Chapter 8.A-C: Natural Deduction
Natural deduction is a method for establishing the validity of symbolized
arguments. It is simpler and more enlightened than truth tables. This
method, though, cannot prove invalidity, only validity. We must use truth
tables to prove invalidity. The way this deduction works is that the
conclusion of an argument is actually derived through a series of steps,
analogous to proofs in geometry. Each step depends on a rule of inference,
or an argument form. Any argument that is a substitution instance of any of
these forms is a valid argument. There are two types of rules of inference:
implication rules and replacement rules. We will have a total of 18 forms.
We’ll learn and use the first 4 implication rules today.
1. Modus ponens (MP)
pq
p
q
Example: If it rained today, then the street is wet. It rained today. Therefore,
the street is wet.
Look at the examples on page 357.
2. Modus tollens (MT)
pq
~q
~p
Example: If it rained today, then the street is wet. The street is not wet.
Therefore, it did not rain today.
Look at the examples on pages 358-359.
3. Hypothetical syllogism (HS)
pq
qr
pr
Example: If I live in Atlanta, then I live in Georgia. If I live in Georgia, then
I live in the United States. Therefore, if I live in Atlanta, then I live in the
United States.
Look at the examples on page 359-360.
4. Disjunctive syllogism (DS)
pvq
~p
q
Example: Either CDs are superior to records, or DVDs are superior to film.
CDs are not superior to records. Therefore, DVDs are superior to film.
Look at the examples on page 360.
We will be using these rules of inference to show that a particular
conclusion can be validly derived from a given set of assumptions
(premises). These four rules are sufficient to derive conclusions from many
simple arguments. (Once we have all 18, we can do any valid argument).
To begin, we symbolize the argument:
If the Astros win the playoff, then the Braves will win the pennant. If the
Astros do not win the playoff, then either Connolly or Davis will be fired.
The Braves will not lose the pennant. Furthermore, Connolly will not be
fired. Therefore, Davis will be fired.
Number the premises. Write the conclusion to the right of the last premise.
1. A  B
2. ~ A  (C v D)
3. ~ B
4. ~ C
/D
Another example:
1. ~R
2. P  S
3. R v ~S
4. ~P  Q
/Q
Look at the “Strategies and Tactics” on page 369:
1. R  (P  Q)
2. ~R  ~L
3. (P  Q)  S
4. ~S
5. L v (M  N)
/M  N
1. F  G
2. F v H
3. ~G
4. H  (G  I)
/FI
1. ~(A  B) v [~(E  F)  (C  D)]
2. ~ ~ (A  B)
3. ~ (E  F)
4. D  G
/CG
1. S  T
2. T  U
3. R  S
/RU
From the exercises
#4
1. S  (P  Q)
2. (P  Q)  R
3. ~R
/ ~S
# 14
1. ~ R v (P  Q)
2. (P  Q)  (Q  ~R)
3. ~ ~R
/~P
# 18
1. ( Q v R)  ~ P
2. ~ P  [P v (Q  P])
3. Q v R
/R
#20
1. (P  S)  ~Q
2. P  R
3. (P  R)  (R  Q)
4. (P  Q)  (R  S)
/~P
Try this one!
1. ~N  [(B  D)  (N v ~E)]
2. (B  E)  ~N
3. B  D
4. D  E
/ ~D