Inquiry Into Inquiry

Pragmatic Traction • 3

Posted on September 25, 2017 by Jon Awbrey

Re: Deborah G. Mayo • Revisiting Popper’s Demarcation of Science

I think Peirce would say that any struggle to pass from the irritation of doubt toward the settlement of belief is a form of inquiry — it’s just that some forms work better than others over the long haul. Instead of a demarcation Peirce describes a spectrum of methods, graded according to their measure of success in achieving the aim of inquiry.

Tenacity, Authority, Plausibility, Inquiry

Posted in Abduction, C.S. Peirce, Control, Cybernetics, Deborah G. Mayo, Deduction, Error, Error-Controlled Regulation, Feedback, Fixation of Belief, Hypothesis, Induction, Inference, Information, Information Theory, Inquiry, Inquiry Driven Systems, Knowledge, Knowledge Representation, Learning, Learning Theory, Likelihood, Logic, Logic of Science, Peirce, Philosophy, Philosophy of Science, Pragmatic Information, Pragmatic Maxim, Pragmatism, Probability, Probable Reasoning, Scientific Inquiry, Scientific Method, Semiotics, Statistical Inference, Statistics, Uncertainty | Tagged Abduction, C.S. Peirce, Control, Cybernetics, Deborah G. Mayo, Deduction, Error, Error-Controlled Regulation, Feedback, Fixation of Belief, Hypothesis, Induction, Inference, Information, Information Theory, Inquiry, Inquiry Driven Systems, Knowledge, Knowledge Representation, Learning, Learning Theory, Likelihood, Logic, Logic of Science, Peirce, Philosophy, Philosophy of Science, Pragmatic Information, Pragmatic Maxim, Pragmatism, Probability, Probable Reasoning, Scientific Inquiry, Scientific Method, Semiotics, Statistical Inference, Statistics, Uncertainty | Leave a comment

Pragmatic Traction • 2

Posted on September 16, 2017 by Jon Awbrey

Re: FB | Ecology of Systems Thinking • Richard Saunders

I’m about to be diverted for a couple of weeks but this is an ever-ongoing question so I know I’ll be coming back to it again. The short shrift goes a bit like this —

The gist of the idea that Peirce dubbed the pragmatic maxim is really a mathematical principle that has always been hard to render in ordinary language, largely due to the Procrustean subject-predicate embeddings most of the languages we know and love impose on its core structure. The primal form is more like one of those bi‑stable gestalts — duck‑rabbit, Necker cube, old‑young woman, etc. One way to get a mental handle on the matter is to mull over the many variations on its underlying theme, such as the ones I quoted and discussed in my blog post —

Pragmatic Maxim

Pragmatic Traction • 1

Posted on September 14, 2017 by Jon Awbrey

Re: Deborah G. Mayo • Peircean Induction and the Error-Correcting Thesis

C.S. Peirce’s pragmatic maxim marks the place where the tire of theory meets the test track of experience — it tells us how general ideas are impacted by practical consequences. If our concept of an object is the sum of its conceivable practical effects then the truth of a concept can be defeated by single outcome outside the sum.

Minimal Negation Operators • 4

Posted on September 1, 2017 by Jon Awbrey

Note. I’m including a more detailed definition of minimal negation operators in terms of conventional logical operations largely because readers of particular tastes have asked for it in the past. But it can easily be skipped until one has a felt need for it. Skimmed lightly, though, it can serve to illustrate a major theme in logic and mathematics, namely, the Relativity of Complexity or the Relativity of Primitivity to the basis we have chosen for constructing our conceptual superstructures.

⁂ ⁂ ⁂

Defining minimal negation operators over a more conventional basis is next in order of exposition, if not necessarily in order of every reader’s reading. For what it’s worth and against the day when it may be needed, here is a definition of minimal negations in terms of $\land,$ $\lor,$ and $\lnot.$

Formal Definition

To express the general form of $\nu_k$ in terms of familiar operations, it helps to introduce an intermediary concept.

