Differential Propositional Calculus • 27
•
https://inquiryintoinquiry.com/2023/12/19/differential-propositional-calcul…
Commentary On Small Models —
One reason for engaging in our present order of extremely reduced
but explicitly controlled case study is to throw light on the general
study of languages, formal and natural, in their full array of syntactic,
semantic, and pragmatic aspects. Propositional calculus is one of the
last points of departure where it is possible to see that trio of aspects
interacting in a non‑trivial way without being immediately and totally
overwhelmed by the complexity they generate. Often that complexity leads
investigators of formal and natural languages to adopt the strategy of
focusing on a single aspect and abandoning all hope of understanding the
whole, whether it’s the still living natural language or the dynamics of
inquiry crystallized in formal logic.
From the perspective I find most useful here, a language is a
syntactic system designed or evolved in part to express a set
of descriptions. When the explicit symbols of a language have
extensions in its object world which are actually infinite, or
when the implicit categories and generative devices of a linguistic
theory have extensions in its subject matter which are potentially
infinite, then the finite characters of terms, statements, arguments,
grammars, logics, and rhetorics force a surplus intension to color all
its symbols and functions, across the spectrum from object language to
metalinguistic reflection.
In the aphorism of W. von Humboldt often cited by Chomsky, language
requires “the infinite use of finite means”. That is necessarily
true when the extensions are infinite, when the referential symbols
and grammatical categories of a language possess infinite sets of
models and instances. But it also voices a practical truth when the
extensions, though finite at every stage, tend to grow at exponential
rates.
Resources —
Differential Logic and Dynamic Systems
•
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part…
Differential Logic • Commentary On Small Models
•
https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part…
Regards,
Jon
cc:
https://www.academia.edu/community/l8m8z5