Re: C.S. Peirce, Spencer Brown, & Me

[Jon Awbrey]

Cf: Charles Sanders Peirce, George Spencer Brown, and Me • 11
http://inquiryintoinquiry.com/2021/01/18/charles-sanders-peirce-george-spencer-brown-and-me-11/

All,

There's a new “Laws of Form” group in town. James Bowery et al. have
just revived the earlier group on a new platform and everything looks
pretty handy so far. There's an honest-to-goodness 60s vibe about it
for me since it's well-known to those in the know how Spencer Brown's
work builds on Peirce's, not just because his calculus of indications
resurrects aspects of Peirce's alpha level logical graphs but because
the broader scope of his interests touched on inductive reasoning and
the whole welter of knotty tangles in the pragmatics of communication,
computation, concept formation which Ashby, Arbib, Bateson, Korzybski,
R.D. Laing, McCulloch, Peirce, Polanyi, Watzlawick, and others probed
in the matrix of quasi-paradoxes and games people play with symbols.

All of which inspires me to revise and extend the series of posts
I shared with the old group a few years back, fixing in passing
the large number of now broken links.

Regards,

Jon

inquiry into inquiry: https://inquiryintoinquiry.com/
academia: https://independent.academia.edu/JonAwbrey
oeiswiki: https://www.oeis.org/wiki/User:Jon_Awbrey
facebook page: https://www.facebook.com/JonnyCache

[Jon Awbrey]

Cf: Charles Sanders Peirce, George Spencer Brown, and Me • 1

https://inquiryintoinquiry.com/2017/07/20/charles-sanders-peirce-george-spencer-brown-and-me-1/

All,

Here's a piece of CSP-GSB related biography I shared with the
previous Laws of Form group when I signed on a few years ago.

⁂

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,
converting graph-theoretic data structures formed on their ideas to
good purpose in the mechanics of its propositional reasoning engine.
I thought it might contribute to a number of ongoing discussions if
I could articulate what I think I learned from that experience.

Regards,

Jon

[Jon Awbrey]

Cf: Charles Sanders Peirce, George Spencer Brown, and Me • 2
https://inquiryintoinquiry.com/2017/07/21/charles-sanders-peirce-george-spencer-brown-and-me-2/

All,

Here's a lightly revised, link-repaired edition of a previous post.

⁂

I’m making an effort to present this material in a more gradual and
logical order than I’ve ever managed to do before. There are issues
about the relationship between episodic and semantic memory that are
giving me trouble as I try to remember how I came to look at things
the way I do … but never mind that now. I’ll eventually get around to
explaining the forces that drove me to generalize the forms of logical
graphs from trees to cacti, as graph theorists call them, and how that
made the transition to differential logic so much easier than it would
have been otherwise, but I think it would be better now to begin at
the beginning with the common core of forms introduced by CSP and GSB.

Here’s a couple of articles I wrote for that purpose:

Logical Graphs
https://oeis.org/wiki/Logical_Graphs

Propositional Equation Reasoning Systems
https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems

There are versions of those articles at several other places on the web
which may be better formatted or more convenient for discussion:

Logical Graphs (Wikiversity)
https://en.wikiversity.org/wiki/Logical_graph

Logical Graphs (Inquiry Blog)
1. https://inquiryintoinquiry.com/2008/07/29/logical-graphs-1/
2. https://inquiryintoinquiry.com/2008/09/19/logical-graphs-2/

One big issue arising at the beginning is the question of “duality”.
Both C.S. Peirce and Spencer Brown understood they were dealing with
a “very abstract calculus”, one which could be interpreted for the
purposes of ordinary propositional logic in two different ways.

Peirce called the two different ways of interpreting the abstract graphs
his “entitative” and “existential” graphs. He started out with a system
of graphs he opted to interpret in the entitative manner but switched over
to the existential choice as he developed his logical graphs beyond the
purely propositional level.

Spencer Brown elected to emphasize the entitative reading in his
main exposition but he was very clear in the terminology he used
that the forms and transformations themselves are independent of
their interpretations.

Table 1 at either of the locations linked below has columns for
the graph-theoretic forms and the parenthesis-string forms of
several basic expressions, reading them under the existential
interpretation.

Table 1. Syntax and Semantics of a Calculus for Propositional Logic

a.
https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Appendices#Table_1._Syntax_and_Semantics_of_a_Calculus_for_Propositional_Logic

b. https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Cactus_Language_for_Propositional_Logic

The Tables linked below serve to compare the existential and entitative
interpretations of logical graphs by providing translations into familiar
notations and English paraphrases for a few of the most basic and commonly
occurring forms.

