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Mar 21, 2020, 10:36:33 AM3/21/20

to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Differential Logic • Overview

At: http://inquiryintoinquiry.com/2020/03/20/differential-logic-%e2%80%a2-overview/

All,

The previous series of posts on Differential Propositional Calculus

( https://inquiryintoinquiry.com/?s=Differential+Propositional+Calculus )

brought us to the threshold of the subject without quite stepping over,

but I wanted to lay out the necessary ingredients in the most concrete,

intuitive, and visual way possible before taking up the abstract forms.

One of my readers on Facebook told me "venn diagrams are obsolete" and

of course we all know they become unwieldy as our universes of discourse

expand beyond four or five dimensions. Indeed, one of the first lessons

I learned when I set about implementing CSP's graphs and GSB's forms on the

computer was that 2-dimensional representations of logic are a death trap

in numerous conceptual and computational ways. Still, venn diagrams do us

good service in visualizing the relationships among extensional, functional,

and intensional aspects of logic. A facility with those relationships is

critical to the computational applications and statistical generalizations

of logic commonly used in mathematical and empirical practice.

At any rate, intrepid readers will have amped up their visual imaginations

well enough at this point to pick their way through the cactus patch ahead.

The link above or the transcript below outlines my last, best introduction

to Differential Logic, which I'll be working to improve as I serialize it

to my blog.

Part 1 ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1 )

Introduction ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Introduction )

Cactus Language for Propositional Logic (

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Cactus_Language_for_Propositional_Logic )

Differential Expansions of Propositions (

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Differential_Expansions_of_Propositions )

Bird's Eye View ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Bird.27s_Eye_View )

Worm's Eye View ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Worm.27s_Eye_View )

Part 2 ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2 )

Propositional Forms on Two Variables (

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2#Propositional_Forms_on_Two_Variables )

Transforms Expanded over Differential Features (

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2#Transforms_Expanded_over_Differential_Features )

Transforms Expanded over Ordinary Features (

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2#Transforms_Expanded_over_Ordinary_Features )

Operational Representation ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2#Operational_Representation )

Part 3 ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3 )

Development • Field Picture (

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3#Development_.E2.80.A2_Field_Picture )

Proposition and Tacit Extension (

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3#Proposition_and_Tacit_Extension )

Enlargement and Difference Maps (

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3#Enlargement_and_Difference_Maps )

Tangent and Remainder Maps ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3#Tangent_and_Remainder_Maps )

Least Action Operators ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3#Least_Action_Operators )

Goal-Oriented Systems ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3#Goal-Oriented_Systems )

Further Reading ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3#Further_Reading )

Document History ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Document_History )

Regards,

Jon

inquiry into inquiry: https://inquiryintoinquiry.com/

academia: https://independent.academia.edu/JonAwbrey

oeiswiki: https://www.oeis.org/wiki/User:Jon_Awbrey

facebook page: https://www.facebook.com/JonnyCache

At: http://inquiryintoinquiry.com/2020/03/20/differential-logic-%e2%80%a2-overview/

All,

The previous series of posts on Differential Propositional Calculus

( https://inquiryintoinquiry.com/?s=Differential+Propositional+Calculus )

brought us to the threshold of the subject without quite stepping over,

but I wanted to lay out the necessary ingredients in the most concrete,

intuitive, and visual way possible before taking up the abstract forms.

One of my readers on Facebook told me "venn diagrams are obsolete" and

of course we all know they become unwieldy as our universes of discourse

expand beyond four or five dimensions. Indeed, one of the first lessons

I learned when I set about implementing CSP's graphs and GSB's forms on the

computer was that 2-dimensional representations of logic are a death trap

in numerous conceptual and computational ways. Still, venn diagrams do us

good service in visualizing the relationships among extensional, functional,

and intensional aspects of logic. A facility with those relationships is

critical to the computational applications and statistical generalizations

of logic commonly used in mathematical and empirical practice.

At any rate, intrepid readers will have amped up their visual imaginations

well enough at this point to pick their way through the cactus patch ahead.

The link above or the transcript below outlines my last, best introduction

to Differential Logic, which I'll be working to improve as I serialize it

to my blog.

Part 1 ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1 )

Introduction ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Introduction )

Cactus Language for Propositional Logic (

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Cactus_Language_for_Propositional_Logic )

Differential Expansions of Propositions (

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Differential_Expansions_of_Propositions )

Bird's Eye View ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Bird.27s_Eye_View )

Worm's Eye View ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Worm.27s_Eye_View )

Part 2 ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2 )

Propositional Forms on Two Variables (

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2#Propositional_Forms_on_Two_Variables )

Transforms Expanded over Differential Features (

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2#Transforms_Expanded_over_Differential_Features )

Transforms Expanded over Ordinary Features (

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2#Transforms_Expanded_over_Ordinary_Features )

Operational Representation ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2#Operational_Representation )

Part 3 ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3 )

Development • Field Picture (

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3#Development_.E2.80.A2_Field_Picture )

Proposition and Tacit Extension (

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3#Proposition_and_Tacit_Extension )

Enlargement and Difference Maps (

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3#Enlargement_and_Difference_Maps )

Tangent and Remainder Maps ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3#Tangent_and_Remainder_Maps )

Least Action Operators ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3#Least_Action_Operators )

Goal-Oriented Systems ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3#Goal-Oriented_Systems )

Further Reading ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_3#Further_Reading )

Document History ( https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Document_History )

Regards,

Jon

inquiry into inquiry: https://inquiryintoinquiry.com/

academia: https://independent.academia.edu/JonAwbrey

oeiswiki: https://www.oeis.org/wiki/User:Jon_Awbrey

facebook page: https://www.facebook.com/JonnyCache

Mar 22, 2020, 12:00:20 PM3/22/20

to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Differential Logic • 1

At: http://inquiryintoinquiry.com/2020/03/22/differential-logic-%e2%80%a2-1/

Introduction

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

Differential logic is the component of logic whose object is the description of variation — for example, the aspects of

change, difference, distribution, and diversity — in universes of discourse subject to logical description. A

definition that broad naturally incorporates any study of variation by way of mathematical models, but differential

logic is especially charged with the qualitative aspects of variation pervading or preceding quantitative models. To

the extent a logical inquiry makes use of a formal system, its differential component treats the principles governing

the use of a “differential logical calculus”, that is, a formal system with the expressive capacity to describe change

and diversity in logical universes of discourse.

Simple examples of differential logical calculi are furnished by “differential propositional calculi”. A differential

propositional calculus is a propositional calculus extended by a set of terms for describing aspects of change and

difference, for example, processes taking place in a universe of discourse or transformations mapping a source universe

to a target universe. Such a calculus augments ordinary propositional calculus in the same way the differential

calculus of Leibniz and Newton augments the analytic geometry of Descartes.

Regards,

Jon

At: http://inquiryintoinquiry.com/2020/03/22/differential-logic-%e2%80%a2-1/

Introduction

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

Differential logic is the component of logic whose object is the description of variation — for example, the aspects of

change, difference, distribution, and diversity — in universes of discourse subject to logical description. A

definition that broad naturally incorporates any study of variation by way of mathematical models, but differential

logic is especially charged with the qualitative aspects of variation pervading or preceding quantitative models. To

the extent a logical inquiry makes use of a formal system, its differential component treats the principles governing

the use of a “differential logical calculus”, that is, a formal system with the expressive capacity to describe change

and diversity in logical universes of discourse.

Simple examples of differential logical calculi are furnished by “differential propositional calculi”. A differential

propositional calculus is a propositional calculus extended by a set of terms for describing aspects of change and

difference, for example, processes taking place in a universe of discourse or transformations mapping a source universe

to a target universe. Such a calculus augments ordinary propositional calculus in the same way the differential

calculus of Leibniz and Newton augments the analytic geometry of Descartes.

Regards,

Jon

Mar 23, 2020, 3:30:27 PM3/23/20

to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Differential Logic ??? 2

At: http://inquiryintoinquiry.com/2020/03/23/differential-logic-%e2%80%a2-2/

Cactus Language for Propositional Logic

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

The development of differential logic is facilitated by having a moderately efficient calculus in place at the level of

boolean-valued functions and elementary logical propositions. One very efficient calculus on both conceptual and

computational grounds is based on just two types of logical connectives, both of variable k-ary scope. The syntactic

formulas of this calculus map into a family of graph-theoretic structures called "painted and rooted cacti" which lend

visual representation to the functional structures of propositions and smooth the path to efficient computation.

The first kind of connective takes the form of a parenthesized sequence of propositional expressions, written (e_1, e_2,

..., e_k) and meaning exactly one of the propositions e_1, e_2, ..., e_k is false, in short, their minimal negation is

true. An expression of this form maps into a cactus structure called a "lobe", in this case, "painted" with the colors

e_1, e_2, ..., e_k as shown below.

