[CG] Re: Theme One Program • Exposition

Theme One Program • Exposition 9 • https://inquiryintoinquiry.com/2024/06/19/theme-one-program-exposition-9-b/ Transformation Rules and Equivalence Classes — The abstract character of the cactus language relative to its logical interpretations makes it possible to give abstract rules of equivalence for transforming cacti among themselves and partitioning the space of cacti into formal equivalence classes. The transformation rules and equivalence classes are “purely formal” in the sense of being indifferent to the logical interpretation, entitative or existential, one happens to choose. Two definitions are useful here: • A “reduction” is a transformation which preserves equivalence classes and reduces the level of graphical complexity. • A “basic reduction” is a reduction which applies to a basic connective, either a node connective or a lobe connective. The two kinds of basic reductions are described as follows. A “node reduction” is permitted just in case every component cactus joined to a node itself reduces to a node, as shown below. • https://inquiryintoinquiry.files.wordpress.com/2018/04/cactus-graph-e280a2-… A “lobe reduction” is permitted just in case exactly one component cactus listed in a lobe reduces to an edge, as shown below. • https://inquiryintoinquiry.files.wordpress.com/2018/04/cactus-graph-e280a2-… That is roughly the gist of the rules. More formal definitions can wait for the day when we have to explain all this to a computer. Regards, Jon cc: https://www.academia.edu/community/Vj8Gg0 cc: https://mathstodon.xyz/@Inquiry/110039613333677509