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Either John & Henry are same age or John is older ...

생성일: 2025년 10월 2일

답변에 사용된 모델 GPT-5 Thinking by Chat01

질문

Either John & Henry are same age or John is older than Henry. If they’re same age then Elizabeth and John are not the same age. If John is older than Henry then John is older than Mary. Therefore either Elizabeth & John are not the same age or John is older than Mary.
Solve using premises and sentential logic

사고

Breaking down the argument

The user wants to validate an argument using propositional logic. First, I need to formalize the premises using variables like P, Q, etc. Then, I could use truth tables, natural deduction, or inference rules to show the conclusion follows logically. The statements involve ages and relationships between John, Henry, and Elizabeth, with the conclusion aiming to prove that either Elizabeth and John are not the same age or John is older than Mary.

Solving with constructive dilemma

This looks like a standard constructive dilemma proof: if we have (A ∨ B), (A→C), (B→D), we can conclude (C ∨ D) by elimination. I’ll keep it simple and formal with clear steps using premises and deduction, focusing on the necessary steps without extra details since the user requested a concise formal solution.

Presenting the proof

We can formalize the argument with propositional variables:

  • S: John and Henry are the same age
  • O: John is older than Henry
  • N: Elizabeth and John are not the same age
  • M: John is older than Mary

Then, the derivation would involve assuming S or O and applying implications to reach the conclusion (N ∨ M). This follows the pattern of constructive dilemma and can also be handled via truth tables or resolution.

Preparing the proof

For a clean and concise proof using propositional logic, let’s define our variables:

  • S: John and Henry are the same age
  • O: John is older than Henry
  • E: Elizabeth and John are not the same age
  • M: John is older than Mary

We’ll proceed step by step, assuming S or O and applying the logical rules (Modus Ponens, Disjunction Introduction) to conclude E ∨ M. I'll keep it structured and straightforward.

Thought for 15s

답변

Validity proof (sentential logic)

Symbolization

Let
S = “John and Henry are the same age.”
O = “John is older than Henry.”
E = “Elizabeth and John are not the same age.”
M = “John is older than Mary.”

Premises

  1. S∨OS \lor O
  2. S→ES \to E
  3. O→MO \to M

Goal: E∨ME \lor M

Natural-deduction proof (by cases / constructive dilemma)

  1. Assume SS.

  2. From 2 and 4, EE (Modus Ponens).

  3. From 5, E∨ME \lor M (∨-Introduction).

  4. Assume OO.

  5. From 3 and 7, MM (Modus Ponens).

  6. From 8, E∨ME \lor M (∨-Introduction).

  7. From 1 together with subproofs (4–6) and (7–9), infer E∨ME \lor M (∨-Elimination).

Thus, from the premises it follows that either Elizabeth & John are not the same age or John is older than Mary.
This is an instance of the constructive dilemma: (P∨Q),(P→R),(Q→S) ⊢ (R∨S)(P\lor Q),(P\to R),(Q\to S)\ \vdash\ (R\lor S).

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