Lobe Connective

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-ej-lobe-connective.jpg

The second kind of connective is a concatenated sequence of propositional expressions, written e_1 e_2 ... e_k and

meaning all of the propositions e_1, e_2, ..., e_k are true, in short, their logical conjunction is true. An expression

of this form maps into a cactus structure called a "node", in this case, "painted" with the colors e_1, e_2, ..., e_k as

shown below.

Node Connective

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-ej-node-connective.jpg

All other propositional connectives can be obtained through combinations of these two forms. As it happens, the

parenthesized form is sufficient to define the concatenated form, making the latter formally dispensable, but it's

convenient to maintain it as a concise way of expressing more complicated combinations of parenthesized forms. While

working with expressions solely in propositional calculus, it's easiest to use plain parentheses for logical

connectives. In contexts where ordinary parentheses are needed for other purposes an alternate typeface (...) may be

used for the logical operators.

Regards,

Jon

At: http://inquiryintoinquiry.com/2020/03/23/differential-logic-%e2%80%a2-2/

Cactus Language for Propositional Logic

The development of differential logic is facilitated by having a moderately efficient calculus in place at the level of

boolean-valued functions and elementary logical propositions. One very efficient calculus on both conceptual and

computational grounds is based on just two types of logical connectives, both of variable k-ary scope. The syntactic

formulas of this calculus map into a family of graph-theoretic structures called "painted and rooted cacti" which lend

visual representation to the functional structures of propositions and smooth the path to efficient computation.

The first kind of connective takes the form of a parenthesized sequence of propositional expressions, written (e_1, e_2,

..., e_k) and meaning exactly one of the propositions e_1, e_2, ..., e_k is false, in short, their minimal negation is

true. An expression of this form maps into a cactus structure called a "lobe", in this case, "painted" with the colors

e_1, e_2, ..., e_k as shown below.

Lobe Connective

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-ej-lobe-connective.jpg

The second kind of connective is a concatenated sequence of propositional expressions, written e_1 e_2 ... e_k and

meaning all of the propositions e_1, e_2, ..., e_k are true, in short, their logical conjunction is true. An expression

of this form maps into a cactus structure called a "node", in this case, "painted" with the colors e_1, e_2, ..., e_k as

shown below.

Node Connective

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-ej-node-connective.jpg

All other propositional connectives can be obtained through combinations of these two forms. As it happens, the

parenthesized form is sufficient to define the concatenated form, making the latter formally dispensable, but it's

convenient to maintain it as a concise way of expressing more complicated combinations of parenthesized forms. While

working with expressions solely in propositional calculus, it's easiest to use plain parentheses for logical

connectives. In contexts where ordinary parentheses are needed for other purposes an alternate typeface (...) may be

used for the logical operators.

Regards,

Jon

Mar 24, 2020, 2:30:27 PM3/24/20

to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Differential Logic : 3

At: http://inquiryintoinquiry.com/2020/03/24/differential-logic-%e2%80%a2-3/

Cactus Language for Propositional Logic

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

Table 1 shows the cactus graphs, the corresponding cactus expressions, their logical meanings under the so-called

"existential interpretation", and their translations into conventional notations for a sample of basic propositional forms.

Table 1. Syntax and Semantics of a Calculus for Propositional Logic

https://inquiryintoinquiry.files.wordpress.com/2020/03/syntax-and-semantics-of-a-calculus-for-propositional-logic-2.0.png

The simplest expression for logical truth is the empty word, typically denoted by epsilon or lambda in formal languages,

where it is the identity element for concatenation. To make it visible in context, it may be denoted by the equivalent

expression "(( ))", or, especially if operating in an algebraic context, by a simple "1". Also when working in an

algebraic mode, the plus sign "+" may be used for exclusive disjunction. Thus we have the following translations of

algebraic expressions into cactus expressions.

* a + b = (a, b)

* a + b + c = (a, (b, c)) = ((a, b), c)

It is important to note the last expressions are not equivalent to the 3-place form (a, b, c).

Regards,

Jon

At: http://inquiryintoinquiry.com/2020/03/24/differential-logic-%e2%80%a2-3/

Cactus Language for Propositional Logic

Table 1 shows the cactus graphs, the corresponding cactus expressions, their logical meanings under the so-called

"existential interpretation", and their translations into conventional notations for a sample of basic propositional forms.

Table 1. Syntax and Semantics of a Calculus for Propositional Logic

https://inquiryintoinquiry.files.wordpress.com/2020/03/syntax-and-semantics-of-a-calculus-for-propositional-logic-2.0.png

The simplest expression for logical truth is the empty word, typically denoted by epsilon or lambda in formal languages,

where it is the identity element for concatenation. To make it visible in context, it may be denoted by the equivalent

expression "(( ))", or, especially if operating in an algebraic context, by a simple "1". Also when working in an

algebraic mode, the plus sign "+" may be used for exclusive disjunction. Thus we have the following translations of

algebraic expressions into cactus expressions.

* a + b = (a, b)

* a + b + c = (a, (b, c)) = ((a, b), c)

It is important to note the last expressions are not equivalent to the 3-place form (a, b, c).

Regards,

Jon

Mar 26, 2020, 1:30:36 PM3/26/20

to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Differential Logic : 4

At: http://inquiryintoinquiry.com/2020/03/26/differential-logic-%e2%80%a2-4/

Differential Expansions of Propositions

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

Bird's Eye View

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

An efficient calculus for the realm of logic represented by boolean functions and elementary propositions makes it

feasible to compute the finite differences and the differentials of those functions and propositions.

For example, consider a proposition of the form "p and q" graphed as two letters attached to a root node:

Cactus Graph Existential p and q

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-existential-p-and-q.jpg

Written as a string, this is just the concatenation p q.

The proposition pq may be taken as a boolean function f(p, q) having the abstract type f : B x B -> B, where B = {0, 1}

is read in such a way that 0 means false and 1 means true.

Imagine yourself standing in a fixed cell of the corresponding venn diagram, say, the cell where the proposition pq is

true, as shown in the following Figure:

Venn Diagram p and q

https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagram-p-and-q.jpg

Now ask yourself: What is the value of the proposition pq at a distance of dp and dq from the cell pq where you are

standing?

Don't think about it -- just compute:

Cactus Graph (p,dp)(q,dq)

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-pdpqdq.jpg

The cactus formula (p, dp)(q, dq) and its corresponding graph arise by substituting p + dp for p and q + dq for q in the

boolean product or logical conjunction pq and writing the result in the two dialects of cactus syntax. This follows

from the fact the boolean sum p + dp is equivalent to the logical operation of exclusive disjunction, which parses to a

cactus graph of the following form:

Cactus Graph (p,dp)

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-pdp.jpg

Next question: What is the difference between the value of the proposition pq over there, at a distance of dp and dq,

and the value of the proposition pq where you are standing, all expressed in the form of a general formula, of course?

Here is the appropriate formulation:

Cactus Graph ((p,dp)(q,dq),pq)

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-pdpqdqpq.jpg

There is one thing I ought to mention at this point: Computed over B, plus and minus are identical operations. This

will make the relation between the differential and the integral parts of the appropriate calculus slightly stranger

than usual, but we will get into that later.

Last question, for now: What is the value of this expression from your current standpoint, that is, evaluated at the

point where pq is true? Well, substituting 1 for p and 1 for q in the graph amounts to erasing the labels p and q, as

shown here:

Cactus Graph (( ,dp)( ,dq), )

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-dp-dq-.jpg

And this is equivalent to the following graph:

Cactus Graph ((dp)(dq))

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-dpdq.jpg

We have just met with the fact that the differential of the "and" is the "or" of the differentials.

* p and q ---Diff--> dp or dq

Cactus Graph pq Diff ((dp)(dq))

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-pq-diff-dpdq.jpg

It will be necessary to develop a more refined analysis of that statement directly, but that is roughly the nub of it.

If the form of the above statement reminds you of De Morgan's rule, it is no accident, as differentiation and negation

turn out to be closely related operations. Indeed, one can find discussions of logical difference calculus in the

Boole-De Morgan correspondence and Peirce also made use of differential operators in a logical context, but the

exploration of these ideas has been hampered by a number of factors, not the least of which has been the lack of a

syntax adequate to handle the complexity of expressions evolving in the process.

Regards,

Jon

At: http://inquiryintoinquiry.com/2020/03/26/differential-logic-%e2%80%a2-4/

Differential Expansions of Propositions

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

Bird's Eye View

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

An efficient calculus for the realm of logic represented by boolean functions and elementary propositions makes it

feasible to compute the finite differences and the differentials of those functions and propositions.

