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Apr 21, 2021, 4:00:14 PM4/21/21

to Cybernetic Communications, Laws of Form, Ontolog Forum, Structural Modeling, SysSciWG

Cf: Logical Graphs, Truth Tables, Venn Diagrams • 1

http://inquiryintoinquiry.com/2021/04/21/logical-graphs-truth-tables-venn-diagrams-1/

Re: Peirce List

https://list.iupui.edu/sympa/arc/peirce-l/2021-04/thrd2.html#00095

::: Mauro Bertani

https://list.iupui.edu/sympa/arc/peirce-l/2021-04/msg00095.html

::: Helmut Raulien

https://list.iupui.edu/sympa/arc/peirce-l/2021-04/msg00096.html

Dear Mauro, Helmut, All ...

I'll be focusing on logical graphs, especially the duality between

entitative and existential interpretations, for quite a while longer,

so this doesn't address your questions about modal logic, but you might

find it useful to compare the representations of logical operators by

means of truth tables with those using logical graphs.

You could start with the top eight entries in the section

headed “Logical Operators” on the following page.

• Logical Syllabus ( https://inquiryintoinquiry.com/logic-syllabus/ )

There's also a page bringing all eight of

those Truth Tables together in one place.

• Truth Tables ( https://oeis.org/wiki/Truth_table )

I had been meaning to include the corresponding Logical Graphs and

Venn Diagrams — I'll spend some of my pandemic time working on that —

It looks like it would be worth the candle reviewing their properties

as representations of basic operations and going over their relative

utilities for various logical purposes.

The following two pages also contain

useful synopses of the boolean basics.

• Zeroth Order Logic

( https://oeis.org/wiki/Zeroth_order_logic )

• Minimal Negation Operators

( https://oeis.org/wiki/Minimal_negation_operator )

Regards,

Jon

Resource

========

• Survey of Animated Logical Graphs

( https://inquiryintoinquiry.com/2020/08/23/survey-of-animated-logical-graphs-3/ )

http://inquiryintoinquiry.com/2021/04/21/logical-graphs-truth-tables-venn-diagrams-1/

Re: Peirce List

https://list.iupui.edu/sympa/arc/peirce-l/2021-04/thrd2.html#00095

::: Mauro Bertani

https://list.iupui.edu/sympa/arc/peirce-l/2021-04/msg00095.html

::: Helmut Raulien

https://list.iupui.edu/sympa/arc/peirce-l/2021-04/msg00096.html

Dear Mauro, Helmut, All ...

I'll be focusing on logical graphs, especially the duality between

entitative and existential interpretations, for quite a while longer,

so this doesn't address your questions about modal logic, but you might

find it useful to compare the representations of logical operators by

means of truth tables with those using logical graphs.

You could start with the top eight entries in the section

headed “Logical Operators” on the following page.

• Logical Syllabus ( https://inquiryintoinquiry.com/logic-syllabus/ )

There's also a page bringing all eight of

those Truth Tables together in one place.

• Truth Tables ( https://oeis.org/wiki/Truth_table )

I had been meaning to include the corresponding Logical Graphs and

Venn Diagrams — I'll spend some of my pandemic time working on that —

It looks like it would be worth the candle reviewing their properties

as representations of basic operations and going over their relative

utilities for various logical purposes.

The following two pages also contain

useful synopses of the boolean basics.

• Zeroth Order Logic

( https://oeis.org/wiki/Zeroth_order_logic )

• Minimal Negation Operators

( https://oeis.org/wiki/Minimal_negation_operator )

Regards,

Jon

Resource

========

• Survey of Animated Logical Graphs

( https://inquiryintoinquiry.com/2020/08/23/survey-of-animated-logical-graphs-3/ )

May 29, 2021, 12:12:32 PM5/29/21

to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Logical Graphs, Truth Tables, Venn Diagrams • 2

https://inquiryintoinquiry.com/2021/05/29/logical-graphs-truth-tables-venn-diagrams-2/

Re: Laws of Form

https://groups.io/g/lawsofform/topic/logical_graphs_truth_tables/82270207

::: John Mingers ( https://groups.io/g/lawsofform/message/273 )

<QUOTE JM:>

Most of the recent discussion is about two-variable logic forms where there is a logical relation between two logical

variables. I want to bring up the subject of three-variable logic which I think is very rich but not much discussed.