Table A. Existential Interpretation
https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Appendices#Table_A._Existential_Interpretation

Table B. Entitative Interpretation
https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Appendices#Table_B._Entitative_Interpretation

Table C. Dualing Interpretations
https://oeis.org/wiki/Theme_One_Program_%E2%80%A2_Appendices#Table_C._Dual_Interpretations

Regards,

Jon

[Jon Awbrey]

Cf: Charles Sanders Peirce, George Spencer Brown, and Me • 3
https://inquiryintoinquiry.com/2017/07/31/charles-sanders-peirce-george-spencer-brown-and-me-3/

All,

Here's the next installment in my bio-graphical intro,
with all the links repaired and the graphics upgraded.

Re: Laws of Form
https://groups.io/g/lawsofform/topic/c_s_peirce_spencer_brown/79916661

There are a number of “difficulties at the beginning” that arise here.
I’ve been trying to get to the point where I can respond to James Bowery’s
initial comments

https://inquiryintoinquiry.com/2016/11/30/time-topology-differential-logic-6/#comment-33608

and also to questions about the relation between Spencer Brown’s
imaginary logical values and the development of differential logic.

The larger issue I see at this point has to do with the relationship
between the “algebra” and the “arithmetic” of logical graphs. Peirce
came right up to the threshold of discovering that relationship several
times in his later work on existential graphs but never quite pushed it
through to full realization. It was left to Spencer Brown to bring it
to light.

The relationship between Primary Arithmetic and Primary Algebra
is discussed in the following article.

Logical Graphs ( https://oeis.org/wiki/Logical_Graphs )

• Primary Arithmetic as Semiotic System
https://oeis.org/wiki/Logical_Graphs#Primary_arithmetic_as_semiotic_system

• Primary Algebra as Pattern Calculus
https://oeis.org/wiki/Logical_Graphs#Primary_algebra_as_pattern_calculus

The other issue has to do with my using a different J₁ than Spencer Brown.
I believe I even called it J₁′ in the early days but eventually lost the
prime as time went by. As far as I can remember, it initially had to do
with negotiating between the systems of C.S. Peirce and Spencer Brown but
I think I stuck with the variant because it sorts the types of change —
modifying structure and moving variables — into different bins.

Image Files
===========

This Blog
I₁ : https://inquiryintoinquiry.files.wordpress.com/2020/09/axiom-i1.jpg
I₂ : https://inquiryintoinquiry.files.wordpress.com/2020/09/axiom-i2.jpg
J₁ : https://inquiryintoinquiry.files.wordpress.com/2020/09/axiom-j1.jpg
J₂ : https://inquiryintoinquiry.files.wordpress.com/2020/09/axiom-j2.jpg

Oeis Wiki
I₁ : https://oeis.org/wiki/File:Initial_I1.jpg
I₂ : https://oeis.org/wiki/File:Initial_I2.jpg
J₁ : https://oeis.org/wiki/File:Initial_J1.jpg
J₂ : https://oeis.org/wiki/File:Initial_J2.jpg

See also the discussions at the following locations.

Logical Graphs • Formal Development
https://inquiryintoinquiry.com/2008/09/19/logical-graphs-2/

Propositional Equation Reasoning Systems
https://oeis.org/wiki/Propositional_Equation_Reasoning_Systems

Regards,

Jon

[Jon Awbrey]

Cf: Charles Sanders Peirce, George Spencer Brown, and Me • 4
https://inquiryintoinquiry.com/2017/08/06/charles-sanders-peirce-george-spencer-brown-and-me-4/

All,

Two things that had a big impact on my studies of Peirce and Spencer Brown
over the years were my parallel studies in mathematics and computer science.
In the overlap between those areas came courses in logic, mathematical
linguistics, and the theory of formal languages, grammars, and automata.
My intellectual wanderings over a nine-year undergraduate career would
take me through a cycle of majors from math and physics, to communication,
psychology, philosophy, and a cross-cultural liberal arts program, then
back to grad school in mathematics. The puzzles Peirce and Spencer Brown
beset my brain with were a big part of what drove me back to math, since
I could see I had no chance of resolving them without learning a lot more
algebra, logic, and topology than I had learned till then.

Resources
=========

Prospects for Inquiry Driven Systems
https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems
https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems#Bibliography

Mathematical Notes
https://oeis.org/wiki/User:Jon_Awbrey/Mathematical_Notes

Regards,

Jon

[Jon Awbrey]

Cf: Charles Sanders Peirce, George Spencer Brown, and Me • 5
https://inquiryintoinquiry.com/2017/08/12/charles-sanders-peirce-george-spencer-brown-and-me-5/

All,

| Continuing the salvage and update of posts lost when Yahoo! Groups defuncted.
| That's just the way the internet cookie crumbles when capitalism incorpulent
| takes over the web that a science-minded ARPA once built at taxpayer expense.

