The truth values of P and Q are false and true respectively. Is P ↔ Q true?
G001 — The truth values of P and Q are false and true respectively. Is P ↔ Q true?
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.
MOD25 uses LOCAL hyphens/named domains,legacy matrices underscores/reals;BASE logic acquires no domain/op.Output tuples,matrices,numeric fields and fraction texts follow THIS source,not a universal/1 rule.Retain originals/English/AST/status.
derived = [[[No, No, Yes], [No, Yes, No], [Yes, No, No], [Yes, Yes, Yes]]]
Conditions: none
P/Q are GIVEN EXACT Booleans;do not infer truth from context.Show all four defining cases.OR is INCLUSIVE;material implication fails ONLY at P true/Q false and is not causation;check BOTH directions for iff.
Substitute THIS P/Q to evaluate the connective;iff needs both directions true.A false-premise implication may be true without establishing causation or proving the underlying propositions.
S003 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
S002 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
S001 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate.
Result
No
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.