Animated Logical Graphs • 3

Posted on January 26, 2015 by Jon Awbrey

Re: Peirce List • Helmut Raulien

I have a little more leisure now to start climbing back into the saddle, so let me see where we left off …

Try looking into the article I linked before:

Or my first couple of blog posts on Logical Graphs:

There are literally decades of thought and work that went into those, and if they do not engage the reader in the excitement of possible future developments then I would sorely appreciate any feedback on where and why they fail to do so.

George Spencer Brown is one of the few writers I’ve run across in the time since my first encounter with Peirce’s logical graphs who truly grasped the full depth of Peirce’s insight into logic, a vision that pierced the veil of logical interpretations, entitative and existential, to the deep formal unity between them.  That is one of the reasons I’ve made an effort to treat the two interpretations in parallel as far as I was able.  It is an extremely enticing research question to me whether that symmetry is necessarily broken as we pass from propositional to quantificational logic, or whether there is some way it may be or need be maintained.

But that is a question for the future …

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 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | 12 Comments

Pragmatism About Theoretical Entities • 1

Posted on January 20, 2015 by Jon Awbrey

By theoretical entities I mean things like classes, properties, qualities, sets, situations, or states of affairs, in general, the putative denotations of theoretical concepts, formulas, sentences, terms, or treatises, in brief, the ostensible objects of signs.

A conventional statement of Ockham’s Razor is —

  • Entities shall not be multiplied beyond necessity.

That is still good advice, as practical maxims go, but a pragmatist will read that as practical necessity or utility, qualifying the things that we need to posit in order to think at all, without getting lost in endless circumlocutions of perfectly good notions.

Nominalistic revolts are well-intentioned when they naturally arise, seeking to clear away the clutter of ostentatious entities ostensibly denoted by signs that do not denote.

But that is no different in its basic intention than what Peirce sought to do, clarifying metaphysics though the application of the Pragmatic Maxim.

Taking the long view, then, pragmatism can be seen as a moderate continuation of Ockham’s revolt, substituting a principled revolution for what tends to descend to a reign of terror.

Posted in Abstraction, C.S. Peirce, Essentialism, Hypostatic Abstraction, Logic, Mathematics, Metaphysics, Method, Nominalism, Ockham, Ockham's Razor, Peirce, Pragmatic Maxim, Pragmatism, Realism, Semiotics, Theory | Tagged , , , , , , , , , , , , , , , , | Leave a comment

Animated Logical Graphs • 2

Posted on January 14, 2015 by Jon Awbrey

Re: Peirce List • Jim Willgoose

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 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | 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

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 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | 12 Comments

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 , | 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 , , , | 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


Venn Diagram PQR		 (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.


Venn Diagram Frankl Figure 5		 (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.

Table 4.0 PQR Difference Map

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:

Table 4.0 PQR Difference Map Col Sum

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 , , , , , , , , | 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 , , | 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


Venn Diagram PQR		 (1)

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


Venn Diagram Frankl Figure 3		 (3)

Table 4.0 PQR Enlargement Map

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


Venn Diagram Frankl Figure 4		 (4)

Table 4.0 PQR Tacit Extension

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


Venn Diagram Frankl Figure 5		 (5)

Table 4.0 PQR Difference Map

To be continued …

Resources

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

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 , , , , , , , , , , , , , , , , , , , , , , | 10 Comments