## Differential Logic • Discussion 5

JA:
The differential proposition $\mathrm{d}A$ is one we use to describe a change of state
(or a state of change) from $A$ to $\texttt{(} A \texttt{)}$ or the reverse.
LA:
Does this mean that if $A$ is the proposition “The sky is blue”, then $\mathrm{d}A$ would be the statement “The sky is not blue”?  Don’t you already have a notation for this in $A$ and $\texttt{(} A \texttt{)} \, ?$  From where does “state” and “change of state” come in relation to a proposition?

Dear Lyle,

The differential variable $\mathrm{d}A : X \to \mathbb{B}$ is a derivative variable, a qualitative analogue of a velocity vector in the quantitative realm.

Let’s say $x \in \mathbb{R}$ is a real value giving the membrane potential in a particular segment of a nerve cell’s axon and $A : \mathbb{R} \to \mathbb{B}$ is a categorical variable predicating whether the site is in the activated state, $A(x) = 1,$ or not, $A(x) = 0.$  We observe the site at discrete intervals, a few milliseconds apart, and obtain the following data.

• At time $t_1$ the site is in a resting state, $A(x) = 0.$
• At time $t_2$ the site is in an active state, $A(x) = 1.$
• At time $t_3$ the site is in a resting state, $A(x) = 0.$

On current information we have no way of predicting the state at time $t_2$ from the state at time $t_1$ but we know action potentials are inherently transient so we can fairly well guess the state of change at time $t_2$ is $\mathrm{d}A = 1,$ in other words, about to be changing from $A$ to $\texttt{(} A \texttt{)}.$  The site’s qualitative “position” and “velocity” at time $t_2$ can now be described by means of the compound proposition $A ~ \mathrm{d}A.$

## Differential Logic • Discussion 4

MB:
About Lobe Connective and Node Connective and their consequences,
I have a question:

You say that genus and species are evaluated by the proposition $\texttt{(} a \texttt{,(} b \texttt{),(} c \texttt{))}.$

The following proposition would no longer be appropriate:  $a \texttt{(} b \texttt{,} c \texttt{)}.$

And another question about differential calculus:

When we talk about $A$ and $\mathrm{d}A$ we talk about $A$ and $\texttt{(} A \texttt{)}$
or is it more similar to $A$ and $B \, ?$

Dear Mauro,

The proposition $\texttt{(} a \texttt{,(} b \texttt{),(} c \texttt{))}$ describes a genus $a$ divided into species $b$ and $c.$

The proposition $a \texttt{(} b \texttt{,} c \texttt{)}$ says $a$ is always true while just one of $b$ or $c$ is true.

The first proposition leaves space between the whole universe and the genus $a$
while the second proposition identifies the genus $a$ with the whole universe.

The differential proposition $\mathrm{d}A$ is one we use to describe a change of state
(or a state of change) from $A$ to $\texttt{(} A \texttt{)}$ or the reverse.

## Differential Logic and Dynamic Systems • Discussion 6

Re: Differential Logic and Dynamic Systems • Discussions • (4)(5)
Re: Laws of FormLyle Anderson

LA:
Have you ever done any statistical analysis of data, or any of the statistical sciences such as the Theory of Error, Experimental Psychology, Thermodynamics, Quantum Theory?  These handle what you are calling differential logic using probability distribution functions of various types over various knowledge domains.  If you are heading toward logic based on vague or imprecise statements, then there is the well established science and application of Fuzzy Logic.

Dear Lyle,

Yes, been there done that.  On top of the usual theory courses in Probability and Statistics, Measure Theory, etc. contributing to a master’s in math (1980) and the canonical year of “Advanced Psychometrics”, really a quaint legacy term for what had long since morphed into statistical methods for experimental design and analysis, sine qua non for a master’s in psych (1989), there was a raft of short courses on statistical computing packages which always seemed to get me a lot more bucks for the head-banging than anything else as the years wore on.

