Inquiry Into Inquiry<p><strong>Cactus Language • Preliminaries 5</strong></p><p>The easiest way to define the language 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.</p><p>To do that we introduce a family of operations called <i>syntactic connectives</i> on the strings of If the strings on which they operate are already sentences of then the operations amount to <i>sentential connectives</i>. 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 <i>propositional connectives</i>.</p><p>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.</p><p>We begin by defining two sets of basic operations on strings of </p> Concatenation <p>The <i>concatenation</i> of one string is the string </p><p>The <i>concatenation</i> of two strings is the string </p><p>The <i>concatenation</i> of strings is the string </p> Surcatenation <p>The <i>surcatenation</i> of one string is the string </p><p>The <i>surcatenation</i> of two strings is the string </p><p>The <i>surcatenation</i> of strings is the string </p> <p><strong>Resources</strong></p><ul><li><a href="https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries" rel="nofollow noopener noreferrer" target="_blank">Cactus Language • Preliminaries</a></li><li><a href="https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/" rel="nofollow noopener noreferrer" target="_blank">Survey of Animated Logical Graphs</a></li><li><a href="https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/" rel="nofollow noopener noreferrer" target="_blank">Survey of Theme One Program</a></li></ul><p>cc: <a href="https://www.academia.edu/community/lavx71" rel="nofollow noopener noreferrer" target="_blank">Academia.edu</a> • BlueSky • <a href="https://groups.io/g/lawsofform/topic/cactus_language/112006094" rel="nofollow noopener noreferrer" 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#MinimalNegationOperators
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Inquiry Into Inquiry<p><strong>Cactus Language • Preliminaries 4</strong></p><p>The informal mechanisms illustrated in the preceding discussion equip us with a description of <i>cactus language</i> adequate to providing conceptual and computational representations for the minimal formal logical system variously known as <i>propositional logic</i> or <i>sentential calculus</i>.</p> <p>The <i>painted cactus language</i> is actually a parameterized family of languages, consisting of one language for each set of <i>paints</i>.</p><p>The alphabet is the disjoint union of the following two sets of symbols.</p><p> is the alphabet of <i>markers</i>, the set of <i>punctuation marks</i>, or the collection of <i>syntactic constants</i> common to all the languages Various ways of representing the elements of are shown in the following display.</p><p><a href="https://inquiryintoinquiry.com/wp-content/uploads/2025/04/cactus-language-display-2-blog.png" rel="nofollow noopener noreferrer" target="_blank"></a></p><p> is the <i>palette</i>, the alphabet of <i>paints</i>, or the collection of <i>syntactic variables</i> peculiar to the language The set of signs in may be enumerated as follows.</p><p></p> <p><strong>Resources</strong></p><ul><li><a href="https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries" rel="nofollow noopener noreferrer" target="_blank">Cactus Language • Preliminaries</a></li><li><a href="https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/" rel="nofollow noopener noreferrer" target="_blank">Survey of Animated Logical Graphs</a></li><li><a href="https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/" rel="nofollow noopener noreferrer" target="_blank">Survey of Theme One Program</a></li></ul><p>cc: <a href="https://www.academia.edu/community/ld23wj" rel="nofollow noopener noreferrer" target="_blank">Academia.edu</a> • BlueSky • <a href="https://groups.io/g/lawsofform/topic/cactus_language/112006094" rel="nofollow noopener noreferrer" target="_blank">Laws of Form</a> • <a href="https://mathstodon.xyz/@Inquiry/114100591406422570" rel="nofollow noopener noreferrer" target="_blank">Mathstodon</a> • <a href="https://www.researchgate.net/post/Cactus_Language_Preliminaries" rel="nofollow noopener noreferrer" target="_blank">Research Gate</a><br>cc: <a href="https://lists.cs.uni-kassel.de/hyperkitty/list/cg@lists.iccs-conference.org/thread/RUFX4CMZQRLVPQKU4X3MK2RAFLCM6GCK/" rel="nofollow noopener noreferrer" target="_blank">Conceptual Graphs</a> • <a href="https://groups.google.com/g/cybcom/c/kqwcuOMDtJo" rel="nofollow noopener noreferrer" target="_blank">Cybernetics</a> • <a href="https://groups.google.com/g/structural-modeling/c/zL3NIx-kawo" rel="nofollow noopener noreferrer" target="_blank">Structural Modeling</a> • <a href="https://groups.google.com/g/syssciwg/c/hWqifzttgiI" rel="nofollow noopener noreferrer" 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Inquiry Into Inquiry<p><strong>Cactus Language • Preliminaries 3</strong></p><p>A few definitions from formal language theory are required at this point.</p> <p>An <i>alphabet</i> is a finite set of signs, typically, </p><p>A <i>string</i> over an alphabet is a finite sequence of signs from </p><p>The <i>length</i> of a string is just its length as a sequence of signs.</p><p>The <i>empty string</i> 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.</p><p>A sequence of length is typically presented in the following concatenated forms.</p><p></p><p>or</p><p></p><p>with for all </p> <p>The following notations provide useful alternatives.</p> <p> = “” = the empty string.</p><p> = = the language consisting of a single empty string.</p> <p>Several operations on strings find sufficient application to motivate the following definitions. </p> <p>To <i>erase</i> an appearance of a sign is to replace it with an appearance of the blank symbol “ ”.</p><p>To <i>delete</i> an appearance of a sign is to replace it with an appearance of the empty string “”.</p><p>If is a string which ends with a sign then is the string which results by <i>deleting</i> the terminal from </p><p>A <i>token</i> is a particular appearance of a sign.</p> <p>Finally —</p> <p>The <i>kleene star</i> of alphabet is the set of all strings over In particular, includes among its elements the empty string </p><p>The <i>kleene plus</i> of an alphabet is the set of all positive length strings over in other words, everything in but the empty string.</p><p>A <i>formal language</i> over an alphabet is a subset of In brief, If is a string over and is an element of then it is customary to call a <i>sentence</i> of Thus, a formal language is defined by specifying its elements, which amounts to saying what it means to be a sentence of </p> <p><strong>Resources</strong></p><ul><li><a href="https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries" rel="nofollow noopener noreferrer" target="_blank">Cactus Language • Preliminaries</a></li><li><a href="https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/" rel="nofollow noopener noreferrer" target="_blank">Survey of Animated Logical Graphs</a></li><li><a href="https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/" rel="nofollow noopener noreferrer" target="_blank">Survey of Theme One Program</a></li></ul><p>cc: <a href="https://www.academia.edu/community/l79q2Q" rel="nofollow noopener noreferrer" target="_blank">Academia.edu</a> • BlueSky • <a href="https://groups.io/g/lawsofform/topic/cactus_language/112006094" rel="nofollow noopener noreferrer" target="_blank">Laws of Form</a> • <a href="https://mathstodon.xyz/@Inquiry/114100591406422570" rel="nofollow noopener noreferrer" target="_blank">Mathstodon</a> • <a