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A system 'works up to n values' if its models are exactly {Nand, Nor} on 2 elements, none on 3, and (for n = 4) exactly the 12 tables isomorphic to the Nand of the 4-element Boolean algebra. The published axiom ((a.b).c).(a.((a.c).a)) = c is rewritten with the variable on the right as (((b.c).a).(b.((b.a).b))) = a. Runtime about 11 minutes, of which 587 s is the 4-valued check of the 296 six-Nand survivors." } ], "@type": [ "https://w3id.org/delphi/ontology/Assessment" ], "http://www.w3.org/2000/01/rdf-schema#comment": [ { "@value": "Measured: 100 six-Nand three-variable single axioms {X = a} (of 288684) whose models on domains of size 2, 3 and 4 are exactly those of Boolean Nand (n = 327042, exact enumeration). Interpretation: Finite-model screening is a necessary condition for axiomatising Boolean algebra (every Boolean algebra is a model, and no 3-element algebra is), not a sufficient one; the sufficiency of the six-Nand axiom rests on the equational proof the book presents on pp. 810-811 (found with automated theorem provers) which standard tooling here cannot re-run. The test therefore checks everything about the claim that is checkable by enumeration: that the published axiom is consistent with being an axiom for logic on all domains up to 4 elements, that six Nands is the minimum passing even the weakest screens, and that a screen-passing system appears within the first few thousand systems of the book's own enumeration. 'Simplest few thousand' is read against the book's own ordering (by total number of Nands, systems {X = a} and {X = a, a.b = b.a})." } ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Claim test of wolfram2002-ch12-42: mixed, confidence 0.85" } ], "http://www.w3.org/2000/01/rdf-schema#seeAlso": [ { "@id": "https://github.com/fractastical/delphisci/blob/bf2af6ee0a84e23ffa6414ad7a0a50a8c276506e/test-data/claim-tests/wolfram-nks/wolfram2002-ch12-42/result.json" } ], "http://www.w3.org/ns/prov#wasAttributedTo": [ { "@id": "https://w3id.org/delphi/resource/agent/delphisci-claim-tester-claude-fable-5-1" } ], "https://w3id.org/delphi/ontology/admissibleDatasets": [ { "@value": "Planned datasets: nand-axiom-enumeration-generator" } ], "https://w3id.org/delphi/ontology/assessmentType": [ { "@value": "claim-test/0.1.0" } ], "https://w3id.org/delphi/ontology/dataKind": [ { "@value": "computational" } ], "https://w3id.org/delphi/ontology/disconfirmingObservation": [ { "@value": "Pre-registered. FAIL (refutes) if the published six-Nand axiom has a model on 3 elements or a non-Boolean model on 4 elements (then it does not axiomatise Boolean logic), or if no system with at most 6 Nands in total passes the screens up to 4 values (then nothing 'among the simplest few thousand' is a candidate). MIXED if (a) holds but (b) fails (a shorter single axiom survives the screens, undermining 'as short as possible'), or if (c) fails only on the count bound." } ], "https://w3id.org/delphi/ontology/evidenceAgainst": [ { "@value": "Criterion (a) holds: the published six-Nand axiom has no 3-element model and exactly the 12 Boolean-Nand models on 4 elements, and every count the book reports for its search (0 of 16896, 296 of 288684, 100 up to 4 values, 32 of 3402) is reproduced exactly." }, { "@value": "Criterion (b) as pre-registered fails: 8 four-Nand and 16 five-Nand single axioms over three variables survive the 2- and 3-valued screens (e.g. ((b.a).(a.