Inquiry Into Inquiry

Cactus Language • Pragmatics 12

Posted on August 10, 2025 by Jon Awbrey

The concatenation $\mathfrak{L}_1 \cdot \mathfrak{L}_2$ of the formal languages $\mathfrak{L}_1$ and $\mathfrak{L}_2$ is just a cartesian product $\mathfrak{L}_1 \times \mathfrak{L}_2$ of the sets $\mathfrak{L}_1$ and $\mathfrak{L}_2$ but the relation of cartesian products to set‑theoretic intersections and thus to logical conjunctions is not immediately clear. One way of seeing a type of relation is to focus on the information needed to specify each construction and thus to reflect on the signs used to bear the information. As a first approach to the topic of information I introduce the following set of ideas, intended to be taken in a very provisional way.

A stricture is a specification of a certain set in a certain place, relative to a number of other sets yet to be specified. It is assumed one knows enough to tell if two strictures are equivalent as pieces of information but any more determinate indications, for instance, names for the places mentioned in the stricture or bounds on the number of places involved, are regarded as extraneous impositions, outside the proper concern of the definition, no matter how convenient they happen to be for a particular discussion. As a schematic form of illustration, a stricture can be pictured in the following shape.

$``\ldots \times X \times Q \times X \times \ldots"$

A strait is the object specified by a stricture, in other words, a certain set in a certain place of an otherwise yet to be specified relation. Somewhat sketchily, the strait corresponding to the stricture just given can be pictured in the following shape.

$\ldots \times X \times Q \times X \times \ldots$

In that picture $Q$ is a certain set and $X$ is the universe of discourse relevant to a given discussion. As a stricture does not contain a sufficient amount of information to specify the number of sets it intends to set in place, or even to pin down the absolute location of the set it does set in place, it appears to place an unspecified number of unspecified sets in a vague and uncertain state of affairs. Taken out of its interpretive context the residual information a stricture is able to bear makes all of the following potentially equivalent as strictures.

$\begin{array}{ccccccc} ``Q" & , & ``X \times Q \times X" & , & ``X \times X \times Q \times X \times X" & , & \ldots \end{array}$

With respect to what those strictures specify, that leaves all of the following equivalent as straits.

$\begin{array}{ccccccc} Q & = & X \times Q \times X & = & X \times X \times Q \times X \times X & = & \ldots \end{array}$

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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 • Pragmatics 11

Posted on August 9, 2025 by Jon Awbrey

I am throwing together a wide variety of different operations into the bins labeled additive and multiplicative but it’s easy to observe a natural organization and even some relations approaching isomorphisms among and between the members of each class.

The relation between logical disjunction and the union of sets and the relation between logical conjunction and the intersection of sets ought to be clear enough for present purposes. But the relation of set‑theoretic union to category‑theoretic co‑product and the relation of set‑theoretic intersection to syntactic concatenation deserve a closer look at this point.

The effect of a co‑product as a disjointed union, in other words, that creates an object tantamount to a disjoint union of sets in the resulting co‑product even if some of those sets intersect non‑trivially and even if some of them are identical in reality, can be achieved in several ways.

The usual conception is that of making a separate copy, for each part of the intended co‑product, of the set assigned to that part. One imagines the set assigned to a particular part of the co‑product as being distinguished by a particular color, in other words, by the attachment of a distinct index, label, or tag, any sort of marker inherited by and passed on to every element of the set in that part. A concrete image of the construction can be achieved by imagining each set and each element of each set is placed in an ordered pair with the sign of its color, index, label, or tag. One describes that as the injection of each set into the corresponding part of the co‑product.

For example, given the sets $P$ and $Q,$ overlapping or not, one defines the indexed or marked sets $P_{[1]}$ and $Q_{[2]},$ amounting to the copy of $P$ into the first part of the co‑product and the copy of $Q$ into the second part of the co‑product, in the following manner.

$\begin{array}{lllll} P_{[1]} & = & (P, 1) & = & \{ (x, 1) : x \in P \}, \\ Q_{[2]} & = & (Q, 2) & = & \{ (x, 2) : x \in Q \}. \end{array}$

Using the co‑product operator ( $\textstyle\coprod$ ) for the construction, the sum, the co‑product, or the disjointed union of $P$ and $Q$ in that order can be represented as the ordinary union of $P_{[1]}$ and $Q_{[2]}.$

$\begin{array}{lll} P \coprod Q & = & P_{[1]} \cup Q_{[2]}. \end{array}$

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Cactus Language • Pragmatics 10

Posted on August 5, 2025 by Jon Awbrey

One insight arising from Peirce’s work on the mathematics underlying logic is that the operations on sets known as complementation, intersection, and union, along with the corresponding logical operations of negation, conjunction, and disjunction, are not as fundamental as they first appear. That is because all of them can be constructed or derived from a smaller set of operations, in fact, taking the logical side of things, from either one of two sole sufficient operators called amphecks by Peirce, strokes by those who re‑discovered them later, and known in computer science as the operators nand and nnor. Thus by virtue of their precedence in the orders of construction and derivation, the sole sufficient operators have to be regarded as the simplest and most primitive in principle, even if they are scarcely recognized as lying among the more familiar elements of logic.

