Natural Deduction Exercise: Proving ¬P from Given Premises

How can we prove the conclusion ¬P from the given premises using natural deduction rules?

Let's break down the steps and apply the rules of natural deduction to reach the conclusion.

Solution:

To prove the conclusion ¬P, we need to use the given premises and apply the rules of natural deduction. Let's go step by step:

1. P⊃(∼E⊃L) 2. P⊃∼E 3. ∼L

From premise 3, we can apply the rule of negation elimination to obtain L:
4. ∼∼L (double negation)
5. L (negation elimination)

Now, let's consider premise 2 and use it to derive ∼P:
6. P⊃∼E (premise 2)

7. ∼E (modus ponens on premises 5 and 6)

From premise 1, we can use modus ponens to get L:
8. P⊃(∼E⊃L) (premise 1)
9. ∼E⊃L (modus ponens on premises 1 and 7)
10. L (modus ponens on premises 5 and 9)

Since we have derived both L and ∼L, there is a contradiction. Therefore, the assumption that P is true must be false, which means ∼P is true.

Hence, we have proved the conclusion ∼P.

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