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Question 34649: Existential Truth on a Finite Domain

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0068-34649 · q-20261003-advance-20261004-r01-quantifier-0049

Original question and givens

Using classical truth semantics, the explicit finite integer domain is U={11, 12, 14, 16} and P(x) means x is an integer multiple of (4). Is ∃x∈U P(x) true? Answer Yes or No.

  • G001 — Using classical truth semantics, the explicit finite integer domain is U={11, 12, 14, 16} and P(x) means x is an integer multiple of (4). Is ∃x∈U P(x) true? Answer Yes or No.

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-quantifier-finite-existential, [U, operation, predicate]]

After

derived = [exists, no-invented-domain]

Conditions: none

Separate source banks:legacy RAW SAFE with unbounded derived BigInts versus BASE Number.isInteger without invented SAFE/million caps.Operations,domains and permissive/exact keys follow THIS bank.Keep ORIGINAL JSON intact.

Step identifiers and dependencies

Rule: R68_BANK_specific_keys_operations_and_numeric_guards · Uses: G001

S002 · implies

Before

source roles = [quantifier ORIGINAL U, [11, 12, 14, 16]]

After

derived = [[11, 12, 14, 16]]

Conditions: none

THIS LDL set has0..100 DISTINCT RAW SAFE labels,with the empty set allowed.Sort only the derived representation;preserve original JSON order.Do not import another BASE set protocol.

Step identifiers and dependencies

Rule: LDL_RAW_distinct_SAFE_set_empty_allowed · Uses: G001, S001

S003 · implies

Before

source roles = [{'kind': 'multiple', 'value': 4}]

After

derived = [Yes, validation only;0 is NOT added to U]

Conditions: none

The SOURCE validates predicateValue(0)EVEN for empty U:EXACT kind/value keys,SAFE value and a NONZERO multiple divisor.This is THIS protocol’s qualification,NOT adding0 to U or a universal theorem about divisibility by0.

Step identifiers and dependencies

Rule: Quantifier_SOURCE_predicate_validity_at0_EVEN_empty_U · Uses: G001, S002

S004 · implies

Before

source roles = [11, {'kind': 'multiple', 'value': 4}]

After

derived = [No]

Conditions: none

Substitute EACH explicitly listed domain element into the SAME qualified predicate and record pass/fail.Do not replace complete finite coverage by a successful sample or extrapolate to unlisted values.

Step identifiers and dependencies

Rule: Quantifier_EACH_original_domain_element_predicate_substitution · Uses: G001, S003

S005 · implies

Before

source roles = [12, {'kind': 'multiple', 'value': 4}]

After

derived = [Yes]

Conditions: none

Substitute EACH explicitly listed domain element into the SAME qualified predicate and record pass/fail.Do not replace complete finite coverage by a successful sample or extrapolate to unlisted values.

Step identifiers and dependencies

Rule: Quantifier_EACH_original_domain_element_predicate_substitution · Uses: G001, S004

S006 · implies

Before

source roles = [14, {'kind': 'multiple', 'value': 4}]

After

derived = [No]

Conditions: none

Substitute EACH explicitly listed domain element into the SAME qualified predicate and record pass/fail.Do not replace complete finite coverage by a successful sample or extrapolate to unlisted values.

Step identifiers and dependencies

Rule: Quantifier_EACH_original_domain_element_predicate_substitution · Uses: G001, S005

S007 · implies

Before

source roles = [16, {'kind': 'multiple', 'value': 4}]

After

derived = [Yes]

Conditions: none

Substitute EACH explicitly listed domain element into the SAME qualified predicate and record pass/fail.Do not replace complete finite coverage by a successful sample or extrapolate to unlisted values.

Step identifiers and dependencies

Rule: Quantifier_EACH_original_domain_element_predicate_substitution · Uses: G001, S006

S008 · implies

Before

source roles = [[11, 12, 14, 16], [[11, No], [12, Yes], [14, No], [16, Yes]], exists]

After

derived = [[12, 16], [11, 14], Yes]

Conditions: none

∀ requires NO failures;∃ requires a passing member;¬∀ means a failing member;¬∃ means NO passing member.For empty U the four results are True/False/False/True,AFTER predicate qualification.No infinite-domain inference follows.

Step identifiers and dependencies

Rule: Quantifier_ALL_finite_truth_rows_negation_and_EMPTY_conventions · Uses: G001, S007

Reverse checks

  • S008 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof.
  • S007 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof.
  • S006 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof.
  • S005 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof.
  • S004 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof.
  • S003 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof.
  • S002 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof.
  • S001 · premise_and_result_audit — Reverse-check THIS question’s ORIGINAL givens,domain and step premises.Preserve originals/English/MKOs/ASTs/status;do not reverse a one-way implication or call finite checks independent review or general formal proof.

Result

Yes

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: bc413c5120262ceab3275bc4c8f5dd56ccbbb3f5f459f9e4428a092c89750b5c
  • candidate record SHA-256: d4a61e64066229e2892eda808c41b649f77554a8d0804517111a84f65b4bd96e
  • semantic review receipt SHA-256: e7a54104fa02c235f7591329bd6db8d626dd74bf0fae29c557b1f269c85066ed
  • question-first witness SHA-256: 8401a3c2064c1275a6a7ea8f00412fa12286cec7b2db6c305ff9dba3724ae229
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON