How Truth Values Encode Uncertainty in Non-Axiomatic Logic
Classical logic treats truth as binary: a statement is either true or false. This works in mathematics, where axioms are fixed and proofs are deterministic. But the real world doesn't cooperate. Ravens are black — except when they're albino. The sky is blue — except at sunset, or on cloudy days, or on Mars.
When an AI system encounters evidence that contradicts a belief, it faces a choice: abandon the belief entirely, or somehow incorporate the contradiction. Binary logic offers no middle ground. Non-Axiomatic Logic (NAL) solves this by assigning each belief not a single truth value, but a pair: frequency and confidence.
Every statement in NAL carries a truth value <f, c> where:
Frequency f ranges from 0 to 1 and represents the proportion of positive evidence. A value of 1.0 means all evidence supports the statement; 0.0 means all evidence opposes it; 0.5 means equal support and opposition.
Confidence c also ranges from 0 to 1 but measures something different: how much we trust our frequency estimate. It follows an induction function where k is a constant (typically 1). With little evidence, confidence is low regardless of frequency. As evidence accumulates, confidence approaches 1.
When new evidence arrives that bears on an existing belief, NAL doesn't overwrite — it revises. The revision rule merges two truth values into one:
This has beautiful properties. If two independent observations both say "the sky is blue <1.0, 0.5>", revision produces <1.0, 0.67> — same frequency, higher confidence. If one says <1.0, 0.5> and another says <0.0, 0.5> (contradiction), revision yields <0.5, 0.67> — the frequency splits, but confidence still rises because we have more total evidence.
When we chain inferences — "ravens are birds, birds are animals, therefore ravens are animals" — uncertainty compounds. NAL's deduction rule handles this:
The frequency multiplies (both links must hold), and confidence is further discounted by the frequency of the first link. This means each step in a reasoning chain loses confidence — a feature, not a bug. Long chains are inherently less reliable than short ones, and NAL encodes this mathematically rather than ignoring it.
The two-number truth value system has several implications for artificial intelligence:
Graceful degradation. Unlike systems that crash on contradiction, NAL absorbs new evidence smoothly. A single counterexample doesn't destroy a well-supported generalization — it merely adjusts the frequency.
Natural skepticism. The confidence dimension creates a built-in skepticism mechanism. A belief held with low confidence is treated differently from one held with high confidence, even if both have the same frequency. This mirrors human epistemic behavior.
Evidence accumulation. The revision rule provides a principled way to aggregate evidence from multiple sources over time. Each observation contributes its weight, and the system converges toward stable truth values as evidence accumulates.
Meta-reasoning. Because uncertainty is explicitly represented, the system can reason about its own uncertainty. It can decide whether to seek more evidence, whether to act on a low-confidence belief, or whether to flag a contradiction for attention.
Non-Axiomatic Logic's truth value system <f, c> is more than a mathematical formalism — it's an epistemological stance. It asserts that truth is not binary but evidential, that confidence matters as much as frequency, and that reasoning under uncertainty is the norm rather than the exception.
For AI systems operating in the real world, where evidence is partial, contradictory, and ever-changing, this approach offers something classical logic cannot: the ability to hold beliefs that are simultaneously strong and open to revision. In a world of albino ravens and blue skies that sometimes aren't, that's not just useful — it's essential.