href="https://www.researchgate.net/post/Cactus_Language_Preliminaries" rel="nofollow noopener noreferrer" target="_blank">Research Gate</a><br>cc: <a href="https://lists.cs.uni-kassel.de/hyperkitty/list/cg@lists.iccs-conference.org/thread/RUFX4CMZQRLVPQKU4X3MK2RAFLCM6GCK/" rel="nofollow noopener noreferrer" target="_blank">Conceptual Graphs</a> • <a href="https://groups.google.com/g/cybcom/c/kqwcuOMDtJo" rel="nofollow noopener noreferrer" target="_blank">Cybernetics</a> • <a 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href="https://inquiryintoinquiry.com/tag/cactus-graphs/" target="_blank">#CactusGraphs</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/differential-logic/" target="_blank">#DifferentialLogic</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/equational-inference/" target="_blank">#EquationalInference</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/formal-grammars/" target="_blank">#FormalGrammars</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/formal-languages/" target="_blank">#FormalLanguages</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/graph-theory/" target="_blank">#GraphTheory</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/logic/" target="_blank">#Logic</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/logical-graphs/" target="_blank">#LogicalGraphs</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/mathematics/" target="_blank">#Mathematics</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/minimal-negation-operators/" target="_blank">#MinimalNegationOperators</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/painted-cacti/" target="_blank">#PaintedCacti</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/propositional-calculus/" target="_blank">#PropositionalCalculus</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/visualization/" target="_blank">#Visualization</a></p>
Inquiry Into Inquiry<p><strong>Cactus Language • Preliminaries 2</strong></p><p>As a temporary notation, let the relationship between a particular sign and a particular object , namely, the fact that denotes or the fact that is denoted by , be symbolized in one of the following two ways.</p><p></p><p>Now consider the following paradigm.</p><p><a href="https://inquiryintoinquiry.com/wp-content/uploads/2025/04/cactus-language-display-1-blog.png" rel="nofollow noopener noreferrer" target="_blank"></a></p><p>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.</p><p></p><p>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.</p><p></p><p>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.</p><p></p><p>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.</p><p></p><p><strong>Resources</strong></p><ul><li><a href="https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries" rel="nofollow noopener noreferrer" target="_blank">Cactus Language • Preliminaries</a></li><li><a href="https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/" rel="nofollow noopener noreferrer" target="_blank">Survey of Animated Logical Graphs</a></li><li><a href="https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/" rel="nofollow noopener noreferrer" target="_blank">Survey of Theme One Program</a></li></ul><p>cc: <a href="https://www.academia.edu/community/lOraYE" rel="nofollow noopener noreferrer" target="_blank">Academia.edu</a> • BlueSky • <a href="https://groups.io/g/lawsofform/topic/cactus_language/112006094" rel="nofollow noopener noreferrer" target="_blank">Laws of Form</a> • <a href="https://mathstodon.xyz/@Inquiry/114100591406422570" rel="nofollow noopener noreferrer" target="_blank">Mathstodon</a> • <a href="https://www.researchgate.net/post/Cactus_Language_Preliminaries" rel="nofollow noopener noreferrer" target="_blank">Research Gate</a><br>cc: <a href="https://lists.cs.uni-kassel.de/hyperkitty/list/cg@lists.iccs-conference.org/thread/RUFX4CMZQRLVPQKU4X3MK2RAFLCM6GCK/" rel="nofollow noopener noreferrer" target="_blank">Conceptual Graphs</a> • <a href="https://groups.google.com/g/cybcom/c/kqwcuOMDtJo" rel="nofollow noopener noreferrer" target="_blank">Cybernetics</a> • <a href="https://groups.google.com/g/structural-modeling/c/zL3NIx-kawo" rel="nofollow noopener noreferrer" target="_blank">Structural Modeling</a> • <a href="https://groups.google.com/g/syssciwg/c/hWqifzttgiI" rel="nofollow noopener noreferrer" target="_blank">Systems Science</a></p><p><a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/automata/" target="_blank">#Automata</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/boolean-algebra/" target="_blank">#BooleanAlgebra</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/boolean-functions/" target="_blank">#BooleanFunctions</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/c-s-peirce/" target="_blank">#CSPeirce</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/cactus-graphs/" target="_blank">#CactusGraphs</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/differential-logic/" target="_blank">#DifferentialLogic</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/equational-inference/" target="_blank">#EquationalInference</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/formal-grammars/" target="_blank">#FormalGrammars</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/formal-languages/" target="_blank">#FormalLanguages</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/graph-theory/" target="_blank">#GraphTheory</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/logic/" target="_blank">#Logic</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/logical-graphs/" target="_blank">#LogicalGraphs</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/mathematics/" target="_blank">#Mathematics</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/minimal-negation-operators/" target="_blank">#MinimalNegationOperators</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/painted-cacti/" target="_blank">#PaintedCacti</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/propositional-calculus/" target="_blank">#PropositionalCalculus</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/visualization/" target="_blank">#Visualization</a></p>
Inquiry Into Inquiry<p><strong>Cactus Language • Preliminaries 1</strong></p><p>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.</p><p>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.</p><p>Herbert J. Bernstein • “Idols of Modern Science”</p><p>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.</p><p>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.</p><p>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.