Peirce's Law Proof Animation
https://inquiryintoinquiry.files.wordpress.com/2012/01/peirces-law-2-0-animation.gif

Here are blog and wiki versions of an article I wrote on Peirce’s Law,
an axiom or theorem (depending on your choice of logical basis) which
distinguishes classical from intuitionistic propositional calculus.
Aside from its pivotal logical status it affords a nice illustration
of several important features of logical graphs in the style of
Peirce and Spencer Brown.

Peirce’s Law
============
• Inquiry Blog ( https://inquiryintoinquiry.com/2008/10/06/peirces-law/ )
• OEIS Wiki ( https://oeis.org/wiki/Peirce%27s_law )
• MyWikiBiz ( http://mywikibiz.com/Peirce's_law )
• Wikiversity ( https://en.wikiversity.org/wiki/Peirce's_law )

Here's another resource on the relationship between
Classical and Intuitionistic Propositional Calculus:

Propositions As Types Analogy
• https://oeis.org/wiki/Propositions_As_Types_Analogy

Regards,

Jon

[Jon Awbrey]

Cf: Charles Sanders Peirce, George Spencer Brown, and Me • 6
https://inquiryintoinquiry.com/2017/08/18/charles-sanders-peirce-george-spencer-brown-and-me-6/

Cf: Logical Graphs
https://oeis.org/wiki/Logical_Graphs

All,

The formal system of logical graphs is defined by a foursome of formal equations,
called “initials” when regarded purely formally, in abstraction from potential
interpretations, and called “axioms” when interpreted as logical equivalences.
There are two “arithmetic initials” and two “algebraic initials”, as shown below.

Arithmetic Initials
===================

Figure I₁
https://inquiryintoinquiry.files.wordpress.com/2020/09/axiom-i1.png

Figure I₂
https://inquiryintoinquiry.files.wordpress.com/2020/09/axiom-i2.png

Algebraic Initials
==================

Figure J₁
https://inquiryintoinquiry.files.wordpress.com/2020/09/axiom-j1.png

Figure J₂
https://inquiryintoinquiry.files.wordpress.com/2020/09/axiom-j2.png

Spencer Brown uses a different formal equation for his first algebraic initial —
where I use “a (a) = ( )” he uses “(a (a)) = ”. For the moment, let’s refer to
my J₁ as J_1a and his J₁ as J_1b and use that notation to examine the relationship
between the two systems.

It is easy to see that the two systems are equivalent, since we have the following
proof of J_1b by way of J_1a and I₂.


a a
o---o
|
@

=======J1a {delete}

o---o
|
@

=======I2 {cancel}

@

=======QED J1b

In choosing between systems I am less concerned with small differences
in the lengths of proofs than I am with other factors. It is difficult
for me to remember all the reasons for decisions I made forty or fifty
years ago — as a general rule, Peirce’s way of looking at the relation
between mathematics and logic has long been a big influence on my thinking
and the other main impact is accountable to the nuts and bolts requirements
of computational representation.

But looking at the choice with present eyes, I think I continue to prefer
the {I₁, I₂, J_1a, J₂} system over the alternative simply for the fact it
treats two different types of operation separately, namely, changes in
graphical structure versus changes in the placement of variables.

Regards,

Jon

[Jon Awbrey]

Cf: Charles Sanders Peirce, George Spencer Brown, and Me • 12
http://inquiryintoinquiry.com/2021/01/25/charles-sanders-peirce-george-spencer-brown-and-me-12/

Re: Laws of Form
https://groups.io/g/lawsofform/topic/sociological_reading_of_lof/79753680
::: Dirk Baecker
https://groups.io/g/lawsofform/message/8

DB:
Thanks to James Bowery for inviting me to this group. Maybe it is not
exactly what I am looking for, since I am interested in a sociological
reading of LoF. Which means that I am searching for a mathematics more
akin to semantics than to physics. I am still not sure whether in this
respect LoF is a wonderful metaphor to understand basic features of an
oscillating communication or whether chapter 11 opens up possibilities
to compute semantic values starting with imaginary states.

