Inner Semantics #4

Some additional thoughts.

Suppes Set-Theoretic Predicates

The idea that a language or formal system might be embedded into a single proposition seems radical at first blush. Traditional philosophy has it that propositions are expressed by, within, or a part of some language system.

Modern mathematical techniques and philosophy informed by modern mathematics have taken the opposite approach:

1. I've discussed Internal Logics within the context of Category Theory.
2. There's the trivial observation that an Object Language can be expressed as a conjunction of various sentences or propositions within a Metalanguage.
3. And, then there's Suppes Set-Theoretic Predicates I allude to in various papers.

I'll focus on the third item above here. The common use of such techniques above should give us more confidence that something Semantic Immuration (Inner Semantics) isn't wrong from the get-go.

A great paper that briefly describes a landmark idea Patrick Suppes (Standford, CSLI) introduced in Representation and Invariance of Scientific Structures (a work that successfully attempts to formalize all major mathematical items used currently in social and hard sciences):

Grab the paper: Suppes predicates

Essentially, we can define certain satisfaction conditions that define or uniquely characterize an algebraic structure into a predicate and express that predicate in the standard way. (The work above extends Suppes' approach to a technique advanced by Bourbaki.)

Two observations:

1. A sentence or proposition can express a predicate that defines the formal system of which it is a part.
2. Such a move might be further justified by the trivial observation I allude to above that an Object Language can be expressed as a conjunction of various sentences or propositions within a Metalanguage. (I don't recollect seeing this as explicit justification that Suppes articulates in the book itself.) We would then see Set-Theoretic Predicates as an extension and refinement of that fundamental kind of operation further licensing the move to justification.

Consequently, a Set-Theoretic Predicate can be expressed for a system with exactly one proposition, the proposition expressing the Set-Theoretic Predicate itself.