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Question 34347: Invalid Affirmation of the Consequent

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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R0067-34347 · q-20260921-proof-invalid-converse-0022

Original question and givens

Let P mean “the integer n is divisible by 3” and Q mean “n² is divisible by 3”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

  • G001 — Let P mean “the integer n is divisible by 3” and Q mean “n² is divisible by 3”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?

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

source roles = [qf-proof-invalid-converse, [conclusion, counterexample_truth_assignment, inference_rule, p, premises, q]]

After

derived = [BASE-no-required-operation, no-invented-domain]

Conditions: none

Check THIS bank’s keys/operation/domain:legacy table underscores,MOD50 local hyphens;permissive BASE data acquires no unknown-field restriction.Keep Number.isInteger and SAFE source guards distinct,while preserving ORIGINAL JSON number representations.

Step identifiers and dependencies

Rule: R67_SOURCE_specific_keys_operation_domain_and_numeric_bank · Uses: G001

S002 · implies

Before

source roles = [invalid_converse, [P→Q, Q], P]

After

derived = [[P, Q, P->Q, second premise, both premises, conclusion], [[No, No, Yes, No, No, No], [No, Yes, Yes, Yes, Yes, No], [Yes, No, No, No, No, Yes], [Yes, Yes, Yes, Yes, Yes, Yes]], No]

Conditions: none

Treat P/Q as GENERAL proposition symbols,without adding extra arithmetic relations between their illustrative texts.Use the displayed COLUMN LABELS to audit ALL four valuations.Validity requires a true conclusion in EVERY row satisfying BOTH premises.This neither proves the premises factually true nor establishes causation.

Step identifiers and dependencies

Rule: Proof_FULL_truth_table_premise_satisfaction_THEN_conclusion · Uses: G001, S001

S003 · implies

Before

source roles = [No, Yes, [P→Q, Q]]

After

derived = [[No, Yes, Yes, Yes, Yes, No], No]

Conditions: none

With P false/Q true,the implication and Q are BOTH true while conclusion P is false.The converse inference is therefore invalid IN GENERAL.A special equivalence theorem for particular P/Q would require an EXTRA proof,not a silently added premise.

Step identifiers and dependencies

Rule: Proof_CONVERSE_explicit_Boolean_countervaluation_BOTH_premises_true · Uses: G001, S002

Reverse checks

  • S003 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof.
  • S002 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof.
  • S001 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof.

Result

Cannot

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: 17db1d039a1fb73174b613b414b7ce30cbe86614b1b99ae2b134c92a6e3f5b47
  • candidate record SHA-256: 76036a267b090a82f2db4179b7cfbd0582fa082c9fe852016f1f08d4dd873de6
  • semantic review receipt SHA-256: c89efa383b67bb2cf25bcf16160b9f5f614bbe5e5e791f4d7f58234d17a3371c
  • question-first witness SHA-256: 12766d451c5375842b4fc2843ae94807ec55f15763a916558026912a13873d57
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON