Jon Awbrey<p>Cactus Language • Overview 3.2<br>• <a href="https://inquiryintoinquiry.com/2025/03/07/cactus-language-overview-3/" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">inquiryintoinquiry.com/2025/03</span><span class="invisible">/07/cactus-language-overview-3/</span></a></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>Resources —</p><p>Cactus Language • Overview<br>• <a href="https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">oeis.org/wiki/Cactus_Language_</span><span class="invisible">%E2%80%A2_Overview</span></a></p><p>Survey of Animated Logical Graphs<br>• <a href="https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">inquiryintoinquiry.com/2024/03</span><span class="invisible">/18/survey-of-animated-logical-graphs-7/</span></a></p><p>Survey of Theme One Program<br>• <a href="https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">inquiryintoinquiry.com/2024/02</span><span class="invisible">/26/survey-of-theme-one-program-6/</span></a></p><p><a href="https://mathstodon.xyz/tags/Peirce" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Peirce</span></a> <a href="https://mathstodon.xyz/tags/Logic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Logic</span></a> <a href="https://mathstodon.xyz/tags/Semiotics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Semiotics</span></a> <a href="https://mathstodon.xyz/tags/LogicalGraphs" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>LogicalGraphs</span></a> <a href="https://mathstodon.xyz/tags/DifferentialLogic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>DifferentialLogic</span></a> <br><a href="https://mathstodon.xyz/tags/AutomataTheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>AutomataTheory</span></a> <a href="https://mathstodon.xyz/tags/FormalLanguages" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>FormalLanguages</span></a> <a href="https://mathstodon.xyz/tags/FormalGrammars" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>FormalGrammars</span></a> <a href="https://mathstodon.xyz/tags/GraphTheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>GraphTheory</span></a></p>
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#FormalGrammars
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Jon Awbrey<p>Cactus Language • Overview 3.1<br>• <a href="https://inquiryintoinquiry.com/2025/03/07/cactus-language-overview-3/" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">inquiryintoinquiry.com/2025/03</span><span class="invisible">/07/cactus-language-overview-3/</span></a></p><p>In the development of Cactus Language to date the following two species of graphs have been instrumental.</p><p>• Painted And Rooted Cacti (PARCAI).<br>• Painted And Rooted Conifers (PARCOI).</p><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 “fibers” or “inverse images” of the propositions.</p><p><a href="https://mathstodon.xyz/tags/Peirce" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Peirce</span></a> <a href="https://mathstodon.xyz/tags/Logic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Logic</span></a> <a href="https://mathstodon.xyz/tags/Semiotics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Semiotics</span></a> <a href="https://mathstodon.xyz/tags/LogicalGraphs" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>LogicalGraphs</span></a> <a href="https://mathstodon.xyz/tags/DifferentialLogic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>DifferentialLogic</span></a> <br><a href="https://mathstodon.xyz/tags/AutomataTheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>AutomataTheory</span></a> <a href="https://mathstodon.xyz/tags/FormalLanguages" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>FormalLanguages</span></a> <a href="https://mathstodon.xyz/tags/FormalGrammars" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>FormalGrammars</span></a> <a href="https://mathstodon.xyz/tags/GraphTheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>GraphTheory</span></a></p>
Jon Awbrey<p>Cactus Language • Overview 1.2<br>• <a href="https://inquiryintoinquiry.com/2025/03/01/cactus-language-overview-1/" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">inquiryintoinquiry.com/2025/03</span><span class="invisible">/01/cactus-language-overview-1/</span></a></p><p>Resource —</p><p>For readers interested and intrepid enough to read ahead, here’s an outline of my work in progress on the OEIS Wiki, which I’ll be revising and serializing to my Inquiry blog.</p><p>Part 1<br>• <a href="https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">oeis.org/wiki/Cactus_Language_</span><span class="invisible">%E2%80%A2_Part_1</span></a></p><p>Cactus Language • Syntax<br>• <a href="https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Syntax" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">oeis.org/wiki/Cactus_Language_</span><span class="invisible">%E2%80%A2_Part_1#Syntax</span></a></p><p>Part 2<br>• <a href="https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_2" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">oeis.org/wiki/Cactus_Language_</span><span class="invisible">%E2%80%A2_Part_2</span></a></p><p>Generalities About Formal Grammars<br>• <a href="https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_2#Generalities" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">oeis.org/wiki/Cactus_Language_</span><span class="invisible">%E2%80%A2_Part_2#Generalities</span></a></p><p>Part 