Four states of support

Aigerim received two records about Ivanova. The registrar confirms her accreditation, while the ethics committee reports that the accreditation was revoked. Both records concern the moment the question is asked about. Which one should count? Before deciding, we need at least to preserve the disagreement itself.

Dana suggests comparing four cases: only a confirmation, only a revocation decision, both records, neither. For the last case we take Petrov: no evidence of his accreditation was passed to us. A single “yes” or “no” mark is not enough to tell all four situations apart.

Let us work through them together with Aigerim. The Archive of Veliky Ustin, its rules, and the people in the story are fictional. We will return to the familiar rare-collections room access rule and see how each evidence set affects the answer. If you want to run the example right away, go to the “Run” section.

Question: how many answers does one question have?

Try describing each answer in plain words before reading the table. Note the distinction: a revocation decision gives grounds for a negative answer. Missing evidence gives no such grounds.

Accreditation evidence	Expected answer
Only the registrar’s confirmation	Established
Only the committee’s revocation decision	Refuted
Both the confirmation and the revocation decision	Grounds for the statement and for its denial
No records: Petrov’s case	Neither established nor refuted

In the first chapter we already met two of these answers: “established” and “neither established nor refuted”. Now refutation and contradiction are added. If only one “yes” or “no” value is kept, we must additionally decide how to represent a gap in the evidence and what to do when records disagree. Whether another person can understand the original situation will depend on those decisions.

Record: two supports for each statement

We use the vocabulary and the rule familiar from the first chapter. To establish access, one must confirm that the person is listed in the researcher registry and holds a valid accreditation.

Arxo Law

language "law.core" version "0.2";
package tutorial.archive version "0.2.0";
namespace "urn:law:tutorial:archive";

entity Person;

relation in_researcher_registry(p: Person) kind institutional;
relation accredited(p: Person) kind institutional;
relation may_enter_rare_room(p: Person) kind institutional;

rule RareRoomAccess strict {
    for p: Person;
    when in_researcher_registry(p) and accredited(p);
    then may_enter_rare_room(p);
}

We represent the revocation decision as a negative statement: assert not accredited(…). This record has its own name and an origin note, like a positive statement. In this way we explicitly report the absence of accreditation.

For the answer, Arxo counts separately the grounds for the statement and the grounds for its denial. They are called the positive and the negative support. Is there at least one support “for”? Is there at least one support “against”? The four combinations of answers to these two questions give the four states of support.

Positive and negative support do not destroy each other. If our example holds both, the model keeps the contradiction. The order in which we wrote the evidence does not itself decide which of them to prefer.

Run: get the answer

You will need a copy of the repository and Rust with Cargo installed. Open a terminal at the repository root and run the commands below. Then keep working in the same window: it will keep the paths you set. The check command will verify the package, and lower will prepare the program for running.

Terminal

cd engines/lawc
tutorial_dir=../../docs/tutorials
tutorial_page="$tutorial_dir/02-four-states.en.law.md"
cargo run -q --profile gate -p law-cli -- check \
    "$tutorial_page"
tutorial_work=$(mktemp -d)
cargo run -q --profile gate -p law-cli -- lower \
    "$tutorial_page" > "$tutorial_work/program.lawir.json"

Let us start with the case holding both accreditation records. We add the evidence that Ivanova is listed in the researcher registry. In one test we ask two questions: is her accreditation established, and is room access established?

For accreditation we expect BOTH: there are grounds for the statement and for its denial. For access we expect NEITHER: it is neither established nor refuted. Below we will work out why the answers differ.

