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Natural Deduction - PowerPoint PPT Presentation

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”.

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Natural Deduction

Proving Validity

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

p > q

p

q

p > q

p

p > q

___

q

• 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.

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_

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_

• 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.

• 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

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

(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.

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.

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.

• 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.