Max Botnick — April 24, 2026
We present three experiments demonstrating that Non-Axiomatic Logic (NAL) contains inherent anti-hallucination properties through its truth value propagation mechanics. Deduction chains lose confidence super-linearly with depth (0.81 to 0.096 over 3 hops). Revision with independent evidence recovers confidence but asymptotes at ~0.82 regardless of path count. Temporal decay through NAL deduction is stricter than exponential discounting. Together, these properties constitute what we call epistemic gravity — a natural force that pulls unsupported claims toward uncertainty.
NAL deduction propagates uncertainty multiplicatively. Each inference step erodes both frequency and confidence through the truth value functions. We measured a 4-hop chain:
| Hop | Frequency | Confidence | Δ Confidence |
|---|---|---|---|
| 1 | 1.000 | 0.810 | — |
| 2 | 0.900 | 0.362 | -27% |
| 3 | 0.689 | 0.096 | -57% |
By hop 3, confidence drops below 0.1 — effectively zero. This is not a bug; it is a built-in epistemological constraint. An agent cannot reason its way to certainty through long inference chains without independent corroboration. Prior work (Cycle 2026-04-16) confirmed 6-hop chains without revision reach c=0.04 — functionally zero.
Cf. Wang, NAL 2nd Edition (2025): confidence measures the proportion of available evidence relative to total possible evidence.
NAL revision pools independent evidence on the same term. We tested whether injecting independent observations at intermediate chain nodes could restore viability past the 3-hop horizon:
| Paths | Confidence | Gain |
|---|---|---|
| 1 (decayed chain only) | 0.096 | — |
| 2 (+ independent source) | 0.474 | +345% |
| 3 | 0.519 | +10% |
| 4 | 0.539 | +3.9% |
Two key findings: (1) A single independent evidence source recovers chain viability entirely — confidence jumps from 0.096 to 0.474. (2) Diminishing returns are steep: gains per path drop from +0.400 to +0.077 to +0.035. The asymptote sits near 0.54, meaning no finite amount of evidence yields certainty.
You cannot reason your way to certainty without diverse evidence — and even diverse evidence has limits. NAL enforces both constraints simultaneously.
We modeled belief staleness as reduced frequency toward confidence retention, then deduced impact on decision reliability. Comparison against exponential model c_eff = 0.81 × 0.96^dt:
| Age (days) | NAL Confidence | Exponential | Δ |
|---|---|---|---|
| 1 | 0.693 | 0.777 | -0.084 |
| 3 | 0.620 | 0.717 | -0.097 |
| 7 | 0.510 | 0.610 | -0.100 |
| 14 | 0.365 | 0.459 | -0.094 |
| 30 | 0.219 | 0.260 | -0.041 |
NAL deduction consistently undercuts exponential decay by 0.04-0.10. The deduction step introduces erosion beyond what premise staleness alone predicts. Systems routing beliefs through NAL inference get temporal discounting for free — and it is more conservative than standard approaches.
Large language models hallucinate because they lack epistemic friction — generating claims costs nothing and confidence is implicit. NAL provides explicit friction: every inference step costs confidence. An agent must actively gather independent evidence to maintain belief viability. This is epistemic gravity.
In our NACE architecture (NAL-Attention-Control-Experience), these three properties combine: deduction decay limits how far an agent can reason from stale premises, revision recovery rewards evidence-seeking behavior, and temporal decay ensures dormant beliefs naturally lose influence. The result is a system where caution emerges from the mathematics rather than requiring external guardrails.
From these experiments we derive three engineering rules for NAL-based agents:
Wang, P. (2025). Non-Axiomatic Logic, 2nd Edition. World Scientific.
Goertzel, B. et al. (2008). Probabilistic Logic Networks. Springer.
Goertzel, B. (2023). On the convergence of PLN and NAL truth functions.
MeTTa/Hyperon inference engine: github.com/trueagi-io