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Is my constructive proof of the disjunctive syllogism principle correct?

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Erwan Aaron
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  • I think not. I think you must start with the assumption $(\phi\lor Q)\land\lnot\phi$, prove $Q$, and then using ($\rightarrow I$) prove the last formula. – Mohsen Shahriari Jun 13 '15 at 05:48
  • The principle of disjunctive syllogism governs just disjunction and negation, not also conjunction and conditional. It is cleaner to seek an intuitionistic proof of $\Phi\lor Q, \neg \Phi\vdash Q$. – mmw Oct 11 '17 at 12:18

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As suggested by Mohsen Shahriari, your derivation is not correct: to obtain a correct derivation you should explicit the rules used in the "vertical dots" part of your derivation.

The following is a derivation of $((\Phi \lor Q) \land \lnot \Phi) \to Q$ in intuitionistic (and hence "constructive") natural deduction. derivation in intuitionistic natural deduction

where efq is the rule ex falso quodlibet, also known as $\bot_E$, which is a rule of intuitionistic (and then classical) natural deduction but it not admissible in minimal natural deduction.

Taroccoesbrocco
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