For example, consider a proposition of the form "p and q" graphed as two letters attached to a root node:

Cactus Graph Existential p and q

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-existential-p-and-q.jpg

Written as a string, this is just the concatenation p q.

The proposition pq may be taken as a boolean function f(p, q) having the abstract type f : B x B -> B, where B = {0, 1}

is read in such a way that 0 means false and 1 means true.

Imagine yourself standing in a fixed cell of the corresponding venn diagram, say, the cell where the proposition pq is

true, as shown in the following Figure:

Venn Diagram p and q

https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagram-p-and-q.jpg

Now ask yourself: What is the value of the proposition pq at a distance of dp and dq from the cell pq where you are

standing?

Don't think about it -- just compute:

Cactus Graph (p,dp)(q,dq)

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-pdpqdq.jpg

The cactus formula (p, dp)(q, dq) and its corresponding graph arise by substituting p + dp for p and q + dq for q in the

boolean product or logical conjunction pq and writing the result in the two dialects of cactus syntax. This follows

from the fact the boolean sum p + dp is equivalent to the logical operation of exclusive disjunction, which parses to a

cactus graph of the following form:

Cactus Graph (p,dp)

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-pdp.jpg

Next question: What is the difference between the value of the proposition pq over there, at a distance of dp and dq,

and the value of the proposition pq where you are standing, all expressed in the form of a general formula, of course?

Here is the appropriate formulation:

Cactus Graph ((p,dp)(q,dq),pq)

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-pdpqdqpq.jpg

There is one thing I ought to mention at this point: Computed over B, plus and minus are identical operations. This

will make the relation between the differential and the integral parts of the appropriate calculus slightly stranger

than usual, but we will get into that later.

Last question, for now: What is the value of this expression from your current standpoint, that is, evaluated at the

point where pq is true? Well, substituting 1 for p and 1 for q in the graph amounts to erasing the labels p and q, as

shown here:

Cactus Graph (( ,dp)( ,dq), )

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-dp-dq-.jpg

And this is equivalent to the following graph:

Cactus Graph ((dp)(dq))

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-dpdq.jpg

We have just met with the fact that the differential of the "and" is the "or" of the differentials.

* p and q ---Diff--> dp or dq

Cactus Graph pq Diff ((dp)(dq))

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-pq-diff-dpdq.jpg

It will be necessary to develop a more refined analysis of that statement directly, but that is roughly the nub of it.

If the form of the above statement reminds you of De Morgan's rule, it is no accident, as differentiation and negation

turn out to be closely related operations. Indeed, one can find discussions of logical difference calculus in the

Boole-De Morgan correspondence and Peirce also made use of differential operators in a logical context, but the

exploration of these ideas has been hampered by a number of factors, not the least of which has been the lack of a

syntax adequate to handle the complexity of expressions evolving in the process.

Regards,

Jon

Mar 29, 2020, 12:34:33 PM3/29/20

to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Differential Logic : 5

At: http://inquiryintoinquiry.com/2020/03/28/differential-logic-%e2%80%a2-5/

Note. I gave it the old college try at transcribing

the following math formulas but I recommend following

the link above for a much more readable copy.

Differential Expansions of Propositions

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

Worm's Eye View

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

Let's run through the initial example again, keeping an eye on the meanings of the formulas which develop along the way.

We begin with a proposition or a boolean function f(p, q) = pq whose venn diagram and cactus graph are shown below.

Venn Diagram f = pq

https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagram-f-p-and-q.jpg

Cactus Graph f = pq

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-f-p-and-q.jpg

A function like this has an abstract type and a concrete type. The abstract type is what we invoke when we write things

like f : B x B -> B or f : B^2 -> B. The concrete type takes into account the qualitative dimensions or the "units" of

the case, which can be explained as follows.

* Let P be the set of values {(p), p} = {not p, p} isomorphic to B = {0, 1}.

* Let Q be the set of values {(q), q} = {not q, q} isomorphic to B = {0, 1}.

Then interpret the usual propositions about p, q as functions of the concrete type f : P x Q -> B.

We are going to consider various operators on these functions. An operator F is a function which takes one function f

into another function Ff.

The first couple of operators we need to consider are logical analogues of two which play a founding role in the

classical finite difference calculus, namely:

* The difference operator Delta, written here as D.

* The enlargement operator Epsilon, written here as E.

These days, E is more often called the "shift operator".

In order to describe the universe in which these operators operate, it is necessary to enlarge the original universe of

discourse. Starting from the initial space X = P x Q, its "(first order) differential extension" EX is constructed

according to the following specifications:

* EX = X x dX

where:

* X = P x Q

* dX = dP x dQ

* dP = {(dp), dp}

* dQ = {(dq), dq}

The interpretations of these new symbols can be diverse, but the easiest option for now is just to say dp means "change

p" and dq means "change q".

Drawing a venn diagram for the differential extension EX = X x dX requires four logical dimensions, P, Q, dP, dQ, but it

is possible to project a suggestion of what the differential features dp and dq are about on the 2-dimensional base

space X = P x Q by drawing arrows that cross the boundaries of the basic circles in the venn diagram for X, reading an

arrow as dp if it crosses the boundary between p and (p) in either direction and reading an arrow as dq if it crosses

the boundary between q and (q) in either direction, as indicated in the following figure.

Venn Diagram p q dp dq

https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagram-p-q-dp-dq.jpg

Propositions are formed on differential variables, or any combination of ordinary logical variables and differential

logical variables, in the same ways propositions are formed on ordinary logical variables alone. For example, the

proposition (dp (dq)) says the same thing as dp => dq, in other words, there is no change in p without a change in q.

Given the proposition f(p, q) over the space X = P x Q, the "(first order) enlargement of f" is the proposition Ef over

the differential extension EX defined by the following formula:

* Ef(p, q, dp, dq)

= f(p + dp, q + dq)

= f(p xor dp, q xor dq)

In the example f(p, q) = pq, the enlargement Ef is computed as follows:

* Ef(p, q, dp, dq)

= (p + dp)(q + dq)

= (p xor dp)(q xor dq)

The corresponding cactus graph is shown below.

Cactus Graph Ef = (p,dp)(q,dq)

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-ef-pdpqdq.jpg

Given the proposition f(p, q) over X = P x Q, the "(first order) difference of f" is the proposition Df over EX defined

by the formula Df = Ef - f, or, written out in full:

* Df(p, q, dp, dq)

= f(p + dp, q + dq) - f(p, q)

= f(p xor dp, q xor dq) xor f(p, q)

In the example f(p, q) = pq, the difference Df is computed as follows:

* Df(p, q, dp, dq)

= (p + dp)(q + dq) - pq

= (p xor dp)(q xor dq) xor pq

The corresponding cactus graph is shown below.

Cactus Graph Df = ((p,dp)(q,dq),pq)

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-df-pdpqdqpq.jpg

At the end of the previous section we evaluated this first order difference of conjunction Df at a single location in

the universe of discourse, namely, at the point picked out by the singular proposition pq, in terms of coordinates, at

the place where p = 1 and q = 1. This evaluation is written in the form Df|_{pq} or Df|_{(1, 1)}, and we arrived at the

locally applicable law which may be stated and illustrated as follows:

* f(p, q) = pq = p and q => Df|_{pq} = ((dp)(dq)) = dp or dq

Venn Diagram Difference pq @ pq

https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagram-pq-difference-pq-40-pq-1.jpg

Cactus Graph Difference pq @ pq

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-pq-difference-pq-40-pq.jpg

The venn diagram shows the analysis of the inclusive disjunction "dp or dq" into the following exclusive disjunction:

* (dp and not dq) xor (dq and not dp) xor (dp and dq)

The resultant differential proposition may be read to say "change p or change q or both". And this can be recognized as

just what you need to do if you happen to find yourself in the center cell and require a complete and detailed

description of ways to escape it.

Regards,

Jon

At: http://inquiryintoinquiry.com/2020/03/28/differential-logic-%e2%80%a2-5/

Note. I gave it the old college try at transcribing

the following math formulas but I recommend following

the link above for a much more readable copy.

Differential Expansions of Propositions

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

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

Let's run through the initial example again, keeping an eye on the meanings of the formulas which develop along the way.

We begin with a proposition or a boolean function f(p, q) = pq whose venn diagram and cactus graph are shown below.

Venn Diagram f = pq

https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagram-f-p-and-q.jpg

Cactus Graph f = pq

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-f-p-and-q.jpg

A function like this has an abstract type and a concrete type. The abstract type is what we invoke when we write things

like f : B x B -> B or f : B^2 -> B. The concrete type takes into account the qualitative dimensions or the "units" of

the case, which can be explained as follows.

