This introduces Paraconsistent Geodesic Modal Homotopy Type Theory (PGM–HoTT), a new logical and semantic framework designed to address one of the most persistent problems in artificial intelligence and natural language understanding: robust reasoning about identity under ambiguity, contradiction, and uncertainty.
Traditional logical systems struggle when identity information is incomplete, conflicting, or context-dependent situations that are common in real language, perception, and commonsense reasoning. Typical workarounds, such as ad hoc distinctions between “specific entities” and “general categories,” tend to be brittle and fail to scale. PGM–HoTT replaces these hacks with a principled mathematical foundation.
The core innovation of PGM–HoTT is a dual, paraconsistent notion of identity. Instead of treating identity as all-or-nothing, the framework allows positive evidence for identity and positive evidence for non-identity to coexist without collapsing the system. Identity proofs are additionally graded by quantitative costs drawn from a logic quantale, making the “shortest” or most economical proofs geodesics central to reasoning. Modalities allow the system to either forget these costs, recovering ordinary Homotopy Type Theory, or to enforce bounded consistency when needed.
The paper establishes the theoretical soundness of this approach by providing:
- A model theory based on quantale-enriched ∞-groupoids
- A guarded version of univalence compatible with modalities and higher inductive types
- A geodesic formulation of definiteness (“the”) that generalizes the standard contractibility-based account
- Computational rules for graded transport and controlled discharge of negative identity evidence
Building on this foundation, the paper introduces Semantic Elegant Normal Form (SENF), a simple-type semantic representation that normalizes sentence meanings into frames, roles, and constraints. SENF explicitly encodes selection, uniqueness, and graded identity, while remaining compatible with standard neo-Davidsonian semantics when costs are forgotten.
To enable reasoning across paraphrases and discourse, the paper presents TransWeave, a geodesic, role-aware mapping between SENF representations. TransWeave aligns entities, events, roles, times, and regions across sentences, assigns identity costs, and preserves alternative interpretations under guarded conditions. This allows rules and facts to be safely transported even when identity is ambiguous.
Crucially, the framework is shown to integrate cleanly with Probabilistic Logic Networks (PLN). TransWeave produces the bridging links needed for PLN’s deduction, abduction, induction, and temporal reasoning, while PGM–HoTT ensures that identity propagation remains controlled and non-explosive. The paper includes MeTTa-inspired pseudocode, complexity analysis, and a detailed commonsense reasoning case involving pronouns, mereology, temporal shifts, and conflicting claims.
Beyond AI engineering, the framework also offers formal tools for philosophical and linguistic phenomena, including non-dual identity (“no fixed self”), definiteness, anaphora, I–Thou relations, generics, and contradiction-tolerant meaning.
In summary, this work provides:
- A mathematically rigorous identity calculus for AI
- A practical semantic normal form and mapping mechanism
- A direct pathway to scalable, identity-aware commonsense reasoning
Together, PGM–HoTT, SENF, and TransWeave form a unified system that replaces fragile identity heuristics with explicit, graded, and modal reasoning—bringing formal foundations closer to the realities of natural language and human cognition.
Reference
Paraconsistent Geodesic Modal HoTT (PGM–HoTT), Semantic Elegant Normal Form (SENF), and TransWeave for Identity–Aware Commonsense Reasoning with PLN, https://docs.google.com/document/d/1bUgBi2pv3moVlmRc56WCJGOIkPcCL4veLp8ZUMaGx-Y/