Dear Dirk,

Thanks for this comment and all the links. I've been working
along the lines of Peirce's logical graphs and Spencer Brown's
calculus of indications in parallel since those heady early days
of the late 60s and Peirce's semiotics or theory of triadic sign
relations is very much a piece of how I understand both. In that
light I'll feel encouraged to share bits of that along with my work
on LoF — and now everyone knows who to blame! 😉

Regards,

Jon

[Jon Awbrey]

Cf: Charles Sanders Peirce, George Spencer Brown, and Me • 7
https://inquiryintoinquiry.com/2017/08/21/charles-sanders-peirce-george-spencer-brown-and-me-7/

All,

Here's an ice-breaker on the question of logical boundaries ...

A statement P that implies both Q and ¬Q is called a false statement,
and anyone can prove anything at all from a false statement, as we
all too frequently observe on the political front these days.

There is however a reasonable way of handling boundaries, for instance,
as illustrated by the circumference of a region in a venn diagram, and
that is by means of differential logic. I’ve been tortoising my way
toward the goal line of explaining all that, and it’s going a bit slow,
but there’s a gentle introduction at the other end of the link below,
if you wish to achilles ahead.

Differential Propositional Calculus • Part 1
https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Part_1

Regards,

Jon

[Jon Awbrey]

Cf: Charles Sanders Peirce, George Spencer Brown, and Me • 13
http://inquiryintoinquiry.com/2021/01/26/charles-sanders-peirce-george-spencer-brown-and-me-13/

Re: Laws of Form
https://groups.io/g/lawsofform/topic/sociological_reading_of_lof/79753680
::: Dirk Baecker

https://groups.io/g/lawsofform/message/10

DB:
I guess you know Fernando Zalamea's work on Peirce.
He thinks that all of GSB's important ideas are already
in Peirce's Existential Graphs.

I think he may be right, but then there is the issue of
elegance, beauty, and clarity, and here, GSB leads the field.

Dear Dirk,

As you may have gleaned from the bio-graphical narrative I've been
salvaging from the old LoF group, I started down this intertwining
Logical Graph / Laws of Form road sometime in the late 60s, all of
which took me pretty far along my own eigenvectors before I hit on
the work of Zalamea and others early on in this millennium.

In most of the things I've written in the past about the relative
contributions of Peirce and those who came after, Spencer Brown in
particular, I tended to give Peirce a lot of credit for anticipating
the developments others clarified or brought to fruition. More lately
I've observed just how bewildered the untutored reader can become when
faced with Peirce's writings on logical graphs and logic generally,
so I've been rethinking my apportionment of credit. At any rate,
I commented on what I thought was added by whom all through my
bio essay, and I will continue doing that as I go.

Regards,

Jon

[Jon Awbrey]

Cf: Charles Sanders Peirce, George Spencer Brown, and Me • 8
https://inquiryintoinquiry.com/2017/08/22/charles-sanders-peirce-george-spencer-brown-and-me-8/

Re: Boundary Logic

For me, the heart of the matter is “what is the purpose of logic and
what is the purpose of mathematics and what is their relationship?”

There are semiotic situations which appear to violate the
initial conditions of logic but there are ways of approaching
them without reducing our brains to jelly from the getgo.
Charles S. Peirce, following on Aristotle’s negotiation of
the boundary between logic and rhetoric, developed his theory
of triadic sign relations in large part to manage just these
sorts of situations.

I’m determined to keep my gnosis close to the grinstone for
the time being but here is a smattering of old notes which
give a hint as to Peirce’s way of approaching the question.

C.S. Peirce on “General” and “Vague”
https://oeis.org/wiki/User:Jon_Awbrey/EXCERPTS#Excerpt_6._Peirce_.28CP_5.448.29
https://cs.nyu.edu/pipermail/fom/2009-March/thread.html#13437
1. https://cs.nyu.edu/pipermail/fom/2009-March/013437.html
2. https://cs.nyu.edu/pipermail/fom/2009-March/013446.html
3. https://cs.nyu.edu/pipermail/fom/2009-March/013448.html

Regards,

Jon

[Jon Awbrey]

Cf: Charles Sanders Peirce, George Spencer Brown, and Me • 9
https://inquiryintoinquiry.com/2017/08/22/charles-sanders-peirce-george-spencer-brown-and-me-9/

Re: Boundary Logic

A wider field of investigation opens up at this point, spanning the diversity of interactions among languages we use,
and systems of signs in general, to the thoughts ever streaming through our heads, to the universes we talk and think
upon, from Plato’s Heaven to Gaia’s Green Earth to the Tumbling Galaxies Beyond.

The complexities in play when we consider a domain of Signs, a domain of Ideas, and a domain of Objects all wound up in
relationship to one another is what Peirce’s “semiotics” or theory of sign relations is all about. Viewing the
enterprise of logic within the broader frame of semiotics not only gives us more insight into its means and ends but
affords us more “elbow room” for carrying out its operations.

