 ### Level 6: Proof By Natural Deduction

Level 6.1: The Basics
Level 6.4: The Biconditional
Level 6.5: The Exclusive Disjunction
Level 6.6: Transposition
Level 6.7: Material Implication
Level 6.8: De Morgan's Law
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# Level 6.1: The Basics

In Progress

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.

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. Others less familiar patterns will be introduced along the way.

#### Example 1

Let’s look at an example.

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

As the normal ordering of the disjunctive syllogism inference has been reversed, it may be a little difficult to see that we are indeed dealing with a disjunctive syllogism inference pattern here. So, here’s how the inference looks with the premises ordered in the more familiar way:

 4 (Q v R) 3 ~R 5 Q DS/3,4

#### Example 2

Now let’s look at another example:

Now let’s look at an example involving disjunctive syllogism and modus tollens (MT). Again, we will be start with three premises and finally end with a conclusion in step 5.

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

The first move is to put premises 1 and 3 together to form a disjunctive syllogism inference with the conclusion appearing in step 4:

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

Next, we run a modus tollens inference

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

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