal aesthetic, but a personal aesthetic is the moral equivalent of a religion, and folks are pretty free about that.

I’ll have more to say about my personal aesthetic … all in good time.

Resources

Prospects for Inquiry Driven Systems • Logic, Ethics, Aesthetics
Wikiversity • Descriptive Science • Normative Science

cc: Cybernetics • Ontolog Forum • Peirce List • Structural Modeling • Systems Science

Problems In Philosophy • 8

Posted on November 4, 2020 by Jon Awbrey

Re: Ontolog Forum • David Whitten

Dear David,

I’ll extend this post tomorrow, apocalypse permitting, but while I wait for the election returns I’ll post just a pair of links to the Wikiversity articles on Descriptive Science and Normative Science, forked over from the Wikipedia articles as I left them a decade and a half ago. I have no idea what, if anything exists on Wikipedia itself these days but this much gives the basic ideas in a couple of nutshells.

Resource

Prospects for Inquiry Driven Systems • Logic, Ethics, Aesthetics

cc: Cybernetics • Ontolog Forum • Peirce List • Structural Modeling • Systems Science

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