Appendix D — The Fourth Value

Why "no evidence" and "evidence against" must never be the same number

This appendix answers a question that turns out to sit underneath the whole book: what kind of thing is a piece of evidence's status, and what happens when a system gets that kind wrong.

It is the most abstract section here. It is also the one with the most direct consequence for anything you build.

The failure, stated concretely first

The Los Angeles Police Department's Operation LASER maintained lists of "chronic offenders." Its own Inspector General found that almost half the people on those lists had zero or one violent-crime arrest, and around 10% had no recorded quality interactions with police at all. (Chapter 6) [ESTABLISHED]

Read that as a logic problem rather than a policy problem. A person with zero arrests and a person with one arrest were both being scored by a system that took records as its input. But zero records is not a low value. Zero records is an absence of information, and an absence of information is a different kind of thing from a low measurement — as different as an empty measuring cup is from a cup measured at zero millilitres.

The system had one channel to express both. So it expressed both as a small number, and small numbers combine with other small numbers to produce a score, and the score went on a list, and the list went to officers.

That is the entire content of this appendix, and everything below is the formal machinery for saying it precisely.

1 · Three values

Classical logic has two: true and false. Every proposition is one or the other, and there is no third option. This is a strong assumption and it has been questioned for a century.

Jan Łukasiewicz (1920) introduced the first systematic three-valued logic, motivated by future contingents — Aristotle's sea battle tomorrow, which seemed to him neither true nor false yet. The third value was possible or indeterminate. [ESTABLISHED]

Stephen Kleene (1938, 1952) gave the third value a different and, for our purposes, far more useful reading: undefined, or unknown — the value of a computation that has not returned. Kleene distinguished a strong and a weak three-valued logic, and the difference is exactly about how ignorance propagates. In the strong logic, "unknown OR true" is true — you do not need to resolve the unknown, because one true disjunct settles it. In the weak logic, any unknown component contaminates the whole formula. [ESTABLISHED] (SEP, Many-Valued Logic)

That distinction is not academic. It is the difference between an investigative system that says "I cannot assess this suspect because one field is missing" and one that says "one confirmed link is enough regardless of what else is missing." Both behaviours are defensible. Choosing between them accidentally is not.

If you have ever written SQL, you have already used three-valued logic. Codd's relational model gave NULL exactly this status, and comparisons against NULL return UNKNOWN rather than false. It is also the most criticised feature of the language, largely because most people writing queries do not realise which logic they are in. [ESTABLISHED] (SQL and the snare of three-valued logic; Nulls, three-valued logic and ambiguity in SQL)

2 · The fourth value, and the man who wanted it for exactly our problem

In 1977, Nuel Belnap published two closely related pieces — A Useful Four-Valued Logic and How a Computer Should Think. [ESTABLISHED] (Belnap, "A Useful Four-Valued Logic," in Dunn & Epstein eds., Modern Uses of Multiple-Valued Logic, Reidel, 1977, pp. 5–37 — Springer; and "How a Computer Should Think," in Ryle ed., Contemporary Aspects of Philosophy, Oriel Press, 1977, pp. 30–56. Citations verified against the reference list of Sutcliffe, Pelletier & Hazen, FLAIRS-31, 2018.)

His motivating scenario was a question-answering machine holding a knowledge base fed by multiple sources — a machine that might be told contradictory things, and that would have topics about which it had been told nothing at all. He observed that classical logic handles both cases disastrously and in opposite directions: it explodes on contradiction and it has nothing sensible to say about silence.

His four values are not degrees. They are states of what the system has been told:

Now look at this book's own grading system, which I designed before reading Belnap and which arrived at nearly the same place because the domain forces it:

Three of the four are Belnap's. The fourth, [CLAIMED], has no counterpart in his scheme because Belnap's machine trusts its sources. Investigation cannot. [CLAIMED] records not the content of what was said but the interest of whoever said it — a vendor selling the tool, an agency reporting its own performance. That is a provenance annotation rather than a truth value, and it is the axis Chapter 7's Move operates on.

The generalisation worth carrying: an evidence record needs at least two independent dimensions — what we have been told and who told us — and collapsing them into a single confidence number destroys both.

