Inquiry Into Inquiry

Cactus Language • Semantics 6

Posted on October 27, 2025 by Jon Awbrey

If one takes the view that PARCEs and PARCs amount to a pair of intertranslatable representations for the same domain of objects then denotation brackets of the form $\downharpoonleft \ldots \downharpoonright$ can be used to indicate the logical denotation $\downharpoonleft s_j \downharpoonright$ of a sentence $s_j$ or the logical denotation $\downharpoonleft C_j \downharpoonright$ of a cactus $C_j.$

The relations connecting sentences, graphs, and propositions are shown in the next two Tables.

$\text{Semantic Translation} \stackrel{_\bullet}{} \text{Functional Form}$

$\text{Semantic Translation} \stackrel{_\bullet}{} \text{Equational Form}$

Between the sentences and the propositions run the graph‑theoretic data structures which arise in the process of parsing sentences and catalyze their potential for expressing logical propositions or indicator functions. The graph‑theoretic medium supplies an intermediate form of representation between the linguistic sentences and the indicator functions, not only rendering the possibilities of connection between them more readily conceivable in fact but facilitating the necessary translations on a practical basis.

In each Table the passage from the first to the middle column articulates the mechanics of parsing cactus language sentences into graph‑theoretic data structures while the passage from the middle to the last column articulates the semantics of interpreting cactus graphs as logical propositions or indicator functions.

Aside from their common topic, the two Tables present slightly different ways of drawing the maps which go to make up the full semantic transformation.

Semantic Translation • Functional Form: The first Table shows the functional associations connecting each domain with the next, taking the triple of a sentence $s_j,$ a cactus $C_j,$ and a proposition $q_j$ as basic data, and fixing the rest by recursion on those ingredients.
Semantic Translation • Equational Form: The second Table records the transitions in the form of equations, treating sentences and graphs as alternative types of signs and generalizing the denotation bracket to indicate the proposition denoted by either type.

It should be clear at this point that either scheme of translation puts the triples of sentences, graphs, and propositions roughly in the roles of signs, interpretants, and objects, respectively, of a triadic sign relation. Indeed, the roughly can be rendered exactly as soon as the domains of a suitable sign relation are specified precisely.

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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 | 2 Comments

Cactus Language • Semantics 5

Posted on October 14, 2025 by Jon Awbrey

Last time we reached the threshold of a potential codomain or target space for the kind of semantic function we need at this point, one able to supply logical meanings for the syntactic strings and graphs of a given cactus language. In that pursuit we came to contemplate the following definitions.

Logical Conjunction: The conjunction $\mathrm{Conj}_j^J q_j$ of a set of propositions $\{ q_j : j \in J \}$ is a proposition which is true if and only if every one of the $q_j$ is true.; $\mathrm{Conj}_j^J q_j$ is true $\Leftrightarrow$ $q_j$ is true for every $j \in J.$
Logical Surjunction: The surjunction $\mathrm{Surj}_j^J q_j$ of a set of propositions $\{ q_j : j \in J \}$ is a proposition which is true if and only if exactly one of the $q_j$ is untrue.; $\mathrm{Surj}_j^J q_j$ is true $\Leftrightarrow$ $q_j$ is untrue for unique $j \in J.$

If the set of propositions $\{ q_j : j \in J \}$ is finite then the logical conjunction and logical surjunction can be represented by means of sentential connectives, incorporating the sentences which represent the propositions into finite strings of symbols.

If $J$ is finite, for instance, if $J$ consists of the integers in the interval $j = 1 ~\text{to}~ k,$ and if each proposition $q_j$ is represented by a sentence $s_j,$ then the following forms of expression are possible.

Logical Conjunction: The conjunction $\mathrm{Conj}_j^J q_j$ can be represented by a sentence which is constructed by concatenating the $s_j$ in the following fashion.; $\mathrm{Conj}_j^J q_j ~\leftrightsquigarrow~ s_1 s_2 \ldots s_k.$
Logical Surjunction: The surjunction $\mathrm{Surj}_j^J q_j$ can be represented by a sentence which is constructed by surcatenating the $s_j$ in the following fashion.; $\mathrm{Surj}_j^J q_j ~\leftrightsquigarrow~ \texttt{(} s_1 \texttt{,} s_2 \texttt{,} \ldots \texttt{,} s_k \texttt{)}.$

If one opts for a mode of interpretation which moves more directly from the parse graph of a sentence to the potential logical meaning of both the PARC and the PARCE then the following specifications are in order.

A cactus graph rooted at a particular node is taken to represent what that node represents, namely, its logical denotation.

