Higher Order Sign Relations

[Jon Awbrey]

Cf: Higher Order Sign Relations • 1
https://inquiryintoinquiry.com/2012/08/01/higher-order-sign-relations-1/

All,

When interpreters reflect on their own use of signs they require an
appropriate technical language in which to pursue their reflections.
For this they need signs referring to sign relations, signs referring
to elements and components of sign relations, and signs referring to
properties and classes of sign relations. The orders of signs developing
as reflection evolves can be placed under the description of “higher order
signs” and the extended sign relations involving them can be referred to as
“higher order sign relations”.

Continue Reading at “Inquiry Driven Systems • Higher Order Sign Relations”
https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations

I’ve been working apace to format my old dissertation proposal on Inquiry Driven Systems
( https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Overview )for the web but I was
reminded of this part when the subject of “signs about signs” came up recently on the
Peirce List.

Regards,

Jon

[Jon Awbrey]

Cf: Higher Order Sign Relations • 2
https://inquiryintoinquiry.com/2019/12/04/higher-order-sign-relations-2/

Re: FB | Charles S. Peirce Society • John Corcoran
https://www.facebook.com/groups/peircesociety/

All,

Questions about use and mention came up recently on Facebook.
In pragmatic semiotics the trade-off between “signs-of-objects”
and “signs-as-objects” opens up the wider space of “higher order
sign relations”. Here is how I introduced the subject in earlier
work on Inquiry Driven Systems.

When interpreters reflect on their use of signs they require an
appropriate technical language in which to pursue the reflections.
They need signs referring to sign relations, signs referring to

elements and components of sign relations, and signs referring
to properties and classes of sign relations. The orders of
signs developing as reflection evolves can be placed under
the description of “higher order signs” and the extended
sign relations involving them can be referred to as
“higher order sign relations”.

Exposition continues at “Higher Order Sign Relations”
https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Higher_Order_Sign_Relations

Regards,

Jon

[Jon Awbrey]

Cf: Higher Order Sign Relations • 3
https://inquiryintoinquiry.com/2019/12/23/higher-order-sign-relations-3/

Re: Ontolog Forum • Joseph Simpson
https://groups.google.com/g/ontolog-forum/c/B9HpfImt3aQ/m/R7shee_3BAAJ
Re: Relations, Types, Functions
https://oeis.org/wiki/Inquiry_Driven_Systems_%E2%80%A2_Part_12#Rel_Typ_Fun

All,

JS refers to the following passage from my text on Inquiry Driven Systems.

<QUOTE JA:>
The subject matters of relations, types, and functions enjoy
a form of recursive involvement with one another which makes
it difficult to know where to get on and where to get off
the circle of explanation. As I currently understand their
relationship, it can be approached in the following order.

• Relations have types.
• Types are functions.
• Functions are relations.

In this setting, a “type” is a function from the places of a relation,
that is, from the index set of its components, to a collection of sets
known as the domains of the relation.
</QUOTE>

My 3-basket mantra recited above harks back to the mid 1980s when
I took a course on “Applications of Lambda Calculus” from John Gray
at Illinois. It was all about categories, combinators, and computation,
focusing especially on cartesian closed categories, one of the hot topics
of the day. We had a packet of readings from the classic sources and used
J. Lambek and P.J. Scott's “Introduction to Higher Order Categorical Logic”
as our main text. I followed that up with a supervised independent study
where I explored various themes of my own.

The directions I pursued and continue to explore
all have to do with mutating category theory just
far enough to encompass Peirce's 3-eyed vision in
a more natural fashion.

I'll make that more explicit when I next get a chance.

Regards,

Jon

[Cliff Joslyn]

Jon: Categorical approaches to systems theory have been very attractive
to me for a long time. My current work is categorically adjacent, and
I'm funding some efforts in this direction. The category of binary
relations is central to our immediate work in hypergraphs and high-order
networks, but is also to any general systems theoretical approach. I've
approached topoi and closed Cartesian categories a few times, but admit
it's challenging. I need something at the level that David Spivak and
crew have been developing to become more fluent, if you're aware of his
work. Any worked examples you could provide would be very useful and
welcome.