3<br>• <a href="https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">oeis.org/wiki/Cactus_Language_</span><span class="invisible">%E2%80%A2_Part_3</span></a></p><p>Cactus Language • Stylistics<br>• <a href="https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Stylistics" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">oeis.org/wiki/Cactus_Language_</span><span class="invisible">%E2%80%A2_Part_3#Stylistics</span></a></p><p>Cactus Language • Mechanics<br>• <a href="https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Mechanics" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">oeis.org/wiki/Cactus_Language_</span><span class="invisible">%E2%80%A2_Part_3#Mechanics</span></a></p><p>Cactus Language • Semantics<br>• <a href="https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Semantics" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">oeis.org/wiki/Cactus_Language_</span><span class="invisible">%E2%80%A2_Part_3#Semantics</span></a></p><p>Stretching Exercises<br>• <a href="https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Stretching_Exercises" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">oeis.org/wiki/Cactus_Language_</span><span class="invisible">%E2%80%A2_Part_3#Stretching_Exercises</span></a></p><p>References<br>• <a href="https://oeis.org/wiki/Cactus_Language_%E2%80%A2_References" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">oeis.org/wiki/Cactus_Language_</span><span class="invisible">%E2%80%A2_References</span></a></p><p>Document History <br>• <a href="https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Document_History" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">oeis.org/wiki/Cactus_Language_</span><span class="invisible">%E2%80%A2_Document_History</span></a></p><p><a href="https://mathstodon.xyz/tags/Peirce" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Peirce</span></a> <a href="https://mathstodon.xyz/tags/Logic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Logic</span></a> <a href="https://mathstodon.xyz/tags/Semiotics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Semiotics</span></a> <a href="https://mathstodon.xyz/tags/LogicalGraphs" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>LogicalGraphs</span></a> <a href="https://mathstodon.xyz/tags/DifferentialLogic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>DifferentialLogic</span></a> <br><a href="https://mathstodon.xyz/tags/Automata" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Automata</span></a> <a href="https://mathstodon.xyz/tags/FormalLanguages" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>FormalLanguages</span></a> <a href="https://mathstodon.xyz/tags/FormalGrammars" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>FormalGrammars</span></a> <a href="https://mathstodon.xyz/tags/GraphTheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>GraphTheory</span></a></p>
Jon Awbrey<p>Cactus Language • Overview 1.1<br>• <a href="https://inquiryintoinquiry.com/2025/03/01/cactus-language-overview-1/" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">inquiryintoinquiry.com/2025/03</span><span class="invisible">/01/cactus-language-overview-1/</span></a></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>— Herbert J. Bernstein • “Idols of Modern Science”</p><p>The following report describes a calculus for representing propositions as sentences, that is, as syntactically defined sequences of signs, and for working with those sentences in light of their semantically defined contents as logical propositions. In their computational representation the expressions of the calculus parse into a class of graph‑theoretic data structures whose underlying graphs are called “painted cacti”.</p><p>Painted cacti are a specialization of what graph‑theorists refer to as “cacti”, which are in turn a generalization of what they call “trees”. The data structures corresponding to painted cacti have especially nice properties, not only useful in computational terms but interesting from a theoretical standpoint. The remainder of the present Overview is devoted to motivating the development of the indicated family of formal languages, going under the generic name of Cactus Language.</p><p><a href="https://mathstodon.xyz/tags/Peirce" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Peirce</span></a> <a href="https://mathstodon.xyz/tags/Logic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Logic</span></a> <a href="https://mathstodon.xyz/tags/Semiotics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Semiotics</span></a> <a href="https://mathstodon.xyz/tags/LogicalGraphs" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>LogicalGraphs</span></a> <a href="https://mathstodon.xyz/tags/DifferentialLogic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>DifferentialLogic</span></a> <br><a href="https://mathstodon.xyz/tags/Automata" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>Automata</span></a> <a href="https://mathstodon.xyz/tags/FormalLanguages" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>FormalLanguages</span></a> <a href="https://mathstodon.xyz/tags/FormalGrammars" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>FormalGrammars</span></a> <a href="https://mathstodon.xyz/tags/GraphTheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>GraphTheory</span></a></p>
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