Inquiry Into Inquiry

Readings On Determination • Discussion 4

Posted on April 29, 2020 by Jon Awbrey

Determination, along with the related concepts of constraint, definition, form, structure, etc., are subjects of recurring discussion. Here are links to readings I collected back when I began approaching inference, information, inquiry, logic, sign processes, sign relations, and all that from more dedicated systems-theoretic and systems-engineering angles.

cc: Cybernetics • Ontolog Forum • Peirce (1) (2) • Structural Modeling • Systems Science

Posted in C.S. Peirce, Comprehension, Constraint, Definition, Determination, Differential Logic, Extension, Form, Indication, Information = Comprehension × Extension, Inquiry, Inquiry Driven Systems, Intension, Leibniz, Logic, Logic of Relatives, Mathematics, Peirce, Prigogine, Relation Theory, Relational Programming, Semiotics, Sign Relations | Tagged C.S. Peirce, Comprehension, Constraint, Definition, Determination, Differential Logic, Extension, Form, Indication, Information = Comprehension × Extension, Inquiry, Inquiry Driven Systems, Intension, Leibniz, Logic, Logic of Relatives, Mathematics, Peirce, Prigogine, Relation Theory, Relational Programming, Semiotics, Sign Relations | 6 Comments

Differential Logic • 10

Posted on April 25, 2020 by Jon Awbrey

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

Table A2. Propositional Forms on Two Variables

We took as our first example the boolean function $f_{8}(p, q) = pq$ corresponding to the logical conjunction $p \land q$ and examined how the differential operators $\mathrm{E}$ and $\mathrm{D}$ 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, $\mathrm{E}f_{8}(p, q, \mathrm{d}p, \mathrm{d}q)$ and $\mathrm{D}f_{8}(p, q, \mathrm{d}p, \mathrm{d}q),$ 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 $\mathrm{E}$ and $\mathrm{D}$ act on that set. There being some advantage to singling out the enlargement or shift operator $\mathrm{E}$ in its own right, we’ll begin by computing $\mathrm{E}f$ for each function $f$ in the above tables.

cc: Category Theory • Cybernetics • Ontolog • Structural Modeling • Systems Science
cc: FB | Differential Logic • Laws of Form • Peirce (1) (2) (3) (4)

Posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Change, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamics, Equational Inference, Frankl Conjecture, Functional Logic, Gradient Descent, Graph Theory, Hologrammautomaton, Indicator Functions, Inquiry Driven Systems, Leibniz, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Surveys, Time, Topology, Visualization, Zeroth Order Logic | Tagged Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Change, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamics, Equational Inference, Frankl Conjecture, Functional Logic, Gradient Descent, Graph Theory, Hologrammautomaton, Indicator Functions, Inquiry Driven Systems, Leibniz, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Surveys, Time, Topology, Visualization, Zeroth Order Logic | 6 Comments

Peirce’s Categories • 14

Posted on April 16, 2020 by Jon Awbrey

Re: Peirce List • Robert Marty

RM:: What do you think of the presuppositions between the levels?
Do they make sense to you?

At this point I have mostly questions, which would take further research to answer, not to mention unpacking many books still in boxes from our move a year and a half ago, none of which I’m at liberty to do right now. So, just off the cuff …

Presupposition is one of those words I tend to avoid, as it has too many uses at odds with each other. There are at least the architectonic, causal, and logical meanings. It it were only a matter of logic, I would say $P ~\mathrm{presupposes}~ Q$ means $P \Rightarrow Q.$ But usually people have something more pragmatic or rhetorical in mind than pure logic would require, something like enthymeme.

It’s also common for people to confound the implication order $P \Rightarrow Q$ with the causal order $P ~\mathrm{causes}~ Q,$ whereas it’s more like the reverse of that. In more complex settings we usually have the architectonic sense in mind, and that is what I sensed in the case of the normative sciences. Viewed with regard to their bases, logic is a special case of ethics and ethics is a special case of aesthetics, but with regard to their level of oversight, aesthetics must submit to ethical control and ethics must submit to logical control.

Early on, Peirce used involution with the meaning it has in arithmetic or number theory, namely, exponentiation, where $x^y$ means $\text{taking}~ x ~\text{to the power of}~ y.$ See the following passage and commentary.

Peirce’s 1870 Logic Of Relatives • The Sign of Involution

As far as the boolean or propositional analogue goes, $x^y ~\text{for}~ x, y ~\text{in}~ \{ 0, 1 \}$ means the same thing as $x \Leftarrow y,$ as one can tell by comparing the following two operation tables.

I haven’t looked into whether Peirce uses “involution” or “involvement” with that sense in his later writings.

