Ask a system whether a claim is true and it hands you a number. The Transformer was introduced in 2017, confidence 0.9. Scaling is all you need, confidence 0.9. Two claims, same score, and the score is lying to you about one of them. The first is a settled historical fact that ten thousand sources agree on. The second is a contested research position that serious people argue about daily. A single number cannot tell them apart, because a single number has no room to say how it knows what it knows. It collapses the strength of a belief and the weight of the evidence behind that belief into one scalar, and those are different things that need to stay separate.
The second dimension is where the rest of this goes. The truth value that a serious knowledge graph attaches to a claim is a pair. A strength estimates how true the claim is, and a confidence says how much evidence stands behind that strength. The idea comes from Probabilistic Logic Networks, the reasoning system Goertzel built for OpenCog, and you do not need the full apparatus to get most of the value. A disciplined subset, what we will call PLN-lite, gives you calibrated belief that propagates through the graph the way evidence accumulates in a courtroom, and it drops into a property graph without much ceremony. The full system is more intricate than what runs here, and this essay marks the line between the two carefully, because the place this material goes wrong is in pretending a simplification is the real thing.
Two numbers, and what each one carries
Start with what the pair means, because the whole system rests on it. The strength, written s, lives in zero to one and behaves like a probability: it is the system's estimate of how true the claim is. A strength of 0.9 says the claim is probably true. A strength of 0.1 says it is probably false. A strength of 0.5 says the claim is as likely true as not. So far this is the scalar everyone already has.
The confidence, written c, also lives in zero to one, and it measures something the strength cannot: how much evidence backs the strength. This is the dimension that pulls apart the two claims from the opening. A strength of 0.5 with high confidence means the system has seen a great deal of evidence and that evidence genuinely points both ways. The claim is contested, and the system knows it is contested, and the 0.5 is a hard-won balance rather than a shrug. A strength of 0.5 with near-zero confidence means the system has no idea, has seen almost nothing, and the 0.5 is a placeholder prior standing in for the absence of information. Same strength. Opposite epistemic states. A scalar probability erases the difference and an agent built on it cannot tell when it is balanced versus when it is blind.
That distinction is the genuine core of PLN and the reason the two-dimensional truth value exists. Strength is probability-like. Confidence is evidential support. The two are not reducible to each other, and a system that tracks only the first is unable to make the decisions that depend on the second: whether to explore or exploit, how fast new evidence should move an estimate, how much weight a claim should carry when it feeds a downstream inference.
Where confidence comes from
The confidence number is derived from an evidence count, and the derivation is the part worth getting exactly right. Let N be the amount of evidence backing a claim, the count of observations, corroborating sources, or supporting inferences the system has accumulated. Let k be a constant, a system-wide parameter that sets how much evidence it takes to feel confident. Then confidence is:
c = N / (N + k)Read what this curve does. With no evidence, N is zero and confidence is zero. As evidence accumulates, confidence climbs toward one but never reaches it, because for any finite N the denominator is always larger than the numerator. You can be very confident. You cannot be certain, which is the correct behavior for a system reasoning under incomplete information. When N equals k, confidence is exactly 0.5, which gives k a clean interpretation: it is the amount of evidence at which the system is halfway to full confidence. Tune k and you tune how evidence-hungry the whole graph is.
The map runs both ways, and you need both directions. Storing the count and computing confidence on read is one move. Going the other way, recovering the effective evidence count from a stated confidence, is the algebra that makes merging work. Solve the formula for N:
N = c / (1 - c) * kAt confidence zero this returns no evidence. As confidence approaches one the implied count runs to infinity, matching the asymptote. In practice you store the truth value internally as strength and count, the pair s and N, and expose the normalized confidence only when something needs to read it. The count is the quantity that composes cleanly under the operations that matter, and confidence is the human-facing projection of it.
One marker before going further, stated plainly. The formula c = N / (N + k) is a faithful engineering instantiation of PLN's general scheme, which requires only that confidence rise monotonically with evidence and saturate below one. The PLN book leaves the exact function open. The rational form captures that behavior, sits comfortably with the Beta-Binomial intuition underneath PLN's truth values, and is simple enough to run on every claim in a large graph. It is PLN-lite, accurate to the spirit and explicit about being a simplification.
Combining evidence
The operation that makes a knowledge graph more than a pile of assertions is revision: two independent observations of the same proposition arrive, and the system has to fold them into one truth value that reflects both. This is where the count earns its place as the stored quantity, because the merge is clean only in terms of counts.
Take two truth values for the same claim, the pair s1 and c1 from one source and s2 and c2 from another, sharing the same k. Recover each effective count through the inversion, N1 and N2. Then the merged truth value follows three steps. The counts add, because independent evidence accumulates:
N = N1 + N2The strength is the average of the two strengths weighted by their counts, because the source that has seen more should pull harder:
s = (N1 * s1 + N2 * s2) / (N1 + N2)And the merged confidence comes back from the combined count through the same forward formula, c = N / (N + k). Under the Beta-Binomial reading, each strength is an estimated success rate for a process, each count is an effective sample size, and the merged strength is the posterior mean after pooling the samples while the merged count is the total sample size. The operator is associative and commutative, which is exactly what you want: the order in which evidence arrives does not change where the graph ends up, and you can fold a hundred sources together one at a time and reach the same place as combining them all at once.