* Let P be the set of values {(p), p} = {not p, p} isomorphic to B = {0, 1}.

* Let Q be the set of values {(q), q} = {not q, q} isomorphic to B = {0, 1}.

Then interpret the usual propositions about p, q as functions of the concrete type f : P x Q -> B.

We are going to consider various operators on these functions. An operator F is a function which takes one function f

into another function Ff.

The first couple of operators we need to consider are logical analogues of two which play a founding role in the

classical finite difference calculus, namely:

* The difference operator Delta, written here as D.

* The enlargement operator Epsilon, written here as E.

These days, E is more often called the "shift operator".

In order to describe the universe in which these operators operate, it is necessary to enlarge the original universe of

discourse. Starting from the initial space X = P x Q, its "(first order) differential extension" EX is constructed

according to the following specifications:

* EX = X x dX

where:

* X = P x Q

* dX = dP x dQ

* dP = {(dp), dp}

* dQ = {(dq), dq}

The interpretations of these new symbols can be diverse, but the easiest option for now is just to say dp means "change

p" and dq means "change q".

Drawing a venn diagram for the differential extension EX = X x dX requires four logical dimensions, P, Q, dP, dQ, but it

is possible to project a suggestion of what the differential features dp and dq are about on the 2-dimensional base

space X = P x Q by drawing arrows that cross the boundaries of the basic circles in the venn diagram for X, reading an

arrow as dp if it crosses the boundary between p and (p) in either direction and reading an arrow as dq if it crosses

the boundary between q and (q) in either direction, as indicated in the following figure.

Venn Diagram p q dp dq

https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagram-p-q-dp-dq.jpg

Propositions are formed on differential variables, or any combination of ordinary logical variables and differential

logical variables, in the same ways propositions are formed on ordinary logical variables alone. For example, the

proposition (dp (dq)) says the same thing as dp => dq, in other words, there is no change in p without a change in q.

Given the proposition f(p, q) over the space X = P x Q, the "(first order) enlargement of f" is the proposition Ef over

the differential extension EX defined by the following formula:

* Ef(p, q, dp, dq)

= f(p + dp, q + dq)

= f(p xor dp, q xor dq)

In the example f(p, q) = pq, the enlargement Ef is computed as follows:

* Ef(p, q, dp, dq)

= (p + dp)(q + dq)

= (p xor dp)(q xor dq)

The corresponding cactus graph is shown below.

Cactus Graph Ef = (p,dp)(q,dq)

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-ef-pdpqdq.jpg

Given the proposition f(p, q) over X = P x Q, the "(first order) difference of f" is the proposition Df over EX defined

by the formula Df = Ef - f, or, written out in full:

* Df(p, q, dp, dq)

= f(p + dp, q + dq) - f(p, q)

= f(p xor dp, q xor dq) xor f(p, q)

In the example f(p, q) = pq, the difference Df is computed as follows:

* Df(p, q, dp, dq)

= (p + dp)(q + dq) - pq

= (p xor dp)(q xor dq) xor pq

The corresponding cactus graph is shown below.

Cactus Graph Df = ((p,dp)(q,dq),pq)

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-df-pdpqdqpq.jpg

At the end of the previous section we evaluated this first order difference of conjunction Df at a single location in

the universe of discourse, namely, at the point picked out by the singular proposition pq, in terms of coordinates, at

the place where p = 1 and q = 1. This evaluation is written in the form Df|_{pq} or Df|_{(1, 1)}, and we arrived at the

locally applicable law which may be stated and illustrated as follows:

* f(p, q) = pq = p and q => Df|_{pq} = ((dp)(dq)) = dp or dq

Venn Diagram Difference pq @ pq

https://inquiryintoinquiry.files.wordpress.com/2020/03/venn-diagram-pq-difference-pq-40-pq-1.jpg

Cactus Graph Difference pq @ pq

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-pq-difference-pq-40-pq.jpg

The venn diagram shows the analysis of the inclusive disjunction "dp or dq" into the following exclusive disjunction:

* (dp and not dq) xor (dq and not dp) xor (dp and dq)

The resultant differential proposition may be read to say "change p or change q or both". And this can be recognized as

just what you need to do if you happen to find yourself in the center cell and require a complete and detailed

description of ways to escape it.

Regards,

Jon

Apr 4, 2020, 11:00:32 AM4/4/20

to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Differential Logic • 6

At: http://inquiryintoinquiry.com/2020/04/03/differential-logic-%e2%80%a2-6/

Differential Expansions of Propositions

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

Panoptic View • Difference Maps

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

In the last section we computed what is variously called the "difference map", the "difference proposition", or the

"local proposition" Df_x of the proposition f(p, q) = pq at the point x where p = 1 and q = 1.

In the universe of discourse X = P × Q, the four propositions pq, p (q), (p) q, (p)(q) indicating the "cells", or the

smallest distinguished regions of the universe, are called "singular propositions". These serve as an alternative

notation for naming the points (1, 1), (1, 0), (0, 1), (0, 0), respectively.

Thus we can write D}f_x = Df|_x = Df|_(1, 1) = Df|_pq, so long as we know the frame of reference in force.

In the example f(p, q) = pq, the value of the difference proposition Df_x at each of the four points x in X may be

computed in graphical fashion as shown below.

Cactus Graph Df = ((p,dp)(q,dq),pq)

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-df-pdpqdqpq-1.jpg

Cactus Graph Difference pq @ pq = ((dp)(dq))

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-difference-pq-40-pq-dpdq.jpg

Cactus Graph Difference pq @ p(q) = (dp)dq

https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-difference-pq-40-pq-dpdq.jpg

Cactus Graph Difference pq @ (p)q = dp(dq)

https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-difference-pq-40-pq-dpdq-1.jpg

Cactus Graph Difference pq @ (p)(q) = dp dq

https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-difference-pq-40-pq-dp-dq.jpg

The easy way to visualize the values of these graphical expressions is just to notice the following equivalents.

Cactus Graph Lobe Rule

https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-ej-lobe-rule.jpg

Cactus Graph Spike Rule

https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-ej-spike-rule.jpg

Laying out the arrows on the augmented venn diagram, one gets a picture of a differential vector field.

Venn Diagram Difference pq

https://inquiryintoinquiry.files.wordpress.com/2020/04/venn-diagram-difference-pq-1.jpg

The Figure shows the points of the extended universe EX = P × Q × dP × dQ indicated by

the difference map Df : EX → B, namely, the following six points or singular propositions.

1. p q dp dq

2. p q dp (dq)

3. p q (dp) dq

4. p (q) (dp) dq

5. (p) q dp (dq)

6. (p)(q) dp dq

The information borne by Df should be clear enough from a survey of these six points — they tell you what you have to do

from each point of X in order to change the value borne by f(p, q), that is, the move you have to make in order to reach

a point where the value of the proposition f(p, q) is different from what it is where you started.

We have been studying the action of the difference operator D on propositions of the form f : P × Q → B, as illustrated

by the example f(p, q) = pq = the conjunction of p and q. The resulting difference map Df is a "(first order)

differential proposition", that is, a proposition of the form Df : P × Q × dP × dQ → B.

The augmented venn diagram shows how the "models" or "satisfying interpretations" of Df distribute over the extended

universe of discourse EX = P × Q × dP × dQ. Abstracting from that picture, the difference map Df can be represented in

the form of a "digraph" or "directed graph", one whose points are labeled with the elements of X = P × Q and whose

arrows are labeled with the elements of dX = dP × dQ, as shown in the following Figure.

Directed Graph Difference pq

https://inquiryintoinquiry.files.wordpress.com/2020/04/directed-graph-difference-pq.jpg

The same 6 points of the extended universe EX = P × Q × dP × dQ given by

the difference map Df : EX → B can be described by the following formula.

Df = p q · ((dp)(dq))

+ p (q) · (dp) dq

+ (p) q · dp (dq)

+ (p)(q) · dp dq

https://en.wikipedia.org/api/rest_v1/media/math/render/svg/347100e1473cdb29b37a928a60eb0661486a1937

Any proposition worth its salt can be analyzed from many different points of view, any one of which has the potential to

reveal previously unsuspected aspects of the proposition’s meaning. We will encounter more and more of these

alternative readings as we go.

At: http://inquiryintoinquiry.com/2020/04/03/differential-logic-%e2%80%a2-6/

Differential Expansions of Propositions

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

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

In the last section we computed what is variously called the "difference map", the "difference proposition", or the

"local proposition" Df_x of the proposition f(p, q) = pq at the point x where p = 1 and q = 1.