To make a long story short, we don’t have to “escape language” because we don’t live inside any language or system of
signs, even if we get so confused sometimes as to think we do. We live in that wider world of reality and only use
languages and other systems of signs to describe what little we can of it.

Regards,

Jon

[Jon Awbrey]

Cf: Charles Sanders Peirce, George Spencer Brown, and Me • 10
https://inquiryintoinquiry.com/2017/08/25/charles-sanders-peirce-george-spencer-brown-and-me-10/

All,

With any formal system it is easy to spend a long time
roughing out primitives and reviewing first principles
before getting on to practical applications, and logical
graphs are no different in that respect. But the promise
of clearer and more efficient methods for solving realistic
problems is what led me to the visual calculi of Peirce and
Spencer Brown in the first place, so my aim through all our
rehearsal of rudiments is to make a bridge to applications
a few steps closer to what the real world throws our way.

I’ve been thinking how to make the transition from basic ingredients
of logical graphs and laws of form to slightly more interesting examples,
still “toy worlds” as AI folk call them but suggestive to some degree of
what might be possible in the long run. I’ll spend a few days gathering
assorted examples I’ve worked up before and try presenting those.

Regards,

Jon

[Michael DeBellis]

Jon, This may be a dumb question because I've just started taking Peirce seriously in the last year or so and some of his more complex ideas still aren't completely clear to me but here goes: Has anyone come up with an OWL upper model (i.e., something like the upper models in Cyc and BFO) based on Peirce's work?  I've come to appreciate Peirce as a major figure in the history of logic, information theory, semiotics, etc. but I've never quite been able to map his ideas into a logical model in OWL. I'm not sure if this is because trying to do so isn't consistent with what Peirce is trying to do or just that I still haven't grasped his ideas completely. Or perhaps the subset of FOL that OWL supports isn't powerful enough to map to Peirce. At an initial reading it seems like there should be a good fit because (at least as I understand it) one of Peirce's core ideas of symbols (as opposed to icons or indexes) seems like a perfect fit to the triple model (Subject Predicate Object) that is the foundation (RDF/RDFS) for OWL. Would like to know your opinions on this. 

Michael

[Igor Toujilov]

Michael,

Just one note: RDF/RDFS is not a foundation for OWL. Historically RDF/RDFS was established before OWL. OWL standard defines a formal mapping from OWL to RDF/RDFS. This is an example of how good standards should take care about compatibility to previous technologies. And OWL adopted RDF/RDFS as one of its serialisation formats. However OWL is not dependent on RDF/RDFS. OWL has other serialisation formats. You might say: OWL was inspired by RDF/RDFS. But RDF/RDFS and OWL can live without each other.

Igor

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[Michael DeBellis]

Igor, thanks, good point. Just to make sure I understand: Is Turtle also ultimately RDF/RDFS or is it independent? And I'm assuming all the other options: OWL/XML, OWL Functional, Manchester OWL, OBO, LaTeX, Json-LD, are independent of RDF/RDFS, correct?

[Tom Tinsley]

Michael,

 

You said: “I've come to appreciate Peirce as a major figure in the history of logic, information theory, semiotics, etc.” I agree and think you might be interested in an application of the Peircean categories to IT.

 

As an enterprise architect, I believe what is needed is an enterprise capable platform implementing the Peircean Categories as:

• Possibility –Tbox layers of imported ontologies as the logical model.
• Actuality – Abox data in the form of patterns in a database for query & update.
• Necessity – Service Component Architecture for processes.

Peirce presented the dependencies of these categories to be processed by human intelligence. In IT, this is done by AI.

 

I am working with a prototype of this platform that executes services and provides visualization of each category in the form of data graphs at multiple levels of detail.

 

Although I have described some of this on my website, http://OtterServer.com. Any comments or recommendations are welcomed either here or on the website. The next step is under consideration: open source, publish, YouTube, incorporate, …?

 

Tom Tinsley

--

[Michael DeBellis]

Thanks Tom, I will check out your web site. 

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[Igor Toujilov]

Michael,

Thanks.

Turtle is not a framework -- it's just one of RDF serialisations.

OWL/XML, OWL Functional, and Manchester are all OWL serialisation syntaxes, which are not dependent logically on RDF/RDFS framework. They borrow some syntactical terms from RDF/RDFS just for convenience and unification purposes.

I am not familiar with OWL serialisation in OBO, so I couldn't tell.

LaTeX is not machine understandable because it's not de-serialisable. It's for publishing only -- one way transformation from OWL in memory to a printing image for humans.

