Differential Logic, Dynamic Systems, Tangent Functors • Discussion 4

Re: Systems ScienceLT

To clarify my previous remark about General Systems Theory, I wasn’t trying to define a whole field but merely to describe my experience in forums like these, where it took me a while to realize that when I use the word “system” a great many people are not thinking what I’m thinking when I use it.  The first thing in my mind is almost always a state space X and the possible trajectories of a representative point through it.  But a lot of people will be thinking of a “system”, like the word says, as a collection of parts “standing together”.  Naturally I’d like to reach the point of discussing such things, it’s just that it takes me a while, and considerable analysis of X, to get there.

It goes without saying this has to do with the boundaries of my own experience and the emphases of my teachers and other influencers in systems, the early ones taking their ground in Ashby, Wiener, and the MIT school, the later ones stressing optimal control and learning organizations, but mostly it has to do with my current objectives and the species of intelligent systems, Inquiry Driven Systems, I want to understand and help to build.

Resources

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Differential Logic, Dynamic Systems, Tangent Functors • Discussion 3

Re: Systems Science • (1)(2)(3)

Another thing to keep in mind here is the difference between General Systems Theory, following on Bertalanffy et al., and what is known as Dynamical Systems Theory (DST) or Mathematical Systems Theory (MST).  GST spends a lot of time studying part-whole hierarchies while DST/MST deals with the state space of a system and the possible trajectories of the system through it.

Category theory is especially useful in the latter application, abstracting or generalizing as it does the concepts of mathematical objects, functions, and transformations.

For my part I have come to take the DST/MST approach as more fundamental since it starts with fewer assumptions about the anatomy or architecture of the as-yet hypothetical agent, making it one of the first and continuing tasks of the agent to discover its own boundaries, potentials, and structures.

Resources

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Posted in Amphecks, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamical Systems, Dynamical Systems, Graph Theory, Hill Climbing, Hologrammautomaton, Information Theory, Inquiry Driven Systems, Intelligent Systems, Knowledge Representation, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Systems, Visualization | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Semiotics, Semiosis, Sign Relations • 3

For ease of reference, here are two variants of Peirce’s 1902 definition of a sign, which he gives in the process of defining logic.

Selections from C.S. Peirce, “Carnegie Application” (1902)

No. 12.  On the Definition of Logic

Logic will here be defined as formal semiotic.  A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time.  Namely, a sign is something, A, which brings something, B, its interpretant sign determined or created by it, into the same sort of correspondence with something, C, its object, as that in which itself stands to C.  It is from this definition, together with a definition of “formal”, that I deduce mathematically the principles of logic.  I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has virtually been quite generally held, though not generally recognized.  (NEM 4, 20–21).

No. 12.  On the Definition of Logic [Earlier Draft]

Logic is formal semiotic.  A sign is something, A, which brings something, B, its interpretant sign, determined or created by it, into the same sort of correspondence (or a lower implied sort) with something, C, its object, as that in which itself stands to C.  This definition no more involves any reference to human thought than does the definition of a line as the place within which a particle lies during a lapse of time.  It is from this definition that I deduce the principles of logic by mathematical reasoning, and by mathematical reasoning that, I aver, will support criticism of Weierstrassian severity, and that is perfectly evident.  The word “formal” in the definition is also defined.  (NEM 4, 54).

Reference

  • Charles S. Peirce (1902), “Parts of Carnegie Application” (L 75), published in Carolyn Eisele (ed., 1976), The New Elements of Mathematics by Charles S. Peirce, vol. 4, 13–73.  Online.

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Semiotics, Semiosis, Sign Relations • 2

Re: CyberneticsOntolog ForumStructural ModelingSystems Science

Here are links to fuller discussions of semiotics.

The approach described here develops from what I regard as the core definition of triadic sign relations, one explicit enough to support a consequential theory of signs.  Peirce gives that definition in the process of defining logic itself, as detailed in the following texts.

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Semiotics, Semiosis, Sign Relations • 1

A first mention of semiotics (and cybersemiotics) in another group gave me a chance to begin a fresh introduction to the subject.  I thought it might be useful to share that here.

Here’s the intro I wrote for Wikipedia many moons ago.  There wasn’t much left of it the last time I looked there but I had previously saved several copies elsewhere.

Readers already registered on Wikipedia would find it easy to use the talk page on Wikiversity if they wanted to engage in discussion there.

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Theme One • A Program Of Inquiry : 17

Re: Ontolog Forum • (1)
Re: Systems Science • (1)
Re: Laws Of Form • (1)(2)(3)(4)

The move is all over but the unpacking, and the time looks ripe to pick up this thread from last spring.  Here, by way of a quick refresher, are a few Tables from earlier discussions.

  • Theme One • A Program Of Inquiry : 11
    • Tables 1 and 2 illustrate the existential and entitative interpretations of cactus graphs and cactus expressions by means of English translations for a few of the most basic forms.

Existential Interpretation

Table 1 illustrates the existential interpretation of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.

