alea-cond-scope

Formal-logic questions about the conditional and scope structure of real U.S. statute and regulation sentences. Each item takes one sentence from the U.S. Code or the Code of Federal Regulations that carries exactly one top-level operator: "if", "only if", "unless", "provided that", "notwithstanding" or "except as provided in". It asks what that sentence, taken alone, establishes:

kind	question	answer
fact	Given a fact (the condition is met / not met, or the consequence holds / does not hold), does a statement about the other part follow?	Yes / No (the opposite follows) / Cannot be determined
necsuff	Is X a sufficient condition, a necessary condition, or neither for Y?	sufficient / necessary / neither
contra	Which option is the contrapositive? (the distractors are the converse, the inverse and the contrary)	letter A-D
follows	Does an if-then conclusion (a restatement, the contrapositive, the converse or the inverse) follow?	Yes / No
ifthen	Rewrite the provision as one "If ..., then ..." statement.	free text
scope_which	Where the provision referred to in a "notwithstanding" / "except as provided in" phrase provides otherwise, which rule governs?	letter A/B

The reading is strict propositional logic:

• "C if A" means A -> C.
• "C only if A" and "C, provided that A" mean C -> A.
• "C unless A" means not-A -> C.
• Contrapositives follow. Converses and inverses do not.
• "Notwithstanding Z, C": C controls where Z would provide otherwise.
• "Except as provided in Z, C": Z controls there.
• Deontic negation is kept exact: the negation of "shall do X" is "is not required to do X", the negation of "may not do X" is "may do X", and the negation of "may do X" is "may not do X".

About 33% of rows carry a short worked explanation that ends in a plain Answer: line; the rest answer directly. About 4% of rows include one or two demonstrations, and a demonstration never shares its answer with the target. Rows are rendered as plain text, Q/A, or chat (about 40 / 30 / 30).

Releases

• Release 1 (data/train-00000-of-00002.parquet, 2,100 rows): U.S. Code sentences (plus a few eCFR sentences), parsed blind by gpt-6-luna and GLM-5.3-Flash.
• Release 2 (data/train-00001-of-00002.parquet, 8,727 rows): eCFR sentences, parsed blind by GLM-5.3-Flash and DeepSeek-V4.1-Flash.

Sources

Public-record U.S. law from the KL3M data collection:

• U.S. Code sections (2023 release), excluding Titles 18 and 34;
• eCFR sections (current Code of Federal Regulations).

Federal Register preamble sentences and contract sentences were parsed too, but were left out after review, for the reasons given under Quality review.

Sentences were split with nupunkt. A leading enumerator or run-in heading may be cut. The sentence text is otherwise never edited.

How gold was made

Language models never produce the answers.

1. Selection by code. Code selected sentences of 12-90 words with exactly one conditional or scope operator and no second condition marker.

2. Deterministic parse. For the explicit patterns, code parses the connective, which side is the condition, the condition and consequence word sets, and the deontic force (from the first modal verb).

3. Two blind LLM parses from different model families. Each sentence was parsed independently, with the same prompt and from the sentence alone, by two models: in release 1, gpt-6-luna (OpenAI) and GLM-5.3-Flash (Z.ai, via OpenRouter); in release 2, GLM-5.3-Flash and DeepSeek-V4.1-Flash (via OpenRouter). Each parse returned the connective type, the condition and consequence spans, the deontic force, and short affirmed / negated phrasings of each clause.

4. Exact agreement or drop. A sentence was kept only when the deterministic parse and both blind LLM parses all agreed on the same canonical form:

   • the same connective type and direction, and the same deontic force;
   • condition and consequence word sets overlapping with Jaccard >= 0.75 pairwise.

   It was also dropped when:

   • either model judged the sentence not a clean single conditional;
   • a parser marked the condition as disjunctive (a few disjunctive conditions were not marked; see Quality review);
   • a clause used "only" / "all" / "every";
   • the consequence had no clear modal force, e.g. "should".

5. Checked phrasings. The affirmed and negated clause phrasings come from a parse and are checked by code:

   • their content words are in the source sentence;
   • the negation parity flips between the affirmed and negated forms;
   • the deontic negation has the right form;
   • no number appears that is not in the source.

6. Answers by code. Code derives every answer from the agreed form, P -> Q, and its contrapositive. The worked explanations are rendered by code from the same form.

Agreement, release 1 (1,182 sentences parsed by both models):

comparison	agreement
luna vs GLM, connective type	98.6%
luna vs GLM, full canonical form	91.5%
deterministic vs luna, type	99.5%
deterministic vs GLM, type	97.9%
deterministic spans vs the agreed LLM spans	87.8%

After every gate, 419 sentences were kept (35%).

