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### Natural Deduction

Proving Validity

The Basics of Deduction

- Argument forms are instances of deduction (true premises guarantee the truth of the conclusion).

p > q

p

q

p > q

p

p > q

___

q

Argument Forms as “Rules”

- A “rule” for deduction tells you what you may do, given the presence of certain kinds of statements or premises.

- Since all argument forms do just this (tell us what we may infer, given certain kinds of premises), they function as rules.

A Simple Example (p. 344, #3)

We use the first line to record the conclusion the “rule” lets us draw.

To do so correctly, we have to know what rule to follow – or what argument form this argument illustrates. Use the second line to record this.

1. R > D

2. E > R

_______ ____

2. E > R

1. R > D

E > D___ _HS_

Another Example (p. 344, #6)

Since our argument forms each have only two premises, look for the two that you can use; ignore the other.

~ J v P

~ J

S > J

_________ ____

~ J v P

~ J

S > J

_________ ____

~ J v P

~ J

S > J

~ S______ _MT_

Your Strategy for Finding Conclusions

- You are just looking for any two premises that follow the “rules” or argument forms; they don’t need to be near each other, or in the standard order.
- Use the “left side/right side” technique for dealing with tildes (statement variables stand for anything on either side of an operator). Example: p. 344, #13.

Multiple-Step Deductions

- You sometimes need to use more than one argument form (rule of implication) in a given argument to derive the conclusion.
- The strategy is to find the conclusion to be derived in one of the given premises, and “work backward” from there.
- More strategy suggestions are on p. 343 and 344

Example 1 – p. 346, #12

Our conclusion is ~ T

1. ~ M v ( B v ~ T)

2. B > W

3. ~ ~ M

4. ~ W / ~ T

1. ~ M v ( B v ~ T)

2. B > W

3. ~ ~ M

4. ~ W / ~ T

1. ~ M v ( B v ~ T)

2. B > W

3. ~ ~ M

4. ~ W / ~ T

If we could “isolate” (B v ~ T) from premise 1, and find a way to get ~ B, then we could deduce ~ T.

B v ~ T

~ B

~ T

p v q (q = ~ T)

~ p

q

Example 1, continued

(B v ~ T) is in a disjunctive statement. To “isolate” it, we need a disjunctive syllogism.

1. ~ M v ( B v ~ T)

2. B > W

3. ~ ~ M

4. ~ W / ~ T

1. ~ M v ( B v ~ T)

2. B > W

3. ~~ M

4. ~ W / ~ T

To get one disjunct, we need to have the negation of the other (DS).

Premise 3 gives us the negation of ~ M, in premise 1.

Example 1 – Partial Deduction

1. ~ M v ( B v ~ T)

2. B > W

3. ~ ~ M

4. ~ W / ~ T

5. B v ~ T 1, 3 DS

1. ~ M v ( B v ~ T)

2. B > W

3. ~ ~ M

4. ~ W / ~ T

Now we need to get ~ B, so we can deduce ~ T through another DS.

Example 1 – Full Deduction

Look at premises 2 and 4. Through MP, we can uses these premises to deduce ~ B.

- 1. ~ M v ( B v ~ T)
- 2. B > W
- 3. ~ ~ M
- 4. ~ W / ~ T
- B v ~ T 1, 3 DS
- ~ B2, 4 MT

- 1. ~ M v ( B v ~ T)
- 2. B > W
- 3. ~ ~ M
- 4. ~ W / ~ T
- B v ~ T 1, 3 DS
- ~ B 2, 4 MT
- ~ T 5, 6 DS

We are ready for our final step – using our previously established conclusions to derive the final, or main, conclusion through DS.

Multiple Step Deductions - Conclusions

- Use your knowledge of argument forms to apply the (first 4) rules of implication to show validity through deduction.
- Be flexible in finding premise pairs; they may be anywhere in the argument.
- Be stringent in your justifications; every line must show premise numbers and the rule through which you connect them to the derived statement.

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