Denotation of a Node: The logical denotation of a node is the logical conjunction of that node’s arguments, which are defined as the logical denotations of that node’s attachments.; The logical denotation of either a blank symbol or empty node is the boolean value $\underline{1} = \mathrm{true}.$; The logical denotation of the paint $\mathfrak{p}_j$ is the proposition $p_j,$ a proposition regarded as primitive, at least, with respect to the level of analysis represented in the current instance of $\mathfrak{C} (\mathfrak{P}).$
Denotation of a Lobe: The logical denotation of a lobe is the logical surjunction of that lobe’s arguments, which are defined as the logical denotations of that lobe’s appendants.; As a corollary, the logical denotation of the parse graph of $\texttt{()},$ also known as a needle, is the boolean value $\underline{0} = \mathrm{false}.$

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Cactus Language • Semantics 4

Posted on October 12, 2025 by Jon Awbrey

Words spoken are symbols or signs (symbola) of affections or impressions (pathemata) of the soul (psyche); written words are the signs of words spoken. As writing, so also is speech not the same for all races of men. But the mental affections themselves, of which these words are primarily signs (semeia), are the same for the whole of mankind, as are also the objects (pragmata) of which those affections are representations or likenesses, images, copies (homoiomata). (Aristotle, De Interp. i. 16^a4).

At this point we have two distinct dialects, scripts, or modes of presentation for the typical cactus language $\mathfrak{C} (\mathfrak{P}),$ each of which needs to be interpreted, that is to say, equipped with a semantic function defined on its domain.

$\textsc{parce} (\mathfrak{P})$: There is the language of strings in $\textsc{parce} (\mathfrak{P}),$ the painted and rooted cactus expressions collectively forming the language $\mathfrak{L} = \mathfrak{C} (\mathfrak{P}) \subseteq \mathfrak{A}^* = (\mathfrak{M} \cup \mathfrak{P})^*.$
$\textsc{parc} (\mathfrak{P})$: There is the language of graphs in $\textsc{parc} (\mathfrak{P}),$ the painted and rooted cacti themselves, a family of graphs or species of data structures formed by parsing the language of strings.

Those two modalities of formal language, like written and spoken natural languages, are meant to have compatible interpretations, which means it is generally sufficient to give the meanings of just one or the other.

All that remains is to provide a codomain or target space for the intended semantic function, that is, to supply a suitable range of logical meanings for the memberships of those languages to map into. One way to do that proceeds by making the following definitions.

Logical Conjunction: The conjunction $\mathrm{Conj}_j^J q_j$ of a set of propositions $\{ q_j : j \in J \}$ is a proposition which is true if and only if every one of the $q_j$ is true.; $\mathrm{Conj}_j^J q_j$ is true $\Leftrightarrow$ $q_j$ is true for every $j \in J.$
Logical Surjunction: The surjunction $\mathrm{Surj}_j^J q_j$ of a set of propositions $\{ q_j : j \in J \}$ is a proposition which is true if and only if exactly one of the $q_j$ is untrue.; $\mathrm{Surj}_j^J q_j$ is true $\Leftrightarrow$ $q_j$ is untrue for unique $j \in J.$

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Cactus Language • Semantics 3

Posted on October 10, 2025 by Jon Awbrey

The task before us is to specify a semantic function for the cactus language $\mathfrak{L} = \mathfrak{C}(\mathfrak{P}),$ in other words, to define a mapping from the space of syntactic expressions to a space of logical statements which “interprets” each expression of $\mathfrak{C}(\mathfrak{P})$ as an expression which says something, an expression which bears a meaning, in short, an expression which denotes a proposition, and is in the end a sign of an indicator function.

When the syntactic expressions of a formal language are given a referent significance in logical terms, for example, as denoting propositions or indicator functions, then each form of syntactic combination takes on a corresponding form of logical significance.

A handy way of providing a logical interpretation for the expressions of any given cactus language is to introduce a family of operators on indicator functions called propositional connectives, to be distinguished from the associated family of syntactic combinations called sentential connectives, where the relationship between the two realms of connection is exactly that between objects on the one hand and their signs on the other.

A propositional connective, as an entity of a well‑defined functional and operational type, can be treated in every way as a logical or mathematical object and thus as the type of object which can be denoted by the corresponding form of syntactic entity, namely, the sentential connective appropriate to the case at hand.

There are two basic types of connectives, called the blank connectives and the bound connectives, respectively, with one connective of each type for each natural number $k = 0, 1, 2, 3, \ldots.$

Blank Connective

The blank connective of $k$ places is signified by the concatenation of the $k$ sentences filling those places.

For the initial case $k = 0,$ the blank connective is an empty string or a blank symbol, both of which have the same denotation among propositions.

For the generic case $k > 0,$ the blank connective takes the form $s_1 \cdot \ldots \cdot s_k.$ In the type of data called a text, the use of the center dot “⋅” is generally supplanted by whatever number of spaces and line breaks serve to improve the readability of the resulting text.