Definition. Let the function $\lnot_j : \mathbb{B}^k \to \mathbb{B}$ be defined for each integer $j$ in the interval $[1, k]$ by the following equation.

$\begin{matrix} \lnot_j (x_1, \ldots, x_j, \ldots, x_k) & = & x_1 \land \ldots \land x_{j-1} \land \lnot x_j \land x_{j+1} \land \ldots \land x_k. \end{matrix}$

Then ${\nu_k : \mathbb{B}^k \to \mathbb{B}}$ is defined by the following equation.

$\begin{matrix} \nu_k (x_1, \ldots, x_k) & = & \lnot_1 (x_1, \ldots, x_k) \lor \ldots \lor \lnot_j (x_1, \ldots, x_k) \lor \ldots \lor \lnot_k (x_1, \ldots, x_k). \end{matrix}$

We may take the boolean product $x_1 \cdot \ldots \cdot x_k$ or the logical conjunction $x_1 \land \ldots \land x_k$ to indicate the point $x = (x_1, \ldots, x_k)$ in the space $\mathbb{B}^k,$ in which case the minimal negation $\texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)}$ indicates the set of points in $\mathbb{B}^k$ which differ from $x$ in exactly one coordinate. This makes $\texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)}$ a discrete functional analogue of a point-omitted neighborhood in ordinary real analysis, more precisely, a point-omitted distance-one neighborhood. Viewed in that light the minimal negation operator can be recognized as a differential construction, an observation opening a very wide field.

The remainder of this discussion proceeds on the algebraic convention making the plus sign $(+)$ and the summation symbol $(\textstyle\sum)$ both refer to addition mod 2. Unless otherwise noted, the boolean domain $\mathbb{B} = \{ 0, 1 \}$ is interpreted for logic in such a way that $0 = \mathrm{false}$ and $1 = \mathrm{true}.$ This has the following consequences.

The operation $x + y$ is a function equivalent to the exclusive disjunction of $x$ and $y,$ while its fiber of $1$ is the relation of inequality between $x$ and $y.$
The operation $\textstyle\sum_{j=1}^k x_j$ maps the bit sequence $(x_1, \ldots, x_k)$ to its parity.

The following properties of the minimal negation operators ${\nu_k : \mathbb{B}^k \to \mathbb{B}}$ may be noted.

The function $\texttt{(} x \texttt{,} y \texttt{)}$ is the same as that associated with the operation $x + y$ and the relation $x \ne y.$
In contrast, $\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}$ is not identical to $x + y + z.$
More generally, the function $\nu_k (x_1, \dots, x_k)$ for $k > 2$ is not identical to the boolean sum $\textstyle\sum_{j=1}^k x_j.$
The inclusive disjunctions indicated for the $\nu_k$ of more than one argument may be replaced with exclusive disjunctions without affecting the meaning since the terms in disjunction are already disjoint.

Resources

cc: Cybernetics • Ontolog Forum • Peirce List • Structural Modeling • Systems Science
cc: FB | Minimal Negation Operators • Laws of Form

Posted in Amphecks, Animata, Boolean Functions, C.S. Peirce, Cactus Graphs, Differential Logic, Functional Logic, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Propositional Equation Reasoning Systems, Semiotics, Venn Diagrams, Visualization | Tagged Amphecks, Animata, Boolean Functions, C.S. Peirce, Cactus Graphs, Differential Logic, Functional Logic, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Propositional Equation Reasoning Systems, Semiotics, Venn Diagrams, Visualization | 10 Comments

Minimal Negation Operators • 3

Posted on August 30, 2017 by Jon Awbrey

It will take a few more rounds of stage-setting before we are able to entertain concrete examples of applications but the following may indicate the direction of generalization embodied in minimal negation operators.

