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Logic and Proof Method Diagnostics

Attempt P1–P5 after clusters 1–2. Allow 4–6 hours for proofs and revision. Use feedback after saving your attempt.

  1. P1 — Scope and witnesses. Over integers, translate and decide both claims: every x has some y greater than x; some y is greater than every x. Negate each. Repeat with both domains empty. State which witnesses may depend on x.
  2. P2 — Implication. Make the complete truth table for (P→Q)↔(¬Q→¬P). Find a counter-row for (P→Q)↔(Q→P). Explain why four rows are exhaustive.
  3. P3 — Direct and contrapositive. Prove the sum of two odd integers is even. Then prove that if an integer's square is odd, the integer is odd. Name each assumption, integer witness and fact about even/odd cases you use.
  4. P4 — Contradiction and cases. Prove there is no greatest integer. Then prove n(n+1) is even for every integer n by exhaustive parity cases. Explain why contradiction is sufficient for the first proof without well-ordering.
  5. P5 — Proof repair. Refute: from ab=0, divide by a to conclude b=0. Repair the conclusion or add a valid hypothesis. Then critique “no request mutates shared state, so all requests commute”: state the observable result and assumptions needed before this becomes a mathematical claim.

For each proof submit a clean argument plus the scratch attempt separately. A method label alone is insufficient; an alternative valid method is acceptable. The former Goldbach exercise is replaced with a bounded task that does not ask a beginning learner to resolve an open research problem.