Inquiry Into Inquiry

Cactus Language • Preliminaries 9

Posted on April 25, 2025 by Jon Awbrey

We now have the materials in place to formulate a definition of our subject.

The painted cactus language with paints in the set $\mathfrak{P} = \{ p_j : j \in J \}$ is the formal language $\mathfrak{L} = \mathfrak{C} (\mathfrak{P}) \subseteq \mathfrak{A}^* = (\mathfrak{M} \cup \mathfrak{P})^*$ defined as follows.

$\begin{array}{ll} \text{PC 1.} & \text{The blank symbol}~ m_1 ~\text{is a sentence.} \\ \text{PC 2.} & \text{The paint}~ p_j ~\text{is a sentence for each}~ j ~\text{in}~ J. \\ \text{PC 3.} & \mathrm{Conc}^0 ~\text{and}~ \mathrm{Surc}^0 ~\text{are sentences.} \\ \text{PC 4.} & \text{For each positive integer}~ n, \\ & \text{if}~ s_1, \ldots, s_n ~\text{are sentences} \\ & \text{then}~ \mathrm{Conc}_{k=1}^n s_k ~\text{is a sentence} \\ & \text{and}~ \mathrm{Surc}_{k=1}^n s_k ~\text{is a sentence.} \end{array}$

In the idiom of formal language theory, a string $s$ is called a sentence of $\mathfrak{L}$ if and only if it belongs to $\mathfrak{L},$ or simply a sentence if the language $\mathfrak{L}$ is understood. A sentence of $\mathfrak{C} (\mathfrak{P})$ is referred to as a painted and rooted cactus expression on the palette $\mathfrak{P},$ or a cactus expression for short.

Resources

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cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Posted in Automata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Differential Logic, Equational Inference, Formal Grammars, Formal Languages, Graph Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Propositional Calculus, Visualization | Tagged Automata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Differential Logic, Equational Inference, Formal Grammars, Formal Languages, Graph Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Propositional Calculus, Visualization | 9 Comments

Cactus Language • Preliminaries 8

Posted on April 22, 2025 by Jon Awbrey

Defining the basic operations of concatenation and surcatenation on arbitrary strings gives them operational meaning for the all‑inclusive language $\mathfrak{L} = \mathfrak{A}^*.$ With that in hand it is time to adjoin the notion of a more discriminating grammaticality, in other words, a more properly restrictive concept of a sentence.

If $\mathfrak{L}$ is an arbitrary formal language over an alphabet of the type we have been discussing, that is, an alphabet of the form $\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P},$ then there are a number of basic structural relations which can be defined on the strings of $\mathfrak{L}.$

Concatenation: $s$ is the concatenation of $s_1$ and $s_2$ in $\mathfrak{L}$
if and only if
$s_1$ is a sentence of $\mathfrak{L},$ $s_2$ is a sentence of $\mathfrak{L},$
and
$s = s_1 \cdot s_2$; $s$ is the concatenation of the $k$ strings $s_1, \ldots, s_k$ in $\mathfrak{L}$
if and only if
$s_j$ is a sentence of $\mathfrak{L}$ for all $j = 1 \ldots k$
and
$s = \mathrm{Conc}_{j=1}^k s_j = s_1 \cdot \ldots \cdot s_k$

Discatenation: $s$ is the discatenation of $s_1$ by $t$
if and only if
$s_1$ is a sentence of $\mathfrak{L},$ $t$ is an element of $\mathfrak{A},$
and
$s_1 = s \cdot t$
in which case we more commonly write
$s = s_1 \cdot t^{-1}$

Subclause: $s$ is a subclause of $\mathfrak{L}$
if and only if
$s$ is a sentence of $\mathfrak{L}$
and
$s$ ends with a $``\text{)}"$

Subcatenation: $s$ is the subcatenation of $s_1$ by $s_2$
if and only if
$s_1$ is a subclause of $\mathfrak{L},$ $s_2$ is a sentence of $\mathfrak{L},$
and
$s = s_1 \cdot (``\text{)}")^{-1} \cdot ``\text{,}" \cdot s_2 \cdot ``\text{)}"$

Surcatenation: $s$ is the surcatenation of the $k$ strings $s_1, \ldots, s_k$ in $\mathfrak{L}$
if and only if
$s_j$ is a sentence of $\mathfrak{L}$ for all ${j = 1 \ldots k}$
and
$s = \mathrm{Surc}_{j=1}^k s_j = ``\text{(}" \cdot s_1 \cdot ``\text{,}" \cdot \ldots \cdot ``\text{,}" \cdot s_k \cdot ``\text{)}"$

The converses of the above decomposition relations amount to the corresponding composition operations. As complementary forms of analysis and synthesis they make it possible to articulate the structures of strings and sentences in two directions.