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Cactus Language • Pragmatics 9

Posted on August 4, 2025 by Jon Awbrey

A moment’s reflection on the issue of style, giving due consideration to the received array of stylistic choices, ought to inspire at least the question: “Are those the only choices there are?”

There are abundant indications that other options, more differentiated varieties of description and more integrated ways of approaching individual languages, are likely to be conceivable, feasible, and even more ultimately viable. If a suitably generic style, one that incorporates the full scope of logical combinations and operations, is broadly available, then it would no longer be necessary, or even apt, to argue in universal terms about which style is best, but more useful to investigate how we might adapt the local styles to the local requirements. The medium of a fully generic style would yield a viable compromise between additive and multiplicative canons and render the choice between parallel and serial a false alternative, at least, when expressed in the globally exclusive terms which are currently and most commonly adopted to pose it.

One set of indications comes from the study of machines, languages, and computation, including theories of their structures and relations. The forms of composition and decomposition known as parallel and serial are merely the limiting special cases in two directions of specialization of a more generic form, commonly known as the cascade form of combination. That is a well‑known fact in the theories dealing with automata and their associated formal languages but its implications do not seem to be widely appreciated outside those fields. In particular, the availability of that option dispells the need to choose one extreme or the other, since most of the natural cases are likely to exist somewhere in between.

Another set of indications appears in algebra and category theory, where forms of composition and decomposition related to the cascade combination, namely, the semi‑direct product and its special case, the wreath product, are encountered at higher levels of generality than the cartesian products of sets or the direct products of spaces.

In those domains of operation, one finds it necessary to consider also the co‑product of sets and spaces, a construction which artificially creates a disjoint union of sets, that is, a union of spaces which are being treated as independent. It does that, in effect, by indexing, coloring, or preparing the otherwise possibly overlapping domains which are being combined. What renders that a chimera or a hybrid form of combination is the fact that the indexing is tantamount to a cartesian product of a singleton set, namely, the conventional index, color, or affix in question, with the individual domain which is entering as a factor, a term, or a participant in the final result.

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Cactus Language • Pragmatics 8

Posted on August 2, 2025 by Jon Awbrey

It is useful to examine the relation between syntactic production $(:>\!)$ and logical implication $(\Rightarrow\!)$ with one eye to what they have in common and another eye to how they differ.

The production $q :> W$ says the appearance of the symbol $q$ in a sentential form implies the possibility of replacing $q$ with $W.$ Although that sounds like a possible implication, to the extent that $q$ implies a possible $W$ or that $q$ possibly implies $W,$ the qualifiers possible and possibly are essential to the meaning of what is actually implied. In effect, those qualifications reverse the direction of implication, making $``q \Leftarrow W"$ the best analogue for the sense of the production.

One way to understand a production of the form $q :> W$ is to realize non‑terminal symbols have the significance of hypotheses. The terminal strings form the empirical matter of the language in question while the non‑terminal symbols mark the patterns or types of substrings which may be recognized in the linguistic corpus. If one observes a portion of a terminal string which fits the pattern of a sentential form $W$ then it is an admissible hypothesis, according to the theory of the language afforded by the formal grammar, that the piece of string not only fits the type $q$ but even comes to be generated under the auspices of the non‑terminal symbol $``q".$

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Cactus Language • Pragmatics 7

Posted on August 1, 2025 by Jon Awbrey

There is a curious sort of diagnostic clue which often serves to reveal the dominance of one mode or the other within an individual thinker’s cognitive style. Examined on the question of what constitutes the natural numbers, an additive thinker tends to start the sequence at 0 while a multiplicative thinker tends to regard it as beginning at 1.

In any style of description, grammar, or theory of a language, it is usually possible to tease out the influence of the contrasting traits, namely, the additive attitude versus the multiplicative tendency which go to make up the style in question, and even to determine the dominant inclination or point of view which establishes its perspective on the target domain.

In each style of formal grammar, the multiplicative aspect is present in the sequential concatenation of signs, in both the augmented strings and the terminal strings. In settings where the non‑terminal symbols classify types of strings, the concatenation of the non-terminal symbols signifies the cartesian product over the corresponding sets of strings.