## Differential Logic and Dynamic Systems • Discussion 5

Last time I mentioned a few old writings I thought might serve to smooth the transition from “Differential Propositional Calculus” to the steeper climbs of “Differential Logic” and “Differential Logic and Dynamic Systems”.  I was pleased to discover LibreOffice made nice PDFs of my ancient Word Docs, so I can make a first installment on my promissory note by linking to a couple of those documents below.

The above selections were written to document the Theme One Program I developed all through the 1980s.  The program and documentation were eventually submitted to fulfill the thesis requirement for my Master’s in Psychology at Michigan State (1989).  Over time I’ll upgrade the formatting and serialize the relevant parts to my blog.

## 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

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?

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.

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.

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 …

### Resources

cc: CyberneticsOntolog • Peirce List (1) (2) (3)Structural ModelingSystems Science
cc: FB | Logical GraphsLaws of Form

## Logical Graphs, Truth Tables, Venn Diagrams • 8

Re: Laws of FormJohn MingersLyle Anderson
Re: Logical Graphs, Truth Tables, Venn Diagrams • (2)(3)

Looking to the day we can make our ascent to logical graphs with increasing numbers of variables, I’d like to flag the following points of departure for future 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.

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.

cc: CyberneticsOntolog • Peirce List (1) (2) (3)Structural ModelingSystems Science
cc: FB | Logical GraphsLaws of Form

## Animated Logical Graphs • 76

Re: Richard J. LiptonThe Art Of Math
Re: Animated Logical Graphs • (30)(45)(57)(58)(59)(60)(61)(62)(63)(64)(65)(66)(69)(70)(71)(72)(73)(74)(75)

Taking from our wallets an old schedule of orbits from Episode 72, let’s review the classes of logical graphs we’ve covered so far.

### Self-Dual Logical Graphs

Four orbits of self-dual logical graphs, $x, y, \texttt{(} x \texttt{)}, \texttt{(} y \texttt{)},$ were discussed in Episode 73.

The logical graphs $x, y, \texttt{(} x \texttt{)}, \texttt{(} y \texttt{)}$ denote the boolean functions $f_{12}, f_{10}, f_{3}, f_{5},$ in that order, and the value of each function $f$ at each point $(x, y)$ in $\mathbb{B} \times \mathbb{B}$ is shown in the Table above.

### Constants and Amphecks

Two orbits of logical graphs called constants and amphecks were discussed in Episode 74.

The constant logical graphs denote the constant functions

$f_{0} : \mathbb{B} \times \mathbb{B} \to 0 \quad \text{and} \quad f_{15} : \mathbb{B} \times \mathbb{B} \to 1.$

• Under $\mathrm{Ex}$ the logical graph whose text form is “  ” denotes the function $f_{15}$
and the logical graph whose text form is $\texttt{(} ~ \texttt{)}"$ denotes the function $f_{0}.$
• Under $\mathrm{En}$ the logical graph whose text form is “  ” denotes the function $f_{0}$
and the logical graph whose text form is $\texttt{(} ~ \texttt{)}"$ denotes the function $f_{15}.$

The ampheck logical graphs denote the ampheck functions

$f_{1}(x, y) = \textsc{nnor}(x, y) \quad \text{and} \quad f_{7}(x, y) = \textsc{nand}(x, y).$

• Under $\mathrm{Ex}$ the logical graph $\texttt{(} xy \texttt{)}$ denotes the function $f_{7}(x, y) = \textsc{nand}(x, y)$
and the logical graph $\texttt{(} x \texttt{)(} y \texttt{)}$ denotes the function $f_{1}(x, y) = \textsc{nnor}(x, y).$
• Under $\mathrm{En}$ the logical graph $\texttt{(} xy \texttt{)}$ denotes the function $f_{1}(x, y) = \textsc{nnor}(x, y)$
and the logical graph $\texttt{(} x \texttt{)(} y \texttt{)}$ denotes the function $f_{7}(x, y) = \textsc{nand}(x, y).$

### Subtractions and Implications

The logical graphs called subtractions and implications were discussed in Episode 75.