</p><p><strong>Resources</strong></p><ul><li><a href="https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Cactus_Language_.E2.80.A2_Preliminaries" rel="nofollow noopener noreferrer" target="_blank">Cactus Language • Preliminaries</a></li><li><a href="https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/" rel="nofollow noopener noreferrer" target="_blank">Survey of Animated Logical Graphs</a></li><li><a href="https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/" rel="nofollow noopener noreferrer" target="_blank">Survey of Theme One Program</a></li></ul><p>cc: <a href="https://www.academia.edu/community/LpWbE7" rel="nofollow noopener noreferrer" target="_blank">Academia.edu</a> • BlueSky • <a href="https://groups.io/g/lawsofform/topic/cactus_language/112006094" rel="nofollow noopener noreferrer" target="_blank">Laws of Form</a> • <a href="https://mathstodon.xyz/@Inquiry/114100591406422570" rel="nofollow noopener noreferrer" target="_blank">Mathstodon</a> • <a href="https://www.researchgate.net/post/Cactus_Language_Preliminaries" rel="nofollow noopener noreferrer" target="_blank">Research Gate</a><br>cc: <a href="https://lists.cs.uni-kassel.de/hyperkitty/list/cg@lists.iccs-conference.org/thread/RUFX4CMZQRLVPQKU4X3MK2RAFLCM6GCK/" rel="nofollow noopener noreferrer" target="_blank">Conceptual Graphs</a> • <a href="https://groups.google.com/g/cybcom/c/kqwcuOMDtJo" rel="nofollow noopener noreferrer" target="_blank">Cybernetics</a> • <a href="https://groups.google.com/g/structural-modeling/c/zL3NIx-kawo" rel="nofollow noopener noreferrer" target="_blank">Structural Modeling</a> • <a href="https://groups.google.com/g/syssciwg/c/hWqifzttgiI" rel="nofollow noopener noreferrer" target="_blank">Systems Science</a></p><p><a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/automata/" target="_blank">#Automata</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/boolean-algebra/" target="_blank">#BooleanAlgebra</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/boolean-functions/" target="_blank">#BooleanFunctions</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/c-s-peirce/" target="_blank">#CSPeirce</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/cactus-graphs/" target="_blank">#CactusGraphs</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/differential-logic/" target="_blank">#DifferentialLogic</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/equational-inference/" target="_blank">#EquationalInference</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/formal-grammars/" target="_blank">#FormalGrammars</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/formal-languages/" target="_blank">#FormalLanguages</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/graph-theory/" target="_blank">#GraphTheory</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/logic/" target="_blank">#Logic</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/logical-graphs/" target="_blank">#LogicalGraphs</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/mathematics/" target="_blank">#Mathematics</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/minimal-negation-operators/" target="_blank">#MinimalNegationOperators</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/painted-cacti/" target="_blank">#PaintedCacti</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/propositional-calculus/" target="_blank">#PropositionalCalculus</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/visualization/" target="_blank">#Visualization</a></p>
Inquiry Into Inquiry<p><strong>Cactus Language • Overview 4</strong></p><p>Depending on whether a formal language is called by the type of sign it enlists or the type of object its signs denote, a cactus language may be called a <i>sentential calculus</i> or a <i>propositional calculus</i>, respectively.</p><p>When the syntactic definition of a language is well enough understood the language can begin to acquire a semantic function. In natural circumstances the syntax and the semantics are likely to be engaged in a process of co‑evolution, whether in ontogeny or in phylogeny, which is to say the two developments tend to form parallel sides of a single bootstrap. But that is not always the easiest way, at least not at first, to formally comprehend the nature of their action or the power of their interaction.</p><p>According to the customary modes of formal reconstruction, a language of the type we are considering is first presented in terms of its syntax, in other words, as a formal language of strings called <i>sentences</i>, and thus amounting to a particular subset of the possible strings which can be formed on a finite alphabet of signs. A syntactic definition of a specific <i>cactus language</i> which proceeds along purely formal lines is carried out in <a href="https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Syntax" rel="nofollow noopener noreferrer" target="_blank">Cactus Language • Syntax</a>. After that, the development of the language’s more concrete aspects can be seen as a matter of defining the following two functions.</p><ul><li>The first is a function which takes each sentence of the language into a computational data structure, namely, a generalized tree‑like parse graph called a <i>painted cactus</i>.</li><li>The second is a function which takes each sentence of the language or its interpolated parse graph into a logical proposition, ending with an indicator function as the object denoted by the sentence.</li></ul><p>The discussion of syntax brings up a number of associated issues which need to be clarified before going on. They may be thought of as questions of <i>style</i>, in other words, the manner of description, grammar, or theory one finds available or chooses as preferable for a given language. Those issues are discussed in <a href="https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Stylistics" rel="nofollow noopener noreferrer" target="_blank">Cactus Language • Stylistics</a>.</p><p>There is an aspect of syntax so schematic in its basic character that it can be conveyed by computational data structures, so algorithmic in its uses that it can be automated by routine mechanisms, and so fixed in its nature that its practical exploitation can be served by the usual devices of computation. Because it involves the transformation of signs it can be recognized as an aspect of semiotics. Since it can be carried out in abstraction from meaning it is not up to the level of semantics, much less a complete pragmatics, though it does incline to the pragmatic aspects of computation which are auxiliary to and incidental to the human use of language. That aspect of formal language use may be described as the <i>algorithmics</i> or <i>mechanics</i> of language processing. A mechanical conversion of cactus languages into their associated data structures is discussed in <a href="https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Mechanics" rel="nofollow noopener noreferrer" target="_blank">Cactus Language • Mechanics</a>.</p><p>In the usual way of proceeding on formal grounds, meaning is added by giving each grammatical sentence, or each syntactically distinguished string, an interpretation as a logically meaningful sentence, in effect, equipping or providing each abstractly well‑formed sentence with a logical proposition for it to denote. A semantic interpretation of cactus language is carried out in <a href="https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Semantics" rel="nofollow noopener noreferrer" target="_blank">Cactus Language • Semantics</a>.