Is it possible to derive Peirce's Law: |-- [(alpha rarr beta) rarr alpha] rarr alpha

Rose Graves

Rose Graves

Answered question

2022-08-10

Is it possible to derive Peirce's Law:
[ ( α β ) α ] α
in a calculus that has modus ponens, the Deduction Theorem, Cut rule, Inconsistency effect and Principle of Indirect Proof?

Answer & Explanation

Ben Logan

Ben Logan

Beginner2022-08-11Added 13 answers

Proof
1) ( ϕ ψ ) ϕ --- premise
2) ¬ ϕ --- premise
3) ¬ ϕ ( ϕ ψ )
4) ϕ ψ --- from 2) and 3) by mp
Up to now we have: ( ϕ ψ ) ϕ , ¬ ϕ ϕ
Obviously: ( ϕ ψ ) ϕ , ¬ ϕ ¬ ϕ
Thus, we can use Indirect proof to get:
6) ( ϕ ψ ) ϕ , ϕ
7) ( ( ϕ ψ ) ϕ ) ϕ --- from 6) by Deduction Theorem.

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