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Question 5971: Material Implication and Biconditional Tables

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0010-05971 · q-20261009-advance-20261013-r05-logical-connectives-0046

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, 0, 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, 0, 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.

Step-by-step derivation

S001 · implies

Before

given8 rows A=[0, 0, 1, 0, 0, 0, 0, 1];B=[1, 0, 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 dependencies

Rule: classical_eight_row_connective_scope · Uses: G001

S002 · implies

Before

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 dependencies

Rule: material_implication_row · Uses: G001, S001

S003 · implies

Before

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 dependencies

Rule: biconditional_both_directions_row · Uses: G001, S002

S004 · implies

Before

row=1;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 dependencies

Rule: material_implication_row · Uses: G001, S003

S005 · implies

Before

row=1;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 dependencies

Rule: biconditional_both_directions_row · Uses: G001, S004

S006 · implies

Before

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 dependencies

Rule: material_implication_row · Uses: G001, S005

S007 · implies

Before

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 dependencies

Rule: biconditional_both_directions_row · Uses: G001, S006

S008 · implies

Before

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 dependencies

Rule: material_implication_row · Uses: G001, S007

S009 · implies

Before

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 dependencies

Rule: biconditional_both_directions_row · Uses: G001, S008

S010 · implies

Before

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 dependencies

Rule: material_implication_row · Uses: G001, S009

S011 · implies

Before

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 dependencies

Rule: biconditional_both_directions_row · Uses: G001, S010

S012 · implies

Before

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 dependencies

Rule: material_implication_row · Uses: G001, S011

S013 · implies

Before

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 dependencies

Rule: biconditional_both_directions_row · Uses: G001, S012

S014 · implies

Before

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 dependencies

Rule: material_implication_row · Uses: G001, S013

S015 · implies

Before

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 dependencies

Rule: biconditional_both_directions_row · Uses: G001, S014

S016 · implies

Before

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 dependencies

Rule: material_implication_row · Uses: G001, S015

S017 · implies

Before

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 dependencies

Rule: biconditional_both_directions_row · Uses: G001, S016

S018 · implies

Before

([True, True, False, True, True, True, True, True], [False, True, 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 dependencies

Rule: 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, True, 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: 2e56a646f9f02a32fd5272464099ab996ec1ced514d6b32ab26bbea381eeea10
  • candidate record SHA-256: 54dc72df5659d10e6138bb3561726c366b3d3c708383238923a57753bec821e6
  • semantic review receipt SHA-256: 0328166a2e335aac7a5fb34556e68c80ba7d16612e822b3999c8ac348a9ca06f
  • question-first witness SHA-256: 0da18e4e6cf71ece5ec5f00dd0cd161217b6d251672bc0b46d79fe7e6253f5fc
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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