Cactus Language • Overview 4 • https://inquiryintoinquiry.com/2025/03/12/cactus-language-overview-4/ 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 “sentential calculus” or a “propositional calculus”, respectively. 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. 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 “sentences”, 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 cactus language which proceeds along purely formal lines is carried out in Cactus Language • Syntax. After that, the development of the language's more concrete aspects can be seen as a matter of defining the following two functions. • 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 “painted cactus”. • 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. 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 “style”, 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 Cactus Language • Stylistics. 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 “algorithmics” or “mechanics” of language processing. A mechanical conversion of cactus languages into their associated data structures is discussed in Cactus Language • Mechanics. 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 Cactus Language • Semantics. Resources — Cactus Language • Overview • https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview Survey of Animated Logical Graphs • https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs… Survey of Theme One Program • https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/ Regards, Jon cc: https://www.academia.edu/community/lJvgBA cc: https://www.researchgate.net/post/Cactus_Language_for_Propositional_Calculus