Two-Stage Interpretation in Formal Logic

The most famous logical process you have never heard of

Two Stage Interpretation. Royalty free image by DALL-E 3 and Victor Morgante

Two-Stage Interpretation (TSI) in formal logic is the most famous process you have never heard of.

Kurt Gödel used TSI in his Incompleteness Theorems of formal logic.

There are three steps and two interpretations:

1. Create a bunch of theorems in the system of Principia Mathematica;
2. Interpretation 1: Interpret instances of those theorems as numbers (Gödel Numbers); and
3. Interpretation 2: Interpret those numbers as theorems (proofs) under the system of Principia Mathematica.

Two-Stage Interpretation not only appears in formal logic, but also the logic of computer science.

Take, for instance, the four-layer architecture of the OMG’s (Object Management Group’s) Meta-Object Facility.

OMG’s Four-Layer Architecture. Royalty Free Image. WikiMedia.org.

Models stored in the M3-layer (the meta-metamodel), as instances of the M2-layer (metamodel) have in turn instances that are models (the M1-layer). But M2-layer artifacts/instances (models) may be expressed as XML, of which is not intended for…