Inquiry Into Inquiry

Animated Logical Graphs • 2

Posted on January 14, 2015 by Jon Awbrey

It’s almost 50 years now since I first encountered the volumes of Peirce’s Collected Papers in the math library at Michigan State, and shortly afterwards a friend called my attention to the entry for Spencer Brown’s Laws of Form in the Whole Earth Catalog and I sent off for it right away. I would spend the next decade just beginning to figure out what either one of them was talking about in the matter of logical graphs and I would spend another decade after that developing a program, first in Lisp and then in Pascal, that turned graph-theoretic data structures formed on their ideas to good purpose as the basis of its reasoning engine. I thought it might contribute to a number of long-running and ongoing discussions if I could articulate what I think I learned from that experience.

So I’ll try to keep focused on that.

Posted in Abstraction, Amphecks, Analogy, Animata, Automated Research Tools, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Deduction, Diagrammatic Reasoning, Duality, Form, Graph Theory, Iconicity, Laws of Form, Leibniz, Logic, Logical Graphs, Mathematics, Model Theory, Peirce, Peirce's Law, Praeclarum Theorema, Pragmatism, Proof Theory, Propositional Calculus, Semiotics, Spencer Brown, Theorem Proving, Visualization | Tagged Abstraction, Amphecks, Analogy, Animata, Automated Research Tools, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Deduction, Diagrammatic Reasoning, Duality, Form, Graph Theory, Iconicity, Laws of Form, Leibniz, Logic, Logical Graphs, Mathematics, Model Theory, Peirce, Peirce's Law, Praeclarum Theorema, Pragmatism, Proof Theory, Propositional Calculus, Semiotics, Spencer Brown, Theorem Proving, Visualization | 12 Comments

Animated Logical Graphs • 1

Posted on January 8, 2015 by Jon Awbrey

For Your Musement …

Here are some animations I made up to illustrate several different styles of proof in an extended topological variant of Peirce’s Alpha Graphs for propositional logic.

☞ Proof Animations

See the following article for a full discussion of this type of logical graph.

☞ Logical Graphs

Additional Resources

Looking Back On 2014

Posted on December 31, 2014 by Jon Awbrey

The WordPress.com stats helper monkeys prepared a 2014 annual report for this blog.

Here's an excerpt:

The concert hall at the Sydney Opera House holds 2,700 people. This blog was viewed about 25,000 times in 2014. If it were a concert at Sydney Opera House, it would take about 9 sold-out performances for that many people to see it.

Click here to see the complete report.

Posted in Annual Report, Year In Review | Tagged Annual Report, Year In Review | Leave a comment

☝ What Ariadne Said To Theseus ☟

Posted on December 9, 2014 by Jon Awbrey

☞ ❝You have to understand, the Minotaur is not clueless —
it just has a different goal than getting out of the maze.❞ ☜

Posted in Mantra, Maxim, Maze, Myth | Tagged Mantra, Maxim, Maze, Myth | Leave a comment

Frankl, My Dear • 12

Posted on November 24, 2014 by Jon Awbrey

It is one of the rules of my system of general harmony, that the present is big with the future, and that he who sees all sees in that which is that which shall be.

Leibniz • Theodicy

Re: Dick Lipton & Ken Regan • (1) • (2)

We continue with the differential analysis of the proposition in Example 1.

Example 1

(1)

Like any moderately complex proposition, the difference map of a proposition has many equivalent logical expressions and can be read in many different ways.

(5)

The expansion of $\mathrm{D}f$ computed in Post 9 and further discussed in Post 10 is shown again below with the terms arranged by number of positive differential features, from lowest to highest.

$\begin{array}{*{4}{l}} \multicolumn{4}{l}{\mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =} \\[10pt] & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & \texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} \cdot \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} \\[10pt] + & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & + & \texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} \cdot \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} \\[10pt] + & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} & + & \texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} \cdot \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} \end{array}$

$\begin{array}{*{4}{l}} + & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & + & \texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} \cdot \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} \\[10pt] + & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} & + & \texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} \cdot \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} \\[10pt] + & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} & + & \texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} \cdot \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \\[10pt] + & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} & + & \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \cdot \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \end{array}$

The terms of the difference map $\mathrm{D}f$ may be obtained from the table below by multiplying the base factor at the head of each column by the differential factor that appears beneath it in the body of the table.

