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Question 34376: Modus Tollens

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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R0067-34376 · q-20260921-proof-modus-tollens-0001

Original question and givens

Let P mean “the integer n is divisible by 4” and Q mean “the integer n is even”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, can the conclusion be derived in general?

  • G001 — Let P mean “the integer n is divisible by 4” and Q mean “the integer n is even”. The stated premises are P→Q and ¬Q, and the proposed conclusion is ¬P. Under modus tollens, 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-modus-tollens, [conclusion, 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 = [modus_tollens, [P→Q, ¬Q], ¬P]

After

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

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 = [modus_tollens, [P→Q, ¬Q], ¬P]

After

derived = [Yes]

Conditions: none

MP:given P and P→Q,infer Q.MT:assuming P would give Q,contradicting¬Q,so infer¬P.Both preserve the DECLARED premises and do not use an unproved converse.

Step identifiers and dependencies

Rule: Proof_VALID_MP_or_MT_natural_inference_under_declared_premises · 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

Can

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: c5d9315f8bf13e44a90a7102a10b8d443052d1dd6efb590115605be3b8788d0f
  • candidate record SHA-256: 819e8b0062cd83ead77fd12f9395d4dc6edb12c5b50699670e785455c2b3c5e3
  • semantic review receipt SHA-256: c89efa383b67bb2cf25bcf16160b9f5f614bbe5e5e791f4d7f58234d17a3371c
  • question-first witness SHA-256: 12766d451c5375842b4fc2843ae94807ec55f15763a916558026912a13873d57
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON