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R0006-03591 · q-20261003-advance-20261004-r02-counterexample-0016
Using classical truth semantics, the explicit finite integer domain is U={3, 4, 5} and P(x) means x is an integer multiple of (2). For the claim ∀x∈U P(x), is c=7 a valid counterexample?
G001 — Using classical truth semantics, the explicit finite integer domain is U={3, 4, 5} and P(x) means x is an integer multiple of (2). For the claim ∀x∈U P(x), is c=7 a valid counterexample?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
U={3, 4, 5};predicate=multiple 2After
a counterexample requires c∈U and P(c)=False
Conditions: none
A counterexample needs both domain membership and predicate failure;an outside integer does not refute this restricted universal statement even if its formula is false.
Rule: finite_counterexample_domain · Uses: G001
Before
test 2 divides 7
After
7=3·2+1
Conditions: none
Use the absolute value of the nonzero divisor;remainder0 is equivalent to divisibility by the original integer.
Rule: predicate_divisibility_remainder · Uses: G001, S001
Before
x=7;predicate=multiple 2
After
P(7)=No
Conditions: none
Substitute this integer into the declared condition;the formula's truth value does not change its domain eligibility.
Rule: explicit_predicate_value · Uses: G001, S002
Before
c=7;c∈U=No;P(c)=No
After
original valid witness=No
Conditions: none
Use conjunction,not just predicate failure or domain membership alone.
Rule: membership_and_predicate_failure · Uses: G001, S003
Before
the explicitly declared finite exercise
After
no inference about an unspecified infinite domain
Conditions: none
Complete enumeration decides this finite universal statement,but finite success or absence of counterexamples is not a universal proof over an infinite domain.
Rule: finite_logic_scope · Uses: G001, S004
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.No
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: eadfa14a8302c5fb02da15acda2814f04b104fecfd73c274d18b23a7768ca480candidate record SHA-256: e62a4076c1dddf35a5dada892130cecb9898a77f3dfe770c302d672b0b2aa442semantic review receipt SHA-256: 568eefd71709ac3dc16a0d5b3b4ad3e83376a8ef300c9e5493eee0721500b29dquestion-first witness SHA-256: d964b91bedb9d4918bbc055f567762a09f5a58af6c876268e719d5100332d855Lean: NOT_RUN · human review: NOT_PERFORMED