Bound Connective

The bound connective of $k$ places is signified by the surcatenation of the $k$ sentences filling those places.

For the initial case $k = 0,$ the bound connective is an empty closure, an expression taking one of the forms $\texttt{()}, \texttt{(~)}, \texttt{(~~)}, \ldots$ with any number of spaces between the parentheses, all of which have the same denotation among propositions.

For the generic case $k > 0,$ the bound connective takes the form $\texttt{(} s_1 \texttt{,} \ldots \texttt{,} s_k \texttt{)}.$

Resources

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Cactus Language • Semantics 2

Posted on October 8, 2025 by Jon Awbrey

It is common in formal settings to speak of interpretation as if it created a direct connection from the signs of a formal language to the objects of the intended domain, in effect, as if it determined the denotative component of a sign relation. But closer attention to what goes on reveals that the process of interpretation is more indirect, that what it does is provide each sign of a prospectively meaningful source language with a translation into an already established target language, where already established means its relationship to pragmatic objects is taken for granted at the moment in question.

With that in mind, it is clear interpretation is an affair of signs which at best respects the objects of all the signs entering into it, and so it is the connotative aspect of semiotics we find to embody the process. There is nothing wrong with our saying we interpret expressions of a formal language as signs referring to functions or propositions or other objects so long as we understand the reference is generally achieved by way of more familiar and perhaps less formal signs we already take to denote those objects.

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Cactus Language • Semantics 1

Posted on October 6, 2025 by Jon Awbrey

Alas, and yet what are you, my written and painted thoughts! It is not long ago that you were still so many‑coloured, young and malicious, so full of thorns and hidden spices you made me sneeze and laugh — and now? You have already taken off your novelty and some of you, I fear, are on the point of becoming truths: they already look so immortal, so pathetically righteous, so boring!

Nietzsche • Beyond Good and Evil

The discussion to follow describes a particular semantics for painted cactus languages, showing one way to link logical meanings with the bare syntactic forms of linguistic expressions. Forging those links between signs and intents gives the parametric family of formal languages in question one of its principal interpretations.

We’ll keep that interpretation in our sights for the time being but it must be remembered it forms just one of many such interpretations which may be conceivable and even viable in the long run. Indeed, the distinction between the sign domain and the object domain can be observed in the fact that many languages can be deployed to depict the same set of objects while any language worth its salt is bound to give rise to a host of salient interpretations.

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Cactus Language • Mechanics 5

Posted on October 4, 2025 by Jon Awbrey

Re: Cactus Language • Mechanics 4

The following Table summaries the mechanics of the parsing rules given in the previous post.

$\text{Algorithmic Translation Rules}$

A substructure of a painted and rooted cactus $C$ is defined recursively as follows. Starting from the root node of the cactus $C,$ each of its attachments is a substructure of $C.$ If a substructure is a blank or a paint then it constitutes a minimal substructure, meaning no further substructures of $C$ arise from it. If a substructure is a lobe then each of its appendants is also a substructure of $C$ and needs to be examined for further substructures.

The concept of substructure can be used to define the varieties of deletion and erasure operations which respect the structure of the abstract graph. In that application a blank symbol “ ” is treated as a primer, in other words, a clear paint or neutral tint, in effect letting $m_1 = p_0.$ In that frame of discussion it is useful to make the following distinction.

To delete a substructure is to replace it with an empty node, in effect, to reduce the whole structure to a trivial point.
To erase a substructure is to replace it with a blank symbol, in effect, to paint it out of the picture or overwrite it.

A bare PARC, loosely referred to as a bare cactus, is a painted and rooted cactus on the empty palette $\mathfrak{P} = \varnothing.$ A bare cactus can be described in various ways, depending on how the form arises in practice.

Leaning on the definition of a bare PARCE, a bare PARC can be described as the type of parse graph which arises from parsing a bare cactus expression, in other words, from parsing a sentence of the bare cactus language $\mathfrak{C}^0 = \mathrm{PARCE}^0.$

To express it more in its own terms, a bare PARC can be defined by tracing the recursive definition of a generic PARC, but then by detaching an independent form of description from the source of that analogy. The method is sufficiently sketched as follows.

A bare PARC is a PARC whose attachments are limited to blanks and bare lobes.
A bare lobe is a lobe whose appendants are limited to bare PARCs.

In practice a bare cactus is usually encountered in the process of analyzing or handling an arbitrary PARC, the circumstances of which frequently call for deleting or erasing its paints. Among other things, that generally makes it easier to observe the unadorned properties of its underlying graphical structure.