In two-variable logic, as we know, there are 16 possible relations. With three variables, there are 8 rows in the truth

table and so 2⁸ = 256 possibilities. Many of these are the same at 2-variable, eg. AND(a,b,c) or OR(a,b,c) but some are

different, eg. IF a THEN b ELSE c. This latter one is really at the heart of all computer programming.

I haven't seen much written about this although William Bricken has done some (see for example “Symmetry in Boolean

Functions with Examples for Two and Three Variables” ( http://iconicmath.com/mypdfs/symmetry-and-figures.020404.pdf )).

Here he shows that when you take into account reflections and rotations there are actually 14 distinct forms within the 256.

</QUOTE>

Dear John,

One of the biggest advantages of the systems of graphical forms derived from C.S. Peirce's logical graphs and Spencer

Brown's calculus of indications is precisely the conceptual and computational efficiencies they afford us in dealing

with propositional forms and boolean functions of many variables. This has been one of my main motivations in pursuing

their development for the last half century and I think we have hopes of enjoying those benefits once we've gotten our

dose of minimum logical requirements and cross the threshold of first principles.

That said, I still have work to do on the logical graphs for two-variable boolean functions since I've been using those

as logical man-in-the-moon marigolds to study the effects of the En ↔ Ex duality. That duality is associated with a

transformation group of order two which partitions the set of sixteen functions into ten orbits. The groups William

Bricken considers have much higher orders at each number of variables and thus partition their spaces of functions into

many fewer orbits in each case. See the first reference below.

Have to break here …

Jon

References

==========

• “Number of Boolean Functions Distinct under Complementation/Permutation”,

A000370 ( https://oeis.org/A000370 ), N.J.A. Sloane (ed.),

The On-Line Encyclopedia of Integer Sequences ( https://oeis.org ).

• “The If..then..else statement”

( https://www.freepascal.org/docs-html/ref/refsu57.html ),

in Michaël Van Canneyt (May 2021), Free Pascal Reference Guide

( https://www.freepascal.org/docs-html/ref/ref.html ).

https://inquiryintoinquiry.com/2021/05/29/logical-graphs-truth-tables-venn-diagrams-2/

Re: Laws of Form

https://groups.io/g/lawsofform/topic/logical_graphs_truth_tables/82270207

::: John Mingers ( https://groups.io/g/lawsofform/message/273 )

<QUOTE JM:>

Most of the recent discussion is about two-variable logic forms where there is a logical relation between two logical

variables. I want to bring up the subject of three-variable logic which I think is very rich but not much discussed.

In two-variable logic, as we know, there are 16 possible relations. With three variables, there are 8 rows in the truth

table and so 2⁸ = 256 possibilities. Many of these are the same at 2-variable, eg. AND(a,b,c) or OR(a,b,c) but some are

different, eg. IF a THEN b ELSE c. This latter one is really at the heart of all computer programming.

I haven't seen much written about this although William Bricken has done some (see for example “Symmetry in Boolean

Functions with Examples for Two and Three Variables” ( http://iconicmath.com/mypdfs/symmetry-and-figures.020404.pdf )).

Here he shows that when you take into account reflections and rotations there are actually 14 distinct forms within the 256.

</QUOTE>

Dear John,

One of the biggest advantages of the systems of graphical forms derived from C.S. Peirce's logical graphs and Spencer

Brown's calculus of indications is precisely the conceptual and computational efficiencies they afford us in dealing

with propositional forms and boolean functions of many variables. This has been one of my main motivations in pursuing

their development for the last half century and I think we have hopes of enjoying those benefits once we've gotten our

dose of minimum logical requirements and cross the threshold of first principles.