Resources

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cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Cactus Language • Preliminaries 7

Posted on April 19, 2025 by Jon Awbrey

The array of syntactic operators may be put in more organized form by making a few additional conventions and auxiliary definitions.

Concatenation

The conception of concatenation permits extension to its natural prequel, the corresponding operator on zero operands.

$\mathrm{Conc}^0 = ``" = \text{the empty string.}$

From that beginning the operation of concatenation may be broken into stages by means of the following conceptions.

The precatenation $\mathrm{Prec}(s_1, s_2)$ of two strings $s_1, s_2$ is defined as follows.

$\mathrm{Prec} (s_1, s_2) = s_1 \cdot s_2.$

The concatenation of $n$ strings $s_1, \ldots, s_n$ may now be given a new definition as the iterated precatenation of $n+1$ strings beginning with $s_0 = \mathrm{Conc}^0 = ``"$ and continuing through the remaining $n$ strings.

$\text{For}~ n = 0, ~\mathrm{Conc}_{k=0}^n s_k = \mathrm{Conc}^0 = ``".$

$\text{For}~ n > 0, ~\mathrm{Conc}_{k=1}^n s_k = \mathrm{Prec}(\mathrm{Conc}_{k=0}^{n-1} s_k, s_n).$

Surcatenation

The conception of surcatenation permits extension to its natural prequel, the corresponding operator on zero operands.

$\mathrm{Surc}^0 = ``()".$

From that beginning the operation of surcatenation may be broken into stages by means of the following conceptions.

A subclause in $\mathfrak{A}^*$ is a string ending with $``)".$

The subcatenation $\mathrm{Subc} (s_1, s_2)$ of a subclause $s_1$ by a string $s_2$ is defined as follows.

$\mathrm{Subc} (s_1, s_2) = s_1 \cdot (``)")^{-1} \cdot ``," \cdot s_2 \cdot ``)".$

The surcatenation of $n$ strings $s_1, \ldots, s_n$ may now be given a new definition as the iterated subcatenation of $n+1$ strings beginning with $s_0 = \mathrm{Surc}^0 = ``()"$ and continuing through the remaining $n$ strings.

$\text{For}~ n = 0, ~\mathrm{Surc}_{k=0}^n s_k = \mathrm{Surc}^0 = ``()".$

$\text{For}~ n > 0, ~\mathrm{Surc}_{k=1}^n s_k = \mathrm{Subc}(\mathrm{Surc}_{k=0}^{n-1} s_k, s_n).$

Notice that the expressions $\mathrm{Conc}_{k=0}^0 s_k$ and $\mathrm{Surc}_{k=0}^0 s_k$ are defined in such a way that the respective operators $\mathrm{Conc}^0$ and $\mathrm{Surc}^0$ simply ignore, in the manner of constants, whatever sequences of strings $s_k$ may be listed as their ostensible arguments.

Resources

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cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Cactus Language • Preliminaries 6

Posted on April 16, 2025 by Jon Awbrey

The definitions of the syntactic connectives can be made a little more succinct by defining the following pair of generic operators on strings.

Concatenation

The concatenation $\mathrm{Conc}_{k=1}^n$ of the sequence of $n$ strings $(s_k)_{k=1}^n$ is defined recursively as follows.

$\text{For}~ n = 1, ~\mathrm{Conc}_{k=1}^n s_k = s_1.$

$\text{For}~ n > 1, ~\mathrm{Conc}_{k=1}^n s_k = \mathrm{Conc}_{k=1}^{n-1} s_k \cdot s_n.$

Surcatenation

The surcatenation $\mathrm{Surc}_{k=1}^n$ of the sequence of $n$ strings $(s_k)_{k=1}^n$ is defined recursively as follows.