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Cactus Language • Mechanics 4

Posted on September 27, 2025 by Jon Awbrey

To develop a parser for cactus languages in a functional programming style takes a way to express the description of a PARC in terms of its nodes, by recursion from the root up. That requires each node to be specified by a functional expression, having a call of the generic function name $``\mathrm{Node}"$ be followed by a list of arguments naming the attachments of the node in question and having a call of the generic function name $``\mathrm{Lobe}"$ be followed by a list of arguments naming the appendants of the lobe in question. Thus one writes expressions of the following forms.

Working from a structural description of the cactus language, or any suitable formal grammar for $\mathfrak{C} (\mathfrak{P}),$ it is possible to give a recursive definition of a function called $``\mathrm{Parse}"$ which maps each expression in $\textsc{parce} (\mathfrak{P})$ to the corresponding graph in $\textsc{parc} (\mathfrak{P}).$ One way to do that proceeds as follows.

The Concatenation $\mathrm{Conc}_{j=1}^k$ of the $k$ sentences $(s_j)_{j=1}^k$ has a Parse defined as follows.

$\mathrm{Parse} (\mathrm{Conc}^0) ~=~ \mathrm{Node}^0.$
For $k > 0,$
$\mathrm{Parse} (\mathrm{Conc}_{j=1}^k s_j) ~=~ \mathrm{Node}_{j=1}^k \mathrm{Parse} (s_j).$

The Surcatenation $\mathrm{Surc}_{j=1}^k$ of the $k$ sentences $(s_j)_{j=1}^k$ has a Parse defined as follows.

$\mathrm{Parse} (\mathrm{Surc}^0) ~=~ \mathrm{Lobe}^0.$
For $k > 0,$
$\mathrm{Parse} (\mathrm{Surc}_{j=1}^k s_j) ~=~ \mathrm{Lobe}_{j=1}^k \mathrm{Parse} (s_j).$

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Cactus Language • Mechanics 3

Posted on September 25, 2025 by Jon Awbrey

Although the definition of a cactus graph lobe in terms of its intrinsic structural components is logically sufficient it is also useful to characterize the structure of a lobe in extrinsic relational terms, that is, to view the structure that typifies a lobe in relation to the structures of other PARCs and to mark the inclusion of the special type within the general run of PARCs.

That approach to the question of types results in a form of description that appears to be a bit more analytic, at least in mnemonic or prima facie terms, if not ultimately more revealing. Working in that vein, a lobe can be characterized as a special type of PARC called an unpainted root plant (UR‑plant).

An UR‑plant is a PARC of a simpler sort, at least, with respect to the recursive ordering of graph‑theoretic cacti being followed here. As a type, it is defined by the presence of two properties, that of being planted and that of having an unpainted root, defined as follows.

A PARC is planted if its list of attachments has just one PARC.
A PARC is UR if its list of attachments has no blanks or paints.

In short, an UR‑planted PARC has a single PARC as its only attachment, and since that attachment is prevented from being a blank or a paint, the single attachment at its root has to be another sort of structure, that which we call a lobe.

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Cactus Language • Mechanics 2

Posted on September 22, 2025 by Jon Awbrey

The structure of a painted cactus, insofar as it presents itself to the visual imagination, can be described as follows. The overall structure, as given by its underlying graph, falls within the species of graph commonly known as a rooted cactus, to which is added the idea that each of its nodes can be painted with a finite sequence of paints, chosen from a palette given by the parametric set $\{ ``\text{~}" \} \cup \mathfrak{P} = \{ m_1 \} \cup \{ p_1, \ldots, p_k \}.$

It is conceivable on purely graph‑theoretic grounds to have a class of cacti which are painted but not rooted, so it may occasionally be necessary, for the sake of precision, to more exactly pinpoint our target species of graphical structure as a painted and rooted cactus (PARC).

A painted cactus, as a rooted graph, has a distinguished node called its root. By starting from the root and working recursively, the rest of its structure can be described in the following fashion.

Each node of a PARC consists of a graphical point or vertex plus a finite sequence of attachments, described in relative terms as the attachments at or to that node. An empty sequence of attachments defines the empty node. Otherwise, each attachment is one of three kinds: a blank, a paint, or a type of PARC called a lobe.

Each lobe of a PARC consists of a directed graphical cycle plus a finite sequence of appendants, described in relative terms as the appendants of or on that lobe. Since every lobe comes already attached to a particular node, exactly one vertex of the corresponding cycle is the vertex at that node. The remaining vertices of the cycle have their definitions filled out according to the appendants of the lobe in question.

An empty sequence of appendants is structurally equivalent to a sequence containing a single empty node as its only appendant. Either way of looking at it defines a graph‑theoretic structure called a needle or a terminal edge. Otherwise, each appendant of a lobe is itself an arbitrary PARC.

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