That said, I still have work to do on the logical graphs for two-variable boolean functions since I've been using those

as logical man-in-the-moon marigolds to study the effects of the En ↔ Ex duality. That duality is associated with a

transformation group of order two which partitions the set of sixteen functions into ten orbits. The groups William

Bricken considers have much higher orders at each number of variables and thus partition their spaces of functions into

many fewer orbits in each case. See the first reference below.

Have to break here …

Jon

References

==========

• “Number of Boolean Functions Distinct under Complementation/Permutation”,

A000370 ( https://oeis.org/A000370 ), N.J.A. Sloane (ed.),

The On-Line Encyclopedia of Integer Sequences ( https://oeis.org ).

• “The If..then..else statement”

( https://www.freepascal.org/docs-html/ref/refsu57.html ),

in Michaël Van Canneyt (May 2021), Free Pascal Reference Guide

( https://www.freepascal.org/docs-html/ref/ref.html ).

May 30, 2021, 11:45:08 AM5/30/21

to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Logical Graphs, Truth Tables, Venn Diagrams • 3

https://inquiryintoinquiry.com/2021/05/30/logical-graphs-truth-tables-venn-diagrams-3/

Re: Laws of Form

https://groups.io/g/lawsofform/topic/logical_graphs_truth_tables/82270207

::: John Mingers ( https://groups.io/g/lawsofform/message/273 )

::: Lyle Anderson ( https://groups.io/g/lawsofform/message/275 )

Dear John, Lyle,

There is nothing simple about the interpretation of If-Then-Else constructions in ordinary language as they combine the

equivocation between formal and material implication at the outset with the vacillation between exclusive and inclusive

disjunction at the final Or-Else.

Nor is there anything straightforward about the implementation of If-Then-Else clauses in half-functional

half-procedural programming languages like Pascal. In settings like that they do not render as pure boolean expressions

but as boolean tests determining a choice between procedural branches. Multiply that by the diversity of evaluation

strategies for boolean expressions — (complete|partial), (eager|greedy|lazy), etc. — and the possibilities are legion.

That is all well and good, those are just the choices that are out there, and we can work with anyone's understanding of

If-Then-Else as a boolean function so long as they give us their intended truth table so we don't have to guess what

they have in mind.

I'll touch on If-Then-Else again when we turn to what I regard as the proper handling of Case Analysis in the systems of

logical graphs evolving from the work of C.S. Peirce and Spencer Brown.

As it happens, I did once write out all 256 boolean functions on three variables in cactus syntax several years ago —

pursuant to discussions in Stephen Wolfram's New Kind of Science (NKS) Forum regarding Elementary Cellular Automaton

Rules (ECARs), which are in effect just that set of boolean functions. I'll have to dig up a passel of ancient links

from the WayBack Machine, but see the following archive page for a hint of how it went.

• Cactus Rules

( https://web.archive.org/web/20041025093703/http://forum.wolframscience.com/archive/topic/256-1.html )

To be continued …

https://inquiryintoinquiry.com/2021/05/30/logical-graphs-truth-tables-venn-diagrams-3/

Re: Laws of Form

https://groups.io/g/lawsofform/topic/logical_graphs_truth_tables/82270207

::: John Mingers ( https://groups.io/g/lawsofform/message/273 )

Dear John, Lyle,

There is nothing simple about the interpretation of If-Then-Else constructions in ordinary language as they combine the

equivocation between formal and material implication at the outset with the vacillation between exclusive and inclusive

disjunction at the final Or-Else.

Nor is there anything straightforward about the implementation of If-Then-Else clauses in half-functional

half-procedural programming languages like Pascal. In settings like that they do not render as pure boolean expressions

but as boolean tests determining a choice between procedural branches. Multiply that by the diversity of evaluation

strategies for boolean expressions — (complete|partial), (eager|greedy|lazy), etc. — and the possibilities are legion.

