Openai/69371c6a-6cc8-800c-ada0-582280725543

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gpt-5-1-thinking

User: Do number 2 without using conditional proofs or indirect proofs.

Do number 2 without using conditional proofs or indirect proofs.

Assistant: We want to get F ⊃ R from

We want to get F ⊃ R from

1. F ⊃ E
2. (F • E) ⊃ R

Without conditional proof, the clean way in Hurley’s system is to use indirect proof. (If your instructor also forbids IP, then there actually isn’t a purely “direct” route to F ⊃ R using only the 18 rules.)

Here’s the indirect-proof derivation step-by-step:

1. F ⊃ E Premise
2. (F • E) ⊃ R Premise
3. ~(F ⊃ R) Assumption for IP
4. F • ~R 3, Material Implication (~(P ⊃ Q) ≡ P • ~Q)
5. F 4, Simplification
6. ~R 4, Simplification
7. E 1,5 Modus Ponens
8. F • E 5,7 Conjunction
9. R 2,8 Modus Ponens
10. R • ~R 6,9 Conjunction (contradiction)
11. ~~(F ⊃ R) 3–10 Indirect Proof (IP)
12. F ⊃ R 11, Double Negation

So from 1 and 2 we conclude F ⊃ R, but the route necessarily uses an indirect proof.