1. Doing Derivation

2. Derivations
As with proving equivalence claims with laws, there is no set procedure to follow when doing natural deduction – no straightforward algorithm to tell us what to do next.
Rather, we have to use a bit of ingenuity and creativity. It’s also a matter of practice – the more derivations you do, the better a feel you will get for what rules to apply when.

3. Derivations
When doing derivation, usually we have the premises and the conclusion already – all we need is to prove that the conclusion does indeed follow from the premises.
So we know where we’re starting off from, and we know where we’re going. And this can give us big clues as to what rules we should start applying.

4. An example
So, imagine we have been asked to prove that the following argument is valid: X ∨ Y Y ∨ Z ~Y X ∧ Z

5. An example
As said, we know where we’re starting from and where we’ve got to get to: X ∨ Y P 2 Y ∨ Z P 3 ~Y P ? ? X ∧ Z

6. An example
We’re starting off with a couple of disjunctions, and a negation. So we know that we could apply maybe the disjunction elimination rule – and perhaps something with negation too, though we don’t have a double negation so the negation elimination rule won’t be of immediate use to us.
And we’re ending up with a conjunction. So we’re most likely going to have to use the conjunction introduction at some point.

7. An example
To use conjunction introduction to get X ∧ Z we’re going to have to derive both X and Z individually.
So we want to get X and Z. And we know that disjunction elimination might be a good option for us. So, is there a way to get X and Z by applying disjunction elimination?

8. An example
Yes there is! Disjunction elimination tells us that if we have X ∨ Y and not ~Y we may conclude that X. So from (1) and (3) it looks like we can get X. And, similarly, from (2) and (3) we can get Z!
Then we just need to apply conjunction introduction and things are looking rosy.
Let’s try all that.

9. An example
1 X ∨ Y P 2 Y ∨ Z P 3 ~Y P 4 X 1, 3, ∨E 5 Z 2, 3, ∨E 6 X ∧ Z 4, 5, ∧I
And there we have it!

10. Another example A ∧ ~B ~~~B
Hint: We’re ending up with a (triple) negation. So negation introduction is probably a good place to start…

11. Another example
1 A ∧ ~B P 2 ~~B A 3 A ∧ ~B 1, R 4 ~B 3, ∧E 5 ~~~B 2-4, ~I

12. A little tip
In general, it’s not a good idea to just start applying rules without any clear idea of where you are heading for. You can end up going nowhere after a lot of work.
But sometimes you can just get stuck – you know you have to get to Y, but you just can’t see how X can get you there. In that case it might be worth just trying a rule or two with Y to see where it gets you – it might just give you the stroke of inspiration you need…

13. And another…
~G ⊃ ~D (E ≡ F) ∧ D G

14. And another…
1 ~G ⊃ ~D P (E ≡ F) ∧ D P 3 ~G A 4 ~G ⊃ ~D 1, R 5 ~D 3, 4, ⊃E 6 (E ≡ F) ∧ D 2, R 7 D 6, ∧E 8 ~~G , ~I 9 G 8, ~E

15. Another little tip
If you can’t work out how you could possibly get to your conclusion from your premises, try thinking of a sentence or two that definitely would be able to get you to your conclusion.
Then try to think of a way to get from your premises to these intermediary sentences. If you can do that, then you can get from there to your conclusion no problem! Breaking the derivation down into manageable steps like this can be a big help.

16. A tricky one… ~P ∨ Q P ∨ ~Q P ≡ Q
We can apply the previous tip to this example. We know that if we had P ⊃ Q and Q ⊃ P then we could conclude P ≡ Q. So now all we need is to work out how to get those two sentences…

17. 1. ~P ∨ Q. P. 2. P ∨ ~Q. P. 3. P. A. 4. ~P. A. 5. P. 3, R. 6. ~~P
1 ~P ∨ Q P 2 P ∨ ~Q P 3 P A ~P A P 3, R 6 ~~P 4-5, ~I 7 ~P ∨ Q 1, R 8 Q 6, 7, ∨E 9 P ⊃ Q , ⊃I Q A ~Q A Q 10, R ~~Q , ~I P ∨ ~Q 2, R P 13, 14, ∨E 16 Q ⊃ P , ⊃I 17 P ≡ Q 9, 16, ≡I

18. Bonus problem…
On page 75 Teller claims that ‘it’s not hard to come up with problems which will stump your instructor’. Let’s test this! Your task is to come up with a valid argument that I can’t prove to be valid with natural deduction. I’ll accept submissions in the next class, and if I don’t have a proof by the class after that…