</p><p><strong>Resources</strong></p><ul><li><a href="https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview" rel="nofollow noopener noreferrer" target="_blank">Cactus Language • Overview</a></li><li><a href="https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/" rel="nofollow noopener noreferrer" target="_blank">Survey of Animated Logical Graphs</a></li><li><a href="https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/" rel="nofollow noopener noreferrer" target="_blank">Survey of Theme One Program</a></li></ul><p>cc: <a href="https://www.academia.edu/community/lJvgBA" rel="nofollow noopener noreferrer" target="_blank">Academia.edu</a> • BlueSky • <a href="https://groups.io/g/lawsofform/topic/cactus_language/111514096" rel="nofollow noopener noreferrer" target="_blank">Laws of Form</a> • <a href="https://mathstodon.xyz/@Inquiry/114100591406422570" rel="nofollow noopener noreferrer" target="_blank">Mathstodon</a> • <a href="https://www.researchgate.net/post/Cactus_Language_for_Propositional_Calculus" rel="nofollow noopener noreferrer" target="_blank">Research Gate</a><br>cc: <a href="https://lists.cs.uni-kassel.de/hyperkitty/list/cg@lists.iccs-conference.org/thread/J2KLOHUSWS4UCZD5Y3RLHVSX3GPNEJX5/" rel="nofollow noopener noreferrer" target="_blank">Conceptual Graphs</a> • <a href="https://groups.google.com/g/cybcom/c/Ord_shd77qs" rel="nofollow noopener noreferrer" target="_blank">Cybernetics</a> • <a href="https://groups.google.com/g/structural-modeling/c/ChzpYWTZg5g" rel="nofollow noopener noreferrer" target="_blank">Structural Modeling</a> • <a href="https://groups.google.com/g/syssciwg/c/G8pqFl9NA04" rel="nofollow noopener noreferrer" target="_blank">Systems Science</a></p><p><a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/automata/" target="_blank">#Automata</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/boolean-algebra/" target="_blank">#BooleanAlgebra</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/boolean-functions/" target="_blank">#BooleanFunctions</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/c-s-peirce/" target="_blank">#CSPeirce</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/cactus-graphs/" target="_blank">#CactusGraphs</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/differential-logic/" target="_blank">#DifferentialLogic</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/equational-inference/" target="_blank">#EquationalInference</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/formal-grammars/" target="_blank">#FormalGrammars</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/formal-languages/" target="_blank">#FormalLanguages</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/graph-theory/" target="_blank">#GraphTheory</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/logic/" target="_blank">#Logic</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/logical-graphs/" target="_blank">#LogicalGraphs</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/mathematics/" target="_blank">#Mathematics</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/minimal-negation-operators/" target="_blank">#MinimalNegationOperators</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/painted-cacti/" target="_blank">#PaintedCacti</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/propositional-calculus/" target="_blank">#PropositionalCalculus</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/visualization/" target="_blank">#Visualization</a></p>
Inquiry Into Inquiry<p><strong>Cactus Language • Overview 3</strong></p><p>In the development of Cactus Language to date the following two species of graphs have been instrumental.</p><ul><li>Painted And Rooted Cacti (PARCAI).</li><li>Painted And Rooted Conifers (PARCOI).</li></ul><p>It suffices to begin with the first class of data structures, developing their properties and uses in full, leaving discussion of the latter class to a part of the project where their distinctive features are key to developments at that stage. Partly because the two species are so closely related and partly for the sake of brevity, we’ll always use the genus name “PARC” to denote the corresponding cacti.</p><p>To provide a computational middle ground between sentences seen as syntactic strings and propositions seen as indicator functions the language designer must not only supply a medium for the expression of propositions but also link the assertion of sentences to a means for inverting the indicator functions, that is, for computing the <i>fibers</i> or <i>inverse images</i> of the propositions.</p><p>Given a body of conceivable propositions we need a way to follow the threads of their indications from their object domain to their values for the mind and a way to follow those same threads back again. Moreover, we need to implement both ways of proceeding in computational form. Thus we need programs for tracing the clues sentences provide from the universe of their objects to the signs of their values and, in turn, from signs to objects. Ultimately, we need to render propositions so functional as indicators of sets and so essential for examining the equality of sets as to give a rule for the practical conceivability of sets. Tackling that task requires us to introduce a number of new definitions and a collection of additional notational devices, to which we now turn.</p><p><strong>Resources</strong></p><ul><li><a href="https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview" rel="nofollow noopener noreferrer" target="_blank">Cactus Language • Overview</a></li><li><a href="https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/" rel="nofollow noopener noreferrer" target="_blank">Survey of Animated Logical Graphs</a></li><li><a href="https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/" rel="nofollow noopener noreferrer" target="_blank">Survey of Theme One Program</a></li></ul><p>cc: <a href="https://www.academia.edu/community/L2XRbj" rel="nofollow noopener noreferrer" target="_blank">Academia.edu</a> • BlueSky • <a href="https://groups.io/g/lawsofform/topic/cactus_language/111514096" rel="nofollow noopener noreferrer" target="_blank">Laws of Form</a> • <a href="https://mathstodon.xyz/@Inquiry/114100591406422570" rel="nofollow noopener noreferrer" target="_blank">Mathstodon</a> • <a href="https://www.researchgate.net/post/Cactus_Language_for_Propositional_Calculus" rel="nofollow noopener noreferrer" target="_blank">Research Gate</a><br>cc: <a href="https://lists.cs.uni-kassel.de/hyperkitty/list/cg@lists.iccs-conference.org/thread/J2KLOHUSWS4UCZD5Y3RLHVSX3GPNEJX5/" rel="nofollow noopener noreferrer" target="_blank">Conceptual Graphs</a> • <a href="https://groups.google.com/g/cybcom/c/Ord_shd77qs" rel="nofollow noopener noreferrer" target="_blank">Cybernetics</a> • <a href="https://groups.google.com/g/structural-modeling/c/ChzpYWTZg5g" rel="nofollow noopener noreferrer" target="_blank">Structural Modeling</a> • <a href="https://groups.google.com/g/syssciwg/c/G8pqFl9NA04" rel="nofollow noopener noreferrer" target="_blank">Systems Science</a></p><p><a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/automata/" target="_blank">#Automata</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/boolean-algebra/" target="_blank">#BooleanAlgebra</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/boolean-functions/" target="_blank">#BooleanFunctions</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/c-s-peirce/" target="_blank">#CSPeirce</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/cactus-graphs/" target="_blank">#CactusGraphs</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/differential-logic/" target="_blank">#DifferentialLogic</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/equational-inference/" target="_blank">#EquationalInference</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/formal-grammars/" target="_blank">#FormalGrammars</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/formal-languages/" target="_blank">#FormalLanguages</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/graph-theory/" target="_blank">#GraphTheory</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/logic/" target="_blank">#Logic</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/logical-graphs/" target="_blank">#LogicalGraphs</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/mathematics/" target="_blank">#Mathematics</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/minimal-negation-operators/" target="_blank">#MinimalNegationOperators</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/painted-cacti/" target="_blank">#PaintedCacti</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/propositional-calculus/" target="_blank">#PropositionalCalculus</a> <a rel="nofollow noopener noreferrer" class="hashtag u-tag u-category" href="https://inquiryintoinquiry.com/tag/visualization/" target="_blank">#Visualization</a></p>
Inquiry Into Inquiry<p><strong>Differential Propositional Calculus • 37</strong></p><p><strong>Foreshadowing Transformations • Extensions and Projections of Discourse</strong></p><p>And, despite the care which she took to look behind her at every moment, she failed to see a shadow which followed her like her own shadow, which stopped when she stopped, which started again when she did, and which made no more noise than a well‑conducted shadow should.</p><p>— Gaston Leroux • <i>The Phantom of the Opera</i></p><p>Many times in our discussion we have occasion to place one universe of discourse in the context of a larger universe of discourse. An embedding of the type is implied any time we make use of one basis which happens to be included in another basis When discussing differential relations we usually have in mind the extended alphabet has a special construction or a specific lexical relation with respect to the initial alphabet one which is marked by characteristic types of accents, indices, or inflected forms.</p><p><strong>Resources</strong></p><ul><li><a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_3" rel="nofollow noopener noreferrer" target="_blank">Differential Logic and Dynamic Systems</a></li><li><a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_3#Foreshadowing_Transformations" rel="nofollow noopener noreferrer" target="_blank">Differential Logic • Foreshadowing Transformations</a></li></ul><p>cc: <a href="https://www.academia.edu/community/LGvKA9" rel="nofollow noopener noreferrer" target="_blank">Academia.edu</a> • <a 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Inquiry Into Inquiry<p><strong>Differential Propositional Calculus • 36</strong></p><p><strong>Transformations of Discourse</strong></p><p>It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth; for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based. The same thing cannot, however, be said about mathematics; for here we have the new method of thought, pure intellect, the very well‑spring of the times, the <i>fons et origo</i> of an unfathomable transformation.</p><p>— Robert Musil • <i>The Man Without Qualities</i></p><p>Here we take up the general study of <i>logical transformations</i>, or maps relating one universe of discourse to another. In many ways, and especially as applied to the subject of intelligent dynamic systems, the argument will develop the antithesis of the statement just quoted. Along the way, if incidental to my ends, I hope the present essay can pose a fittingly irenic epitaph to the frankly ironic epigraph inscribed at its head.</p><p>The goal is to answer a single question: <i>What is a propositional tangent functor?</i> In other words, the aim is to develop a clear conception of what manner of thing would pass in the logical realm for a genuine analogue of the tangent functor, an object conceived to generalize as far as possible in the abstract terms of category theory the ordinary notions of functional differentiation and the all too familiar operations of taking derivatives.</p><p>As a first step we examine the types of transformations we already know as <i>extensions</i> and <i>projections</i> and we use their special cases to illustrate several styles of logical and visual representation which figure in the sequel.</p><p><strong>Resources</strong></p><ul><li><a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_3" rel="nofollow noopener noreferrer" target="_blank">Differential Logic and Dynamic Systems</a></li><li><a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_3#Transformations_of_Discourse" rel="nofollow noopener noreferrer" target="_blank">Differential Logic • Transformations of Discourse</a></li></ul><p>cc: <a href="https://www.academia.edu/community/lnY2aN" rel="nofollow noopener noreferrer" target="_blank">Academia.edu</a> • <a href="https://groups.google.com/g/cybcom/c/PFFoQfjcrpE" rel="nofollow noopener noreferrer" target="_blank">Cybernetics</a> • <a href="https://groups.google.com/g/structural-modeling/c/1f2HOxj9Fpc" rel="nofollow noopener noreferrer" target="_blank">Structural Modeling</a> • <a href="https://groups.google.com/g/syssciwg/c/V7-N3e0fCFk" rel="nofollow noopener noreferrer" target="_blank">Systems Science</a><br>cc: <a 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Inquiry Into Inquiry<p><strong>Differential Propositional Calculus • 35</strong></p><p><strong>Example 2. Drives and Their Vicissitudes (concl.)</strong></p><p>Applied to the example of ‑gear curves, the indexing scheme results in the data of the next two Tables, showing one period for each orbit.</p><p><a href="https://inquiryintoinquiry.com/wp-content/uploads/2023/12/differential-logic-e280a2-fourth-gear-orbits-e280a2-tablestitles.png" rel="nofollow noopener noreferrer" target="_blank"></a></p><p>The states in each orbit are listed as ordered pairs where may be read as a temporal parameter indicating the present time of the state and where is the decimal equivalent of the binary numeral </p><p>Grasped more intuitively, the Tables show each state with a subscript equal to the numerator of its rational index, taking for granted the constant denominator In that way the temporal succession of states can be reckoned by a <i>parallel round‑up rule</i>. Namely, if is any pair of adjacent digits in the state index then the value of in the next state is </p><p><strong>Resources</strong></p><ul><li><a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2" rel="nofollow noopener noreferrer" target="_blank">Differential Logic and Dynamic Systems</a></li><li><a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#Drives" rel="nofollow noopener noreferrer" target="_blank">Differential Logic • Drives and Their Vicissitudes</a></li></ul><p>cc: <a href="https://www.academia.edu/community/V96Dbj" rel="nofollow noopener noreferrer" target="_blank">Academia.edu</a> • <a href="https://groups.google.com/g/cybcom/c/PFFoQfjcrpE" rel="nofollow noopener noreferrer" target="_blank">Cybernetics</a> • <a href="https://groups.google.com/g/structural-modeling/c/1f2HOxj9Fpc" rel="nofollow noopener noreferrer" target="_blank">Structural Modeling</a> • <a 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Inquiry Into Inquiry<p><strong>Differential Propositional Calculus • 34</strong></p><p><strong>Example 2. Drives and Their Vicissitudes (cont.)</strong></p><p>With a little thought it is possible to devise a canonical indexing scheme for the states in differential logical systems. A scheme of that order allows for comparing changes of state in universes of discourse that weigh in on different scales of observation.</p><p>To that purpose, let us index the states with the dyadic rationals (or the binary fractions) in the half-open interval Formally and canonically, a state is indexed by a fraction whose denominator is the power of two and whose numerator is a binary numeral formed from the coefficients of state in a manner to be described next.</p><p>The <i>differential coefficients</i> of the state are the values for where is defined as identical to To form the binary index of the state the coefficient is read off as the binary digit associated with the place value Expressed in algebraic terms, the rational index of the state is given by the following equivalent formulations.