(b.c))) = a). All 24 fail on 4 elements (they admit non-Boolean 4-element models), so none is an axiom for logic; the pre-registered 3-valued screen was too weak to separate 'passes a screen' from 'is as short as possible'. Under the 4-valued screen six Nands is indeed the smallest size with any survivor among single axioms {X = a}." }, { "@value": "Criterion (c) fails only on the count bound, and only because the pre-registered enumeration order places every single axiom with fewer Nands before the two-axiom family: the first family-II system that works up to 4 values is number 1248 of the 3402 family-II systems with 6 Nands (number 1716 counting family-II systems of all sizes), but up to 36204 when all 34956 shorter single axioms are counted first. Whether that is 'the first few thousand' depends on the order chosen; the book's own text counts only the family it searches." }, { "@value": "Finite-model screening is necessary but not sufficient for axiomatising Boolean algebra; the sufficiency of the published axiom (that it proves all of Boolean algebra) rests on the equational proof the book reports and was not re-derived here. Per the pre-registered rule, (a) true with (b) and (c) failing as described gives 'mixed'; every failure is a calibration artefact of the pre-registered criteria rather than a disagreement with the book's reported search results." } ], "https://w3id.org/delphi/ontology/expectedObservation": [ { "@value": "Pre-registered. Enumerate all Nand expression trees X with k Nands (k+1 leaves) over variables a, b, c and consider (I) single axioms {X = a} for k <= 6 (34488 systems for k <= 5, 288684 for k = 6, plus the 16896 two-variable systems for k = 6) and (II) two-axiom systems {X = a, a.b = b.a} with k = 4 (3402 systems, 6 Nands in total). Screen each system against all 16 binary operations on {0,1}, all 19683 on {0,1,2} and, by backtracking search, all operations on {0,1,2,3}; a system 'works up to n values' if its models on each domain of size <= n are exactly the operation tables isomorphic to the Nand (or, on 2 elements, Nand or Nor) of a Boolean algebra of that size, i.e. {Nand, Nor} on 2, none on 3, the 12 tables isomorphic to Nand_4 on 4. PASS (supports) if (a) the published axiom works up to 4 values; (b) no single axiom {X = a} with k <= 5 over <= 3 variables works up to 3 values; (c) at least one system of family (II) works up to 4 values, and the number of systems of forms (I) and (II) with at most 6 Nands in total that precede it in the enumeration (fewer Nands first) is below 10000. The book's counts (0 of 16896 two-variable six-Nand systems up to 3 values; 296 of 288684 up to 3 values and 100 up to 4 values; 32 of 3402 in family II) are recorded as cross-checks and do not decide the verdict. If the 4-value search over all 3-value survivors of (I) exceeds 20 minutes it is applied only to the published axiom and to family (II), and the 100-count cross-check is dropped and reported." } ], "https://w3id.org/delphi/ontology/failCriterion": [ { "@value": "Pre-registered. FAIL (refutes) if the published six-Nand axiom has a model on 3 elements or a non-Boolean model on 4 elements (then it does not axiomatise Boolean logic), or if no system with at most 6 Nands in total passes the screens up to 4 values (then nothing 'among the simplest few thousand' is a candidate). MIXED if (a) holds but (b) fails (a shorter single axiom survives the screens, undermining 'as short as possible'), or if (c) fails only on the count bound." } ], "https://w3id.org/delphi/ontology/hasConfidence": [ { "@value": "0.85", "@type": "http://www.w3.org/2001/XMLSchema#double" } ], "https://w3id.org/delphi/ontology/hasEvidence": [ { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch12-42--claim-test--dataset-1" } ], "https://w3id.org/delphi/ontology/hasVerdict": [ { "@value": "mixed" } ], "https://w3id.org/delphi/ontology/hypothesis": [ { "@value": "If the claim holds, the published six-Nand single axiom ((a.b).c).