That is all well and good, those are just the choices that are out there, and we can work with anyone's understanding of

If-Then-Else as a boolean function so long as they give us their intended truth table so we don't have to guess what

they have in mind.

I'll touch on If-Then-Else again when we turn to what I regard as the proper handling of Case Analysis in the systems of

logical graphs evolving from the work of C.S. Peirce and Spencer Brown.

As it happens, I did once write out all 256 boolean functions on three variables in cactus syntax several years ago —

pursuant to discussions in Stephen Wolfram's New Kind of Science (NKS) Forum regarding Elementary Cellular Automaton

Rules (ECARs), which are in effect just that set of boolean functions. I'll have to dig up a passel of ancient links

from the WayBack Machine, but see the following archive page for a hint of how it went.

• Cactus Rules

( https://web.archive.org/web/20041025093703/http://forum.wolframscience.com/archive/topic/256-1.html )

To be continued …

May 31, 2021, 8:16:26 AM5/31/21

to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Logical Graphs, Truth Tables, Venn Diagrams • 4

https://inquiryintoinquiry.com/2021/05/30/logical-graphs-truth-tables-venn-diagrams-4/

... All we are saying is give Peirce a chance ...

• John Mingers ( https://groups.io/g/lawsofform/message/278 )

• Lyle Anderson ( https://groups.io/g/lawsofform/message/279 )

Dear John, Lyle,

I’ve seen too many ways of interpreting and implementing If‑Then‑Else clauses to know what any one person or processor

means by them until they give me the truth table they have in mind, so if you write out the truth table you like for

them I’ll be able to work with that and say something more definite about it.

More importantly, once we get the full power of Peirce’s logical graphs, Spencer Brown’s calculus of indications, and

the extensions to cactus graphs and differential logic in gear we’ll find there are better, clearer, more efficient ways

of handling Boolean Expansions and Case Analysis and more generally applying propositional logic to real problems.

Here’s the NKS Forum link again:

[NKS Forum] Cactus Rules

https://web.archive.org/web/20041025093703/http://forum.wolframscience.com/archive/topic/256-1.html

The anchor post of that series used to have a file attached with the full set of cactus graphs for propositions on three

variables … but it looks like the file was not preserved. There’s a couple of links to other copies below.

[Inquiry List] Cactus Rules

https://web.archive.org/web/20141210144230/http://stderr.org/pipermail/inquiry/2004-April/001322.html

[Ontology List] Cactus Rules

https://web.archive.org/web/20081012033302/http://suo.ieee.org/ontology/msg05518.html

Regards,

Jon

https://inquiryintoinquiry.com/2021/05/30/logical-graphs-truth-tables-venn-diagrams-4/

... All we are saying is give Peirce a chance ...

• John Mingers ( https://groups.io/g/lawsofform/message/278 )

• Lyle Anderson ( https://groups.io/g/lawsofform/message/279 )

Dear John, Lyle,

I’ve seen too many ways of interpreting and implementing If‑Then‑Else clauses to know what any one person or processor

means by them until they give me the truth table they have in mind, so if you write out the truth table you like for

them I’ll be able to work with that and say something more definite about it.

More importantly, once we get the full power of Peirce’s logical graphs, Spencer Brown’s calculus of indications, and

the extensions to cactus graphs and differential logic in gear we’ll find there are better, clearer, more efficient ways

of handling Boolean Expansions and Case Analysis and more generally applying propositional logic to real problems.

Here’s the NKS Forum link again:

[NKS Forum] Cactus Rules

https://web.archive.org/web/20041025093703/http://forum.wolframscience.com/archive/topic/256-1.html

The anchor post of that series used to have a file attached with the full set of cactus graphs for propositions on three

variables … but it looks like the file was not preserved. There’s a couple of links to other copies below.