Json-LD is a kind of RDF/RDFS-based format, which is a JSON alternative to RDF/XML.

On Fri, 29 Jan 2021 at 19:56, Michael DeBellis <mdebe...@gmail.com> wrote:

Igor, thanks, good point. Just to make sure I understand: Is Turtle also ultimately RDF/RDFS or is it independent? And I'm assuming all the other options: OWL/XML, OWL Functional, Manchester OWL, OBO, LaTeX, Json-LD, are independent of RDF/RDFS, correct?

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[Jon Awbrey]

Cf: Charles Sanders Peirce, George Spencer Brown, and Me • 14
https://inquiryintoinquiry.com/2021/01/31/charles-sanders-peirce-george-spencer-brown-and-me-14/

Re: C.S. Peirce, Spencer Brown, and Me • 11
https://inquiryintoinquiry.com/2021/01/18/charles-sanders-peirce-george-spencer-brown-and-me-11/

Re: Laws of Form
https://groups.io/g/lawsofform/topic/c_s_peirce_spencer_brown/79946926
::: Dirk Baecker
https://groups.io/g/lawsofform/message/24

<QUOTE DB:>

Watzlawick's request for a pragmatic calculus of communication up to now
was never appropriately answered. W. Barnett Pearce and Vernon E. Cronen
(Communication, Action, and Meaning : The Creation of Social Realities, 1980)
did important studies on this as did Anthony Wilden (System and Structure :
Essays in Communication and Exchange, 1972), but we still lack it.

</QUOTE>

Dear Dirk,

Watzlawick's request for a pragmatic calculus of communication recalls
McCulloch's earlier question whether the human capacity for insightful
learning and reasoning demands a grasp of trans-dyadic relations, or not.

<QUOTE McCulloch>

But the problem of insight, or intuition, or invention — call it what you will —
we do not understand, although many of us are having a go at it. […] Tarski
thinks that what we lack is a fertile calculus of relations of more than two relata.
I am inclined to agree with him, and if I were now the age I was in 1917, that is
the problem I would tackle.

⁂

That process of insight by which a child learns at least one logical particle,
“neither” or “not both”, when it is given only ostensively — and one must be
so learned — is still a little beyond us. It may perhaps have to wait for
a potent logic of triadic relations, but I now doubt it. (McCulloch, p. 15).

</QUOTE>

The way I see things today, my motto would be “Context Precedes Calculus”
if I had to sum it up as briefly as possible. In other words, the first
order of business is finding the right context for understanding the
phenomena and problems at hand. As far as the human capacity for
conversing with nature and our fellows goes, pragmatic thinkers
informed by Peirce would no doubt point to the context of triadic
sign relations and declare, “Eureka! This Must Be the Place.”

References
==========

• McCulloch, Warren S. (1961), “What Is a Number that a Man May Know It,
and a Man, that He May Know a Number?”, Ninth Alfred Korzybski Memorial
Lecture, General Semantics Bulletin, Numbers 26 and 27, pp. 7–18,
Institute of General Semantics, Lakeville, CT.
Reprinted in Embodiments of Mind, pp. 1–18. Online:
1. http://www.vordenker.de/ggphilosophy/mcculloch_what-is-a-number.pdf
2. http://www.generalsemantics.org/wp-content/uploads/2011/04/gsb-26-27-mcculloch.pdf

• McCulloch, Warren S. (1965), Embodiments of Mind, MIT Press, Cambridge, MA.

[Jon Awbrey]

Cf: Charles Sanders Peirce, George Spencer Brown, and Me • 15
http://inquiryintoinquiry.com/2021/02/05/charles-sanders-peirce-george-spencer-brown-and-me-15/

Re: C.S. Peirce, Spencer Brown, and Me • 11
https://inquiryintoinquiry.com/2021/01/18/charles-sanders-peirce-george-spencer-brown-and-me-11/

Re: Ontolog Forum
https://groups.google.com/g/ontolog-forum/c/8HfnSonF-rY
::: Michael DeBellis
https://groups.google.com/g/ontolog-forum/c/8HfnSonF-rY/m/h97uRlBEAgAJ

<QUOTE MDB:>

I've just started taking Peirce seriously in the last year or so and some of his more complex ideas still aren't
completely clear to me but here goes: Has anyone come up with an OWL upper model (i.e., something like the upper models
in Cyc and BFO) based on Peirce's work? I've come to appreciate Peirce as a major figure in the history of logic,
information theory, semiotics, etc. but I've never quite been able to map his ideas into a logical model in OWL. I'm
not sure if this is because trying to do so isn't consistent with what Peirce is trying to do or just that I still
haven't grasped his ideas completely. Or perhaps the subset of FOL that OWL supports isn't powerful enough to map to
Peirce. At an initial reading it seems like there should be a good fit because (at least as I understand it) one of
Peirce's core ideas of symbols (as opposed to icons or indexes) seems like a perfect fit to the triple model (Subject
Predicate Object) that is the foundation (RDF/RDFS) for OWL. Would like to know your opinions on this.

