On language yet again

 What is language, after all ? Consider a system.  Let this system have subsystems capable of epistemic, volitional and doxastic states. The epistemic and doxastic states can refer to (partial) aspects of the system, its subsystems and their relations, both present, past, future, possible or alternative, counterfactual. Between two subsystems there are channels. The structure of a channel at a given time or interval is called communication data. The meaning of the data is the communion data plus the context which roughly speaking is nothing but the global present and past history of the system, but more specifically the present and past history of the system as it relates to the sender and receiver. This history will of course in turn involve communication.  But subsystems' doxastic, epistemic and volition states also employ data in the configuration of their internal states - thus the symbolic mirroring of the macrocosm in the microcosm - which must bear at least analogy to the communication data of channels. These states evolve, are learned or discovered in the context of communication.  I am happy must include both a state for happiness and a state for awareness with a parameter which represents the happiness state and the time in which the actual happiness state obtains. Of course internal representation can depend on context too. The ideal of mathematical and scientific language and knowledge is that it is maximally context-independent. A major feature that cannot be overlooked involves the growth, disappearance and reproduction of subsystems. UDIL is a good tool for formalizing the objects of knowledge, the states of affairs of subsystems and the total system. Another problem is the existence of a plurality of languages, including modes of expression and interpretation. One can even wonder if there are as many languages as subsystems.

Thus the meaning of communication data or representation data is in general  a function of particular subsystems (for instance the sender and receiver) as well as their state and state-history which will implicate large parts of the whole system and its history.

A very good formalism for these kind of systems is found in the Object Specification Logic of Sernadas et al. Takahara's general systems theory, which is clearly inspired by classical control theory, needs to be improved and reworked in the following form. We are given an input space $X$ with special do-nothing element $\bullet$, a state space $Q$ and an output space $Y$ also with a show-nothing element $\bullet$. Let $T$ represent linearly-ordered time.

Then a (in general non-deterministic) system is a correspondence $\mathcal{S}:  X^T \rightarrow \mathcal{P}((Q\times Y)^T)$. We can easily define all the core notions as well as feedback and interconnection of systems.  But of course each $Q_i$ will be very complex to be able to represent states of the entire system, etc. Also $Q$ will in general have a part $Q'$ which will have spatial determination $Q' \rightarrow \mathbb{R}^3$. If the spatial extent of the system varies then in a way $Q$ also varies. Thus there are reasons to extend the definition of system to sheaves.

Hegel is not an easy or even very clear writer. But therein are many important concepts and observations couched in an unfamiliar language that can be studied in a perhaps more through and lucid way both in previous (there is a huge amount of implicit reference and borrowing in Hegel)  and posterior philosophy. Aristotle, Sextus Empiricus, Augustine, Proclus,  Cusanus, Spinoza, Malebranche, the 'esoteric' Leibniz of Russell and Gödel,  Lotze, Bolzano, Brentano, McTaggart's commentary of Hegel's Logic and  Hofstadter's Gödel, Escher, Bach are to be noted. Hegel's core notions and valid observations are found in a more thorough and lucid form both before and afterwards, even rediscovered in mathematical logic,  category theory, computer science and general systems theory.