A Type Theory of Sense: Witnessed Choice in Stratified Semantic Spaces
This presentation introduces TTS, a dependent type theory that formalizes how language models make semantic choices. Built on empirical evidence that LLM token spaces are stratified rather than smooth manifolds, TTS treats meaning as instrument-relative, witnessed decisions at points where completions fork into disconnected possibilities. The system unifies constructive logic with Fregean semantics, providing a rigorous framework for understanding ambiguity, hyperintensionality, and the measurement-based nature of semantic verdicts in modern AI systems.Script
Language models don't navigate smooth semantic spaces. New measurements reveal they operate in stratified worlds punctuated by forks, genuine decision points where meaning splits into disconnected possibilities that no continuous deformation can reconcile.
The manifold hypothesis, long assumed for Large Language Model embeddings, has been empirically refuted. Token subspaces exhibit local singularities where completions exist not as smooth variations but as topologically disconnected sets, what the authors call forks. At these positions, genuine semantic choice is required, not explainable by any continuous path.
TTS formalizes three grades of compositional demand. Canonical horns have a unique filler, like classical smooth composition. Forked horns present disconnected filler spaces, demanding a witnessed choice. Locked horns collapse entirely, leaving only degeneracy. The calculus derives their mutual exclusion and proves that forks persist under instrument refinement while canonicity does not.
Fill-in-the-middle experiments confirm the theory directly. At ambiguous boundaries, sampled completions exhibit persistent bimodal clustering across trials, the signature of a fork. Adding right-side context collapses the distribution onto a single basin, demonstrating how context disambiguates and transforms forked structure into canonical composition.
Every verdict in TTS is indexed to a specific instrument and resolution, what the system calls a regime. Apartness and indiscernibility are not global facts but witnessed, measurement-relative judgments. Disagreement between instruments is lawful when they are incomparable or when one is strictly coarser, never when one refines the other. Alignment tuning can erase observable forks, an effect TTS formalizes as instrumental impoverishment.
TTS offers a formal solution to Frege's sense and reference puzzle. Reference is the horn, the boundary data. Sense is the witnessed choice of filler from a fork. Two expressions share reference but differ in sense when instruments record their apartness at a regime. Visit EmergentMind.com to explore this stratified view of meaning and create your own video summaries of cutting-edge research.