3 · Why a case file breaks classical logic

Classical logic contains a principle usually given in Latin: ex falso quodlibet. From a contradiction, anything follows. If a system holds both P and not-P, every proposition in its language becomes derivable — the knowledge base does not merely contain an error, it becomes formally worthless.

Paraconsistent logics — Graham Priest's LP, Newton da Costa's systems — are constructed specifically to block that explosion, so that a contradiction stays local instead of poisoning everything. [ESTABLISHED] (SEP, Paraconsistent Logic; IEP)

Why this matters here, plainly: every real case file is inconsistent. Two witnesses give incompatible timings. A suspect's alibi contradicts a cell-site record. A complaint says the caller was female and another describing the identical script says male. This is the normal condition of investigative material, not a defect to be cleaned before reasoning begins.

Classical logic offers two responses, both bad. Resolve every contradiction before reasoning — which means deciding who is lying before you have investigated, which is the tunnel vision of the closing chapter, formalised. Or reason anyway and derive nonsense.

The paraconsistent answer is the one good investigators already use informally: hold the contradiction as a marked, live state attached to that specific proposition, keep reasoning about everything else, and treat the contradiction as an object of investigation in its own right. The contradiction is frequently the most informative thing in the file — it is where somebody is wrong, and finding out who is often the case.

4 · The assumption that turns silence into a lie

Raymond Reiter (1978) named the closed-world assumption: in a closed-world database, anything not derivable from what is stored is inferred to be false. Keith Clark's negation as failure (1978) is the corresponding proof-theoretic rule — failing to prove P licenses concluding not-P. [ESTABLISHED] (closed-world assumption, overview and primary references)

For an airline reservation system this is correct and necessary. If the database has no seat booked for you, you have no seat.

For a criminal intelligence system it is a catastrophe wearing the clothes of a convention. Under a closed-world assumption:

• "No prior arrests" becomes "has not offended."
• "Not flagged in the fraud registry" becomes "not a mule account."
• "No record of this person" becomes "this person is clean."

Each of those inferences is invalid, and each is made constantly, usually invisibly, because the assumption is buried in the query rather than stated in the design.

The open-world assumption — what is not known is simply not known — is the correct posture for investigation, and it has a cost that must be stated: under it, you cannot conclude innocence from absence either. That is uncomfortable and it is also true. It is the formal version of a rule you already know: absence of a log is not evidence that nothing happened.

Note what this implies about Thailand's Central Fraud Registry (Chapters 5 and 7). A system that suspends a person's accounts at every bank on the strength of a flag is operating on presence. The harder question is what its silence means — and the documentation gap at the end of Chapter 6 says nobody has published an answer.

5 · The formalism built for exactly this, and its known failure

If you want ignorance to be representable rather than merely flagged, the standard machinery is Dempster–Shafer theory — Arthur Dempster (1967), formalised in Glenn Shafer's A Mathematical Theory of Evidence (1976). [ESTABLISHED] (Dempster, Upper and Lower Probabilities Induced by a Multivalued Mapping, Ann. Math. Statist. 38(2); Shafer 1976)

Its structural gift is that a claim carries a pair: a lower bound called belief and an upper bound called plausibility. Total ignorance is representable as belief 0, plausibility 1 — "I have no evidence for it and no evidence against it." A single probability cannot express that at all: a probability of 0.5 means balanced evidence, which is an entirely different epistemic state from no evidence, and this is the single most consequential confusion in applied evidence handling.

And now the failure, which must travel with it. Lotfi Zadeh's counterexample (1979) is the standard demolition of naive use of Dempster's rule of combination:

Two doctors examine a patient and agree the diagnosis is meningitis (M), concussion (C), or brain tumour (T). The first assigns m(M) = 0.99 and m(T) = 0.01. The second assigns m(C) = 0.99 and m(T) = 0.01. Both regard a tumour as nearly impossible.

Dempster's rule returns m(T) = 1. Certain brain tumour.

The mechanism is normalisation: the rule discards the roughly 0.98 of mass where the two sources flatly contradict each other, and everything that survives lands on the single hypothesis both happened to touch. [ESTABLISHED] (Zadeh, On the Validity of Dempster's Rule of Combination of Evidence, UCB/ERL Memorandum M79/24, 1979 — Berkeley EECS technical reports; counter-examples to Dempster's rule, review)

Two experts who agree an outcome is nearly impossible, combined, produce certainty in that outcome. That is what happens when a combination rule is applied to sources that disagree. Anyone building an evidence-fusion system should be able to reproduce this example from memory before writing a line of code.

