[CG] Cactus Language

Cactus Language • Overview 1 • https://inquiryintoinquiry.com/2025/03/01/cactus-language-overview-1/ ❝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.❞ — Herbert J. Bernstein • “Idols of Modern Science” 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”. Painted cacti are a specialization of what graph‑theorists call “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. Resource — 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. Part 1 • https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1 Cactus Language • Syntax • https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Syntax Part 2 • https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_2 Generalities About Formal Grammars • https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_2#Generalities Part 3 • https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3 Cactus Language • Stylistics • https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Stylistics Cactus Language • Mechanics • https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Mechanics Cactus Language • Semantics • https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Semantics Stretching Exercises • https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Stretching_Exercises References • https://oeis.org/wiki/Cactus_Language_%E2%80%A2_References Document History • https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Document_History Regards, Jon cc: https://www.academia.edu/community/VXZjbz cc: https://mathstodon.xyz/@Inquiry/114100591406422570 cc: https://www.researchgate.net/post/Cactus_Language_for_Propositional_Calculus