</p><p><a href="https://inquiryintoinquiry.com/wp-content/uploads/2023/12/differential-logic-e280a2-differential-coefficients-e280a2-state-coordinates.png" rel="nofollow noopener noreferrer" target="_blank"></a></p><p><strong>Resources</strong></p><ul><li><a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2" rel="nofollow noopener noreferrer" target="_blank">Differential Logic and Dynamic Systems</a></li><li><a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#Drives" rel="nofollow noopener noreferrer" target="_blank">Differential Logic • Drives and Their Vicissitudes</a></li></ul><p>cc: <a href="https://www.academia.edu/community/lJAaKx" rel="nofollow noopener noreferrer" target="_blank">Academia.edu</a> • <a href="https://groups.google.com/g/cybcom/c/PFFoQfjcrpE" rel="nofollow noopener noreferrer" target="_blank">Cybernetics</a> • <a href="https://groups.google.com/g/structural-modeling/c/1f2HOxj9Fpc" rel="nofollow noopener noreferrer" target="_blank">Structural Modeling</a> • <a href="https://groups.google.com/g/syssciwg/c/V7-N3e0fCFk" rel="nofollow noopener noreferrer" target="_blank">Systems 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Inquiry Into Inquiry<p><strong>Differential Propositional Calculus • 33</strong></p><p><strong>Example 2. Drives and Their Vicissitudes (cont.)</strong></p><p>Expressed in the language of <i>drives</i> and <i>gears</i> our next Example may be described as the family of fourth‑gear curves through the fourth extension Those are the trajectories generated subject to the dynamic law where it’s understood all higher order differences are equal to </p><p>Because and all higher differences are fixed, the state vectors vary only with respect to their projections as points of Thus there is just enough space in a planar venn diagram to plot the orbits and show how they partition the points of It turns out there are just two possible orbits, of eight points each, as shown in the following Figure.</p><p><a href="https://inquiryintoinquiry.com/wp-content/uploads/2023/12/differential-logic-e280a2-fourth-gear-orbits.gif" rel="nofollow noopener noreferrer" target="_blank"></a> <br> </p><p><strong>Resources</strong></p><ul><li><a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2" rel="nofollow noopener noreferrer" target="_blank">Differential Logic and Dynamic Systems</a></li><li><a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#Drives" rel="nofollow noopener noreferrer" target="_blank">Differential Logic • Drives and Their Vicissitudes</a></li></ul><p>cc: <a href="https://www.academia.edu/community/lP14eK" rel="nofollow noopener noreferrer" target="_blank">Academia.edu</a> • <a href="https://groups.google.com/g/cybcom/c/PFFoQfjcrpE" rel="nofollow noopener noreferrer" target="_blank">Cybernetics</a> • <a href="https://groups.google.com/g/structural-modeling/c/1f2HOxj9Fpc" rel="nofollow noopener noreferrer" target="_blank">Structural Modeling</a> • <a href="https://groups.google.com/g/syssciwg/c/V7-N3e0fCFk" rel="nofollow noopener noreferrer" target="_blank">Systems Science</a><br>cc: <a 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Inquiry Into Inquiry<p><strong>Differential Propositional Calculus • 32</strong></p><p><i><br>I open my scuttle at night and see the far‑sprinkled systems,<br>And all I see, multiplied as high as I can cipher, edge but<br> the rim of the farther systems.</i></p><p>— Walt Whitman • <i>Leaves of Grass</i></p><p><strong>Example 2. Drives and Their Vicissitudes</strong></p><p>Before we leave the one‑feature case let’s look at a more substantial example, one which illustrates a general class of curves through the extended feature spaces and affords an opportunity to discuss important themes concerning their structure and dynamics.</p><p>As before let The discussion to follow considers a class of trajectories having the property that for all greater than a fixed value and indulges in the use of a picturesque vocabulary to describe salient classes of those curves.</p><p>Given the above finite order condition, there is a highest order non‑zero difference exhibited at each point of any trajectory one may consider. With respect to any point of the corresponding curve let us call that highest order differential feature the <i>drive</i> at that point. Curves of constant drive are then referred to as ‑<i>gear curves</i>.</p><ul><li><b>Note.</b> The fact that a difference calculus can be developed for boolean functions is well known and was probably familiar to Boole, who was an expert in difference equations before he turned to logic. And of course there is the strange but true story of how the Turin machines of the 1840s prefigured the Turing machines of the 1940s. At the very outset of general purpose mechanized computing we find the motive power driving the Analytical Engine of Babbage, the kernel of an idea behind all of his wheels, was exactly his notion that difference operations, suitably trained, can serve as universal joints for any conceivable computation.</li></ul><p><strong>Resources</strong></p><ul><li><a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2" rel="nofollow noopener noreferrer" target="_blank">Differential Logic and Dynamic Systems</a></li><li><a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#Drives" rel="nofollow noopener noreferrer" target="_blank">Differential Logic • Drives and Their Vicissitudes</a></li></ul><p>cc: <a href="https://www.academia.edu/community/VBqpez" rel="nofollow noopener noreferrer" target="_blank">Academia.edu</a> • <a href="https://groups.google.com/g/cybcom/c/PFFoQfjcrpE" rel="nofollow noopener noreferrer" 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Inquiry Into Inquiry<p><strong>Differential Propositional Calculus • 31</strong></p><p><strong>Tacit Extensions</strong></p><p>Returning to the <a href="https://inquiryintoinquiry.com/wp-content/uploads/2023/12/differential-logic-e280a2-differential-propositions.png" rel="nofollow noopener noreferrer" target="_blank">Table of Differential Propositions</a>, let’s examine how the general concept of a tacit extension applies to the differential extension of a one‑dimensional universe of discourse, where and </p><p>Each proposition has a canonical expression in the set The tacit extension may then be expressed as a logical conjunction where is a logical tautology using all the variables in The following Table shows how the tacit extensions of the propositions may be expressed in terms of the extended basis </p><p> <br> <a href="https://inquiryintoinquiry.com/wp-content/uploads/2023/12/differential-logic-e280a2-tacit-extensions.png" rel="nofollow noopener noreferrer" target="_blank"></a></p><p>In its bearing on the singular propositions over a universe of discourse the above analysis has an interesting interpretation. The tacit extension takes us from thinking about a particular state, like or to considering the collection of outcomes, the outgoing changes or singular dispositions springing or stemming from that state.</p><p><strong>Resources</strong></p><ul><li><a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2" rel="nofollow noopener noreferrer" target="_blank">Differential Logic and Dynamic Systems</a></li><li><a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#Tacit_Extensions" rel="nofollow noopener noreferrer" target="_blank">Differential Logic • Tacit Extensions</a></li></ul><p>cc: <a href="https://www.academia.edu/community/Lg70pv" rel="nofollow noopener noreferrer" target="_blank">Academia.edu</a> • <a href="https://groups.google.com/g/cybcom/c/PFFoQfjcrpE" rel="nofollow noopener noreferrer" target="_blank">Cybernetics</a> • <a href="https://groups.google.com/g/structural-modeling/c/1f2HOxj9Fpc" rel="nofollow noopener 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Inquiry Into Inquiry<p><strong>Differential Propositional Calculus • 30</strong></p><p><i><br>I would really like to have slipped imperceptibly into this lecture, as into all the others I shall be delivering, perhaps over the years ahead.