Agreement, release 2 (5,850 eCFR sentences):

comparison	agreement
DeepSeek vs GLM, connective type	99.3%
DeepSeek vs GLM, full canonical form	90.1-90.9%
deterministic vs DeepSeek, type	99.7-99.8%
deterministic vs GLM, type	98.9-99.1%
deterministic spans vs the agreed LLM spans	88.3-94.0%

Release 2 also drops, by code and without editing any text:

• sentences whose operator sits after a reporting verb ("stated that ... unless");
• "notwithstanding" / "except as" in mid-sentence;
• consequences that start with "no / neither" (an If-then rewrite would shift the quantifier);
• consequences without a finite verb or modal (list fragments);
• status consequences ("entitled", "deemed", "null and void"), whose negation the duty-based reading gets wrong;
• "no such" phrasings.

Of 1,772 agreed eCFR sentences, 972 were used.

Composition

• Size: 10,827 rows (about 1.89M tokens), from 229 + 972 sentences.

• Sources: eCFR 81.4%, U.S. Code 18.6%. This departs from our usual cap of at most 30-40% of rows per source. It is accepted because both sources are primary statute and regulation text, and the meaningful stratum is the title: no CFR or U.S. Code title is above 5% of rows (largest: 46 CFR 4.4%, 26 CFR 3.9%, 41 CFR 3.8%, 7 CFR 3.8%).

• Rows by kind:

  kind	share
  fact	31.9%
  necsuff	22.9%
  follows	22.3%
  contra	11.2%
  ifthen	11.0%
  scope_which	0.8%

• Rows by connective:

  connective	share
  unless	37.2%
  if	29.4%
  only if	19.1%
  provided that	13.5%
  notwithstanding / except as	0.7%

• Labels:

  • fact: Yes 874, No 860, Cannot be determined 1,717.
  • follows: Yes 1,198, No 1,215.
  • necsuff: 816 / 814 / 849.
  • contra: letters 291-309 each.

• Other shares: 33.4% of rows carry worked explanations; 4.0% carry demonstrations. Presentation is 40 / 30 / 30.

• Twins: each sentence appears with opposite facts, as minimal pairs. "if" sentences are grouped with "only if" / "unless" sentences, so the same question meets different connectives; group_id marks these groups.

Surface-cue checks

Each check is a 5-fold grouped bag-of-n-gram logistic regression on the cue-only input, compared with the majority-class prior (lift in points):

kind	question block only (rel. 1 / rel. 2)	provision sentence only	connective only
fact	+1.8 / +4.0	0.0 / -0.1	0.0 / 0.0
follows	+0.2 / -1.1	-1.5 / -0.3	+0.2 / +0.4
necsuff	-3.9 / -0.8	-4.1 / -1.3	+1.8 / +0.6
contra	-4.1 / -1.4	-0.5 / +0.4	+1.8 / +4.0

Before release, whole sentence groups whose question block alone predicted the label were removed (no text edited). For release 2 this mostly hit fact rows (the lift fell from +17.6 to +4.4 before the final measurement).

Quality review

• Release 1.
  • The builder read 62 rows and re-solved each by hand; this found two rendering defects, both fixed.
  • A blind reviewer checked 60 rows: all answers correct, 1 format error (that instruction wording was removed), 8 style notes.
• Release 2: three blind rounds of 60 fresh rows each. No round found a wrong final answer; every finding concerned how a sentence was parsed or reworded.
  • Round 1 (eCFR, contracts, Federal Register preambles): 7 problems.
    • Federal Register preamble sentences often put the condition inside reported or narrative content ("the Guide directs them to ... if"; "VA proposed ... if"), so the extracted conditional is not what the sentence asserts. That source was dropped.
    • Contract requests ("please sign") were also dropped.
  • Round 2 (eCFR and contracts): 5 problems, 4 of them in contracts: an "if" inside an object clause, an "unless" scoped to part of a clause, and "entitled to" negated as a duty. Contract sentences were dropped.
  • Round 3 (eCFR only, after the filters above): 1 problem in 60 rows. A disjunctive condition was negated as "not A or not B" instead of "not A and not B"; the answer was still right. 10 style notes: stiff "is not required to be ..." wording on descriptive clauses, and an unresolved "such" / "they".
• Decontamination. Rows overlapping ALEA evaluation and held-out sets were removed.

Known issues (2026-10-31)

In a 30-item check where a reviewer guessed the label from part of the input, 9 of 30 (30%) were guessed correctly, against a bar of 20%. With only 30 items this is borderline and could be noise; it has not yet been re-run with a larger sample. The set has not been changed.

Limitations

• The answers concern what one provision establishes on its own terms. They are not legal conclusions: other provisions, definitions and case law may change the legal outcome.
• The affirmed / negated phrasings are LLM-written, verified paraphrases close to the source wording. Some sound stiff ("If the Commission shall make ...").
• Only statute and regulation sentences are included. Narrative and contract sentences proved too error-prone for automatic logical-form extraction.