[Inquiry List] Cactus Rules

https://web.archive.org/web/20141210144230/http://stderr.org/pipermail/inquiry/2004-April/001322.html

[Ontology List] Cactus Rules

https://web.archive.org/web/20081012033302/http://suo.ieee.org/ontology/msg05518.html

Regards,

Jon

Jun 1, 2021, 11:30:15 AM6/1/21

Cf: Logical Graphs, Truth Tables, Venn Diagrams • 5

https://inquiryintoinquiry.com/2021/06/01/logical-graphs-truth-tables-venn-diagrams-5/

::: Lyle Anderson ( https://groups.io/g/lawsofform/message/284 )

Re: Anderson, Lyle A. III (1981),

“Systematic Analysis of Algorithms”,

Open Access Master's Theses, Paper 1167.

https://digitalcommons.uri.edu/theses/1167

https://digitalcommons.uri.edu/cgi/viewcontent.cgi?article=2175&context=theses

Thanks, Lyle, your Chapter 4, “Dealing With Conditional Statements”,

provides a detailed treatment of algorithmic branching constructs in

general purpose programming languages but as you note, “we are already

way outside the realm of truth tables with only 1s and 0s”, it tangos

with a higher maintenance date than the one John Mingers brought to

the dance.

I think we are making this problem harder than it needs to be.

Let’s go back to the original question and try to view it with

fresh eyes. All we have to decide is which candidate among

the three-variable boolean functions f : B³ → B provides a

reasonable mathematical proxy for what we mean when we say,

“if p is true then q is true else r is true”.

Experience with informal-to-formal translation tasks tells us

there may be no functional form capturing every nuance of a

natural language idiom but there is usually one serving all

practical purposes in empirical and mathematical contexts.

With that in mind, I’ll munch on it over lunch ...

Resources • Cactus Rules

========================

[NKS Forum]

https://web.archive.org/web/20041025093703/http://forum.wolframscience.com/archive/topic/256-1.html

[Inquiry List]

https://inquiryintoinquiry.com/2021/06/01/logical-graphs-truth-tables-venn-diagrams-5/

::: Lyle Anderson ( https://groups.io/g/lawsofform/message/284 )

Re: Anderson, Lyle A. III (1981),

“Systematic Analysis of Algorithms”,

Open Access Master's Theses, Paper 1167.

https://digitalcommons.uri.edu/theses/1167

https://digitalcommons.uri.edu/cgi/viewcontent.cgi?article=2175&context=theses

Thanks, Lyle, your Chapter 4, “Dealing With Conditional Statements”,

provides a detailed treatment of algorithmic branching constructs in

general purpose programming languages but as you note, “we are already

way outside the realm of truth tables with only 1s and 0s”, it tangos

with a higher maintenance date than the one John Mingers brought to

the dance.

I think we are making this problem harder than it needs to be.

Let’s go back to the original question and try to view it with

fresh eyes. All we have to decide is which candidate among

the three-variable boolean functions f : B³ → B provides a

reasonable mathematical proxy for what we mean when we say,

“if p is true then q is true else r is true”.

Experience with informal-to-formal translation tasks tells us

there may be no functional form capturing every nuance of a

natural language idiom but there is usually one serving all

practical purposes in empirical and mathematical contexts.

With that in mind, I’ll munch on it over lunch ...

Resources • Cactus Rules

========================

[NKS Forum]

https://web.archive.org/web/20041025093703/http://forum.wolframscience.com/archive/topic/256-1.html

[Inquiry List]

https://web.archive.org/web/20141210144230/http://stderr.org/pipermail/inquiry/2004-April/001322.html

[Ontology List]

https://web.archive.org/web/20081012033302/http://suo.ieee.org/ontology/msg05518.html
[Ontology List]

Jun 2, 2021, 5:00:12 PM6/2/21

Dear John, Lyle,

See: Ampheck https://oeis.org/wiki/Ampheck

Peirce discovered this about 1880 but did not publish it,

leaving it to be claimed by Sheffer at a much later date.

In one place he used simple concatenation for the abstract

operation which can be interpreted in either one of two ways:

Both Not (joint denial, NNOR) or Not Both (alternate denial, NAND).