In the universe of discourse X = P × Q, the four propositions pq, p (q), (p) q, (p)(q) indicating the "cells", or the

smallest distinguished regions of the universe, are called "singular propositions". These serve as an alternative

notation for naming the points (1, 1), (1, 0), (0, 1), (0, 0), respectively.

Thus we can write D}f_x = Df|_x = Df|_(1, 1) = Df|_pq, so long as we know the frame of reference in force.

In the example f(p, q) = pq, the value of the difference proposition Df_x at each of the four points x in X may be

computed in graphical fashion as shown below.

Cactus Graph Df = ((p,dp)(q,dq),pq)

Cactus Graph Difference pq @ pq = ((dp)(dq))

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-difference-pq-40-pq-dpdq.jpg

Cactus Graph Difference pq @ p(q) = (dp)dq

https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-difference-pq-40-pq-dpdq.jpg

Cactus Graph Difference pq @ (p)q = dp(dq)

https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-difference-pq-40-pq-dpdq-1.jpg

Cactus Graph Difference pq @ (p)(q) = dp dq

https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-difference-pq-40-pq-dp-dq.jpg

The easy way to visualize the values of these graphical expressions is just to notice the following equivalents.

Cactus Graph Lobe Rule

https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-ej-lobe-rule.jpg

Cactus Graph Spike Rule

https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-ej-spike-rule.jpg

Laying out the arrows on the augmented venn diagram, one gets a picture of a differential vector field.

Venn Diagram Difference pq

https://inquiryintoinquiry.files.wordpress.com/2020/04/venn-diagram-difference-pq-1.jpg

The Figure shows the points of the extended universe EX = P × Q × dP × dQ indicated by

the difference map Df : EX → B, namely, the following six points or singular propositions.

1. p q dp dq

2. p q dp (dq)

3. p q (dp) dq

4. p (q) (dp) dq

5. (p) q dp (dq)

6. (p)(q) dp dq

The information borne by Df should be clear enough from a survey of these six points — they tell you what you have to do

from each point of X in order to change the value borne by f(p, q), that is, the move you have to make in order to reach

a point where the value of the proposition f(p, q) is different from what it is where you started.

We have been studying the action of the difference operator D on propositions of the form f : P × Q → B, as illustrated

by the example f(p, q) = pq = the conjunction of p and q. The resulting difference map Df is a "(first order)

differential proposition", that is, a proposition of the form Df : P × Q × dP × dQ → B.

The augmented venn diagram shows how the "models" or "satisfying interpretations" of Df distribute over the extended

universe of discourse EX = P × Q × dP × dQ. Abstracting from that picture, the difference map Df can be represented in

the form of a "digraph" or "directed graph", one whose points are labeled with the elements of X = P × Q and whose

arrows are labeled with the elements of dX = dP × dQ, as shown in the following Figure.

Directed Graph Difference pq

https://inquiryintoinquiry.files.wordpress.com/2020/04/directed-graph-difference-pq.jpg

The same 6 points of the extended universe EX = P × Q × dP × dQ given by

the difference map Df : EX → B can be described by the following formula.

Df = p q · ((dp)(dq))

+ p (q) · (dp) dq

+ (p) q · dp (dq)

+ (p)(q) · dp dq

https://en.wikipedia.org/api/rest_v1/media/math/render/svg/347100e1473cdb29b37a928a60eb0661486a1937

Any proposition worth its salt can be analyzed from many different points of view, any one of which has the potential to

reveal previously unsuspected aspects of the proposition’s meaning. We will encounter more and more of these

alternative readings as we go.

Apr 6, 2020, 10:24:20 AM4/6/20

to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Differential Logic • 7

At: http://inquiryintoinquiry.com/2020/04/05/differential-logic-%e2%80%a2-7/

Differential Expansions of Propositions

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

Panoptic View • Enlargement Maps

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

The "enlargement" or "shift" operator E exhibits a wealth of interesting and useful properties in its own right, so it

pays to examine a few of the more salient features playing out on the surface of our initial example, f(p, q) = pq.

A suitably generic definition of the extended universe of discourse is afforded by the following set-up.

* Let X = X_1 × ... × X_k.

* Let dX = dX_1 × ... × dX_k.

* Then EX = X × dX

= X_1 × ... × X_k × dX_1 × ... × dX_k

For a proposition of the form f : X_1 × ... × X_k → B, the "(first order) enlargement" of f is the proposition Ef : EX →

B defined by the following equation.

* Ef(x_1, ..., x_k, dx_1, ..., dx_k)

= f(x_1 + dx_1, ..., x_k + dx_k)

= f(x_1 xor dx_1, ..., x_k xor dx_k)

The "differential variables" dx_j are boolean variables of the same type as the ordinary variables x_j. Although it is

conventional to distinguish the (first order) differential variables with the operational prefix "d" this way of

notating differential variables is entirely optional. It is their existence in particular relations to the initial

variables, not their names, which defines them as differential variables.

In the example of logical conjunction, f(p, q) = pq, the enlargement Ef is formulated as follows.

* Ef(p, q, dp, dq) = (p + dp)(q + dq) = (p xor dp)(q xor dq)

Given that this expression uses nothing more than the boolean ring operations of addition and multiplication, it is

permissible to “multiply things out” in the usual manner to arrive at the following result.

* Ef(p, q, dp, dq) = p·q + p·dq + q·dp + dp·dq

To understand what the enlarged or shifted proposition means in logical terms, it serves to go back and analyze the

above expression for Ef in the same way we did for Df. To that end, the value of Ef_x at each x in X may be computed in

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-ef-pdpqdq-1.jpg

Cactus Graph Enlargement pq @ pq = (dp)(dq)

https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-enlargement-pq-40-pq-dpdq.jpg

Cactus Graph Enlargement pq @ p(q) = (dp)dq

https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-enlargement-pq-40-pq-dpdq-1.jpg

Cactus Graph Enlargement pq @ (p)q = dp(dq)

https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-enlargement-pq-40-pq-dpdq-2.jpg

Cactus Graph Enlargement pq @ (p)(q) = dp dq

https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-enlargement-pq-40-pq-dp-dq.jpg

Collating the data of this analysis yields a boolean expansion or disjunctive normal form (DNF) equivalent to the

enlarged proposition Ef.

* Ef = p q · Ef @ p q

+ p (q) · Ef @ p (q)

+ (p) q · Ef @ (p) q}

+ (p)(q) · Ef @ (p)(q)

Here is a summary of the result, illustrated by means of a digraph picture, where the “no change” element ¬dp∧¬dq is

drawn as a loop at the point pq.

Directed Graph Enlargement pq

https://inquiryintoinquiry.files.wordpress.com/2020/04/directed-graph-enlargement-pq.jpg

https://en.wikipedia.org/api/rest_v1/media/math/render/svg/6b32eadd66a62c7cefc8c31d52f46708d68f3adc

We may understand the enlarged proposition Ef as telling us all the ways of reaching a model of the proposition f from

the points of the universe X.

Regards,

Jon

At: http://inquiryintoinquiry.com/2020/04/05/differential-logic-%e2%80%a2-7/

Differential Expansions of Propositions

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

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

The "enlargement" or "shift" operator E exhibits a wealth of interesting and useful properties in its own right, so it

pays to examine a few of the more salient features playing out on the surface of our initial example, f(p, q) = pq.

A suitably generic definition of the extended universe of discourse is afforded by the following set-up.

* Let X = X_1 × ... × X_k.

* Let dX = dX_1 × ... × dX_k.

* Then EX = X × dX

= X_1 × ... × X_k × dX_1 × ... × dX_k

For a proposition of the form f : X_1 × ... × X_k → B, the "(first order) enlargement" of f is the proposition Ef : EX →

B defined by the following equation.

* Ef(x_1, ..., x_k, dx_1, ..., dx_k)

= f(x_1 + dx_1, ..., x_k + dx_k)

= f(x_1 xor dx_1, ..., x_k xor dx_k)

The "differential variables" dx_j are boolean variables of the same type as the ordinary variables x_j. Although it is

conventional to distinguish the (first order) differential variables with the operational prefix "d" this way of

notating differential variables is entirely optional. It is their existence in particular relations to the initial

variables, not their names, which defines them as differential variables.

In the example of logical conjunction, f(p, q) = pq, the enlargement Ef is formulated as follows.

* Ef(p, q, dp, dq) = (p + dp)(q + dq) = (p xor dp)(q xor dq)

permissible to “multiply things out” in the usual manner to arrive at the following result.

* Ef(p, q, dp, dq) = p·q + p·dq + q·dp + dp·dq

To understand what the enlarged or shifted proposition means in logical terms, it serves to go back and analyze the

above expression for Ef in the same way we did for Df. To that end, the value of Ef_x at each x in X may be computed in

graphical fashion as shown below.