</QUOTE>

Dear Michael,

Google still reminds me I spent some time on the RDF-Logic List

( https://lists.w3.org/Archives/Public/www-rdf-logic/ )

back around the turn of the millennium (January 2001 )

(
https://www.w3.org/Search/Mail/Public/search?hdr-1-name=from&hdr-1-query=Jon+Awbrey&index-grp=Public_FULL&index-type=t&type-index=www-rdf-logic&resultsperpage=40&sortby=date-asc
).

I was especially intrigued by the prospect of using triples as
a fundamental data structure. Now the (subject, verb, object)
triples of RDF and the (object, sign, interpretant) triples of
Peirce's semiotics are ostensibly different data types in their
concrete descriptions but that may not obstruct integration too
much if the triples are defined abstractly enough and implemented
polymorphically enough. As far as I can remember, though, the
concrete connotations tended to get in the way of cross-cultural
or trans-silo communication at that time.

That is not, however, the largest obstacle to harmonizing the
logic of Peirce with the ways of FOL as she is spoke today.
I'll take that up when I next get a chance ...

Regards,

Jon

[Jon Awbrey]

Cf: Charles Sanders Peirce, George Spencer Brown, and Me • 16
http://inquiryintoinquiry.com/2022/01/15/charles-sanders-peirce-george-spencer-brown-and-me-16/

Re: Conceptual Graphs
https://lists.cs.uni-kassel.de/hyperkitty/list/c...@lists.iccs-conference.org/thread/JB7MRBGGOSKTPSSBCWHK22GK5BIHFPCH/
::: Gary Zhu
https://lists.cs.uni-kassel.de/hyperkitty/list/c...@lists.iccs-conference.org/message/JB7MRBGGOSKTPSSBCWHK22GK5BIHFPCH/

<QUOTE GZ:>
I'm quite confused on why people are interested in Laws of Form.
What is LOF trying to do? Is it just rewriting logic or is there
something more fundamental. e.g. a universal algebraic system?
What does GSB has to do with DNA, or DNA computing?
What does Lou's work in topology has to do with GSB?
What does GSB's theory has to do with knot theory?
What does GSB's theory has to do with quaternions?
How can GSB's theory be used for designing circuits?
What's wrong with Frege?
</QUOTE>

Dear Gary,

I am deep in the middle of other work right now,
but here's a smattering of resources relevant to
the relation between Peirce's logical graphs and
Spencer Brown's calculus of indications, at least
so far as the core subjects of boolean functions
and propositional calculus are concerned.

Survey of Animated Logical Graphs
https://inquiryintoinquiry.com/2021/05/01/survey-of-animated-logical-graphs-4/

As far as the extension to relations and quantification,
I start from where Peirce started in 1870 and follow up
several of his more radical ideas, ones he himself did
not fully develop. That is what I'm doing on the 1870
Logic of Relatives thread.

Peirce's 1870 “Logic of Relatives” • Overview
https://inquiryintoinquiry.com/2019/09/24/peirces-1870-logic-of-relatives-overview/

Regards,

Jon

[Michael DeBellis]

Jon, thanks for the feedback. One other issue that I think is relevant to all this is the question of if and how humans use concepts such as sets and subsumption to model the world. For a while after reading some of Lakoff's work and even more so the work Ellen Rosche, I started to be convinced that while set theory is obviously the foundation for mathematics and hence science (although Lakoff would disagree) Rosche's work showed that humans don't use it to model their everyday world. 

Although, after discussing this with some of my friends from the semantic AI camp I came back to thinking that Rosche's work, while clearly something that can't be ignored, doesn't necessarily have the wide ranging implications that Lakoff and people who follow him think it does. The experiments that Rosche and others do are very contrived. They present people with odd shapes or collections of dots and require them to generate new categories on the fly in a very short amount of time. There is other evidence such as the work of Frank Keil and to some extent Scott Atran that show people do use subsumption and the equivalent of class hierarchies in many types of reasoning and learning. 

I'm not proposing any answer, I think it is very much an open question, curious if you have any thoughts on it. 

Michael

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[dr.matt...@gmail.com]

Dear Michael,

I think the proper question is if not set theory, then what else?