To begin, let’s observe two ways of generalizing the logical operation commonly known as exclusive disjunction $(\textsc{xor})$ or symmetric difference $(\Delta).$

Let $\mathbb{B}$ = the boolean domain $\{ 0, 1 \}.$

Exclusive disjunction is a boolean function $\Delta : \mathbb{B} \times \mathbb{B} \to \mathbb{B}$ isomorphic to the algebraic field addition $+ : \mathbb{B} \times \mathbb{B} \to \mathbb{B},$ also known as addition mod 2. Adding the language of minimal negation operators to the mix we have the following equivalent expressions.

$\begin{matrix} \textsc{xor}(p, q) & = & \Delta (p, q) & = & p + q & = & \nu (p, q) & = & \texttt{(} p \texttt{,} q \texttt{)} \end{matrix}$

Minimal Negation $\texttt{(} p \texttt{,} q \texttt{)}$ as Parity Indicator

Generalizing the function $p + q$ of two variables to more variables extends the sequence of functions in the fashion $p\!+\!q\!+\!r,$ $p\!+\!q\!+\!r\!+\!s,$ $p\!+\!q\!+\!r\!+\!s\!+\!t,$ and so on. These are known as parity sums, returning a value of $0$ when there are an even number of $1$ ’s in the sum and returning a value of $1$ when there are an odd number of $1$ ’s in the sum.

Minimal Negation $\texttt{(} p \texttt{,} q \texttt{)}$ as Border Indicator

The equivalent expressions $\texttt{(} p \texttt{,} q \texttt{)} = \nu(p, q) = p + q = p \,\Delta\, q = p ~\textsc{xor}~ q$ may be read with a different connotation, indicating the venn diagram cells adjacent to the conjunction $p \land q.$ Generalizing the function $\texttt{(} p \texttt{,} q \texttt{)}$ of two variables to more variables extends the sequence of functions in the fashion $\texttt{(} p \texttt{,} q \texttt{,} r \texttt{)},$ $\texttt{(} p \texttt{,} q \texttt{,} r \texttt{,} s \texttt{)},$ $\texttt{(} p \texttt{,} q \texttt{,} r \texttt{,} s \texttt{,} t \texttt{)},$ and so on. That sequence of operators differs from the sequence of parity sums once it passes the 2-variable case.

The triple sum may be written in terms of 2-place minimal negations as follows.

$\begin{matrix} p + q + r & = & \texttt{((} p \texttt{,} q \texttt{)}\!\texttt{,} r \texttt{)} & = & \texttt{(} p \texttt{,} \texttt{(} q \texttt{,} r \texttt{))} \end{matrix}$

It is important to recognize the triple sum expressions and the 3-place minimal negation $\texttt{(} p \texttt{,} q \texttt{,} r \texttt{)}$ have very different meanings.

Resources

cc: Cybernetics • Ontolog Forum • Peirce List • Structural Modeling • Systems Science
cc: FB | Minimal Negation Operators • Laws of Form

Minimal Negation Operators • 2

Posted on August 30, 2017 by Jon Awbrey

Re: Minimal Negation Operators • 1

The brief description of minimal negation operators given in the previous post is enough to convey the rule of their construction. For future reference, a more formal definition is given below.

Initial Definition

The minimal negation operator $\nu$ is a multigrade operator $(\nu_k)_{k \in \mathbb{N}}$ where each $\nu_k$ is a $k$ -ary boolean function defined by the rule that $\nu_k (x_1, \ldots, x_k) = 1$ if and only if exactly one of the arguments $x_j$ is $0.$

In contexts where the initial letter $\nu$ is understood, minimal negation operators may be indicated by argument lists in parentheses. In the discussion that follows a distinctive typeface will be used for logical expressions based on minimal negation operators, for example, $\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)} = \nu (x, y, z).$

The first four members of this family of operators are shown below. The third and fourth columns give paraphrases in two other notations, where tildes and primes, respectively, indicate logical negation.