Resources

cc: Cybernetics • Ontolog Forum • Peirce List • Structural Modeling • Systems Science

Posted in Abstraction, Aristotle, C.S. Peirce, Category Theory, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Phenomenology, Pragmatic Maxim, Relation Theory, Semiotics, Triadic Relations, Triadicity, Type Theory | Tagged Abstraction, Aristotle, C.S. Peirce, Category Theory, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Phenomenology, Pragmatic Maxim, Relation Theory, Semiotics, Triadic Relations, Triadicity, Type Theory | 7 Comments

Peirce’s Categories • 13

Posted on April 15, 2020 by Jon Awbrey

Re: Peirce List • Robert Marty

With a few choice exceptions I have always found Peirce’s earlier writings on categories, relations, and semiotics to be more clear, exact, and fruitful in practice than his last attempts to explain himself without the requisite logical and mathematical supports.

Still, I like Robert Marty’s “podium” picture of the universal categories, comprehend it all or not, and I found myself once using a similar picture to explain the relationships among the big 3 normative sciences of aesthetics, ethics, and logic. I called this “The Pragmatic Cosmos”, using cosmos in the sense of a global order. It looks like most of this stuff has fallen off the live web but here’s a few links I found.

I’ll copy and format pieces of this to my blog as I get time.

Resources

cc: Cybernetics • Ontolog Forum • Peirce List • Structural Modeling • Systems Science

Differential Logic • Discussion 2

Posted on April 14, 2020 by Jon Awbrey

Re: Peirce List • Edwina Taborsky

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 talking about.

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.

cc: Category Theory • Cybernetics • Ontolog • Structural Modeling • Systems Science
cc: FB | Differential Logic • Laws of Form • Peirce (1) (2) (3) (4)

Semiotics, Semiosis, Sign Relations • Discussion 6

Posted on April 12, 2020 by Jon Awbrey

Re: Peirce List • Jon Alan Schmidt

Just a note to record this citation by Jon Alan Schmidt of an important theme in Peirce. It touches on one of those recurring questions which has come up time and again on the Peirce List over the last twenty years, more acutely in recent Facebook discussions about universes of discourse, and more obliquely in the Ontolog Forum in connection with the AI/CogSci/DataBase issue of open vs. closed worlds. In another life, under another hat, I might have mentioned the Central Limit Theorem at this point as that would have brought us nearer Peirce’s core insight into the matter, but maybe another time …

JAS:: Every proposition is collective and copulative; as I stated in a recent post, its dynamical object is “the entire universe” (CP 5.448n, EP 2:394, 1906), which is “the totality of all real objects” (CP 5.152, EP 2:209, 1903), while its immediate object is “the logical universe of discourse” (CP 2.323, EP 2:283, 1903).

Thanks are due to JAS for calling attention to a critical point. I’m occupied with another train of thought at the moment so I’ll just stop to flag it for a later discussion. Incidentally, or synchronistically, lack of care in distinguishing different objects of the same signs, in particular, immediate and ultimate objects and their corresponding universes or object domains, has been the source of many misunderstandings in scattered discussions on Facebook of late.

Another issue arising here has to do with the difference between the “dimensionality of a relation” and the “number of correlates”. Signs may have any number of correlates in the object domain without requiring the dimensionality of the relevant sign relation to be greater than three. This is one of the consequences of “triadic relation irreducibility”.

Resources

Logic Syllabus	Sign Relation
Semeiotic	Triadic Relation
Universe of Discourse	Relation Theory
Peirce’s 1870 Logic Of Relatives	Relation Reduction

cc: Cybernetics • Ontolog Forum • Peirce List • Structural Modeling • Systems Science

Posted in C.S. Peirce, Category Theory, Logic, Relation Theory, Semiosis, Semiotics, Sign Relations, Triadic Relations | Tagged C.S. Peirce, Category Theory, Logic, Relation Theory, Semiosis, Semiotics, Sign Relations, Triadic Relations | 12 Comments

Differential Logic • 9

Posted on April 11, 2020 by Jon Awbrey

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

cc: Category Theory • Cybernetics • Ontolog • Structural Modeling • Systems Science
cc: FB | Differential Logic • Laws of Form • Peirce (1) (2) (3) (4)

Differential Logic • Discussion 1

Posted on April 9, 2020 by Jon Awbrey

Re: Structural Modeling • Joseph Simpson

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? — the efficiency of Peirce’s logical graphs, augmented by Spencer Brown and generalized to cactus graphs, made it possible for the first time to take on the extra complexities of differential propositional calculus. This makes Peircean themes a constant throughout the development of differential logic.

Hope you and yours are safe and sound,

Jon

cc: Category Theory • Cybernetics • Ontolog • Structural Modeling • Systems Science
cc: FB | Differential Logic • Laws of Form • Peirce (1) (2) (3) (4)

Differential Logic • 8

Posted on April 8, 2020 by Jon Awbrey

Propositional Forms on Two Variables

To broaden our experience with simple examples, let’s examine the sixteen functions of concrete type $P \times Q \to \mathbb{B}$ and abstract type $\mathbb{B} \times \mathbb{B} \to \mathbb{B}.$ Our inquiry into the differential aspects of logical conjunction will pay dividends as we study the actions of $\mathrm{E}$ and $\mathrm{D}$ on this family of forms.