The full boolean expansion of $\mathrm{D}f$ may be condensed to a degree by collecting terms that share the same base factors, as shown in the following display:

$\begin{array}{*{4}{c}} \multicolumn{4}{l}{\mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =} \\[10pt] & \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} & \cdot & \texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{))} \\[4pt] + & \texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} & \cdot & \texttt{~~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)~} \\[4pt] + & \texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} & \cdot & \texttt{~(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)~} \\[4pt] + & \texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} & \cdot & \texttt{~(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~~} \\[4pt] + & \texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} & \cdot & \texttt{~~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)~} \\[4pt] + & \texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} & \cdot & \texttt{~~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~~} \\[4pt] + & \texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} & \cdot & \texttt{~(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~~} \\[4pt] + & \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} & \cdot & \texttt{~~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~~} \end{array}$

This amounts to summing terms along columns of the previous table, as shown at the bottom margin of the next table:

Collecting terms with the same differential factors produces the following expression:

$\begin{array}{*{4}{c}} \multicolumn{4}{l}{\mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =} \\[10pt] & \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & \cdot & q \texttt{~} r \\[4pt] + & \texttt{~} \mathrm{d}q \texttt{~(} \mathrm{d}p \texttt{)(} \mathrm{d}r \texttt{)} & \cdot & p \texttt{~} r \\[4pt] + & \texttt{~} \mathrm{d}r \texttt{~(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} & \cdot & p \texttt{~} q \\[4pt] + & \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & \cdot & \texttt{((} p \texttt{,} q \texttt{))} ~ r \\[4pt] + & \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}r \texttt{~(} \mathrm{d}q \texttt{)} & \cdot & \texttt{((} p \texttt{,} r \texttt{))} ~ q \\[4pt] + & \texttt{~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~(} \mathrm{d}p \texttt{)} & \cdot & \texttt{((} q \texttt{,} r \texttt{))} ~ p \\[4pt] + & \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} & \cdot & p \texttt{~} q \texttt{~} r \\[4pt] + & \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} & \cdot & \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \end{array}$

This is roughly what one would get by summing along rows of the previous tables.

To be continued …

Resources

Posted in Boolean Algebra, Boolean Functions, Computational Complexity, Differential Logic, Frankl Conjecture, Logic, Logical Graphs, Mathematics, Péter Frankl | Tagged Boolean Algebra, Boolean Functions, Computational Complexity, Differential Logic, Frankl Conjecture, Logic, Logical Graphs, Mathematics, Péter Frankl | 10 Comments

❦ Pyramus & Thisbe ❦

Posted on November 12, 2014 by Jon Awbrey

It’s all about love
And the knots thereof
I have known beauty
I’ll bring it to you

Jon Awbrey • 12 Nov 2014

Posted in Anthem, Mantra, Verse | Tagged Anthem, Mantra, Verse | Leave a comment

Frankl, My Dear • 11

Posted on November 10, 2014 by Jon Awbrey

Re: Dick Lipton & Ken Regan • (1) • (2)

Let’s take a moment from the differential analysis of the proposition in Example 1 to form a handy compendium of the results obtained so far.

Example 1

(1)

Enlargement Map $\mathrm{E}(pqr)$ of the Conjunction $pqr$

(3)

Tacit Extension $\boldsymbol\varepsilon(pqr)$ of the Conjunction $pqr$

(4)

Difference Map $\mathrm{D}(pqr)$ of the Conjunction $pqr$

(5)

To be continued …

Resources

Continuity, Generality, Infinity, Law, Synechism • 1

Posted on November 9, 2014 by Jon Awbrey

The concept of continuity Peirce highlights in his synechism is a logical principle somewhat more general than the concepts of either mathematical or physical continua.

Peirce’s concept of continuity is better understood as a concept of lawful regularity or parametric variation. As such, it is basic to the coherence and utility of science, whether classical, relativistic, quantum mechanical, or any conceivable future science deserving the name. (As Aristotle already knew.)

Perhaps the most pervasive examples of this brand of continuity in physics are the “correspondence principles” describing the relationship between classical and contemporary paradigms.

The importance of lawful regularities and parametric variations is not diminished one bit in passing from continuous mathematics to discrete mathematics, nor from theory to application.

It would be good to keep in mind two further points of information, the missing of which seems to lie at the root of many recent disputes on the Peirce List.

It is necessary to distinguish the mathematical concepts of continuity and infinity from the question of their physical realization. The mathematical concepts retain their practical utility for modeling empirical phenomena quite independently of the (meta-)physical question of whether those continua and cardinalities are literally realized in the physical universe. That is equally true of any other domain or level of phenomena — chemical, biological, mental, social, or whatever.
As far as the mathematical concept goes, continuity is relative to topology. Thus what counts as a continuous function or transformation between spaces is relative to the topology under which those spaces are considered and the same spaces may be considered under many different topologies. What topology makes the most sense in a given application is an abductive question.

