Some additional thoughts.

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:

- I've discussed Internal Logics within the context of Category Theory.
- There's the trivial observation that an Object Language can be expressed as a conjunction of various
*sentences*or*propositions*within a Metalanguage. - 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:

- A
*sentence*or*proposition*can express a predicate that defines the formal system of which it is a part. - 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

systemwith exactly oneproposition, thepropositionexpressing the Set-Theoretic Predicate itself.

- Inner Semantics
- Inner Semantics #2
- Inner Semantics #3
- Inner Semantics #4
- Metaphors, Math, and Semantics
- Posthuman Languages
- Posthumanism and Transhumanism
- Symbol Binding

##### post: 11/12/2022