$\text{For}~ n = 1, ~\mathrm{Surc}_{k=1}^n s_k = ``\text{(}" \cdot s_1 \cdot ``\text{)}".$

$\text{For}~ n > 1, ~\mathrm{Surc}_{k=1}^n s_k = \mathrm{Surc}_{k=1}^{n-1} s_k \cdot (``\text{)}")^{-1} \cdot ``\text{,}" \cdot s_n \cdot ``\text{)}".$

Resources

cc: Academia.edu • BlueSky • Laws of Form • Ma^thstodon • Research Gate
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Cactus Language • Preliminaries 5

Posted on April 13, 2025 by Jon Awbrey

The easiest way to define the language $\mathfrak{C}(\mathfrak{P})$ is to indicate the general run of operations required to construct the greater share of its sentences from the designated few which require a special election.

To do that we introduce a family of operations called syntactic connectives on the strings of $\mathfrak{A}^*.$ If the strings on which they operate are already sentences of $\mathfrak{C}(\mathfrak{P})$ then the operations amount to sentential connectives. If the syntactic sentences, viewed as abstract strings of uninterpreted signs, are provided with a semantics where they denote propositions, in other words, indicator functions on a universe of discourse, then the operations amount to propositional connectives.

Rather than presenting the most concise description of cactus languages right from the beginning, it aids comprehension to develop a picture of their forms in gradual stages, starting with the most natural ways of viewing their elements, if somewhat at a distance, and working through the most easily grasped impressions of their structures, if not always the sharpest acquaintances with their details.

We begin by defining two sets of basic operations on strings of $\mathfrak{A}^*.$

Concatenation

The concatenation of one string $s_1$ is the string $s_1.$

The concatenation of two strings $s_1, s_2$ is the string ${s_1 \cdot s_2}.$

The concatenation of $k$ strings $(s_j)_{j = 1}^k$ is the string ${s_1 \cdot \ldots \cdot s_k}.$

Surcatenation

The surcatenation of one string $s_1$ is the string $``\text{(}" \cdot s_1 \cdot ``\text{)}".$

The surcatenation of two strings $s_1, s_2$ is the string $``\text{(}" \cdot s_1 \cdot ``\text{,}" \cdot s_2 \cdot ``\text{)}".$

The surcatenation of $k$ strings $(s_j)_{j = 1}^k$ is the string $``\text{(}" \cdot s_1 \cdot ``\text{,}" \cdot \ldots \cdot ``\text{,}" \cdot s_k \cdot ``\text{)}".$

Resources

cc: Academia.edu • BlueSky • Laws of Form • Ma^thstodon • Research Gate
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Cactus Language • Preliminaries 4

Posted on April 11, 2025 by Jon Awbrey

The informal mechanisms illustrated in the preceding discussion equip us with a description of cactus language adequate to providing conceptual and computational representations for the minimal formal logical system variously known as propositional logic or sentential calculus.

The painted cactus language $\mathfrak{C}$ is actually a parameterized family of languages, consisting of one language $\mathfrak{C}(\mathfrak{P})$ for each set $\mathfrak{P}$ of paints.

The alphabet $\mathfrak{A} = \mathfrak{M} \cup \mathfrak{P}$ is the disjoint union of the following two sets of symbols.

$\mathfrak{M}$ is the alphabet of markers, the set of punctuation marks, or the collection of syntactic constants common to all the languages $\mathfrak{C}(\mathfrak{P}).$ Various ways of representing the elements of $\mathfrak{M}$ are shown in the following display.

$\mathfrak{P}$ is the palette, the alphabet of paints, or the collection of syntactic variables peculiar to the language $\mathfrak{C}(\mathfrak{P}).$ The set of signs in $\mathfrak{P}$ may be enumerated as follows.

$\mathfrak{P} = \{ \mathfrak{p}_j : j \in J \}.$

Resources

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Cactus Language • Preliminaries 3

Posted on April 9, 2025 by Jon Awbrey

A few definitions from formal language theory are required at this point.