In the passage linked above he uses a symbol for NNOR whose closest

HTML facsimiles are ⋏ ⋏ or ⥿ ⥿, with a bar over it

for NAND. He gave 2 × 2 matrix forms for all 16 boolean operators

representing their truth tables, then converted those matrices into

cursive symbols for the operators. Warren S. McCulloch mentioned

Peirce's discovery and his matrices, referring to NAND and NNOR

collectively as “amphecks” on account of their abstract duality.

Regards,

Jon

See: Ampheck https://oeis.org/wiki/Ampheck

Peirce discovered this about 1880 but did not publish it,

leaving it to be claimed by Sheffer at a much later date.

In one place he used simple concatenation for the abstract

operation which can be interpreted in either one of two ways:

Both Not (joint denial, NNOR) or Not Both (alternate denial, NAND).

In the passage linked above he uses a symbol for NNOR whose closest

HTML facsimiles are ⋏ ⋏ or ⥿ ⥿, with a bar over it

for NAND. He gave 2 × 2 matrix forms for all 16 boolean operators

representing their truth tables, then converted those matrices into

cursive symbols for the operators. Warren S. McCulloch mentioned

Peirce's discovery and his matrices, referring to NAND and NNOR

collectively as “amphecks” on account of their abstract duality.

Regards,

Jon

Jun 4, 2021, 12:22:17 PM6/4/21

Cf: Logical Graphs, Truth Tables, Venn Diagrams • 7

https://inquiryintoinquiry.com/2021/06/04/logical-graphs-truth-tables-venn-diagrams-7/

All,

On the subject of Peirce’s ampheck operators

( https://oeis.org/wiki/Ampheck ), see also

our earlier discussion of their duality under

entitative and existential interpretations.

Cf: Animated Logical Graphs • 74

https://inquiryintoinquiry.com/2021/04/30/animated-logical-graphs-74/

The ampheck operators are duals with respect

to entitative and existential interpretations:

• f₁ = f₀₀₀₁ = both not : B × B → B

• f₇ = f₀₁₁₁ = not both : B × B → B

Under the existential interpretation:

• f₁ = f₀₀₀₁ = both not = (x)(y)

• f₇ = f₀₁₁₁ = not both = (xy)

Under the entitative interpretation:

• f₁ = f₀₀₀₁ = both not = (xy)

• f₇ = f₀₁₁₁ = not both = (x)(y)

Regards,

Jon

https://inquiryintoinquiry.com/2021/06/04/logical-graphs-truth-tables-venn-diagrams-7/

All,

On the subject of Peirce’s ampheck operators

( https://oeis.org/wiki/Ampheck ), see also

our earlier discussion of their duality under

entitative and existential interpretations.

Cf: Animated Logical Graphs • 74

https://inquiryintoinquiry.com/2021/04/30/animated-logical-graphs-74/

The ampheck operators are duals with respect

to entitative and existential interpretations:

• f₁ = f₀₀₀₁ = both not : B × B → B

• f₇ = f₀₁₁₁ = not both : B × B → B

Under the existential interpretation:

• f₁ = f₀₀₀₁ = both not = (x)(y)

• f₇ = f₀₁₁₁ = not both = (xy)

Under the entitative interpretation:

• f₁ = f₀₀₀₁ = both not = (xy)

• f₇ = f₀₁₁₁ = not both = (x)(y)

Regards,

Jon

Jun 8, 2021, 2:36:25 PM6/8/21

Cf: Logical Graphs, Truth Tables, Venn Diagrams • 8

https://inquiryintoinquiry.com/2021/06/08/logical-graphs-truth-tables-venn-diagrams-8/

Re: Logical Graphs, Truth Tables, Venn Diagrams

https://inquiryintoinquiry.com/2021/05/29/logical-graphs-truth-tables-venn-diagrams-2/

https://inquiryintoinquiry.com/2021/05/30/logical-graphs-truth-tables-venn-diagrams-3/

All,

Looking to the day we can make our ascent on logical graphs

with increasing numbers of variables, I'd like to flag the

following points of discussion for further development.

