# Question 5974: Material Implication and Biconditional Tables [Return to catalogue page 30](https://ocme.evemisslab.com/en/questions/page/30/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261009-advance-20261013-r05-logical-connectives-foundations/#q-20261009-advance-20261013-r05-logical-connectives-0049) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0010-05974 · q-20261009-advance-20261013-r05-logical-connectives-0049 ## Original question and givens Complete classical eight-row tables have indices 0,…,7, with 1=True and 0=False: A=[0, 0, 1, 0, 0, 0, 0, 1], B=[1, 1, 0, 0, 0, 0, 0, 1]. Give (A→B truth table, A↔B truth table). - G001 — Complete classical eight-row tables have indices 0,…,7, with 1=True and 0=False: A=[0, 0, 1, 0, 0, 0, 0, 1], B=[1, 1, 0, 0, 0, 0, 0, 1]. Give (A→B truth table, A↔B truth table). ## 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-logical-connectives](https://ocme.evemisslab.com/en/mko/mko-logical-connectives/) ## Step-by-step derivation S001 · impliesBefore given8 rows A=[0, 0, 1, 0, 0, 0, 0, 1];B=[1, 1, 0, 0, 0, 0, 0, 1] After classical Boolean values;read all supplied rows in order0,...,7 Conditions: none Interpret1 as True and0 as False and read all eight supplied rows.Do not assume arbitrary eight rows exhaust all valuations or prove every inference rule.Material implication is not causation. Step identifiers and dependenciesRule: classical_eight_row_connective_scope · Uses: G001 S002 · impliesBefore row=0;A=False;B=True After ¬A=True;A→B=True Conditions: none Material implication is defined as ¬A∨B and is false only for a true antecedent and false consequent.A false antecedent supplies no causal support for the consequent. Step identifiers and dependenciesRule: material_implication_row · Uses: G001, S001 S003 · impliesBefore row=0;A=False;B=True After A→B=True;B→A=False;A↔B=False Conditions: none A biconditional requires both material implications,equivalently matching truth values;it does not establish identical meaning or causation. Step identifiers and dependenciesRule: biconditional_both_directions_row · Uses: G001, S002 S004 · impliesBefore row=1;A=False;B=True After ¬A=True;A→B=True Conditions: none Material implication is defined as ¬A∨B and is false only for a true antecedent and false consequent.A false antecedent supplies no causal support for the consequent. Step identifiers and dependenciesRule: material_implication_row · Uses: G001, S003 S005 · impliesBefore row=1;A=False;B=True After A→B=True;B→A=False;A↔B=False Conditions: none A biconditional requires both material implications,equivalently matching truth values;it does not establish identical meaning or causation. Step identifiers and dependenciesRule: biconditional_both_directions_row · Uses: G001, S004 S006 · impliesBefore row=2;A=True;B=False After ¬A=False;A→B=False Conditions: none Material implication is defined as ¬A∨B and is false only for a true antecedent and false consequent.A false antecedent supplies no causal support for the consequent. Step identifiers and dependenciesRule: material_implication_row · Uses: G001, S005 S007 · impliesBefore row=2;A=True;B=False After A→B=False;B→A=True;A↔B=False Conditions: none A biconditional requires both material implications,equivalently matching truth values;it does not establish identical meaning or causation. Step identifiers and dependenciesRule: biconditional_both_directions_row · Uses: G001, S006 S008 · impliesBefore row=3;A=False;B=False After ¬A=True;A→B=True Conditions: none Material implication is defined as ¬A∨B and is false only for a true antecedent and false consequent.A false antecedent supplies no causal support for the consequent. Step identifiers and dependenciesRule: material_implication_row · Uses: G001, S007 S009 · impliesBefore row=3;A=False;B=False After A→B=True;B→A=True;A↔B=True Conditions: none A biconditional requires both material implications,equivalently matching truth values;it does not establish identical meaning or causation. Step identifiers and dependenciesRule: biconditional_both_directions_row · Uses: G001, S008 S010 · impliesBefore row=4;A=False;B=False After ¬A=True;A→B=True Conditions: none Material implication is defined as ¬A∨B and is false only for a true antecedent and false consequent.A false antecedent supplies no causal support for the consequent. Step identifiers and dependenciesRule: material_implication_row · Uses: G001, S009 S011 · impliesBefore row=4;A=False;B=False After A→B=True;B→A=True;A↔B=True Conditions: none A biconditional requires both material implications,equivalently matching truth values;it does not establish identical meaning or causation. Step identifiers and dependenciesRule: biconditional_both_directions_row · Uses: G001, S010 S012 · impliesBefore row=5;A=False;B=False After ¬A=True;A→B=True Conditions: none Material implication is defined as ¬A∨B and is false only for a true antecedent and false consequent.A false antecedent supplies no causal support for the consequent. Step identifiers and dependenciesRule: material_implication_row · Uses: G001, S011 S013 · impliesBefore row=5;A=False;B=False After A→B=True;B→A=True;A↔B=True Conditions: none A biconditional requires both material implications,equivalently matching truth values;it does not establish identical meaning or causation. Step identifiers and dependenciesRule: biconditional_both_directions_row · Uses: G001, S012 S014 · impliesBefore row=6;A=False;B=False After ¬A=True;A→B=True Conditions: none Material implication is defined as ¬A∨B and is false only for a true antecedent and false consequent.A false antecedent supplies no causal support for the consequent. Step identifiers and dependenciesRule: material_implication_row · Uses: G001, S013 S015 · impliesBefore row=6;A=False;B=False After A→B=True;B→A=True;A↔B=True Conditions: none A biconditional requires both material implications,equivalently matching truth values;it does not establish identical meaning or causation. Step identifiers and dependenciesRule: biconditional_both_directions_row · Uses: G001, S014 S016 · impliesBefore row=7;A=True;B=True After ¬A=False;A→B=True Conditions: none Material implication is defined as ¬A∨B and is false only for a true antecedent and false consequent.A false antecedent supplies no causal support for the consequent. Step identifiers and dependenciesRule: material_implication_row · Uses: G001, S015 S017 · impliesBefore row=7;A=True;B=True After A→B=True;B→A=True;A↔B=True Conditions: none A biconditional requires both material implications,equivalently matching truth values;it does not establish identical meaning or causation. Step identifiers and dependenciesRule: biconditional_both_directions_row · Uses: G001, S016 S018 · impliesBefore ([True, True, False, True, True, True, True, True], [False, False, False, True, True, True, True, True]) After complete ordered results of the two requested connectives Conditions: none Each output column retains all eight original rows.This checks the given classical table rowwise,not a general formal proof inferred from matching values or table labels. Step identifiers and dependenciesRule: complete_given_connective_rows · Uses: G001, S017 ## Reverse checks - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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. ## Result ([True, True, False, True, True, True, True, True], [False, False, False, True, True, True, True, True]) 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: 1183dd54f28c53b926f3977cc8003e674e290a6a53435c1a76c57c6a68d544d4 - candidate record SHA-256: 99dc71138768ee11277757a8994e8ad8de15cdb4f67d88aebf2c9ffd67f8673c - semantic review receipt SHA-256: 0328166a2e335aac7a5fb34556e68c80ba7d16612e822b3999c8ac348a9ca06f - question-first witness SHA-256: 0da18e4e6cf71ece5ec5f00dd0cd161217b6d251672bc0b46d79fe7e6253f5fc - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261009-advance-20261013-r05-logical-connectives-0049.json)