Resources

cc: Cybernetics • Ontolog Forum • Peirce List • Structural Modeling • Systems Science
cc: FB | Minimal Negation Operators • Laws of Form

Minimal Negation Operators • 1

Posted on August 27, 2017 by Jon Awbrey

To accommodate moderate levels of complexity in the application of logical graphs to practical problems our Organon requires a class of organules called “minimal negation operators”. I outlined the history of their early development from Peirce’s alpha graphs for propositional calculus in a previous series of posts. The next order of business is to sketch their properties in a systematic fashion and to illustrate their uses. As it turns out, taking minimal negations as primitive operators enables efficient expressions for many natural constructs and affords a bridge between boolean domains of two values and domains with finite numbers of values, for example, finite sets of individuals.

Brief Introduction

A minimal negation operator $(\nu)$ is a logical connective which says “just one false” of its logical arguments. The first four cases are described below.

If the list of arguments is empty, as expressed in the form $\nu(),$ then it cannot be true exactly one of the arguments is false, so $\nu() = \mathrm{false}.$
If $p$ is the only argument then $\nu(p)$ says $p$ is false, so $\nu(p)$ expresses the negation of the proposition $p.$ Written in several common notations we have the following equivalent expressions.

$\nu(p) ~=~ \mathrm{not}(p) ~=~ \lnot p ~=~ \tilde{p} ~=~ p^{\prime}$
If $p$ and $q$ are the only two arguments then $\nu(p, q)$ says exactly one of $p, q$ is false, so $\nu(p, q)$ says the same thing as $p \neq q.$ Expressing $\nu(p, q)$ in terms of ands $(\cdot),$ ors $(\lor),$ and nots $(\tilde{~})$ gives the following form.

$\nu(p, q) ~=~ \tilde{p} \cdot q ~\lor~ p \cdot \tilde{q}$

It is permissible to omit the dot $(\cdot)$ in contexts where it is understood, giving the following form.

$\nu(p, q) ~=~ \tilde{p}q \lor p\tilde{q}$

The venn diagram for $\nu(p, q)$ is shown in Figure 1.

$\text{Figure 1.} ~~ \nu(p, q)$
The venn diagram for $\nu(p, q, r)$ is shown in Figure 2.

$\text{Figure 2.} ~~ \nu(p, q, r)$

The center cell is the region where all three arguments $p, q, r$ hold true, so $\nu(p, q, r)$ holds true in just the three neighboring cells. In other words:

$\nu(p, q, r) ~=~ \tilde{p}qr \lor p\tilde{q}r \lor pq\tilde{r}$

Resources

cc: Cybernetics • Ontolog Forum • Peirce List • Structural Modeling • Systems Science
cc: FB | Minimal Negation Operators • Laws of Form

Charles Sanders Peirce, George Spencer Brown, and Me • 10

Posted on August 25, 2017 by Jon Awbrey

With any formal system it is easy to spend a long time roughing out primitives and reviewing first principles before getting on to practical applications, and logical graphs are no different in that respect. But the promise of clearer and more efficient methods for solving realistic problems is what led me to the visual calculi of C.S. Peirce and Spencer Brown in the first place, so my aim throughout our rehearsal of rudiments is to make a bridge to applications a few steps nearer what the real world throws our way.

I’ve been thinking how to make the transition from basic ingredients of logical graphs and laws of form to slightly more interesting examples — still “toy worlds” as AI folk call them but suggestive to some degree of what might be possible in the long run. I’ll spend a few days gathering assorted examples I’ve worked up previously and try presenting those.

cc: Cybernetics • Laws of Form • Ontolog • Peirce List • Structural Modeling • Systems Science