The subtraction logical graphs denote the subtraction functions

$f_{2}(x, y) = y \lnot x \quad \text{and} \quad f_{4}(x, y) = x \lnot y.$

The implication logical graphs denote the implication functions

$f_{11}(x, y) = x \Rightarrow y \quad \text{and} \quad f_{13}(x, y) = y \Rightarrow x.$

Under the action of the $\mathrm{En} \leftrightarrow \mathrm{Ex}$ duality the logical graphs for the subtraction $f_{2}$ and the implication $f_{11}$ fall into one orbit while the logical graphs for the subtraction $f_{4}$ and the implication $f_{13}$ fall into another orbit, making these two partitions of the four functions orthogonal or transversal to each other.

### Resources

cc: Peirce List (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17)
cc: Structural Modeling (1) (2) • Systems Science (1) (2)
cc: Cybernetics (1) (2) • Ontolog Forum (1) (2)
cc: FB | Logical GraphsLaws of Form

## Logical Graphs, Truth Tables, Venn Diagrams • 7

On the subject of Peirce’s ampheck operators, see the earlier discussion of their duality under entitative and existential interpretations.

The ampheck operators are dual with respect to entitative and existential interpretations:

• $f_{1} = f_{0001} = \text{both not} : \mathbb{B} \times \mathbb{B} \to \mathbb{B}$
• $f_{7} = f_{0111} = \text{not both} : \mathbb{B} \times \mathbb{B} \to \mathbb{B}$

Under the existential interpretation:

• $f_{1} = f_{0001} = \text{both not} = \texttt{(} x \texttt{)(} y \texttt{)}$
• $f_{7} = f_{0111} = \text{not both} = \texttt{(} xy \texttt{)}$

Under the entitative interpretation:

• $f_{1} = f_{0001} = \text{both not} = \texttt{(} xy \texttt{)}$
• $f_{7} = f_{0111} = \text{not both} = \texttt{(} x \texttt{)(} y \texttt{)}$

cc: CyberneticsOntolog • Peirce List (1) (2) (3)Structural ModelingSystems Science
cc: FB | Logical GraphsLaws of Form

## Logical Graphs, Truth Tables, Venn Diagrams • 6

Dear John, Lyle,

See:  Ampheck

Peirce discovered this about 1880 but did not publish it, leaving it to be named after Sheffer at a much later date.  In one discussion Peirce used simple concatenation for the abstract operation which can be interpreted in two ways:  “Both Not” (joint denial, Nnor) or “Not Both” (alternate denial, Nand).  In the passage linked above Peirce used a symbol for Nnor whose nearest facsimiles in HTML are $\curlywedge"$ (&curlywedge;) and “⥿” (&dfisht;), adding an overbar for Nand.  Peirce used 2 × 2 matrices to represent the truth tables of all 16 boolean operators then converted the 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

cc: CyberneticsOntolog • Peirce List (1) (2) (3)Structural ModelingSystems Science
cc: FB | Logical GraphsLaws of Form

## Logical Graphs, Truth Tables, Venn Diagrams • 5

Re: Laws of FormLyle Anderson
Re: Anderson, Lyle A. III (1981), “Systematic Analysis of Algorithms”,
Open Access Master’s Theses, Paper 1167, (1) (2).

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 noted in saying, “we are already way outside the realm of truth tables with only $1 \text{s}$ and $0 \text{s}",$ it tangos with a much-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 $f : \mathbb{B}^3 \to \mathbb{B}$ provides a reasonable mathematical proxy for what we mean when we say, $\text{if}~ p ~\text{is true then}~ q ~\text{is true else}~ r ~\text{is true}".$  Experience with informal-to-formal translation 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.

### Resources

cc: CyberneticsOntolog • Peirce List (1) (2) (3)Structural ModelingSystems Science
cc: FB | Logical GraphsLaws of Form