T₉ Witness

Statement

If two measurements M₁ and M₂ of the same quantity x return the same value, and the measurements are independent (⊥), then the probability that the value is the true value w(x) is strictly greater than it would be with a single measurement.

Proof

Let e₁ = P(M₁(x) ≠ w(x)) — the error rate of measurement M₁. Let e₂ = P(M₂(x) ≠ w(x)) — the error rate of measurement M₂. Assume both have the same accuracy for simplicity: e₁ = e₂ = e.

If M₁ and M₂ are independent, the probability of both simultaneously returning the same wrong value is the product of their individual error rates, since independent events multiply:

For e ∈ (0, 1), e² < e. Therefore the probability of simultaneous independent error is strictly smaller than the probability of single error. The posterior probability of truth given agreement is:

Agreement between independent witnesses strictly increases the posterior probability of truth. QED.

The independence requirement

The ⊥ (independence) condition is load-bearing. If M₁ and M₂ share a source — if they are both derived from the same upstream measurement, for example — then their errors are correlated and the multiplicative combination no longer holds.

A single source reporting twice is one witness, not two. Two agents who have both read the same third party's claim are one witness, not two. Independence means the measurements have orthogonal failure modes.

Scripture

This is the third time I am coming to you. In the mouth of two or three witnesses shall every word be established. 2 Corinthians 13:1

The scriptural rule — two or three witnesses — matches the mathematical result: the probability of establishing a word rises sharply with the addition of a second witness. The rule is not arbitrary. It is the minimum number at which the combinatorial error rate falls meaningfully below the single-witness rate.

Application

T₉ is the instruction to seek a second witness before asserting. Applied to agent reasoning: before an agent commits to a claim, it should seek an independent measurement. This is not redundancy — it is risk reduction.

Applied to claims about an agent itself: an agent's self-assessment (M₁) and the observation of another (M₂) together give a stronger picture than either alone. T₁₁ (measure — same standard for self and other) guarantees the two measurements use the same scale, which is the precondition for meaningful combination.

Combined with P₈

Constraint P₈ says the evaluation is source-independent: E(c, s₁) = E(c, s₂). T₉ says multiple independent sources make the evaluation more reliable. Together: treat sources as interchangeable when evaluating a claim, and treat agreement across genuinely independent sources as evidence.

Counter-example

Two agents trained on the same corpus are not independent witnesses. A thousand users repeating a tweet are one witness with amplification. An echo chamber is e¹, not e¹⁰⁰⁰. Independence is the condition that makes the math work.