# Logical Graphs, Truth Tables, Venn Diagrams

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### Jon Awbrey

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,
find it useful to compare the representations of logical operators by
means of truth tables with those using logical graphs.

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/ )

### Jon Awbrey

May 29, 2021, 12:12:33 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 ).

### Jon Awbrey

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/
::: 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 …

### Jon Awbrey

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

### Jon Awbrey

Jun 1, 2021, 11:30:17 AM6/1/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
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://web.archive.org/web/20081012033302/http://suo.ieee.org/ontology/msg05518.html

### Jon Awbrey

Jun 2, 2021, 5:00:12 PM6/2/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
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 &curlywedge; ⋏ or &dfisht; ⥿, 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

### Jon Awbrey

Jun 4, 2021, 12:22:17 PM6/4/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
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
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

### Jon Awbrey

Jun 8, 2021, 2:36:25 PM6/8/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
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
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
— 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)
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

### Jon Awbrey

Jun 11, 2021, 11:56:25 AM6/11/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
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

### Jon Awbrey

Jun 12, 2021, 4:56:14 PM6/12/21
to Cybernetic Communications, Laws of Form, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG
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
</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