--
O------------------------------------->
| Cliff Joslyn, Cybernetician at Large
V cajo...@gmail.com

[Jon Awbrey]

Cliff,

Just off the top of my head, here's a few sources I recall
most vividly for the way they captured the attractions of
categories, plus a few I still hope to get back to someday.
These come from a bibliography I assembled early in the 90s
plus a few more I added that decade.

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

• https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems#Arbib
• https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems#Hindley
• https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems#Lambek
• https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems#Lie
• https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems#Manes
• https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems#Smullyan
• https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems#Stoy

The following may also be of interest ...

Texts for a course on the relational approach to program construction and semantics,
using Tarski's approach to binary / dyadic relations (Fatma Mili taught this at OU).
• https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems#Mili

Freyd and Scedrov's work on Allegories, a category-theoretic take on binary relations.
• https://oeis.org/wiki/User:Jon_Awbrey/Prospects_for_Inquiry_Driven_Systems#Allegories

Regards,

Jon

[Jon Awbrey]

Cf: Higher Order Sign Relations • 5
http://inquiryintoinquiry.com/2022/08/09/higher-order-sign-relations-5/

Re: Higher Order Sign Relations • 4
https://inquiryintoinquiry.com/2022/08/06/higher-order-sign-relations-4/
Re: Conceptual Graphs • Gary Zhu

<QUOTE GZ:>
Is there any good contemporary reading of Peirce & James that you recommend?
Their original works have been quite challenging for me.

Dear Gary,

As fortune would have it, I haven't found much to recommend in the secondary literature on Peirce over the last couple
of decades. Most of it looks bent on assimilating Peirce to the conventional wits of analytic and continental
philosophy. As a result, I hew pretty close to Peirce himself in my current reading. You could try the two volumes of
the Essential Peirce for general orientation, if a trifle light on the math side of Peirce.

The last contemporary work I read with anything like the spirit of Peirce about it would probably be Sowa's Conceptual
Structures, so try that if you haven't already. Still worth reading are Pragmatism by William James and How We Think by
John Dewey. James and Dewey lacked the mathematical perspective needed to take in Peirce's full scope and Dewey was a
little slow getting up to speed with Peirce's message but he kept at it and had the benefit of living long enough to
become an able expositor of pragmatic and scientific ways. Plus he understood people and society far better than Peirce
ever did.

There are a few references at the end of the following paper.

• Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking
Across the Disciplines 15(1), 40–52. Archive (
https://web.archive.org/web/20001210162300/http://chss.montclair.edu/inquiry/fall95/awbrey.html ) . Journal (
https://www.pdcnet.org/inquiryct/content/inquiryct_1995_0015_0001_0040_0052 ) . Online (doc) (
https://www.academia.edu/1266493/Interpretation_as_Action_The_Risk_of_Inquiry ) (pdf) (
https://www.academia.edu/57812482/Interpretation_as_Action_The_Risk_of_Inquiry ) .

Regards,

Jon

[Cliff Joslyn]

Thanks, I'm partial to these these days:

https://www.amazon.com/Category-Theory-Sciences-MIT-Press/dp/0262028131

https://www.amazon.com/Invitation-Applied-Category-Theory-Compositionality/dp/1108711820

[Jon Awbrey]

Cf: Higher Order Sign Relations • 6
http://inquiryintoinquiry.com/2022/08/14/higher-order-sign-relations-6/

Re: Cybernetics ( https://groups.google.com/g/cybcom/c/UwInmSnuv0c )
::: Cliff Joslyn ( https://groups.google.com/g/cybcom/c/UwInmSnuv0c/m/w9ncBWKUAAAJ )

Cliff Joslyn recommended the following books.

• Spivak, David I. (2014), Category Theory for the Sciences
( https://www.amazon.com/Category-Theory-Sciences-MIT-Press/dp/0262028131 )

• Fong, Brendan, and Spivak, David I. (2019),
An Invitation to Applied Category Theory : Seven Sketches in Compositionality
( https://www.amazon.com/Invitation-Applied-Category-Theory-Compositionality/dp/1108711820 )

Dear Cliff, All,

The following Survey page gives a hint of the tack I've been taking with
category theory since the early days but definitely moving into higher gear
during my year at Illinois in the mid 1980s. John Gray taught a course joint
between math and computer science on the Applications of Lambda Calculus and
David Plaisted taught a course on Resolution-Unification Theorem Proving, both
of which I took and followed up with independent studies. I spent a heady year
making the circuit between math, computer science, and psychology departments
and a lot of what I work on today goes back to issues raised in those days.

• Survey of Precursors Of Category Theory
( https://inquiryintoinquiry.com/2020/09/20/survey-of-precursors-of-category-theory-2/ )

I know that Survey from a couple years ago still looks a little sketchy
but I'll be working to make it less so as time goes by, especially if
I ever get around to unpacking my notes from the basement boxes.

I have managed to sample contemporary approaches to categories at sundry
sites around the web over the last couple of decades — John Baez, nCafe,
nLab, Zulip Category Chat, Topos Group, etc. As great as all that is
there's a reason why it bears but tangentially on the issues I've been
pursuing. That has to do with the Peirce Factor and how far a given
line of inquiry takes account of it.

As luck would have it, one of the texts John Gray used for his course,
Lambek and Scott's “Introduction to Higher Order Categorical Logic”,
resonated strongly with themes I knew from Peirce and that led me
to many adventures of ideas still in progress. The following set
of excerpts I shared with the Standard Upper Ontology Group back
in the day may suggest the character of that work.

• Lambek, J. and Scott, P.J. (1986),
Introduction To Higher Order Categorical Logic
• Excerpts ( https://oeis.org/wiki/User:Jon_Awbrey/Mathematical_Notes#HOC )
• Discussion ( https://oeis.org/wiki/User:Jon_Awbrey/Mathematical_Notes#HOC_Discuss )

There's a lot more to say, but that's all I have time for today …

Regards,

Jon

[Cliff Joslyn]

Very interesting, thank you. I got into all this stuff in grad school in
the late 80's and early 90's through things like these:

@Book{ArM69,  
        Author="Arbib, Michael A",
        Year="1969",
        Title="Theories of Abstract Automata",
        Publisher="Prentic Hall",
        Address="Engelwood Cliff",
}

@Incollection{ArMMaE77, 
        Author="Arbib, Michael A and Manes, Ernst",
        Editor="WE Hartnett",
        Year="1977",
        Title="Category Theoretic Approach to Systems in a Fuzzy World",
        Booktitle="Systems: Approahces, Theories, Applications",
        Pages="1-26",
        Publisher="D. Reidel",
}

@Book{EiS74, 
        Author="Eilenberg, S",
        Year="1974",
        Title="Automata, Languages and Semi-Groups", 
        Volume="A", 
        Publisher="Academic Press", 
        Address="New York", 
}

@Article{GoJ91,
        Author="Goguen, JA",
        Year="1991",
        Title="Categorical Manifesto",
        Journal="Mathematical Structures in Computer Science",
        Volume="1:1",
        Pages="49-67",
        URL="http://www-cse.ucsd.edu/users/goguen/ps/manif.ps.gz", 
}

@Article{MoJ91,
        Author="Mo\v{c}kor, Ji\v{r}i",
        Year="1991",
        Title="Category of Fuzzy Automata",
        Journal="Int. J. of General Systems",
        Volume="20",
        Pages="73-82",
}

@Article{TaYTaT85,
        Author="Takahara, Y and Takai, T",
        Year="1985",
        Title="Category Theoretical Framework of General Systems",
        Journal="Int. J. of General Systems",
        Volume="11:1",
        Pages="1-33",
}

And then of course there's the classics:

@Book{GoR84,  
        Author="Goldblatt, Robert",
        Year="1984",
        Title="Topoi: The Categorical Analysis of Logic",
        Publisher="Dover",
        Address="Mineolo, NY", 
}

@Book{MaS78,  
        Author="MacLane, Saunders", 
        Year="1978",
        Title="Categories for the Working Mathematician",
        Publisher="Springer-Verlag",
        Address="Heidelberg",