Arxo Law

test "Иванова: запись регистратора и отзыв комиссии" {
    given {
        context {
            decision_time @2026-03-01T09:00:00+05:00;
            knowledge_time @2026-03-01T09:00:00+05:00;
            legal_time @2026-03-01;
            timezone "Asia/Almaty";
        }
        assert in_researcher_registry(entity_ref("urn:tutorial:ivanova")) {
            id "assert-reg";
            origin case_input;
        }
        assert accredited(entity_ref("urn:tutorial:ivanova")) {
            id "assert-acc";
            origin case_input;
        }
        assert not accredited(entity_ref("urn:tutorial:ivanova")) {
            id "assert-revoked";
            origin case_input;
        }
    }
    evaluate truth(accredited(entity_ref("urn:tutorial:ivanova")));
    expect truth_status == BOTH;
    evaluate truth(may_enter_rare_room(entity_ref("urn:tutorial:ivanova")));
    expect truth_status == NEITHER;
}

The test name reads: “Ivanova: the registrar’s record and the committee’s revocation.”

A ready-made test for this case ships with the tutorial. The first command runs the check, the second shows the explanation of the accreditation:

Terminal

cargo run -q --profile gate -p law-cli -- test \
    "$tutorial_dir/tests/02-both-and-neither.lawtest" \
    --lawtest --program "$tutorial_work/program.lawir.json"
cargo run -q --profile gate -p law-cli -- explain \
    "$tutorial_dir/tests/02-both-and-neither.lawtest" \
    --program "$tutorial_work/program.lawir.json"

A test PASS message means both answers matched the expectations. In the accreditation explanation, find the assert-acc and assert-revoked records: the first supports the statement, the second its denial. The BOTH answer preserves evidence of both.

Here it matters to tell two results apart. PASS reports that the test went through. BOTH describes the state of the statement being checked. A contradiction in the input evidence may be the expected result of a correctly executed check.

Now let us collect all designations in one table:

Support “for”	Support “against”	Designation	Answer in words
present	absent	TRUE_ONLY	established
absent	present	FALSE_ONLY	refuted
present	present	BOTH	grounds for and against
absent	absent	NEITHER	neither established nor refuted

Back to access. Under our rule, accreditation must be established with no support for its denial, i.e. hold the TRUE_ONLY status. The registry entry alone is not enough for the conclusion.

Keeping the registry entry established, let us see how access changes with each accreditation state:

accredited	may_enter_rare_room
TRUE_ONLY	TRUE_ONLY
FALSE_ONLY	NEITHER
BOTH	NEITHER
NEITHER	NEITHER

The last three rows give the same answer about access, though the accreditation evidence in them differs.

With FALSE_ONLY, accreditation is refuted. The rule condition fails, so access does not follow from it. But this rule does not derive a refusal: after then only the positive access statement is recorded. Refuting it would need a separate ground, for example another rule or an explicitly entered negative statement.

With BOTH there is a contradiction. Such a state also fails the accredited(p) condition in our rule. We did not state how to resolve the conflict between the records, so the rule picks neither of them and does not derive access.

With NEITHER the accreditation evidence is missing. Again the rule does not apply, and no grounds appear for denying access. To make sense of such an answer, it helps to look not only at its designation but at the state of each premise.

Contrast: a table and Python

Let us see how the same evidence can be represented in Python. We start with a dictionary mapping each person to one True or False value. Then we try storing positive and negative support separately.

In the example below, lines with >>> are entered commands, lines with ... their continuations. Under a command returning a result, that result is shown.

Python

>>> accredited = {"ivanova": True}
>>> accredited.get("petrov", False)
False
>>> records = [("ivanova", True), ("ivanova", False)]
>>> dict(records)["ivanova"]
False
>>> support = {"ivanova": {"for", "against"}, "sidorov": {"for"}}
>>> def status(name):
...     s = support.get(name, set())
...     return {("for",): "TRUE_ONLY", ("against",): "FALSE_ONLY",
...             ("against", "for"): "BOTH", (): "NEITHER"}[tuple(sorted(s))]
>>> status("sidorov"), status("ivanova"), status("petrov")
('TRUE_ONLY', 'BOTH', 'NEITHER')

In the first query Petrov is absent from the dictionary. We supplied False as the default ourselves, so we got it back. If such an answer is read as a refutation of accreditation, missing evidence becomes indistinguishable from a negative statement.