Cactus Graph Ef = (p,dp)(q,dq)
https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-graph-ef-pdpqdq-1.jpg

Cactus Graph Enlargement pq @ pq = (dp)(dq)

https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-enlargement-pq-40-pq-dpdq.jpg

Cactus Graph Enlargement pq @ p(q) = (dp)dq

https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-enlargement-pq-40-pq-dpdq-1.jpg

Cactus Graph Enlargement pq @ (p)q = dp(dq)

https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-enlargement-pq-40-pq-dpdq-2.jpg

Cactus Graph Enlargement pq @ (p)(q) = dp dq

https://inquiryintoinquiry.files.wordpress.com/2020/04/cactus-graph-enlargement-pq-40-pq-dp-dq.jpg

Collating the data of this analysis yields a boolean expansion or disjunctive normal form (DNF) equivalent to the

enlarged proposition Ef.

* Ef = p q · Ef @ p q

+ p (q) · Ef @ p (q)

+ (p) q · Ef @ (p) q}

+ (p)(q) · Ef @ (p)(q)

Here is a summary of the result, illustrated by means of a digraph picture, where the “no change” element ¬dp∧¬dq is

drawn as a loop at the point pq.

Directed Graph Enlargement pq

https://inquiryintoinquiry.files.wordpress.com/2020/04/directed-graph-enlargement-pq.jpg

https://en.wikipedia.org/api/rest_v1/media/math/render/svg/6b32eadd66a62c7cefc8c31d52f46708d68f3adc

We may understand the enlarged proposition Ef as telling us all the ways of reaching a model of the proposition f from

the points of the universe X.

Regards,

Jon

Apr 8, 2020, 10:12:32 AM4/8/20

to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Differential Logic • 8

At: http://inquiryintoinquiry.com/2020/04/08/differential-logic-%e2%80%a2-8/

Propositional Forms on Two Variables

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

To broaden our experience with simple examples, let's examine the sixteen functions of concrete type P × Q → B and

abstract type B × B → B. The time we took contemplating logical conjunction from a variety of differential angles will

pay dividends as we study its kindred family of forms in the same lights.

Table A1 arranges the propositional forms on two variables in a convenient order, giving equivalent expressions for each

boolean function in several systems of notation.

Table A1. Propositional Forms on Two Variables

https://inquiryintoinquiry.files.wordpress.com/2020/04/table-a1.-propositional-forms-on-two-variables.png

Regards,

Jon

At: http://inquiryintoinquiry.com/2020/04/08/differential-logic-%e2%80%a2-8/

Propositional Forms on Two Variables

To broaden our experience with simple examples, let's examine the sixteen functions of concrete type P × Q → B and

abstract type B × B → B. The time we took contemplating logical conjunction from a variety of differential angles will

pay dividends as we study its kindred family of forms in the same lights.

Table A1 arranges the propositional forms on two variables in a convenient order, giving equivalent expressions for each

boolean function in several systems of notation.

Table A1. Propositional Forms on Two Variables

https://inquiryintoinquiry.files.wordpress.com/2020/04/table-a1.-propositional-forms-on-two-variables.png

Regards,

Jon

Apr 8, 2020, 4:38:19 PM4/8/20

to structura...@googlegroups.com, Cybernetic Communications, Ontolog Forum, Peirce List, SysSciWG

Jon:

I like the chart, things are starting to make a little more sense to me.

Further, I read your paper," An Architecture for Inquiry
: Building Computer Platforms for Discovery" from Research Gate.

The examples in the paper help to add additional context.

Take care, be good to yourself and have fun,

Joe

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To view this discussion on the web visit https://groups.google.com/d/msgid/structural-modeling/a5022d2b-8ca0-d75d-7947-3cffdd94de26%40att.net.

Joe Simpson

# “Reasonable people adapt themselves to the world.

# Unreasonable people attempt to adapt the world to themselves.

# All progress, therefore, depends on unreasonable people.”

- George Bernard Shaw
- Git Hub link:
- Research Gate link:
- YouTube Channel
- Web Site:

Apr 9, 2020, 10:54:15 AM4/9/20

to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Differential Logic • Discussion 1

At: http://inquiryintoinquiry.com/2020/04/09/differential-logic-%e2%80%a2-discussion-1/

Re: Joseph Simpson

At: https://groups.google.com/d/msg/structural-modeling/xB5tRt4mcEM/IfaF8YlLBgAJ

Thanks, Joe, glad you liked the table, I've got a million of 'em! I'll be setting another mess of tables directly as we

continue studying the effects of differential operators on families of propositional forms.

For anyone wondering, "Where's the Peirce?" — he is the Atlas on whose shoulders the whole world of differential logic

turns. The elegant efficiency of Peirce's logical graphs, augmented by Spencer Brown and extended to cactus graphs,

made it feasible for the first time to take on the extra complexities of differential propositional calculus. So that

theme is a constant throughout the development of differential logic.

Hope you and yours are safe and sound,

Jon

On 4/8/2020 4:38 PM, joseph simpson wrote:

> Jon:

>

> I like the chart, things are starting to make a little more sense to me.

>

> Further, I read your paper, "An Architecture for Inquiry :

> Building Computer Platforms for Discovery" from Research Gate.

[ https://www.academia.edu/1270327/An_Architecture_for_Inquiry_Building_Computer_Platforms_for_Discovery ]

At: http://inquiryintoinquiry.com/2020/04/09/differential-logic-%e2%80%a2-discussion-1/

Re: Joseph Simpson

At: https://groups.google.com/d/msg/structural-modeling/xB5tRt4mcEM/IfaF8YlLBgAJ

Thanks, Joe, glad you liked the table, I've got a million of 'em! I'll be setting another mess of tables directly as we

continue studying the effects of differential operators on families of propositional forms.

For anyone wondering, "Where's the Peirce?" — he is the Atlas on whose shoulders the whole world of differential logic

turns. The elegant efficiency of Peirce's logical graphs, augmented by Spencer Brown and extended to cactus graphs,

made it feasible for the first time to take on the extra complexities of differential propositional calculus. So that

theme is a constant throughout the development of differential logic.

Hope you and yours are safe and sound,

Jon

On 4/8/2020 4:38 PM, joseph simpson wrote:

> Jon:

>

> I like the chart, things are starting to make a little more sense to me.

>

> Further, I read your paper, "An Architecture for Inquiry :

> Building Computer Platforms for Discovery" from Research Gate.

Apr 11, 2020, 10:13:11 AM4/11/20

to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Differential Logic • 9

At: http://inquiryintoinquiry.com/2020/04/11/differential-logic-%e2%80%a2-9/

Propositional Forms on Two Variables

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

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2#Propositional_Forms_on_Two_Variables

Table A2 arranges the propositional forms on two variables according to another plan, sorting propositions with similar

shapes into seven subclasses. Thereby hangs many a tale, to be told in time.

Table A2. Propositional Forms on Two Variables

https://inquiryintoinquiry.files.wordpress.com/2020/04/table-a2.-propositional-forms-on-two-variables-1.png

Regards,

Jon

At: http://inquiryintoinquiry.com/2020/04/11/differential-logic-%e2%80%a2-9/

Propositional Forms on Two Variables

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_2#Propositional_Forms_on_Two_Variables

Table A2 arranges the propositional forms on two variables according to another plan, sorting propositions with similar

shapes into seven subclasses. Thereby hangs many a tale, to be told in time.

Table A2. Propositional Forms on Two Variables

https://inquiryintoinquiry.files.wordpress.com/2020/04/table-a2.-propositional-forms-on-two-variables-1.png

Regards,

Jon

Apr 13, 2020, 9:30:31 PM4/13/20

to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

On 4/9/2020 11:02 AM, Edwina Taborsky wrote:

>

> Yes - I think Peirce and Spencer Brown work very well together.

>

> Thanks for all your work.

>

> Edwina

Thanks, Edwina. I first encountered Peirce's Collected Papers

sometime during my freshman year in one of the quieter corners

of the Michigan State Math Library where I used to hide out to

study and shortly after a friend showed me the description of

Spencer Brown's "Laws of Form" in the 1st Whole Earth Catalog

and I sent off for it right away. I would spend the next ten

years trying to figure out what either one of them was saying.

In my view, Spencer Brown penetrated to the deepest strata of

Peirce's core ideas about logic, recognizing its operational

aspect and relational power in a way we've seldom seen since.

Not too coincidentally, those aspects and powers were a big

part of what I wrote my Senior Thesis on at the end of my

undergrad years.

Regards,

Jon

>

> Yes - I think Peirce and Spencer Brown work very well together.

>

> Thanks for all your work.