Modern work under imprecise probability — Peter Walley's Statistical Reasoning with Imprecise Probabilities (1991), lower and upper previsions, credal sets — offers a more conservative treatment of the same intuition. [ESTABLISHED] (imprecise probability, overview)

6 · Where this is actually used

Now the honest part, and it is short.

In forensic science, courtroom evidence, or named police case-management systems: no documented application found. [ABSENT]

I searched specifically for Dempster–Shafer, paraconsistent, or many-valued reasoning in forensic practice, DNA evidence interpretation, court use, or deployed criminal-intelligence platforms. The searches returned nothing deployed.

Two adjacent items exist and are worth naming precisely so the absence is not overstated:

• Coulson and May (2015), at the IMA Conference on Mathematics in Defence, applied

Dempster–Shafer to intelligence analysis — and describe it as the first time the method had been so applied. A proposal tested on a toy scenario, not adopted practice. [ESTABLISHED] as a paper. (Coulson & May, IMA Conference on Mathematics in Defence, 2015)

• A 2025 paper applying paraconsistent logic to cargo-theft and logistics security in São Paulo.

Adjacent, not case files or courts. [ESTABLISHED] as a paper. (Logistics 9(1):37, 2025)

Meanwhile the field's actual working formalism, as Chapters 3 and 5 describe at length, is the likelihood ratio — and it has judicial history, professional standards, and a court that has already told forensic scientists when they may not use it.

So the verdict has two halves and both must be said.

The mathematics is settled and the diagnosis is correct. Ignorance is not a low probability. Contradiction is not noise. A closed-world query on an open world is a fallacy with a name and a date. These are not open questions.

The application is essentially unbuilt. If you build on it, you are not adopting best practice. You are ahead of the literature, which means you carry the burden of demonstrating it works — and the sober reading of Chapter 6 is that being ahead of the evaluation literature is precisely the condition under which harm has occurred.

7 · What to actually do

The synthesis is one rule, and it does not require adopting any of the exotic formalisms above:

**Use discrete states for the status of evidence. Use likelihood ratios for the weight of evidence. Never use one number for both.**

Two fields, not one. A record carries status ∈ {established, contested, claimed, absent} — what we have been told and by whom — and separately, where and only where the data supports it, a likelihood ratio with its stated proposition level (Chapter 5).

What this buys, concretely:

• A person with no record cannot be scored as low-risk, because "absent" is not a number and will not

multiply into a total. LASER's failure becomes structurally impossible rather than merely discouraged.

• Conflicting witness accounts are held as contested and stay visible as an open question, instead

of being averaged into a middling score that hides the disagreement — which is the failure mode Zadeh's example dramatises.

• A vendor's efficacy figure and a court docket cannot be added together, because claimed and

established are different states rather than different confidences.

• Querying "who has no flag" returns unknown, not innocent.

None of this requires Dempster–Shafer, Belnap's truth tables, or a paraconsistent inference engine. It requires refusing to collapse four distinguishable states into one float — which is a schema decision, made once, at the start, by someone who understood that the four states were different.

And to answer the question that prompted this appendix directly: ternary logic is not an alternative kind of AI, and balanced-ternary hardware — the Soviet Setun machine of 1958 — is a historical curiosity with no bearing on any of this [ESTABLISHED] (Setun, Russian Virtual Computer Museum). The value of many-valued logic here is not computational. It is that it gives you the vocabulary to notice that your system has been lying to you about what it knows.

The Move — Give absence its own field. In any record, table, or schema you build about evidence, make "we have no information" a distinct, storable state that is not zero, not null-as-false, and not a low score — and make it impossible for that state to enter an arithmetic operation. If a value can be absent, the code must be unable to average it. This is the single design decision that separates a system that reports what it knows from one that reports what it happens to have.

What would change my mind: a deployed, independently evaluated investigative system using four-valued or belief-function representation, showing better outcomes than a conventional confidence score, would move this appendix from reasoning to evidence. As of August 2026 no such system is documented, and I am recommending a design principle whose diagnosis is proven and whose remedy is not.