An alphabet is a finite set of signs, typically, $\mathfrak{A} = \{ \mathfrak{a}_1, \ldots, \mathfrak{a}_n \}.$

A string over an alphabet $\mathfrak{A}$ is a finite sequence of signs from $\mathfrak{A}.$

The length of a string is just its length as a sequence of signs.

The empty string is the unique sequence of length 0. It is sometimes denoted by an empty pair of quotation marks, “”, but more often by the Greek symbols epsilon or lambda.

A sequence of length $k > 0$ is typically presented in the following concatenated forms.

$s_1 s_2 \ldots s_{k-1} s_k$

$s_1 \cdot s_2 \cdot \ldots \cdot s_{k-1} \cdot s_k$

with $s_j \in \mathfrak{A}$ for all $j = 1 \ldots k.$

The following notations provide useful alternatives.

$\varepsilon$ = “” = the empty string.

$\underline\varepsilon$ = $\{ \varepsilon \}$ = the language consisting of a single empty string.

Several operations on strings find sufficient application to motivate the following definitions.

To erase an appearance of a sign is to replace it with an appearance of the blank symbol “ ”.

To delete an appearance of a sign is to replace it with an appearance of the empty string “”.

If $s$ is a string which ends with a sign $t$ then $s \cdot t^{-1}$ is the string which results by deleting the terminal $t$ from $s.$

A token is a particular appearance of a sign.

Finally —

The kleene star $\mathfrak{A}^*$ of alphabet $\mathfrak{A}$ is the set of all strings over $\mathfrak{A}.$ In particular, $\mathfrak{A}^*$ includes among its elements the empty string $\varepsilon.$

The kleene plus $\mathfrak{A}^+$ of an alphabet $\mathfrak{A}$ is the set of all positive length strings over $\mathfrak{A},$ in other words, everything in $\mathfrak{A}^*$ but the empty string.

A formal language $\mathfrak{L}$ over an alphabet $\mathfrak{A}$ is a subset of $\mathfrak{A}^*.$ In brief, $\mathfrak{L} \subseteq \mathfrak{A}^*.$ If $s$ is a string over $\mathfrak{A}$ and $s$ is an element of $\mathfrak{L}$ then it is customary to call $s$ a sentence of $\mathfrak{L}.$ Thus, a formal language $\mathfrak{L}$ is defined by specifying its elements, which amounts to saying what it means to be a sentence of $\mathfrak{L}.$

Resources

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cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Cactus Language • Preliminaries 2

Posted on April 5, 2025 by Jon Awbrey

As a temporary notation, let the relationship between a particular sign $s$ and a particular object $o$ , namely, the fact that $s$ denotes $o$ or the fact that $o$ is denoted by $s$ , be symbolized in one of the following two ways.

$\begin{array}{lccc} 1. & s & \rightarrow & o \\[6pt] 2. & o & \leftarrow & s \end{array}$

Now consider the following paradigm.

In the same vein, if we let the sign “blank” denote the sign “ ” then the string of characters inside the first pair of quotation marks will serve as another name for the string of characters inside the second pair of quotation marks. In other words, “blank” is a higher order sign whose object is the sign “ ” and the string of five characters inside the first pair of quotation marks is a sign at a higher level of signification than the string of one character inside the second pair of quotation marks. The relation in question can be abbreviated in either one of the following two ways.

$\begin{array}{ccc} ``\text{blank}" & \rightarrow & ``\text{ }" \\[6pt] ``\text{ }" & \leftarrow & ``\text{blank}" \end{array}$

Using the raised dot “∙” as a sign to mark the articulation of a quoted string into a sequence of possibly shorter quoted strings, and thus to mark the concatenation of a sequence of quoted strings into a possibly larger quoted string, one can write the following equation.

$\begin{array}{lllll} ``\text{ }" & \leftarrow & ``\text{blank}" & = & ``\text{b}" \cdot ``\text{l}" \cdot ``\text{a}" \cdot ``\text{n}" \cdot ``\text{k}" \end{array}$

The above tactic lets us refer to the blank as a type of character and refer to any blank we choose as a token of that type, denoting either in a markèd way, but without the use of quotation marks. As a blank is just what the name “blank” names, it is possible to represent the denoting of the sign “ ” by the name “blank” in the form of an identity between the named objects, as follows.