Posted in Abduction, Aristotle, C.S. Peirce, Cardinality, Constraint, Continua, Continuity, Discreteness, Discretion, Epistemology, Generality, Infinity, Knowledge, Logic, Logic of Science, Mathematical Models, Mathematics, Natural Law, Physics, Quanta, Quantum Mechanics, Synechism, Topology | Tagged Abduction, Aristotle, C.S. Peirce, Cardinality, Constraint, Continua, Continuity, Discreteness, Discretion, Epistemology, Generality, Infinity, Knowledge, Logic, Logic of Science, Mathematical Models, Mathematics, Natural Law, Physics, Quanta, Quantum Mechanics, Synechism, Topology | 10 Comments

Frankl, My Dear • 10

Posted on November 4, 2014 by Jon Awbrey

Re: Dick Lipton & Ken Regan • (1) • (2)

(5)

Figure 5 shows the 14 terms of the difference map $\mathrm{D}f$ as arcs, arrows, or directed edges in the venn diagram of the original proposition $f(p, q, r) = pqr.$ The arcs of $\mathrm{E}f$ are directed into the cell where $f$ is true from each of the other cells. The arcs of $\boldsymbol\varepsilon f$ are directed from the cell where $f$ is true into each of the other cells.

The expansion of $\mathrm{D}f$ computed in the previous post is shown again below with the terms arranged by number of positive differential features, from lowest to highest.

To be continued …

Resources

Frankl, My Dear • 9

Posted on November 3, 2014 by Jon Awbrey

“It doesn’t matter what one does,” the Man Without Qualities said to himself, shrugging his shoulders. “In a tangle of forces like this it doesn’t make a scrap of difference.” He turned away like a man who has learned renunciation, almost indeed like a sick man who shrinks from any intensity of contact. And then, striding through his adjacent dressing-room, he passed a punching-ball that hung there; he gave it a blow far swifter and harder than is usual in moods of resignation or states of weakness.

Robert Musil • The Man Without Qualities

Re: Dick Lipton & Ken Regan • (1) • (2)

We continue with the differential analysis of the proposition in Example 1.

Example 1

(1)

The difference operator $\mathrm{D}$ is defined as the difference $\mathrm{E} - \boldsymbol\varepsilon$ between the enlargement operator $\mathrm{E}$ and the tacit extension operator $\boldsymbol\varepsilon.$

The difference map $\mathrm{D}f$ is the result of applying the difference operator $\mathrm{D}$ to the function $f.$ When the sense is clear, we may refer to $\mathrm{D}f$ simply as the difference of $f.$

In boolean spaces there is no difference between the sum $(+)$ and the difference $(-)$ so the difference operator $\mathrm{D}$ is equally well expressed as the exclusive disjunction or symmetric difference $\mathrm{E} + \boldsymbol\varepsilon.$ In this case the difference map $\mathrm{D}f$ can be computed according to the formula $\mathrm{D}f = (\mathrm{E} + \boldsymbol\varepsilon)f = \mathrm{E}f + \boldsymbol\varepsilon f.$

The action of $\mathrm{D}$ on our present example, $f(p, q, r) = pqr,$ can be computed from the data on hand according to the following prescription.

The enlargement map $\mathrm{E}f,$ computed in Post 5 and graphed in Post 6, is shown again here:

$\mathrm{E}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =$

$\begin{smallmatrix} & p q r \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & p q \texttt{(} r \texttt{)} \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r & + & p \texttt{(} q \texttt{)} r \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} & + & p \texttt{(} q \texttt{)(} r \texttt{)} \,\cdot\, \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \, \mathrm{d}r \\[4pt] + & \texttt{(} p \texttt{)} q r \,\cdot\, \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & \texttt{(} p \texttt{)} q \texttt{(} r \texttt{)} \,\cdot\, \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r & + & \texttt{(} p \texttt{)(} q \texttt{)} r \,\cdot\, \mathrm{d}p \, \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)} & + & \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \,\cdot\, \mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r \end{smallmatrix}$

The tacit extension $\boldsymbol\varepsilon f,$ computed in Post 7 and graphed in Post 8, is shown again here:

$\boldsymbol\varepsilon f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =$

$\begin{array}{*{8}{l}} & p q r ~ \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \\[4pt] + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)} & + & p q r ~ \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~} \end{array}$

The difference map $\mathrm{D}f$ is the sum of the enlargement map $\mathrm{E}f$ and the tacit extension $\boldsymbol\varepsilon f.$

Here we adopt a paradigm of computation for $\mathrm{D}f$ that aids not only in organizing the stages of the work but also in highlighting the diverse facets of logical meaning that may be read off the result.

The terms of the enlargement map $\mathrm{E}f$ are obtained from the table below by multiplying the base factor at the head of each column by the differential factor that appears beneath it in the body of the table.

The terms of the tacit extension $\boldsymbol\varepsilon f$ are obtained from the next table below by multiplying the base factor at the head of the first column by each of the differential factors that appear beneath it in the body of the table.

Finally, the terms of the difference map $\mathrm{D}f$ are obtained by overlaying the displays for $\mathrm{E}f$ and $\boldsymbol\varepsilon f$ and taking their boolean sum entry by entry.

Notice that the “loop” or “no change” term $p q r \cdot \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}$ cancels out, leaving 14 terms in the end.

To be continued …

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Enlargement Map $\mathrm{E}(pqr)$ of the Conjunction $pqr$

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Difference Map $\mathrm{D}(pqr)$ of the Conjunction $pqr$