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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”.

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natural deduction

Natural Deduction

Proving Validity

the basics of deduction
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
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
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
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
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
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
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
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
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
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
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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