$\begin{array}{lll}``\text{ }" & = & \mathrm{blank}\end{array}$

Given the above identities it is possible to extend the use of the “∙” sign to mark the articulation of either named or quoted strings into both named and quoted strings. For example, we have the following equations.

$\begin{array}{ccccc} ``\mathrm{~~}" & = & ``\text{ }" \cdot ``\text{ }" & = & \text{blank} \cdot \text{blank} \\[6pt] ``\mathrm{~blank}" & = & ``\text{ }" \cdot ``\text{blank}" & = & \text{blank} \cdot ``\text{blank}" \\[6pt] ``\mathrm{blank~}" & = & ``\text{blank}" \cdot ``\text{ }" & = & ``\text{blank}" \cdot \text{blank} \end{array}$

Resources

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Cactus Language • Preliminaries 1

Posted on March 30, 2025 by Jon Awbrey

Thus, what looks to us like a sphere of scientific knowledge more accurately should be represented as the inside of a highly irregular and spiky object, like a pincushion or porcupine, with very sharp extensions in certain directions, and virtually no knowledge in immediately adjacent areas. If our intellectual gaze could shift slightly, it would alter each quill’s direction, and suddenly our entire reality would change.

Picture two different configurations of such an irregular shape, superimposed on each other in space, like a double exposure photograph. Of the two images, the only part which coincides is the body. The two different sets of quills stick out into very different regions of space. The objective reality we see from within the first position, seemingly so full and spherical, actually agrees with the shifted reality only in the body of common knowledge. In every direction in which we look at all deeply, the realm of discovered scientific truth could be quite different. Yet in each of those two different situations, we would have thought the world complete, firmly known, and rather round in its penetration of the space of possible knowledge.

Herbert J. Bernstein • “Idols of Modern Science”

The task before us is to describe the syntax of a family of formal languages intended for use as a sentential calculus, and thus interpreted for the purpose of reasoning about propositions and their logical relations.

To carry out our discussion we need a way of referring to signs as if they were objects like any others, in other words, as the sorts of things which can be named, indicated, described, discussed, and renamed if necessary, which can be placed, arranged, and rearranged within a suitable medium of expression — or else manipulated in the mind — which can be articulated and decomposed into their elementary signs, and which can be strung together in sequences to form complex signs.

Signs having signs as their objects are known as “higher order signs”, a topic which demands an adequate level of formalization, but in due time. The present discussion needs a quicker way to get into the subject, even if it settles for informal means which cannot be rendered absolutely precise.

Resources

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Higher Order Sign Relations • Discussion 1

Posted on March 27, 2025 by Jon Awbrey

Re: FB | Charles S. Peirce Society • John Corcoran

Questions about the proper treatment of use and mention from the standpoint of Peirce’s theory of signs came up recently in discussions on Facebook. In pragmatic semiotics the trade‑off between “signs‑of‑objects” and “signs‑as‑objects” opens up the wider space of higher order sign relations. In previous work on Inquiry Driven Systems I introduced the subject in the following way.

When interpreters reflect on their use of signs they require an appropriate technical language in which to pursue their reflections. They need signs referring to sign relations, signs referring to elements and components of sign relations, and signs referring to properties and classes of sign relations. The orders of signs developing as reflection evolves can be organized under the heading of “higher order signs” and the reflective sign relations involving them can be referred to as “higher order sign relations”.

References

John Corcoran

Resources

Inquiry Driven Systems

Reflective Interpretive Frameworks

Higher Order Sign Relations

cc: Academia.edu • BlueSky • Laws of Form • Ma^thstodon • Research Gate
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Posted in Arithmetization, C.S. Peirce, Gödel Numbers, Higher Order Sign Relations, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Logic, Mathematics, Quotation, Reflection, Reflective Interpretive Frameworks, Semiotics, Sign Relations, Triadic Relations, Use and Mention, Visualization | Tagged Arithmetization, C.S. Peirce, Gödel Numbers, Higher Order Sign Relations, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Logic, Mathematics, Quotation, Reflection, Reflective Interpretive Frameworks, Semiotics, Sign Relations, Triadic Relations, Use and Mention, Visualization | 6 Comments

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