>

> Edwina

Thanks, Edwina. I first encountered Peirce's Collected Papers

sometime during my freshman year in one of the quieter corners

of the Michigan State Math Library where I used to hide out to

study and shortly after a friend showed me the description of

Spencer Brown's "Laws of Form" in the 1st Whole Earth Catalog

and I sent off for it right away. I would spend the next ten

years trying to figure out what either one of them was saying.

In my view, Spencer Brown penetrated to the deepest strata of

Peirce's core ideas about logic, recognizing its operational

aspect and relational power in a way we've seldom seen since.

Not too coincidentally, those aspects and powers were a big

part of what I wrote my Senior Thesis on at the end of my

undergrad years.

Regards,

Jon

Apr 26, 2020, 7:30:41 AM4/26/20

to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Differential Logic • 10

At: http://inquiryintoinquiry.com/2020/04/25/differential-logic-%e2%80%a2-10/

It’s been a while, so let’s review …

Tables A1 and A2 showed two ways of organizing the sixteen boolean functions or propositional forms on two variables, as

expressed in several notations. For ease of reference, here are fresh copies of those tables.

Table A1. Propositional Forms on Two Variables

https://inquiryintoinquiry.files.wordpress.com/2020/04/table-a1.-propositional-forms-on-two-variables.png

Table A2. Propositional Forms on Two Variables

https://inquiryintoinquiry.files.wordpress.com/2020/04/table-a2.-propositional-forms-on-two-variables-1.png

We took as our first example the boolean function f_8(p, q) = pq corresponding to the logical conjunction p ∧ q and

examined how the differential operators E and D act on f_8. Each differential operator takes a boolean function of two

variables f_8(p, q) and gives back a boolean function of four variables, Ef_8(p, q, dp, dq) or Df_8(p, q, dp, dq),

respectively.

In the next several posts we’ll extend our scope to the full set of boolean functions on two variables and examine how

the differential operators E and D act on that set. There being some advantage to singling out the enlargement or shift

operator E in its own right, we’ll begin by computing Ef for each function f in the above tables.

Regards,

Jon

At: http://inquiryintoinquiry.com/2020/04/25/differential-logic-%e2%80%a2-10/

It’s been a while, so let’s review …

Tables A1 and A2 showed two ways of organizing the sixteen boolean functions or propositional forms on two variables, as

expressed in several notations. For ease of reference, here are fresh copies of those tables.

Table A1. Propositional Forms on Two Variables

https://inquiryintoinquiry.files.wordpress.com/2020/04/table-a1.-propositional-forms-on-two-variables.png

Table A2. Propositional Forms on Two Variables

https://inquiryintoinquiry.files.wordpress.com/2020/04/table-a2.-propositional-forms-on-two-variables-1.png

We took as our first example the boolean function f_8(p, q) = pq corresponding to the logical conjunction p ∧ q and

examined how the differential operators E and D act on f_8. Each differential operator takes a boolean function of two

variables f_8(p, q) and gives back a boolean function of four variables, Ef_8(p, q, dp, dq) or Df_8(p, q, dp, dq),

respectively.

In the next several posts we’ll extend our scope to the full set of boolean functions on two variables and examine how

the differential operators E and D act on that set. There being some advantage to singling out the enlargement or shift

operator E in its own right, we’ll begin by computing Ef for each function f in the above tables.

Regards,

Jon

Jun 17, 2020, 2:45:30 PM6/17/20

to Cybernetic Communications, Ontolog Forum, Peirce List, Structural Modeling, SysSciWG

Cf: Differential Logic • Discussion 3

At: http://inquiryintoinquiry.com/2020/06/17/differential-logic-%e2%80%a2-discussion-3/

Re: R.J. Lipton

https://rjlipton.wordpress.com/about-me/

Re: P<NP

https://rjlipton.wordpress.com/2020/06/16/pnp/

Instead of boolean circuit complexity I would look at logical graph complexity, where those logical graphs are

constructed from minimal negation operators.

Physics once had a frame problem (complexity of dynamic updating) long before AI did but physics learned to reduce

complexity through the use of differential equations and group symmetries (combined in Lie groups).

One of the promising features of minimal negation operators is their relationship to differential operators. So I've

been looking into that. Here’s a link, a bit in medias res, but what I've got for now.

• Differential Logic • Cactus Language

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Cactus_Language_for_Propositional_Logic

Resources

=========

• Logical Graphs

https://oeis.org/wiki/Logical_Graphs

• Minimal Negation Operators

https://oeis.org/wiki/Minimal_negation_operator

• Survey of Animated Logical Graphs

https://inquiryintoinquiry.com/2019/05/22/survey-of-animated-logical-graphs-%e2%80%a2-2/

• Survey of Differential Logic

https://inquiryintoinquiry.com/2020/02/08/survey-of-differential-logic-%e2%80%a2-2/

• Survey of Theme One Program

https://inquiryintoinquiry.com/2018/02/25/survey-of-theme-one-program-%e2%80%a2-2/

Regards,

Jon

At: http://inquiryintoinquiry.com/2020/06/17/differential-logic-%e2%80%a2-discussion-3/

Re: R.J. Lipton

https://rjlipton.wordpress.com/about-me/

Re: P<NP

https://rjlipton.wordpress.com/2020/06/16/pnp/

Instead of boolean circuit complexity I would look at logical graph complexity, where those logical graphs are

constructed from minimal negation operators.

Physics once had a frame problem (complexity of dynamic updating) long before AI did but physics learned to reduce

complexity through the use of differential equations and group symmetries (combined in Lie groups).

One of the promising features of minimal negation operators is their relationship to differential operators. So I've

been looking into that. Here’s a link, a bit in medias res, but what I've got for now.

• Differential Logic • Cactus Language

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Cactus_Language_for_Propositional_Logic

Resources

=========

• Logical Graphs

https://oeis.org/wiki/Logical_Graphs

• Minimal Negation Operators

https://oeis.org/wiki/Minimal_negation_operator

• Survey of Animated Logical Graphs

https://inquiryintoinquiry.com/2019/05/22/survey-of-animated-logical-graphs-%e2%80%a2-2/

• Survey of Differential Logic

https://inquiryintoinquiry.com/2020/02/08/survey-of-differential-logic-%e2%80%a2-2/

• Survey of Theme One Program

https://inquiryintoinquiry.com/2018/02/25/survey-of-theme-one-program-%e2%80%a2-2/

Regards,

Jon

Jun 15, 2021, 10:54:38 AM6/15/21

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

Cf: Differential Logic • Overview

https://inquiryintoinquiry.com/2020/03/20/differential-logic-overview/

LoF Group,

| The following series of posts on Differential Logic were

| shared to my other lists back in 2020 when the LoF Group

| was experiencing its bout of “listlessness”. I'll copy

| them here partly by way of general background and also

| for context in answering Lyle's last set of questions.

The previous series of posts on Differential Propositional Calculus

( https://inquiryintoinquiry.com/?s=Differential+Propositional+Calculus )

brought us to the threshold of the subject without quite stepping over,

but I wanted to lay out the necessary ingredients in the most concrete,

intuitive, and visual way possible before taking up the abstract forms.

One of my readers on Facebook told me “venn diagrams are obsolete” and

of course we all know they become unwieldy as our universes of discourse

expand beyond four or five dimensions. Indeed, one of the first lessons

I learned when I set about implementing CSP's graphs and GSB's forms on the

computer was that 2-dimensional representations of logic are a death trap

in numerous conceptual and computational ways. Still, venn diagrams do us

good service in visualizing the relationships among extensional, functional,

and intensional aspects of logic. A facility with those relationships is

critical to the computational applications and statistical generalizations

of logic commonly used in mathematical and empirical practice.

At any rate, intrepid readers will have amped up their visual imaginations

well enough at this point to pick their way through the cactus patch ahead.

The link above or the transcript below outlines my last, best introduction

to Differential Logic, which I'll be working to improve as I serialize it

to my blog.

Resource

========

Differential Logic

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Overview

https://inquiryintoinquiry.com/2020/03/20/differential-logic-overview/

LoF Group,

| The following series of posts on Differential Logic were

| shared to my other lists back in 2020 when the LoF Group

| was experiencing its bout of “listlessness”. I'll copy

| them here partly by way of general background and also

| for context in answering Lyle's last set of questions.

The previous series of posts on Differential Propositional Calculus

( https://inquiryintoinquiry.com/?s=Differential+Propositional+Calculus )

brought us to the threshold of the subject without quite stepping over,

but I wanted to lay out the necessary ingredients in the most concrete,

intuitive, and visual way possible before taking up the abstract forms.

