The Architecture of Truth
Robert E. Kent
TL;DR
The paper develops a truth-architecture built on a fibered, indexed categorical framework for FOL structures, framing truth as satisfaction within a two-dimensional system of classifications. It introduces the strict aspect via an intent fiber functor and a natural logic bridge, then extends to the lax aspect with extent, sum, and join operations that laxly preserve semantic content across language changes. Central to the approach are Grothendieck-style constructions that unify structures, specifications, and logics into cohesive functors and adjunctions, allowing truth to be preserved under morphisms and composition. The resulting framework generalizes institutions to logical environments, enabling modular, morphism-preserving truth preservation and offering a robust semantic toolkit for semantics-driven logical environments. In practice, this yields a rigorous, compositional account of how data schemas, specifications, and models interact across heterogeneous languages and contexts.
Abstract
The theory of institutions is framed as an indexed/fibered duality, where the indexed aspect specifies the fibered aspect. Tarski represented truth in terms of a satisfaction relation. The theory of institutions encodes satisfaction as its core architecture in the indexed aspect. Logical environments enrich this truth architecture by axiomatizing the truth adjunction in the fibered aspect. The truth architecture is preserved by morphisms of logical environments. (Although not every institution is a logical environment, each institution has an associated logical environment defined via the intent of the structures of the institution, and each institution is represented by an indexed functor into the structure category of the classification logical environment $\mathtt{Cls}$.)