\text{Table 1.} ~~ \text{Existential Interpretation}
\text{Graph} \text{Expression} \text{Interpretation}

“ ”
~ \mathrm{true}

( )
\texttt{(} ~ \texttt{)} \mathrm{false}

a
a a

(a)
\texttt{(} a \texttt{)} \begin{matrix}  \tilde{a}  \\[2pt]  a^\prime  \\[2pt]  \lnot a  \\[2pt]  \mathrm{not}~ a  \end{matrix}

a b c
a~b~c \begin{matrix}  a \land b \land c  \\[6pt]  a ~\mathrm{and}~ b ~\mathrm{and}~ c  \end{matrix}

((a)(b)(c))
\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))} \begin{matrix}  a \lor b \lor c  \\[6pt]  a ~\mathrm{or}~ b ~\mathrm{or}~ c  \end{matrix}

(a(b))
\texttt{(} a \texttt{(} b \texttt{))} \begin{matrix}  a \Rightarrow b  \\[2pt]  a ~\mathrm{implies}~ b  \\[2pt]  \mathrm{if}~ a ~\mathrm{then}~ b  \\[2pt]  \mathrm{not}~ a ~\mathrm{without}~ b  \end{matrix}

(a, b)
\texttt{(} a, b \texttt{)} \begin{matrix}  a + b  \\[2pt]  a \neq b  \\[2pt]  a ~\mathrm{exclusive~or}~ b  \\[2pt]  a ~\mathrm{not~equal~to}~ b  \end{matrix}

((a, b))
\texttt{((} a, b \texttt{))} \begin{matrix}  a = b  \\[2pt]  a \iff b  \\[2pt]  a ~\mathrm{equals}~ b  \\[2pt]  a ~\mathrm{if~and~only~if}~ b  \end{matrix}

(a, b, c)
\texttt{(} a, b, c \texttt{)} \begin{matrix}  \mathrm{just~one~of}  \\  a, b, c  \\  \mathrm{is~false}  \end{matrix}

((a),(b),(c))
\texttt{((} a \texttt{)}, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))} \begin{matrix}  \mathrm{just~one~of}  \\  a, b, c  \\  \mathrm{is~true}  \end{matrix}

(a, (b),(c))
\texttt{(} a, \texttt{(} b \texttt{)}, \texttt{(} c \texttt{))} \begin{matrix}  \mathrm{genus}~ a ~\mathrm{of~species}~ b, c  \\[6pt]  \mathrm{partition}~ a ~\mathrm{into}~ b, c  \\[6pt]  \mathrm{pie}~ a ~\mathrm{of~slices}~ b, c  \end{matrix}

Entitative Interpretation

Table 2 illustrates the entitative interpretation of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.

\text{Table 2.} ~~ \text{Entitative Interpretation}
\text{Graph} \text{Expression} \text{Interpretation}

“ ”
~ \mathrm{false}

( )
\texttt{(} ~ \texttt{)} \mathrm{true}

a
a a

(a)
\texttt{(} a \texttt{)} \begin{matrix}  \tilde{a}  \\[2pt]  a^\prime  \\[2pt]  \lnot a  \\[2pt]  \mathrm{not}~ a  \end{matrix}

a b c
a~b~c \begin{matrix}  a \lor b \lor c  \\[6pt]  a ~\mathrm{or}~ b ~\mathrm{or}~ c  \end{matrix}

((a)(b)(c))
\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))} \begin{matrix}  a \land b \land c  \\[6pt]  a ~\mathrm{and}~ b ~\mathrm{and}~ c  \end{matrix}

(a)b
\texttt{(} a \texttt{)} b \begin{matrix}  a \Rightarrow b  \\[2pt]  a ~\mathrm{implies}~ b  \\[2pt]  \mathrm{if}~ a ~\mathrm{then}~ b  \\[2pt]  \mathrm{not}~ a, \mathrm{or}~ b  \end{matrix}

(a, b)
\texttt{(} a, b \texttt{)} \begin{matrix}  a = b  \\[2pt]  a \iff b  \\[2pt]  a ~\mathrm{equals}~ b  \\[2pt]  a ~\mathrm{if~and~only~if}~ b  \end{matrix}

((a, b))
\texttt{((} a, b \texttt{))} \begin{matrix}  a + b  \\[2pt]  a \neq b  \\[2pt]  a ~\mathrm{exclusive~or}~ b  \\[2pt]  a ~\mathrm{not~equal~to}~ b  \end{matrix}

(a, b, c)
\texttt{(} a, b, c \texttt{)} \begin{matrix}  \mathrm{not~just~one~of}  \\  a, b, c  \\  \mathrm{is~true}  \end{matrix}

((a, b, c))
\texttt{((} a, b, c \texttt{))} \begin{matrix}  \mathrm{just~one~of}  \\  a, b, c  \\  \mathrm{is~true}  \end{matrix}

(((a), b, c))
\texttt{(((} a \texttt{)}, b, c \texttt{))} \begin{matrix}  \mathrm{genus}~ a ~\mathrm{of~species}~ b, c  \\[6pt]  \mathrm{partition}~ a ~\mathrm{into}~ b, c  \\[6pt]  \mathrm{pie}~ a ~\mathrm{of~slices}~ b, c  \end{matrix}
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Pragmatic Semiotic Information • Discussion 11

Re: Ontolog ForumFerenc Kovacs
Re: Anja-Karina Pahl • Contradiction and Analogy as the Basis for Inventive Thinking

One of the insights coming out of Peirce’s logical work is the fact that negative operations are more powerful than positive operations in the sense that negative operations can generate all possible operations while positive operations by themselves do not suffice.  This is epitomized by his discovery of the amphecks as sole sufficient operators for propositional logic.

The propositional logic algorithm I wrote for my Theme One Program turns this principle to good effect in two ways:

  • The graph-theoretic syntax is based on a graph-theoretic operator, a type of controlled negation called the minimal negation operator, that generalizes Peirce’s graph-theoretic operator for negation.
  • It turns out that recognizing contradictions quickly makes for a high degree of efficiency in finding the “models” or satisfying interpretations of a propositional formula.

Relations of contradiction are also critical in statistical inference, but I’ll need to save that for another day.

Resources

  • Ampheck • (1)(2)
  • Minimal Negation Operator • (1)(2)

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