One of my readers on Facebook told me “venn diagrams are obsolete” and

of course we all know they become unwieldy as our universes of discourse

expand beyond four or five dimensions. Indeed, one of the first lessons

I learned when I set about implementing CSP's graphs and GSB's forms on the

computer was that 2-dimensional representations of logic are a death trap

in numerous conceptual and computational ways. Still, venn diagrams do us

good service in visualizing the relationships among extensional, functional,

and intensional aspects of logic. A facility with those relationships is

critical to the computational applications and statistical generalizations

of logic commonly used in mathematical and empirical practice.

At any rate, intrepid readers will have amped up their visual imaginations

well enough at this point to pick their way through the cactus patch ahead.

The link above or the transcript below outlines my last, best introduction

to Differential Logic, which I'll be working to improve as I serialize it

to my blog.

========

Differential Logic

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Overview

Jun 15, 2021, 2:36:20 PM6/15/21

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

Cf: Differential Logic • 1

https://inquiryintoinquiry.com/2020/03/22/differential-logic-1/

Introduction

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

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Introduction

Differential logic is the component of logic whose object is

the description of variation — for example, the aspects of change,

difference, distribution, and diversity — in universes of discourse

subject to logical description. A definition that broad naturally

incorporates any study of variation by way of mathematical models,

but differential logic is especially charged with the qualitative

aspects of variation pervading or preceding quantitative models.

To the extent a logical inquiry makes use of a formal system,

its differential component treats the principles governing the

use of a differential logical calculus, that is, a formal system

[1] https://oeis.org/wiki/Universe_of_discourse

[2] https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Overview

[3] https://oeis.org/wiki/Propositional_calculus

https://inquiryintoinquiry.com/2020/03/22/differential-logic-1/

Introduction

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

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Introduction

Differential logic is the component of logic whose object is

the description of variation — for example, the aspects of change,

difference, distribution, and diversity — in universes of discourse

subject to logical description. A definition that broad naturally

incorporates any study of variation by way of mathematical models,

but differential logic is especially charged with the qualitative

aspects of variation pervading or preceding quantitative models.

To the extent a logical inquiry makes use of a formal system,

its differential component treats the principles governing the

with the expressive capacity to describe change and diversity in

logical universes of discourse.

Simple examples of differential logical calculi are furnished by

differential propositional calculi. A differential propositional
logical universes of discourse.

Simple examples of differential logical calculi are furnished by

calculus is a propositional calculus extended by a set of terms for

describing aspects of change and difference, for example, processes

taking place in a universe of discourse or transformations mapping

a source universe to a target universe. Such a calculus augments

ordinary propositional calculus in the same way the differential

calculus of Leibniz and Newton augments the analytic geometry

of Descartes.

References
describing aspects of change and difference, for example, processes

taking place in a universe of discourse or transformations mapping

a source universe to a target universe. Such a calculus augments

ordinary propositional calculus in the same way the differential

calculus of Leibniz and Newton augments the analytic geometry

of Descartes.

[1] https://oeis.org/wiki/Universe_of_discourse

[2] https://oeis.org/wiki/Differential_Propositional_Calculus_%E2%80%A2_Overview

[3] https://oeis.org/wiki/Propositional_calculus

Jun 15, 2021, 4:56:26 PM6/15/21

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

Cf: Differential Logic • 2

https://inquiryintoinquiry.com/2020/03/23/differential-logic-2/

Cactus Language for Propositional Logic

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

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Cactus_Language_for_Propositional_Logic

The development of differential logic is facilitated by having a moderately

efficient calculus in place at the level of boolean-valued functions and

elementary logical propositions. One very efficient calculus on both

conceptual and computational grounds is based on just two types of

logical connectives, both of variable k-ary scope. The syntactic

formulas of this calculus map into a family of graph-theoretic

structures called “painted and rooted cacti” which lend visual

representation to the functional structures of propositions

and smooth the path to efficient computation.

The first kind of connective takes the form of a parenthesized sequence

of propositional expressions, written (e₁, e₂, …, eₖ) and meaning exactly

one of the propositions e₁, e₂, …, eₖ is false, in short, their “minimal

shown below.

Figure 1. Lobe Connective

[1] https://oeis.org/wiki/Boolean-valued_function

[2] https://oeis.org/wiki/Minimal_negation_operator

[3] https://oeis.org/wiki/Logical_conjunction

https://inquiryintoinquiry.com/2020/03/23/differential-logic-2/

Cactus Language for Propositional Logic

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

The development of differential logic is facilitated by having a moderately

efficient calculus in place at the level of boolean-valued functions and

elementary logical propositions. One very efficient calculus on both

conceptual and computational grounds is based on just two types of

logical connectives, both of variable k-ary scope. The syntactic

formulas of this calculus map into a family of graph-theoretic

structures called “painted and rooted cacti” which lend visual

representation to the functional structures of propositions

and smooth the path to efficient computation.

The first kind of connective takes the form of a parenthesized sequence

one of the propositions e₁, e₂, …, eₖ is false, in short, their “minimal

negation” is true. An expression of this form maps into a cactus structure

called a “lobe”, in this case, “painted” with the colors e₁, e₂, …, eₖ as
shown below.

Figure 1. Lobe Connective

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-ej-lobe-connective.jpg

The second kind of connective is a concatenated sequence of propositional expressions,

written e₁ e₂ … eₖ and meaning all of the propositions e₁, e₂, …, eₖ are true, in short,
The second kind of connective is a concatenated sequence of propositional expressions,

their logical conjunction is true. An expression of this form maps into a cactus structure

called a “node”, in this case, “painted” with the colors e_1, e_2, ..., e_k as shown below.

Figure 2. Node Connective
called a “node”, in this case, “painted” with the colors e_1, e_2, ..., e_k as shown below.

https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-ej-node-connective.jpg

All other propositional connectives can be obtained through combinations

of these two forms. As it happens, the parenthesized form is sufficient

to define the concatenated form, making the latter formally dispensable,

but it's convenient to maintain it as a concise way of expressing more

complicated combinations of parenthesized forms. While working with

expressions solely in propositional calculus, it's easiest to use

plain parentheses for logical connectives. In contexts where

ordinary parentheses are needed for other purposes an alternate

typeface (...) may be used for the logical operators.

References
All other propositional connectives can be obtained through combinations

of these two forms. As it happens, the parenthesized form is sufficient

to define the concatenated form, making the latter formally dispensable,

but it's convenient to maintain it as a concise way of expressing more

complicated combinations of parenthesized forms. While working with

expressions solely in propositional calculus, it's easiest to use

plain parentheses for logical connectives. In contexts where

ordinary parentheses are needed for other purposes an alternate

typeface (...) may be used for the logical operators.

[1] https://oeis.org/wiki/Boolean-valued_function

[2] https://oeis.org/wiki/Minimal_negation_operator

[3] https://oeis.org/wiki/Logical_conjunction

Jun 16, 2021, 11:10:48 AM6/16/21

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

Cf: Differential Logic • 3

https://inquiryintoinquiry.com/2020/03/24/differential-logic-3/

Cactus Language for Propositional Logic

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

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Cactus_Language_for_Propositional_Logic

The simplest expression for logical truth is the empty word,

typically denoted by ε or λ in formal languages, where it is

• a + b + c = (a, (b, c)) = ((a, b), c)

https://inquiryintoinquiry.com/2020/03/24/differential-logic-3/

Cactus Language for Propositional Logic

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

https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Cactus_Language_for_Propositional_Logic

Table 1 shows the cactus graphs, the corresponding cactus expressions,

their logical meanings under the so-called “existential interpretation”,

and their translations into conventional notations for a sample of

basic propositional forms.

Table 1. Syntax and Semantics of a Calculus for Propositional Logic

https://inquiryintoinquiry.files.wordpress.com/2021/03/syntax-and-semantics-of-a-calculus-for-propositional-logic-3.0.png
their logical meanings under the so-called “existential interpretation”,

and their translations into conventional notations for a sample of

basic propositional forms.

Table 1. Syntax and Semantics of a Calculus for Propositional Logic

The simplest expression for logical truth is the empty word,

the identity element for concatenation. To make it visible

in context, it may be denoted by the equivalent expression

“(())”, or, especially if operating in an algebraic context,
in context, it may be denoted by the equivalent expression

by a simple “1”. Also when working in an algebraic mode, the

plus sign “+” may be used for exclusive disjunction. Thus we

have the following translations of algebraic expressions into

cactus expressions.

• a + b = (a, b)
plus sign “+” may be used for exclusive disjunction. Thus we

have the following translations of algebraic expressions into

cactus expressions.

• a + b + c = (a, (b, c)) = ((a, b), c)

Jun 16, 2021, 4:32:16 PM6/16/21