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
• Differential Logic • Commentary On Small Models

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