JM: Most of the recent discussion is about two-variable logic forms where there is a logical relation between two

logical variables. I want to bring up the subject of three-variable logic which I think is very rich but not much

discussed.

JA: One of the biggest advantages of the systems of graphical forms derived from C.S. Peirce's logical graphs and

https://web.archive.org/web/20041025093703/http://forum.wolframscience.com/archive/topic/256-1.html

There is now a copy of the above content at the following location and

I'll be working to improve the formatting and graphics as time goes on.

• Cactus Rules

https://oeis.org/wiki/User:Jon_Awbrey/Cactus_Rules

• Propositional Forms on Three Variables

https://oeis.org/wiki/User:Jon_Awbrey/Cactus_Rules#Propositional_Forms_on_Three_Variables

Regards,

Jon

https://inquiryintoinquiry.com/2021/06/08/logical-graphs-truth-tables-venn-diagrams-8/

Re: Logical Graphs, Truth Tables, Venn Diagrams

https://inquiryintoinquiry.com/2021/05/29/logical-graphs-truth-tables-venn-diagrams-2/

https://inquiryintoinquiry.com/2021/05/30/logical-graphs-truth-tables-venn-diagrams-3/

All,

Looking to the day we can make our ascent on logical graphs

with increasing numbers of variables, I'd like to flag the

following points of discussion for further development.

JM: Most of the recent discussion is about two-variable logic forms where there is a logical relation between two

logical variables. I want to bring up the subject of three-variable logic which I think is very rich but not much

discussed.

Spencer Brown's calculus of indications is precisely the conceptual and computational efficiencies they afford us in

dealing with propositional forms and boolean functions of many variables.

JA: As it happens, I did once write out all 256 boolean functions on three variables in cactus syntax several years ago
dealing with propositional forms and boolean functions of many variables.

— pursuant to discussions in Stephen Wolfram's New Kind of Science (NKS) Forum regarding Elementary Cellular Automaton

Rules (ECARs), which are in effect just that set of boolean functions. I'll have to dig up a passel of ancient links

from the WayBack Machine, but see the following archive page for a hint of how it went.

• Cactus Rules (NKS Forum)
Rules (ECARs), which are in effect just that set of boolean functions. I'll have to dig up a passel of ancient links

from the WayBack Machine, but see the following archive page for a hint of how it went.

https://web.archive.org/web/20041025093703/http://forum.wolframscience.com/archive/topic/256-1.html

There is now a copy of the above content at the following location and

I'll be working to improve the formatting and graphics as time goes on.

• Cactus Rules

https://oeis.org/wiki/User:Jon_Awbrey/Cactus_Rules

• Propositional Forms on Three Variables

https://oeis.org/wiki/User:Jon_Awbrey/Cactus_Rules#Propositional_Forms_on_Three_Variables

Regards,

Jon

Jun 11, 2021, 11:56:25 AM6/11/21

All,

Just a note to say I've finished the first pass of formatting on the

Cactus Rules page and the content will be a little more readable now.

I'll be working on upgrading the Tables and eventually converting the

ASCII graphics to JPEGs or PNGs.

• https://oeis.org/wiki/User:Jon_Awbrey/Cactus_Rules

• https://oeis.org/wiki/User:Jon_Awbrey/Cactus_Rules#Cactus_Graphs_on_Three_Variables

• https://oeis.org/wiki/User:Jon_Awbrey/Cactus_Rules#Propositional_Forms_on_Three_Variables

Regards,

Jon

Just a note to say I've finished the first pass of formatting on the

Cactus Rules page and the content will be a little more readable now.

I'll be working on upgrading the Tables and eventually converting the

ASCII graphics to JPEGs or PNGs.

