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R0007-04332 · q-20261008-advance-20261010-r07-quantifier-0032
The explicitly finite integer domain is U=[-3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.
G001 — The explicitly finite integer domain is U=[-3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.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.
Before
complete finite U=[-3, -2, -1]
After
universal on empty domain=True;existential on empty domain=False
Conditions: none
Enumerate only the declared finite domain;empty-domain universals are true and existentials false.Complete finite enumeration does not extend to an infinite domain.
Rule: finite_quantifier_scope · Uses: G001
Before
x=-3;predicate=ge 0
After
P(x)=No
Conditions: none
Evaluate the declared decidable integer predicate for each element,retaining both true and false cases.
Rule: finite_predicate_row · Uses: G001, S001
Before
x=-2;predicate=ge 0
After
P(x)=No
Conditions: none
Evaluate the declared decidable integer predicate for each element,retaining both true and false cases.
Rule: finite_predicate_row · Uses: G001, S002
Before
x=-1;predicate=ge 0
After
P(x)=No
Conditions: none
Evaluate the declared decidable integer predicate for each element,retaining both true and false cases.
Rule: finite_predicate_row · Uses: G001, S003
Before
[[-3, No], [-2, No], [-1, No]]
After
T=[];F=[-3, -2, -1];T∪F=U and T∩F=∅
Conditions: none
The true and false parts are disjoint and cover the deduplicated domain;unlisted integers are not counterexamples.
Rule: complete_truth_partition · Uses: G001, S004
Before
false-element count=3;true-element count=0
After
(True, True)
Conditions: none
Not-all P means at least one false element,and not-any P means no true element;the empty domain therefore gives(False,True).Do not confuse the universal with its negation.
Rule: finite_quantifier_negations · Uses: G001, S005
S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.(True, True)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 831dcffee96e3c43d7b474e154adc772ccf87cf4a0e74b99850bfae98f28f1e8candidate record SHA-256: d06b0defbd0182d823556274e884b6e43b29b62c0dc448df0dd4394c3565fccfsemantic review receipt SHA-256: faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485aquestion-first witness SHA-256: 4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42Lean: NOT_RUN · human review: NOT_PERFORMED