Paradisaical Logic and the After Math

[Jon Awbrey]

Cf: Paradisaical Logic and the After Math • Comment 1
https://inquiryintoinquiry.com/2021/04/04/paradisaical-logic-and-the-after-math-comment-1/

Re: Peter Cameron • Cultures, Tribes, or Just an Illusion?
https://cameroncounts.wordpress.com/2012/04/01/cultures-tribes-or-just-an-illusion/

Re: Peirce List
https://list.iupui.edu/sympa/arc/peirce-l/2012-04/thrd1.html#00004
https://list.iupui.edu/sympa/arc/peirce-l/2021-03/thrd7.html#00205
https://list.iupui.edu/sympa/arc/peirce-l/2021-04/thrd1.html#00001

All,

One of many recurring themes — you might call it “The Power of
Negative Thinking” — arose this time on the Peirce List and it
took me back to a piece I wrote nine Aprils ago and that took
me even further back to the very doors I first walked through
into the wonderland of logic à la Peirce.

I fixed the links broken by the ravages of time and the impings
of web developers and I added more links to the original context
of discussion. A partial transcript follows.

Paradisaical Logic and the After Math
=====================================
https://inquiryintoinquiry.com/2012/04/09/paradisaical-logic-and-the-after-math/

Not too coincidentally with the mention of Peirce’s existential graphs,
a tangent of discussion elsewhere brought to mind an old favorite passage
from Peirce, where he is using his entitative graphs to expound the logic
of relatives. Here is the observation I was led to make.

Paradisaical Logic
==================

Negative operations (NOs), if not more important than
positive operations (POs), are at least more powerful
or generative, because the right NOs can generate all
POs, but the reverse is not so.

Which brings us to Peirce’s amphecks, NAND and NNOR,
either of which is a sole sufficient operator for
all boolean operations.

Amphecks ( https://oeis.org/wiki/Ampheck )
NAND ( https://oeis.org/wiki/Logical_NAND )
NNOR ( https://oeis.org/wiki/Logical_NNOR )

In one of his developments of a graphical syntax for logic,
that described in passing an application of the Neither-Nor
operator, Peirce referred to the stage of reasoning before
the encounter with falsehood as “paradisaical logic, because
it represents the state of Man’s cognition before the Fall.”

Here’s a bit of what he wrote there —

C.S. Peirce • Relatives of Second Intention
https://inquiryintoinquiry.com/2012/04/07/c-s-peirce-relatives-of-second-intention/

Resources
=========

Logic Syllabus ( https://oeis.org/wiki/Logic_Syllabus )

Peirce’s 1870 Logic Of Relatives
https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Overview

Regards,

Jon

[Jon Awbrey]

Hi Mauro,

I won’t be getting my computer back till my wife gets done with her zoom conference so I’ll just post this link to a bit about “sole sufficient operators” in boolean algebra or propositional calculus.

Sole Sufficient Operator

http://oeis.org/wiki/Sole_sufficient_operator

See also —

Ampheck 

http://oeis.org/wiki/Ampheck

Regards,

Jon

http://inquiryintoinquiry.com

On Apr 6, 2021, at 10:11 AM, Mauro Bertani <bertan...@gmail.com> wrote:

Dear Jon,

I'm not sure if I understand, but in an old post you said:

" " is "true".

"( )" is "false".

"a b" is "a and b"

"(a)" is "not a"

"((a)(b)) is "a or b"

"(a(b))" is "a implies b"

etc.

By these we can append:

NAND = (a b)

NNOR= (a)(b)

and then we can construct the duality, with the sign /=, like this:

NAND/=(NNOR)

So the power of negative thinking is NAND if all the variable are true the solution is false or say in an other way just a false variable make true the solution. In NNOR if all variable are false the solution is trueor in an other way if just a variable is true the solution is false.

So:

a  b NAND NNOR (NNOR)

V V   F          F           V

V F   V          F           V  

F V   V          F           V

F F   V          V           F

regards

Mauro 

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"[..] events are primarily linguistic or cognitive in nature. That is, the world does not really contain events. Rather, events are the way by which agents classify certain useful and relevant patterns of change."
Allen and Fergusson

"No, no. History of Eternity. At first I wanted to find every single one of the buyers to apologize because of the book and also to thank them for what they had done. There is an explanation for that. If you think of thirty-seven people—those people are real, I mean every one of them has a face of his own, a family, he lives on his own particular street. Why, if you sell, say two thousand copies, it is the same thing as if you had sold nothing at all because two thousand is too vast—I mean, for the imagination to grasp. While thirty-seven people—perhaps thirty-seven are too many, perhaps seventeen would have been better or even seven—but still thirty-seven are still within the scope of one's imagination."

http://www.theparisreview.org/interviews/4331/the-art-of-fiction-no-39-jorge-luis-borges

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[Jon Awbrey]

Cf: Paradisaical Logic and the After Math • Comment 2
http://inquiryintoinquiry.com/2021/04/07/paradisaical-logic-and-the-after-math-comment-2/

Re: Peirce List
https://list.iupui.edu/sympa/arc/peirce-l/2021-04/thrd1.html#00005
::: Mauro Bertani
https://pilot.list.iupui.edu/sympa/arc/peirce-l/2021-04/msg00014.html

[Note: Cleaned up yesterday's copy and added a few resource links.]

Dear Mauro,

My access to the internet is limited today — maybe
I can make a start toward addressing your comments
by linking to an article on “sole sufficient operators”
in boolean algebra and propositional calculus.

• Sole Sufficient Operator
https://oeis.org/wiki/Sole_sufficient_operator

There's more information about Peirce's “amphecks”,
tantamount to what we now call Nand and Nnor in
the following article.

• Ampheck ( https://oeis.org/wiki/Ampheck )

Resources
=========

• Logic Syllabus ( https://oeis.org/wiki/Logic_Syllabus )

• Logical NAND ( https://oeis.org/wiki/Logical_NAND )
• Logical NNOR ( https://oeis.org/wiki/Logical_NNOR )

Regards,

Jon