Question 34710: Negation of a Finite Universal Statement

Original question and givens

Using classical truth semantics, the explicit finite integer domain is U={-3, -2, 0, 2} and P(x) means x≤0. Is ¬(∀x∈U P(x)) true? Answer Yes or No.

• G001 — Using classical truth semantics, the explicit finite integer domain is U={-3, -2, 0, 2} and P(x) means x≤0. 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.

• mko-quantifier

Step-by-step derivation

source roles = [qf-quantifier-negated-universal, [U, operation, predicate]]

derived = [not_forall, no-invented-domain]

source roles = [quantifier ORIGINAL U, [-3, -2, 0, 2]]

derived = [[-3, -2, 0, 2]]

source roles = [{'kind': 'le', 'value': 0}]

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

source roles = [-3, {'kind': 'le', 'value': 0}]

derived = [Yes]

source roles = [-2, {'kind': 'le', 'value': 0}]

derived = [Yes]

source roles = [0, {'kind': 'le', 'value': 0}]

derived = [Yes]

source roles = [2, {'kind': 'le', 'value': 0}]

derived = [No]

source roles = [[-3, -2, 0, 2], [[-3, Yes], [-2, Yes], [0, Yes], [2, No]], not_forall]

derived = [[-3, -2, 0], [2], Yes]

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.