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

cc: Category Theory • Cybernetics • Ontolog • Structural Modeling • Systems Science
cc: FB | Differential Logic • Laws of Form • Peirce (1) (2) (3) (4)

Differential Logic • 7

Posted on April 5, 2020 by Jon Awbrey

Differential Expansions of Propositions

Panoptic View • Enlargement Maps

The enlargement or shift operator $\mathrm{E}$ 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.

$\begin{array}{cccl} \text{Let} & X & = & X_1 \times \ldots \times X_k. \\[6pt] \text{Let} & \mathrm{d}X & = & \mathrm{d}X_1 \times \ldots \times \mathrm{d}X_k. \\[6pt] \text{Then} & \mathrm{E}X & = & X \times \mathrm{d}X \\[6pt] & & = & X_1 \times \ldots \times X_k ~\times~ \mathrm{d}X_1 \times \ldots \times \mathrm{d}X_k \end{array}$

For a proposition of the form $f : X_1 \times \ldots \times X_k \to \mathbb{B},$ the (first order) enlargement of $f$ is the proposition $\mathrm{E}f : \mathrm{E}X \to \mathbb{B}$ defined by the following equation.

$\mathrm{E}f(x_1, \ldots, x_k, \mathrm{d}x_1, \ldots, \mathrm{d}x_k) ~=~ f(x_1 + \mathrm{d}x_1, \ldots, x_k + \mathrm{d}x_k) ~=~ f(\texttt{(} x_1 \texttt{,} \mathrm{d}x_1 \texttt{)}, \ldots, \texttt{(} x_k \texttt{,} \mathrm{d}x_k \texttt{)})$

The differential variables $\mathrm{d}x_j$ 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 ${}^{\backprime\backprime} \mathrm{d} {}^{\prime\prime}$ 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 $\mathrm{E}f$ is formulated as follows.

$\begin{matrix} \mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q) & = & (p + \mathrm{d}p)(q + \mathrm{d}q) & = & \texttt{(} p \texttt{,} \mathrm{d}p \texttt{)(} q \texttt{,} \mathrm{d}q \texttt{)} \end{matrix}$

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.

$\begin{matrix} \mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q) & = & p~q & + & p~\mathrm{d}q & + & q~\mathrm{d}p & + & \mathrm{d}p~\mathrm{d}q \end{matrix}$

To understand what the enlarged or shifted proposition means in logical terms, it serves to go back and analyze the above expression for $\mathrm{E}f$ in the same way we did for $\mathrm{D}f.$ To that end, the value of $\mathrm{E}f_x$ at each $x \in X$ may be computed in graphical fashion as shown below.

Collating the data of this analysis yields a boolean expansion or disjunctive normal form (DNF) equivalent to the enlarged proposition $\mathrm{E}f.$

$\begin{matrix} \mathrm{E}f & = & pq \cdot \mathrm{E}f_{pq} & + & p(q) \cdot \mathrm{E}f_{p(q)} & + & (p)q \cdot \mathrm{E}f_{(p)q} & + & (p)(q) \cdot \mathrm{E}f_{(p)(q)} \end{matrix}$

Here is a summary of the result, illustrated by means of a digraph picture, where the “no change” element $\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)}$ is drawn as a loop at the point $p~q.$

$\begin{array}{rcccccc} f & = & p & \cdot & q \\[4pt] \mathrm{E}f & = & p & \cdot & q & \cdot & \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \\[4pt] & + & p & \cdot & \texttt{(} q \texttt{)} & \cdot & \texttt{(} \mathrm{d}p \texttt{)} \texttt{~} \mathrm{d}q \texttt{~} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & q & \cdot & \texttt{~} \mathrm{d}p \texttt{~} \texttt{(} \mathrm{d}q \texttt{)} \\[4pt] & + & \texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot & \mathrm{d}p \texttt{~~} \mathrm{d}q \end{array}$

We may understand the enlarged proposition $\mathrm{E}f$ as telling us all the ways of reaching a model of the proposition $f$ from the points of the universe $X.$

cc: Category Theory • Cybernetics • Ontolog • Structural Modeling • Systems Science
cc: FB | Differential Logic • Laws of Form • Peirce (1) (2) (3) (4)

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Inquiry Into Inquiry

Readings On Determination • Discussion 4

Differential Logic • 10

Peirce’s Categories • 14

Resources

Peirce’s Categories • 13

Resources

Differential Logic • Discussion 2

Semiotics, Semiosis, Sign Relations • Discussion 6

Resources

Differential Logic • 9

Propositional Forms on Two Variables

Differential Logic • Discussion 1

Differential Logic • 8

Propositional Forms on Two Variables

Differential Logic • 7

Differential Expansions of Propositions

Panoptic View • Enlargement Maps

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