Posted in Abstraction, Amphecks, Analogy, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Cybernetics, Deduction, Differential Analytic Turing Automata, Differential Logic, Duality, Form, Graph Theory, Inquiry, Inquiry Driven Systems, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Proof Theory, Propositional Calculus, Semiotics, Sign Relational Manifolds, Sign Relations, Spencer Brown, Theorem Proving, Time, Topology | Tagged Abstraction, Amphecks, Analogy, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Cybernetics, Deduction, Differential Analytic Turing Automata, Differential Logic, Duality, Form, Graph Theory, Inquiry, Inquiry Driven Systems, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Proof Theory, Propositional Calculus, Semiotics, Sign Relational Manifolds, Sign Relations, Spencer Brown, Theorem Proving, Time, Topology | 6 Comments

Charles Sanders Peirce, George Spencer Brown, and Me • 9

Posted on August 22, 2017 by Jon Awbrey

Re: Boundary Logic

A wider field of investigation opens up at this point, spanning the diversity of interactions among languages we use, and systems of signs in general, to the thoughts ever streaming through our heads, to the universes we talk and think upon, from Plato’s Heaven to Gaia’s Green Earth to the Tumbling Galaxies Beyond.

The complexities in play when we consider a domain of Signs, a domain of Ideas, and a domain of Objects all wound up in relationship to one another is what Peirce’s “semiotics” or theory of sign relations is all about. Viewing the enterprise of logic within the broader frame of semiotics not only gives us more insight into its means and ends but affords us more “elbow room” for carrying out its operations.

To make a long story short, we don’t have to “escape language” because we don’t live inside any language or system of signs, even if we get so confused sometimes as to think we do. We live in that wider world of reality and only use languages and other systems of signs to describe what little we can of it.

cc: Cybernetics • Laws of Form • Ontolog • Peirce • Structural Modeling • Systems Science

Posted in Abstraction, Amphecks, Analogy, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Cybernetics, Deduction, Differential Logic, Duality, Form, Graph Theory, Inquiry, Inquiry Driven Systems, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Peirce, Proof Theory, Propositional Calculus, Semiotics, Sign Relational Manifolds, Sign Relations, Spencer Brown, Theorem Proving, Time, Topology | Tagged Abstraction, Amphecks, Analogy, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Cybernetics, Deduction, Differential Logic, Duality, Form, Graph Theory, Inquiry, Inquiry Driven Systems, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Peirce, Proof Theory, Propositional Calculus, Semiotics, Sign Relational Manifolds, Sign Relations, Spencer Brown, Theorem Proving, Time, Topology | 6 Comments

Charles Sanders Peirce, George Spencer Brown, and Me • 8

Posted on August 22, 2017 by Jon Awbrey

Re: Boundary Logic

For me, the heart of the matter is “what is the purpose of logic and what is the purpose of mathematics and what is their relationship?”

There are semiotic situations which appear to violate the initial conditions of logic but there are ways of approaching them without reducing our brains to jelly from the getgo. Charles S. Peirce, following on Aristotle’s negotiation of the boundary between logic and rhetoric, developed his theory of triadic sign relations in large part to manage just those sorts of situations.

I’m determined to keep my gnosis close to the grinstone for the time being but here is a smattering of old notes which give a hint as to Peirce’s way of approaching the question.

C.S. Peirce on “General” and “Vague”

Collected Papers • CP 5.448
Foundations of Mathematics (FOM) • (1) • (2) • (3)

cc: Cybernetics • Laws of Form • Ontolog • Peirce • Structural Modeling • Systems Science

	Differential Logic •… on Differential Logic • 4
	Survey of Differenti… on Differential Logic • 6
	Differential Logic •… on Survey of Animated Logical Gra…
	Differential Logic •… on Survey of Differential Logic •…
	Differential Logic •… on Logic Syllabus
	Differential Logic •… on Differential Logic • 5
	Survey of Differenti… on Differential Logic • 5
	Differential Logic •… on Differential Logic • 4
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Minimal Negation as Parity Indicator

Minimal Negation as Border Indicator

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Initial Definition

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Brief Introduction

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Minimal Negation $\texttt{(} p \texttt{,} q \texttt{)}$ as Parity Indicator

Minimal Negation $\texttt{(} p \texttt{,} q \texttt{)}$ as Border Indicator