• https://oeis.org/wiki/User:Jon_Awbrey/Cactus_Rules

• https://oeis.org/wiki/User:Jon_Awbrey/Cactus_Rules#Cactus_Graphs_on_Three_Variables

• https://oeis.org/wiki/User:Jon_Awbrey/Cactus_Rules#Propositional_Forms_on_Three_Variables

Regards,

Jon

Jun 12, 2021, 4:56:14 PM6/12/21

Cf: Logical Graphs, Truth Tables, Venn Diagrams • 9

https://inquiryintoinquiry.com/2021/06/12/logical-graphs-truth-tables-venn-diagrams-9/

|| In November 1619, I had a dream involving the Seventh Ode of Ausonius,

|| which begins Quod vitae sectabor iter [“What road in life shall I follow”].

||

|| René Descartes • Experimenta

::: Lyle Anderson ( https://groups.io/g/lawsofform/message/307

<QUOTE LA:>

As I write this on a machine that does its logic 64-bits at a time,

I am finding it hard to imagine where the “ascent” to logical graphs

with increasing numbers of variables will take us that the engineers

haven't already gone. Could you enlighten us on where you think this

is headed?

</QUOTE>

Dear Lyle,

But now it's come to directions and things we must decide.

Here's a passage from Robert Musil I often use as a guide.

<QUOTE>

It is understandable that an engineer should be completely

absorbed in his speciality, instead of pouring himself out

into the freedom and vastness of the world of thought, even

though his machines are being sent off to the ends of the

earth; for he no more needs to be capable of applying to

his own personal soul what is daring and new in the soul

of his subject than a machine is in fact capable of applying

to itself the differential calculus on which it is based.

The same thing cannot, however, be said about mathematics;

for here we have the new method of thought, pure intellect,

the very well-spring of the times, the fons et origo of an

unfathomable transformation.

Robert Musil • The Man Without Qualities

https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_3#Transformations_of_Discourse

</QUOTE>

Just so I won't be misunderstood, there is nothing axiomatic about

Musil's differentiation of mathematics from engineering, much less

human souls from machines. For my part I have oscillated over time

between taking his distinctions at face value and challenging them

with more integral projects of my own. With that in mind the question

becomes: “What degrees of reflection on practice are essential to the

roles of mathematicians and engineers, respectively?”

To be continued …

Jon

https://inquiryintoinquiry.com/2021/06/12/logical-graphs-truth-tables-venn-diagrams-9/

|| In November 1619, I had a dream involving the Seventh Ode of Ausonius,

|| which begins Quod vitae sectabor iter [“What road in life shall I follow”].

||

|| René Descartes • Experimenta

::: Lyle Anderson ( https://groups.io/g/lawsofform/message/307

<QUOTE LA:>

As I write this on a machine that does its logic 64-bits at a time,

I am finding it hard to imagine where the “ascent” to logical graphs

with increasing numbers of variables will take us that the engineers

haven't already gone. Could you enlighten us on where you think this

is headed?

</QUOTE>

Dear Lyle,

But now it's come to directions and things we must decide.

Here's a passage from Robert Musil I often use as a guide.

<QUOTE>

It is understandable that an engineer should be completely

absorbed in his speciality, instead of pouring himself out

into the freedom and vastness of the world of thought, even

though his machines are being sent off to the ends of the

earth; for he no more needs to be capable of applying to

his own personal soul what is daring and new in the soul

of his subject than a machine is in fact capable of applying

to itself the differential calculus on which it is based.

The same thing cannot, however, be said about mathematics;

for here we have the new method of thought, pure intellect,

the very well-spring of the times, the fons et origo of an

unfathomable transformation.

Robert Musil • The Man Without Qualities

https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_3#Transformations_of_Discourse

</QUOTE>

Just so I won't be misunderstood, there is nothing axiomatic about

Musil's differentiation of mathematics from engineering, much less

human souls from machines. For my part I have oscillated over time

between taking his distinctions at face value and challenging them

with more integral projects of my own. With that in mind the question

becomes: “What degrees of reflection on practice are essential to the

roles of mathematicians and engineers, respectively?”

To be continued …

Jon

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