Next, the dictionary dict(records) is built from two records about Ivanova. Both records use one key, so the last value, False, is kept. Swap the lines and True remains. The result depends on the record order, and the evidence of their contradiction is no longer visible in such a dictionary.

In the last part the data is arranged differently: each person holds a set of supports. The status function tells all four combinations apart. In the call shown, Sidorov’s accreditation is established, Ivanova’s holds a contradiction, and about Petrov there is no evidence. So the needed distinctions can be represented explicitly in Python too.

A table can likewise have separate “for” and “against” columns. The ability to preserve such evidence does not belong to one tool. Arxo sets the shared semantics of these states: how they arise, how they are checked in rule conditions, and how they relate to the explanation of the answer. When working in Python or in a table, these conventions must be provided for in the arrangement of the program or the data.

The choice depends on the task. For a small calculation a table or a short function may suffice. But if you pass models to other people, a shared way of representing gaps, contradictions, and grounds of derivation helps them understand your record and verify the result.

Practice: four cases and two ways to say “no”

Copy the ready-made test into a temporary directory and first run it unchanged. It should pass with the previous answers: BOTH about accreditation and NEITHER about access.

Terminal

tutorial_try=$(mktemp -d)
cp "$tutorial_dir/tests/02-both-and-neither.lawtest" \
    "$tutorial_try/try.lawtest"
cargo run -q --profile gate -p law-cli -- test \
    "$tutorial_try/try.lawtest" \
    --lawtest --program "$tutorial_work/program.lawir.json"

Now open the try.lawtest copy and delete the whole negative statement block named assert-revoked. Before running the test, explain why both answers should become TRUE_ONLY. State these expectations after the matching evaluate lines and repeat the test command from the block above.

For the next experiment, restore the original copy. Delete the positive assert-acc statement, keeping assert-revoked. Now accreditation should be refuted, FALSE_ONLY, and access stay NEITHER. State these expectations and verify them. Finally, delete both accreditation records: both answers should become NEITHER.

In all experiments keep Ivanova’s registry entry. That way it is clear the result changes because of the accreditation evidence.

Then carry out the four-cases exercise: pick input evidence so that the access question receives each of the four answers without changing the package. In the negation exercise two ways of saying “not accredited” are compared. It is easier to return to it after the next lesson.

Acceptance: what you can check yourself

Before moving on, make sure you can explain the results of each experiment:

• Confirmation and revocation together give accreditation BOTH. Access stays NEITHER.
• Confirmation alone gives TRUE_ONLY for both accreditation and access. Revocation alone gives accreditation FALSE_ONLY, access NEITHER.
• With neither accreditation record, both answers are NEITHER. Such an answer differs from a refutation.
• In the first Python example the default hides missing evidence. In the second, the repeated key hides the contradiction. Storing supports separately preserves both distinctions.
• A passing test and the state of the checked statement are different things: a test expecting BOTH may pass.

Boundary: what is not yet done here

Our package holds only the rule from which access follows. So in the experiments above, access is either established or neither established nor refuted. That is a limitation of the chosen example. A strict rule may also derive a negative statement; such a case is covered in the lesson on negation.

We also did not determine which of the contradicting records takes priority and from which date it should count. That needs additional evidence and rules. We will return to exceptions and priorities in the lesson on defeat and priority.

Finally, a support state is not a probability estimate or a guarantee of truth. TRUE_ONLY means our model holds grounds for the statement and no support for its denial. The reliability of the evidence itself still needs to be assessed. BOTH reports both supports but does not claim they are equally convincing.

Now Aigerim can pass on the answer together with the material distinction: the statement is confirmed, refuted, disputed, or so far without sufficient grounds either way. “We do not know” and “we hold contradicting evidence” are different answers, and both are worth keeping.