1. Introducing Natural Deduction

2. The story so far…
So, by now we have a precise definition of validity for formal logic.
And a rigorous method for checking validity – drawing out truth tables and checking each line to see whether there are any cases for which each premise is true, but the conclusion false.
But there are a couple of downsides to this method.

3. What’s the problem with truth tables?
First, the truth table method of checking validity is cumbersome. You have to write out a whole truth table, and then go through and check each line. This is fairly laborious, especially when we have more than 4 atomic sentence letters in play.
Secondly, it’s not a particularly good model of how we actually think about validity outside of the formal logic context. It’d be nicer to get something that does a better job of reflecting our everyday patterns of reasoning.

4. Enter natural deduction
This is where natural deduction comes in. It’s just another method for checking validity in formal logic.
But it’s supposed to be a little less cumbersome than truth tables for more complex arguments, and most importantly a better reflection of our actual practice of reasoning – it’s supposed to be more natural.

5. Natural deduction
The basic idea is simple. When we reason, there are certain general patterns of reasoning that we follow. From ‘if it’s cloudy today, then it will rain tomorrow’ and ‘it will rain tomorrow’, we all agree that it is legitimate to conclude that it will rain tomorrow.
Likewise, from ‘Either Theresa will win or Jeremy will win’ and ‘Theresa won’t win’ we all agree that it is legitimate to conclude that ‘Jeremy will win’.

6. Natural deduction Natural deduction tries to mirror these patterns.
First, we draw up a series of ‘rules of inference’. These are formally stated rules that tell us what we are allowed to conclude on the basis of what. By applying these rules of inference to sentences of sentence logic we can get other sentences that we know follow logically, as they follow according to our valid rules of inference.

7. Natural deduction
Showing that a particular argument is valid is now just a matter of showing that we can get to the conclusion by applying valid rules of inference to the premises.
We call this process of applying the rules of inference derivation. We derive the conclusion from the premises.

8. Natural deduction
We’ll get a full list of the rules of inference together in due course. For now, let’s just go through an illustrative example.
We’ll introduce some key concepts along the way: we’ve already met derivation; we’ll also introduce licensing, scope lines, and truth preservation.

9. An example: conditional elimination
Above we used an example of an ordinarily accepted rule of inference: from ‘if it’s cloudy today, then it will rain tomorrow’ and ‘it will rain tomorrow’, we all agree that it is legitimate to conclude that it will rain tomorrow.
Or, schematically and more generally: from ‘P ⊃ Q’ and ‘P’, it is legitimate to conclude that ‘Q’.

10. An example: conditional elimination
We call this rule conditional elimination, and we can codify it like this: P ⊃ Q P Q ⊃E
This just says, that from anything of the form P ⊃ Q and anything of the form P, we may conclude that Q.
⊃E is just an abbreviation for ‘conditional elimination’.

11. Licensing
We say that this rule licenses the inference from P ⊃ Q and P to Q, or that the conclusion of Q on the grounds of P ⊃ Q and P is licensed by the rule.
Notice that we can check with a truth table that if P ⊃ Q and P are both true, then Q must be true. That’s why this is a valid rule of inference – if the premises are true, it is impossible for the conclusion to be false.

12. Truth preservation
We describe this property of the rule by saying that it is truth preserving. That is, it preserves the truth of the premises in the conclusion. If you apply the rule to true premises, you are guaranteed to get a true conclusion.
All of the rules of inference that we will use in natural deduction are truth preserving. This is why natural deduction is a test of validity – if a conclusion follows from some premises in natural deduction, then it is impossible for those premises to be true and the conclusion false.

13. An example: conditional elimination
Ok, back to our example. We’ve now got our conditional elimination rule. How do we apply the rule?
The idea is simple. First we write out all the premises. Then we draw a line under them. And then we apply any rule of inference we want to yield things that follow from the premises, and write those under the line.

14. An example: conditional elimination
Like this:
1 P P 2 Q P 3 P ⊃ R P 4 Q ⊃ S P
5 R 1, 3 ⊃ E 6 S 2, 4, ⊃ E

15. An example: conditional elimination
This is a natural deduction proof of the following argument: P, Q, P ⊃ R, Q ⊃ S therefore R and S or
P Q P ⊃ R Q ⊃ S R S

16. An example: conditional elimination
Let’s break this down a bit.
1 P P 2 Q P 3 P ⊃ R P 4 Q ⊃ S P
5 R 1, 3, ⊃ E 6 S 2, 4, ⊃ E
Remember to number every line of the proof!

17. An example: conditional elimination
Then list all your premises, draw a line, and then write anything you are inferring from the premises underneath.
1 P P 2 Q P 3 P ⊃ R P 4 Q ⊃ S P 5 R 1, 3, ⊃ E 6 S 2, 4, ⊃ E

18. An example: conditional elimination
1 P P 2 Q P 3 P ⊃ R P 4 Q ⊃ S P 5 R 1, 3, ⊃ E 6 S 2, 4, ⊃ E
The ‘P’ indicates that these lines of the proof are premises.
These are very important labels. They tell you how the conclusions were reached – by applying which rules to which premises.

19. An example: conditional elimination
1 P P 2 Q P 3 P ⊃ R P 4 Q ⊃ S P 5 R 1, 3, ⊃ E 6 S 2, 4, ⊃ E
E.g. this tells us that R was inferred from premise 1 (P) and premise 3 (P → R) in accordance with the conditional elimination rule.

20. An example: conditional elimination
1 P P 2 Q P 3 P ⊃ R P 4 Q ⊃ S P 5 R 1, 3, ⊃ E 6 S 2, 4, ⊃ E
This line down here just tells us that we’re at the end of the derivation.

21. An example: conditional elimination
1 P P 2 Q P 3 P ⊃ R P 4 Q ⊃ S P 5 R 1, 3, ⊃ E 6 S 2, 4, ⊃ E
And this line here is called the scope line. We’ll worry about this more later on – for now on, just remember that this is its name.

22. Disjunction We have some similar rules for disjunction:
Introduction: X X ∨ Y ∨I
Elimination: X ∨ Y ~X Y ∨E

23. Disjunction And the rationale is pretty obvious!
Let’s go through it together now.

24. Exercise Provide a derivation for the following in natural deduction:
(P ∧ ~Q) ⊃ ~~~R (P ∧ ~Q)
~~~R