My experience is that something like 90% (not in any sense a precise figure) of what we have to say about the world is (or can be) expressed through some sort of set theory or mereology, and even more if we go to mereotopology.

I have a simple guide as to whether I am looking at a set or an aggregate:

• It’s a set if I am talking about the properties that are common to each of the members (each car has a steering wheel) and of course we have subtypes and powersets and so on.
• It’s an aggregate if I am talking about a property of the whole rather than the parts (there are five tanks of oil and the total quantity of oil in the tanks was 500,000 tonnes) and of course we have parts and various ways things can be assembled.

It is quite surprising just how much comes down to these key theories.

Regards

Matthew

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[John F Sowa]

Dear Matthew, Michel, Eric, and Jon,

 

Every theory of mathematics is completely specified by its formal notation and the axioms stated in that notation.  99.9% of the mathematics that is most widely used today was developed *before* Cantor stated his version of set theory.   Even today working mathematicians have no use for any of the theoretical talk about foundations.  They may have taken a course on that topic at one point, but it is totally irrelevant to any task or problem they have to solve.

 

Furthermore, mathematicians discover, develop, and prove their theories with informal diagrams.  Only *after* they are satisfied that their ideas are correct do they write down a formal proof.  More often than not they discover that their informal diagrams were correct.  Even so, the formal proof is helpful for suggesting further directions.  For quotations about those issues by professional mathematicians, see the first 10 slides of http://jfsowa.com/talks/ppe.pdf .

 

But no working mathematician ever continues the exercise of translating that proof into any version of set theory.    In any case, category theory  can be used as a more general foundation.  And other (meta-)mathematicians are continuing to explore other directions, all of which are completely ignored by mathematicians who have work to do.

 

Furthermore, there are many excellent mathematicians who are anti-foundationalists.  Two of my favorite mathematician-philosophers, Peirce and Wittgenstein, are among them.  They claim that the axioms for any theory are a sufficient foundation for that theory   There is no need to find a single foundation for *all* of mathematics.  Many leading mathematicians today admit that there are so many unsolved problems and questions about foundations that the anti-foundationalists have a strong case.

 

Finally, I don't believe that anybody subscribed to Ontolog Forum (including myself) is qualified to do anything better than to quote or cite people who are experts on these issues.   For a criticism of some brilliant mathematicians whose ideas about foundations were hopelessly misguided, see "The ignorance of the Bourbaki": https://www.dpmms.cam.ac.uk/~ardm/bourbaki.pdf

---

Matthew:  I think the proper question is if not set theory, then what else? My experience is that something like 90% (not in any sense a precise figure) of what we have to say about the world is (or can be) expressed through some sort of set theory or mereology, and even more if we go to mereotopology.

 

I agree that set theory is useful for working mathematicians and ontologists.  But that is totally different from claiming that everything should be mapped all the way down to sets.  As I keep saying in many different ways, any version of mathematics that anybody has found useful for any subject should be available for anybody writing an ontology about that subject.

 

That is the major reason why I keep insisting on having *ALL* of mathematics available at the top level of any ontology that includes every version anybody has defined as well as any version that may be defined in the future. 

 

But somebody said that it too much to put in the top level.   And of course, they are right.  The node labeled Mathematics must be a *branch* off the top, from which any node at any lower level may borrow whatever kind of math is useful.  If your dynamic ontology is organized as a lattice, all such branches may be added whenever needed.  See http://jfsowa.com/pubs/dynonto.pdf .

 

Michael:  One other issue that I think is relevant to all this is the question of if and how humans use concepts such as sets and subsumption to model the world. For a while after reading some of Lakoff's work and even more so the work Ellen Rosche, I started to be convinced that while set theory is obviously the foundation for mathematics and hence science (although Lakoff would disagree) Rosche's work showed that humans don't use it to model their everyday world.

 

Lakoff wrote that article in 1987, and Rosche wrote her articles in the 1970s.  They made some interesting points, but cognitive science has made much further progress in the past 30 or 40 years.  In any case, if you want to prove that humans don't use set theory to model the world, just ask a three-year-old child.  For an example, see the quotations by Laura on page 1 of the following article: "Five questions on epistemic logic", http://jfsowa.com/pubs/5qelogic.pdf .

 

But a proof that humans do or do not use some version of mathematics in their everyday thinking is irrelevant to the question of what theories should be used in  computation.  That a basketball player does not use a certain kind of mathematics in making a 3-point shot is not an argument against using it in a robot that plays basketball.

 

For a review of more recent work by Lakoff, see http://jfsowa.com/pubs/lakoff.pdf .

 

John