The same caveat applies here as to the confidence formula. The count-weighted average with additive counts is a PLN-inspired simplification, not the only revision rule the full book uses. Real PLN includes more careful operators for combining evidence that flowed through structurally different inference chains, for tracking dependence between sources that are not truly independent, and for keeping positive and negative evidence on separate books. For a production graph pooling independent observations, the simple rule is faithful enough and it is the right default. Where it stops being safe is when the two sources are not independent, when both ultimately trace to the same upstream claim, at which point adding their counts double-counts the evidence and the merge overstates confidence. Knowing that boundary is the difference between using the rule and being used by it.
Why contradiction lowers the number
The truth-value system is what makes the contradiction engine from essay three more than bookkeeping. When two claims about the same subject are detected as conflicting, their truth values do not just sit there flagged. They move. Both strengths shift toward the midpoint, weighted by the credibility of the opposing source, and both confidences come down, weighted by how structural the contradiction is. A claim that stood alone at strength 0.85 and high confidence, once a credible contradiction lands and links to it, drops to something lower on both axes, and the drop is deterministic, auditable, and reversible if the contradiction is later resolved or itself invalidated.
This is the mechanism behind a property essay three promised and this essay delivers: an agent's output becomes calibrated without anyone prompting it to hedge. An agent querying a claim at strength 0.85 and confidence 0.9 presents it as established. The same agent querying the same claim after a credible contradiction has pulled it to strength 0.62 and lower confidence presents it as the dominant view in a field that has not reached consensus. The hedging is not bolted on by clever instructions. It falls out of the numbers, because the numbers changed when the epistemic situation changed. The substrate got better calibrated and the answer inherited the calibration.
Propagating through inference
The deepest reason to carry two numbers instead of one shows up when claims feed other claims. Knowledge graphs do not only store facts; they derive new facts from existing ones, and a derivation is only as trustworthy as the weakest link in the chain it ran through. A system that propagates only strength through an inference loses the ability to discount a conclusion that was reached from shaky premises, and after a few hops it is asserting derived claims with the same swagger as direct observations. That is how a knowledge system talks itself into confident nonsense.
A PLN-lite deduction keeps both numbers moving. Take a premise A with its truth value and an implication from A to B with its truth value, and derive a truth value for B. The strength composes the way conditional probability composes, the strength of A times the strength of the implication, mirroring that the probability of B given the chain runs through both. The confidence is the conservative part: the derived claim cannot be more confident than its weakest input, so the effective count of the conclusion is bounded by the smaller of the two input counts, and the confidence comes back from that bounded count through the usual formula. The effect is that a chain through any low-confidence premise yields a low-confidence conclusion, and confidence degrades over long chains rather than surviving intact, which is the correct accumulation of uncertainty.
Be clear about what this is and is not. The real PLN deduction rules operate in term logic over inclusions between concepts rather than plain propositions, they carry separate positive and negative evidence, and the actual equations include normalization factors and overlap estimates between concepts that the simple product-and-minimum shape does not capture. The simplified rule preserves the qualitative behavior PLN cares about, strength composed probabilistically and confidence constrained by the weakest link and bounded by total evidence, without claiming to be the full machinery. Run the lite version, get calibrated propagation, and reach for the book's equations only when a domain genuinely needs the structural precision they buy.
| Operator | What runs here | What full PLN carries | When the difference bites |
|---|---|---|---|
| Confidence from evidence | c = N / (N + k), a rational form that rises monotonically with evidence and saturates below one | The book leaves the exact function open and requires only that behaviour, sitting on a Beta-Binomial intuition underneath | Rarely. The rational form is a faithful instantiation, cheap enough to run on every claim in a large graph |
| Revision | Counts add, strength is the count-weighted average, confidence returns through the forward formula | Separate operators for evidence that flowed through structurally different inference chains, dependence tracking between sources, and positive and negative evidence kept on separate books | The moment two sources are not independent. Both tracing to one upstream claim makes the addition double-count and the merge overstate |
| Deduction | Strength composes as a product; the conclusion’s effective count is bounded by the smaller input count | Term logic over inclusions between concepts rather than plain propositions, separate positive and negative evidence, normalisation factors and overlap estimates between concepts | When a domain genuinely needs the structural precision those equations buy. The simple shape keeps the qualitative behaviour without claiming to be the machinery |
What the agent does with it
All of this serves a query, and the agent on the other end should receive the truth values as ordinary parts of the payload rather than as something it has to compute. A claim comes back with its strength and confidence already shaped by every piece of evidence, every contradiction, and every inference that touched it. The agent reads the pair and calibrates its language to it: confident where confidence is high, hedged where it is low, transparent about contestedness where the strength sits balanced against a high confidence. It can also write back. When the agent reaches a conclusion, it asserts a new claim with its own truth value, and that assertion becomes evidence the next query will fold in through revision.
This closes a loop the series has been building toward. The metagraph from essay one gives claims a place to live. The bi-temporal model from essay two gives them validity windows. The contradiction engine from essay three structures their disagreements. The edges-of-edges patterns from essay four let facts carry facts about themselves. The truth value is the quantity riding on top of all of it, the scalar pair that turns structure into calibrated belief, the thing that makes the difference between a system that knows what it knows and a system that asserts everything with the same flat confidence. Two numbers instead of one is a small change in the schema and a large change in how well calibrated the answers come out.
The next essay leaves epistemics for engineering. The ingestion pipeline, where books and transcripts and papers and posts become resolved entities and extracted claims, and where most knowledge systems quietly die before any of this truth-value machinery ever gets a chance to run.
END OF ESSAY 05 · CONTINUE TO ESSAY 06 →