Level 6: Proof By Natural Deduction

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Estimated Learning Time = 9 hours

What is natural deduction, and why should I learn it?

In this course we will master yet another method of demonstrating the validity of arguments. Of all the methods of proof introduced on this website this method is the closest to the one employed in intuitive reasoning, hence its name: Natural Deduction. Let’s look at an example.

1 (P → (Q v R)) A/1
2 P A/2
3 ~R A/3
5 Q

In this example, we start with three premises (or “assumptions”) and eventually arrive at a conclusion in step 5. To get to our conclusion we will use two familiar inference patterns: modus ponens and disjunctive syllogism. The first move is to put together premises 1 and 2 to arrive at step 4, “(Q v R)”, as follows (the colour coding should help you see that we are indeed dealing with a modus ponens inference):

1 (P → (Q v R)) A/1
2 P A/2
3 ~R A/3
4  (Q v R)  MP/1,2
5 Q

The output or conclusion of the inference is included in the middle column as step 4, while in the right-hand column we indicate the steps from which the conclusion is derived (steps 1 & 2) and the inference pattern via which the conclusion has been derived (“MP” for modus ponens). To complete our proof we simply need to indicate how our final conclusion, “Q”, is derived. Here’s how that looks:

1 (P → (Q v R)) A/1
2 P A/2
3 ~R A/3
4 (Q v R)  MP/1,2
5 Q  DS/3,4

The method of natural deduction involves moving step by obviously valid step from a premise or set of premises to a conclusion. Some of the inference patterns/rules that will be used are already familiar from the previous courses. Other less familiar patterns will be introduced along the way.

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