</i></p><p>— Michel Foucault • <i>The Discourse on Language</i></p><p><strong>Tacit Extensions</strong></p><p>In viewing the previous <a href="https://inquiryintoinquiry.com/wp-content/uploads/2023/12/differential-logic-e280a2-differential-propositions.png" rel="nofollow noopener noreferrer" target="_blank">Table of Differential Propositions</a> it is important to notice the subtle distinction in type between a function and its inclusion as a function even though they share the same logical expression. Naturally, we want to maintain the logical equivalence of expressions representing the same proposition while appreciating the full diversity of a proposition’s functional and typical representatives. Both perspectives, and all the levels of abstraction extending through them, have their reasons, as will develop in time.</p><p>Because this special circumstance points to a broader theme, it’s a good idea to discuss it more generally. Whenever there arises a situation like that above, where one basis is a subset of another basis we say any proposition has a <i>tacit extension</i> to a proposition and we say the space has an <i>automatic embedding</i> within the space </p><p>The <i>tacit extension operator</i> is defined in such a way that puts the same constraint on the variables of within as the proposition initially put on while it puts no constraint on the variables of beyond in effect, conjoining the two constraints.</p><p>Indexing the variables as and the tacit extension from to may be expressed by the following equation.</p><p></p><p>On formal occasions, such as the present context of definition, the tacit extension from to is explicitly symbolized by the operator where the bases and are set in context, but it’s normally understood the may be silent.</p><p><strong>Resources</strong></p><ul><li><a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2" rel="nofollow noopener noreferrer" target="_blank">Differential Logic and Dynamic Systems</a></li><li><a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#Tacit_Extensions" rel="nofollow noopener noreferrer" target="_blank">Differential Logic • Tacit Extensions</a></li></ul><p>cc: <a href="https://www.academia.edu/community/5wQD11" rel="nofollow noopener noreferrer" target="_blank">Academia.edu</a> • <a href="https://groups.google.com/g/cybcom/c/PFFoQfjcrpE" rel="nofollow noopener noreferrer" target="_blank">Cybernetics</a> • <a href="https://groups.google.com/g/structural-modeling/c/1f2HOxj9Fpc" rel="nofollow noopener noreferrer" target="_blank">Structural Modeling</a> • <a href="https://groups.google.com/g/syssciwg/c/V7-N3e0fCFk" rel="nofollow noopener noreferrer" target="_blank">Systems Science</a><br>cc: <a href="https://lists.cs.uni-kassel.de/hyperkitty/list/cg@lists.iccs-conference.org/thread/AAUG4A25KF6DRXCJ6WAJVAXWFHTPBKYZ/" rel="nofollow noopener noreferrer" 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Inquiry Into Inquiry<p><strong>Differential Propositional Calculus • 29</strong></p><p><i><br>I guess it must be the flag of my disposition, out of hopeful<br> green stuff woven.</i></p><p>— Walt Whitman • <i>Leaves of Grass</i></p><p><strong>Back to the Feature</strong></p><p>Let’s assume the sense intended for differential features is well enough established in the intuition for now to continue outlining the structure of the differential extension </p><p>The extended alphabet of cardinality generates the terms of description for the extended space of cardinality according to the following series of equations.</p><p></p><p>The space may be given the nominal type at root isomorphic to An element of may be regarded as a <i>disposition at a point</i> or a <i>situated direction</i>, in effect, a singular mode of change occurring at a single point in the universe of discourse. In practice the modality of those changes may be interpreted in various ways, for example, as expectations, intentions, or observations with respect to the behavior of a system.</p><p>To construct the extended universe of discourse the basic dispositions in need to be extended to the full set of differential propositions each of type There are propositions in as detailed in the following Table.</p><p> <br> <a href="https://inquiryintoinquiry.com/wp-content/uploads/2023/12/differential-logic-e280a2-differential-propositions.png" rel="nofollow noopener noreferrer" target="_blank"></a></p><p>Aside from changing the names of variables and shuffling the order of rows, the Table follows the format previously used for boolean functions of two variables. The rows are grouped to reflect natural similarity classes holding among the propositions. In a future discussion the classes will be given additional explanation and motivation as the orbits of a certain transformation group acting on the set of 16 propositions. Notice that four of the propositions, in their logical expressions, resemble those given in the table for Thus the first set of propositions is automatically embedded in the present set and the corresponding inclusions are indicated at the far left margin of the Table.</p><p><strong>Resources</strong></p><ul><li><a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2" rel="nofollow noopener noreferrer" target="_blank">Differential Logic and Dynamic Systems</a></li><li><a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#Feature" rel="nofollow noopener noreferrer" target="_blank">Differential Logic • Back to the Feature</a></li></ul><p>cc: <a href="https://www.academia.edu/community/VBqD3R" rel="nofollow noopener noreferrer" target="_blank">Academia.edu</a> • <a href="https://groups.google.com/g/cybcom/c/PFFoQfjcrpE" rel="nofollow noopener noreferrer" target="_blank">Cybernetics</a> • <a 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Inquiry Into Inquiry<p><strong>Differential Propositional Calculus • 28</strong></p><p><strong>Commentary On Small Models • 2</strong></p><p>The consequence of dealing with “practically infinite extensions” becomes crucial in building neural network systems capable of learning and adapting, since the adaptive competence of any intelligent system is limited to the objects and domains it is able to represent. If we seek to design systems which operate intelligently with the full deck of propositions dealt by intact universes of discourse then we must supply those systems with succinct representations and efficient transformations in that domain.</p><p>Beyond the ability to learn and adapt, which taken at the ebb so often devolves into bare conformity and confirmation bias, the ability to inquire and reason makes even more demands on propositional representation. The project of constructing inquiry driven systems forces us to contemplate the level of generality embodied in logical propositions. We can see that because the progress of inquiry is driven by the manifest discrepancies occurring among expectations, intentions, and observations, and each of those components of systematic knowledge takes on the fully generic character of an empirical summary or an axiomatic theory.</p><p>A compression scheme by any other name is a symbolic representation — and that is what the differential extension of propositional calculus is intended to supply. But why is that particular program of mental calisthenics worth carrying out in general?