(a.((a.c).a)) = c has exactly the models that Nand-logic has on small domains (on 2 elements only Nand and Nor, on 3 elements none, on 4 elements only the 12 operation tables isomorphic to the Nand of the four-element Boolean algebra); no shorter single axiom of the form X = variable passes even the 2- and 3-element screens; and in a natural enumeration of axiom systems by number of Nands a system passing all screens appears within the first few thousand." } ], "https://w3id.org/delphi/ontology/inferenceMethod": [ { "@value": "analysis.py enumerates every full binary Nand tree with k internal nodes and every leaf labelling over {a,b,c} (or {a,b}), builds the single-axiom systems {X = a} for k = 1..6 (34488 systems for k <= 5, 288684 for k = 6, 16896 two-variable systems for k = 6) and the two-axiom systems {X = a, a.b = b.a} with k = 4 (3402 systems), and screens each against all 16 binary operations on {0,1} and all 19683 on {0,1,2}; survivors of the 3-valued screen (and the published axiom) are then checked on {0,1,2,3} by backtracking over the 4^16 operation tables. A system 'works up to n values' if its models are exactly {Nand, Nor} on 2 elements, none on 3, and (for n = 4) exactly the 12 tables isomorphic to the Nand of the 4-element Boolean algebra. The published axiom ((a.b).c).(a.((a.c).a)) = c is rewritten with the variable on the right as (((b.c).a).(b.((b.a).b))) = a. Runtime about 11 minutes, of which 587 s is the 4-valued check of the 296 six-Nand survivors." } ], "https://w3id.org/delphi/ontology/passCriterion": [ { "@value": "Pre-registered. Enumerate all Nand expression trees X with k Nands (k+1 leaves) over variables a, b, c and consider (I) single axioms {X = a} for k <= 6 (34488 systems for k <= 5, 288684 for k = 6, plus the 16896 two-variable systems for k = 6) and (II) two-axiom systems {X = a, a.b = b.a} with k = 4 (3402 systems, 6 Nands in total). Screen each system against all 16 binary operations on {0,1}, all 19683 on {0,1,2} and, by backtracking search, all operations on {0,1,2,3}; a system 'works up to n values' if its models on each domain of size <= n are exactly the operation tables isomorphic to the Nand (or, on 2 elements, Nand or Nor) of a Boolean algebra of that size, i.e. {Nand, Nor} on 2, none on 3, the 12 tables isomorphic to Nand_4 on 4. PASS (supports) if (a) the published axiom works up to 4 values; (b) no single axiom {X = a} with k <= 5 over <= 3 variables works up to 3 values; (c) at least one system of family (II) works up to 4 values, and the number of systems of forms (I) and (II) with at most 6 Nands in total that precede it in the enumeration (fewer Nands first) is below 10000. The book's counts (0 of 16896 two-variable six-Nand systems up to 3 values; 296 of 288684 up to 3 values and 100 up to 4 values; 32 of 3402 in family II) are recorded as cross-checks and do not decide the verdict. If the 4-value search over all 3-value survivors of (I) exceeds 20 minutes it is applied only to the published axiom and to family (II), and the 100-count cross-check is dropped and reported." } ], "https://w3id.org/delphi/ontology/preregistered": [ { "@value": "false", "@type": "http://www.w3.org/2001/XMLSchema#boolean" } ] }, { "@id": "https://w3id.org/delphi/resource/evidence/wolfram2002-ch12-42--claim-test--dataset-1", "http://purl.org/dc/terms/description": [ { "@value": "the finite objects the claim's search statements are about" } ], "http://purl.org/dc/terms/license": [ { "@value": "generated by this test" } ], "@type": [ "https://w3id.org/delphi/ontology/EvidenceItem" ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Dataset: Enumeration of Nand equational axiom systems with up to 6 Nands over 3 variables, with finite-model screening on domains of size 2, 3, 4, generated by analysis.py" } ], "http://www.w3.org/ns/prov#wasDerivedFrom": [ { "@id": "https://www.wolframscience.com/nks/p808--implications-for-mathematics-and-its-foundations/" } ] }, { "@id": "https://w3id.org/delphi/resource/claim/wolfram2002-ch12-42", "https://w3id.org/delphi/ontology/hasAssessment": [ { "@id": "https://w3id.org/delphi/resource/assessment/wolfram2002-ch12-42--claim-test" } ] } ] }, { "@id": "https://w3id.org/np/RAKfq1Ec8g-FpEfM2-PBV09TWfp_NR3msOV4uxrQkmRx4/pubinfo", "@graph": [ { "@id": "https://w3id.org/np/RAKfq1Ec8g-FpEfM2-PBV09TWfp_NR3msOV4uxrQkmRx4", "http://purl.org/dc/terms/created": [ { "@value": "2026-11-08T16:24:31.945Z", "@type": "http://www.w3.org/2001/XMLSchema#dateTime" } ], "http://purl.org/dc/terms/creator": [ { "@id": "https://orcid.org/0009-0007-4948-9192" } ], "http://purl.org/dc/terms/license": [ { "@id": "https://creativecommons.org/licenses/by/4.0/" } ], "http://purl.org/nanopub/x/hasNanopubType": [ { "@id": "https://w3id.org/delphi/ontology/Assessment" } ], "http://purl.org/nanopub/x/introduces": [ { "@id": "https://w3id.org/delphi/resource/assessment/wolfram2002-ch12-42--claim-test" } ], "http://www.w3.org/2000/01/rdf-schema#label": [ { "@value": "Claim test of wolfram2002-ch12-42: mixed, confidence 0.85" } ], "https://w3id.org/delphi/ontology/domain": [ { "@id": "https://w3id.org/delphi/resource/domain/wolfram-nks-2002" } ] }, { "@id": "https://w3id.org/np/RAKfq1Ec8g-FpEfM2-PBV09TWfp_NR3msOV4uxrQkmRx4/sig", "http://purl.org/nanopub/x/hasAlgorithm": [ { "@value": "RSA" } ], "http://purl.org/nanopub/x/hasPublicKey": [ { "@value": "MIIBIjANBgkqhkiG9w0BAQEFAAOCAQ8AMIIBCgKCAQEAyFHtPB6p0mCjDozxlOY9CmO3nBm7Te3jFbXyLmM+G1tGe47Da0jYIbhkl9Oo1ZjxmCa3LCwlEqK8gj6hEIcL4ahW30JXh9HycfKVk1bugkzMEV15tJdTzAU3sGMFnbNsn43xY2LFLy45tApkRm5byEiM7oRRrooIdwoQKnP4Wzq/bfAoJ9qvW+MGkelyOTd1jbSt+KfmT7xKmDfWzAHddUlWwMX2QJOLxDaffOteTop6ZRgPiNmPJKA/S62Gac9XtSYT3oDkZ6/nr5cxxNLpOyxTf5FPISiCe11xYO/unYjY+mwnOfA8yhJIyx1l0O9I0vpOpn+E5TvyHJV+oSTrvwIDAQAB" } ], "http://purl.org/nanopub/x/hasSignature": [ { "@value": "KZDYesVCWkXAGBO8lhP8LZLv01MHEUl62DGOSrC2xyzMGgbSQPr4ueNnSWod2PJXiQKu97LtqMEqHfw05Ivez4pLKRX2a0z6YoFEtJ0RYJkJf6wKN+I6EWc+h7l7cXcfsCZb/vS2HekgVHnE7Nptx9jVIHkQkYygrzye7E6KUxfjV4tZoyHP9oEreppXP/JYN2fMA2vDdIvbm0zeQBOzWxa5Z/wG313T95vN7F+gy8bLfe40JPsoLOhCA1P05NLaf3+lUNZJ/Wd/AEfma4TzewJe3XKVBb8MbGaczoh4mY7Z6y8wLWE9zbwK6jy6FGHCWva9dmsk5gZYnfypid4sLQ==" } ], "http://purl.org/nanopub/x/hasSignatureTarget": [ { "@id": "https://w3id.org/np/RAKfq1Ec8g-FpEfM2-PBV09TWfp_NR3msOV4uxrQkmRx4" } ], "http://purl.org/nanopub/x/signedBy": [ { "@id": "https://orcid.org/0009-0007-4948-9192" } ] } ] }, { "@id": "https://w3id.org/np/RAKfq1Ec8g-FpEfM2-PBV09TWfp_NR3msOV4uxrQkmRx4/Head", "@graph": [ { "@id": "https://w3id.org/np/RAKfq1Ec8g-FpEfM2-PBV09TWfp_NR3msOV4uxrQkmRx4", "http://www.nanopub.org/nschema#hasAssertion": [ { "@id": "https://w3id.org/np/RAKfq1Ec8g-FpEfM2-PBV09TWfp_NR3msOV4uxrQkmRx4/assertion" } ], "http://www.nanopub.org/nschema#hasProvenance": [ { "@id": "https://w3id.org/np/RAKfq1Ec8g-FpEfM2-PBV09TWfp_NR3msOV4uxrQkmRx4/provenance" } ], "http://www.nanopub.org/nschema#hasPublicationInfo": [ { "@id": "https://w3id.org/np/RAKfq1Ec8g-FpEfM2-PBV09TWfp_NR3msOV4uxrQkmRx4/pubinfo" } ], "@type": [ "http://www.nanopub.org/nschema#Nanopublication" ] } ] } ]