In the context‑free style of formal grammar, the additive aspect is easy to spot. It is signaled by the parallel covering of many augmented strings or sentential forms by the same non‑terminal symbol. In active terms, it calls for the independent rewriting of that non‑terminal symbol by a number of different successors, as in the following scheme.

$\begin{matrix} q & :> & W_1 \\[4pt] \cdots & \cdots & \cdots \\[4pt] q & :> & W_k \end{matrix}$

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Cactus Language • Pragmatics 6

Posted on July 30, 2025 by Jon Awbrey

It is possible to trace the divergence of formal grammar styles to an even more primitive division, distinguishing between the additive or parallel styles and the multiplicative or serial styles. The issue is somewhat confused by the fact that an additive analysis is typically expressed in the form of a series, in other words, a disjoint union of sets or a linear sum of their independent effects. But it is easy enough to sort things out if one observes the more telling connection between parallel and independent. Another way to keep the right associations straight is to employ the term sequential in preference to the more misleading term serial. Whatever one calls the broad division of styles, the scope and sweep of their dimensions of variation can be delineated in the following way.

Additive or parallel styles favor sums of products $(\textstyle \sum\prod)$ as canonical forms of expression, pulling sums, unions, co‑products, and logical disjunctions to the outermost layers of analysis and synthesis, while pushing products, intersections, concatenations, and logical conjunctions to the innermost levels of articulation and generation. The analogous style in propositional logic leads to the disjunctive normal form (DNF).
Multiplicative or serial styles favor products of sums $(\textstyle \prod\sum)$ as canonical forms of expression, pulling products, intersections, concatenations, and logical conjunctions to the outermost layers of analysis and synthesis, while pushing sums, unions, co‑products, and logical disjunctions to the innermost levels of articulation and generation. The analogous style in propositional logic leads to the conjunctive normal form (CNF).

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Cactus Language • Pragmatics 5

Posted on July 29, 2025 by Jon Awbrey

Along with the distinctions we see evolving among different styles of grammar and the preferences different observers display toward them, there naturally arises the question: What is the root of that evolution?

One dimension of variation in formal grammar style can be seen by treating a union of languages, and especially a disjoint union of languages, as a sum $(\textstyle \sum),$ by treating a concatenation of languages as a product $(\textstyle \prod),$ and then by distinguishing the styles of analysis favoring sums of products $(\textstyle \sum\prod)$ from those favoring products of sums $(\textstyle \prod\sum)$ as their canonical forms of description.

If one examines the relationship between grammars and languages closely enough to detect the influence of the above two styles and comes to appreciate how different grammar styles may be used with different degrees of success for different purposes then one begins to see the possibility that alternative styles of description might be based on altogether different linguistic and logical operations.

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Cactus Language • Pragmatics 4

Posted on July 28, 2025 by Jon Awbrey

The questions about boundary conditions we keep encountering betray a more general issue. Already by this point in the discussion the limits of a purely syntactic approach to language are becoming visible. It is not that one cannot go a long way by that road in the analysis of a particular language and the study of languages in general but when it comes to understanding the purpose of a language, extending its use in a chosen direction, or designing a language for a particular set of uses, what matters above all are the pragmatic equivalence classes of signs demanded by the application and intended by the designer and not so much the peculiar characters of signs representing the classes of practical meaning.

Any description of a language is bound to have alternative descriptions. In particular, a formally circumscribed description of a formal language, as any effectively finite description is bound to be, is certain to suggest the equally likely existence and possible utility of other descriptions. A single formal grammar describes but a single formal language, but any formal language is described by many formal grammars, not all of which afford the same grasp of its structure, provide equivalent comprehensions of its character, or yield interchangeable views of its aspects. Even with respect to the same formal language, different formal grammars are typically better for different purposes.

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Cactus Language • Pragmatics 3

Posted on July 27, 2025 by Jon Awbrey

Having broached the distinction between objective propositions and syntactic sentences, its analogy to the distinction between numbers and numerals becomes clear. What are the implications of that distinction for the realm of reasoning about propositions and its representation in sentential logic?

If the purpose of a sentence is precisely to denote a proposition then the proposition is simply the object of whatever sign is taken for the sentence. The computational manifestation of a piece of reasoning about propositions thus amounts to a process taking place entirely within a language of sentences, being a procedure which can rationalize its account by referring to the denominations of sentences among propositions.

As far as it bears on our current context of problems, the upshot is this: Do not worry too much about what roles the empty string $\varepsilon = ``"$ and the blank symbol $m_1 = ``~"$ are supposed to play in a given species of formal language. As it happens, it is far less important to wonder whether those types of formal tokens actually constitute genuine sentences than it is to decide what equivalence classes it makes sense to form over all the sentences in the resulting language, and only then to bother about what equivalence classes those limiting cases of sentences are most conveniently taken to represent.

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