Animated Logical Graphs : 3

Re: Peirce List DiscussionHelmut Raulien

I have a little more leisure now to begin the process of 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 or need be maintained.

But that is a question for the future …

Posted in Amphecks, Analogy, Animata, Automated Research Tools, Boolean Algebra, Boolean Functions, 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 , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Pragmatism About Theoretical Entities : 1

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 Essentialism, Logic, Metaphysics, Method, Nominalism, Ockham, Ockham's Razor, Peirce, Philosophy, Philosophy of Science, Pragmatic Maxim, Pragmatism | Tagged , , , , , , , , , , , | Leave a comment

Animated Logical Graphs : 2

Re: Peirce List DiscussionJim 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 Amphecks, Analogy, Animata, Automated Research Tools, Boolean Algebra, Boolean Functions, 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 , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Animated Logical Graphs : 1

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 Amphecks, Analogy, Animata, Automated Research Tools, Boolean Algebra, Boolean Functions, 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 , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Looking Back On 2014

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 ☟

☞ 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

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 , , , , , , , , | Leave a comment