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

Question

Is it possible to derive Peirce&#039;s Law:  ⊢  ∗  [  (  α  →  β  )  →  α  ]  →  αin a calculus that has modus ponens, the Deduction Theorem, Cut rule, Inconsistency effect and Principle of Indirect Proof?

Ben Logan · Accepted Answer

Proof1)   (      ϕ    →      ψ    )  →      ϕ   --- premise2)   ¬  ϕ --- premise3)       ⊢    ∗    ¬  ϕ  →  (  ϕ  →  ψ  )4)   ϕ  →  ψ --- from 2) and 3) by mpUp to now we have:   (      ϕ    →      ψ    )  →      ϕ    ,  ¬  ϕ      ⊢    ∗    ϕObviously:   (      ϕ    →      ψ    )  →      ϕ    ,  ¬  ϕ      ⊢    ∗    ¬  ϕThus, we can use Indirect proof to get:6)   (      ϕ    →      ψ    )  →      ϕ    ,      ⊢    ∗    ϕ7)      ⊢    ∗    (  (      ϕ    →      ψ    )  →      ϕ    )  →      ϕ   --- from 6) by Deduction Theorem.

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

Answered question

Answer & Explanation

New Questions in High school geometry