</p><p>The provision of a uniform logical framework for describing time‑evolving systems makes the task of understanding complex systems easier than it would otherwise be when we try to tackle each new system <i>de novo</i>, “from scratch” as we say. Having a uniform medium ready to hand helps both in looking for invariant representations of individual cases and also in finding points of comparison among diverse structures otherwise appearing to be isolated systems. All that goes to facilitate the search for compact knowledge, to apply what is learned from individual cases to the general realm.</p><p><strong>Resources</strong></p><ul><li><a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2" rel="nofollow noopener noreferrer" target="_blank">Differential Logic and Dynamic Systems</a></li><li><a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#COSM" rel="nofollow noopener noreferrer" target="_blank">Differential Logic • Commentary On Small Models</a></li></ul><p>cc: <a href="https://www.academia.edu/community/VvWJME" rel="nofollow noopener noreferrer" target="_blank">Academia.edu</a> • <a href="https://groups.google.com/g/cybcom/c/PFFoQfjcrpE" rel="nofollow noopener noreferrer" target="_blank">Cybernetics</a> • <a href="https://groups.google.com/g/structural-modeling/c/1f2HOxj9Fpc" rel="nofollow noopener noreferrer" target="_blank">Structural Modeling</a> • <a 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Inquiry Into Inquiry<p><strong>Differential Propositional Calculus • 27</strong></p><p><strong>Commentary On Small Models • 1</strong></p><p>One reason for engaging in our present order of extremely reduced but explicitly controlled case study is to throw light on the general study of languages, formal and natural, in their full array of syntactic, semantic, and pragmatic aspects. Propositional calculus is one of the last points of departure where it is possible to see that trio of aspects interacting in a non‑trivial way without being immediately and totally overwhelmed by the complexity they generate.</p><p>The generative complexity of formal and natural languages tends to lead investigators to adopt the strategy of focusing on a single aspect of the domain, abandoning hope of understanding the whole, whether it is the still living natural language or the dynamics of inquiry crystallized in formal logic.</p><p>In the perspective adopted here, a language is a syntactic system evolved or designed to express a set of descriptions. If the explicit symbols of a language have extensions in its object world which are actually infinite, or if the implicit categories and generative devices of a linguistic theory have extensions in its subject matter which are potentially infinite, then the finite characters of terms, statements, arguments, grammars, logics, and rhetorics force a surplus intension to color the symbols and functions of that language, all across the spectrum from object language to metalinguistic reflection.</p><p>In the aphorism of Wilhelm von Humboldt often cited by Chomsky, language requires “the infinite use of finite means”. That is necessarily true when the extensions are infinite, when the referential symbols and grammatical categories of a language possess infinite sets of models and instances. But it also voices a practical truth when the extensions, though finite at every stage, tend to grow at exponential rates.</p><p><strong>Resources</strong></p><ul><li><a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2" rel="nofollow noopener noreferrer" target="_blank">Differential Logic and Dynamic Systems</a></li><li><a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#COSM" rel="nofollow noopener noreferrer" target="_blank">Differential Logic • Commentary On Small Models</a></li></ul><p>cc: <a href="https://www.academia.edu/community/lnYpm1" rel="nofollow noopener noreferrer" target="_blank">Academia.edu</a> • <a href="https://groups.google.com/g/cybcom/c/PFFoQfjcrpE" rel="nofollow noopener noreferrer" target="_blank">Cybernetics</a> • <a href="https://groups.google.com/g/structural-modeling/c/1f2HOxj9Fpc" rel="nofollow noopener noreferrer" target="_blank">Structural Modeling</a> • <a 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Inquiry Into Inquiry<p><strong>Differential Propositional Calculus • 26</strong></p><p><strong>Example 1. A Square Rigging (concl.)</strong></p><p>If we eliminate from view the regions of ruled out by the dynamic law then what remains is the quotient structure shown in the following Figure. The picture makes it easy to see how the dynamically allowable portion of the universe is partitioned between the respective holdings of and As it happens, the fact might have been expressed “right off the bat” by an equivalent formulation of the differential law, one which uses the exclusive disjunction to state the law as </p><p><a href="https://inquiryintoinquiry.com/wp-content/uploads/2023/12/differential-logic-e280a2-tiller.png" rel="nofollow noopener noreferrer" target="_blank"></a> <br> </p><p>What we have achieved in this example is to give a differential description of a simple dynamic process. We did this by embedding a directed graph, representing the state transitions of a finite automaton, in the share of a boolean lattice or <i>n</i>‑cube cut out by nullifying all the regions the dynamics outlaws.</p><p>With growth in the dimensions of our contemplated universes it becomes essential, both for human comprehension and for computer implementation, that dynamic structures of interest be represented not actually, by acquaintance, but virtually, by description. In our present study we are using the language of propositional calculus to express the relevant descriptions, and to grasp the structures embodied in subsets of <i>n</i>‑cubes without being forced to actualize all their points.</p><p><strong>Resources</strong></p><ul><li><a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2" rel="nofollow noopener noreferrer" target="_blank">Differential Logic and Dynamic Systems</a></li><li><a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#Example_1" rel="nofollow noopener noreferrer" target="_blank">Differential Logic • A Square Rigging</a></li></ul><p>cc: <a href="https://www.academia.edu/community/lnYp70" rel="nofollow noopener noreferrer" target="_blank">Academia.edu</a> • <a href="https://groups.google.com/g/cybcom/c/PFFoQfjcrpE" rel="nofollow noopener noreferrer" target="_blank">Cybernetics</a> • <a href="https://groups.google.com/g/structural-modeling/c/1f2HOxj9Fpc" 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Inquiry Into Inquiry<p><strong>Differential Propositional Calculus • 25</strong></p><p><strong>Example 1. A Square Rigging (cont.)</strong></p><p>Because the initial space is one‑dimensional we can easily fit the second order extension within the compass of a single venn diagram, charting the pair of converging trajectories as shown in the following Figure.</p><p><a href="https://inquiryintoinquiry.com/wp-content/uploads/2023/12/differential-logic-e280a2-anchor.png" rel="nofollow noopener noreferrer" target="_blank"></a> <br> </p><p><strong>Resources</strong></p><ul><li><a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2" rel="nofollow noopener noreferrer" target="_blank">Differential Logic and Dynamic Systems</a></li><li><a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_2#Example_1" rel="nofollow noopener noreferrer" target="_blank">Differential Logic • A Square Rigging</a></li></ul><p>cc: <a href="https://www.academia.edu/community/5NAKA3" rel="nofollow noopener noreferrer" target="_blank">Academia.edu</a> • <a 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