1. Natural Deduction:
Using simple valid argument forms –as demonstrated by
truth-tables—as rules of inference. A rule of inference
is a rule stating that whenever premises of certain forms
occur, conclusions of a certain form follow necessarily.

2. If it’s sunny we’ll be able to go to the mountains
for a hike, but if it rains we’ll have to stay
home and watch tv all day.
Weatherunderground says there’s only
a 10% chance of rain tomorrow.
What follows?
We’ll be able to go to the mountains
for a hike

3. 1. V v F 2. ~V / F Either apples are vegetables or they’re fruits.
But they’re not vegetables. So…
They are fruits
1. V v F
2. ~V / F
DS, 1,2

4. Either apples and bananas are both
vegetables or else they are both fruit.
But it’s false that Apples and Bananas
are vegetables, so they must be fruit.
1. (A ∙ B) v (F ∙R)
2. ~(A ∙ B) / F ∙ R
How many lines will this truth table require?

5. Focus on the main operator of the statements
p . q
p > q
(A v B) . (C v D)
__ > __
1 > 2
(S v B) > ~W
((W . Y) v X) . (H > B)
[(~K > P) . N] > (A . C)
[F . (G v B)] . (J > I)
Focus on the main operator of the statements
to be clear on what kind of statement each is.

6. Modus Ponens MP
p > q p
----- q
(B v N) > M
(B v N) M
A > C A
------ C
(L > N) > (B > V)
L > N
(H . (J v N)) > (N . L)
H . (J v N)
N . L
B > V

7. Modus Tollens MT
p > q
~ q
-----
~ p
F > H ~H
----- ~F
~(J v L)
(M . B) > (J v L)
~( M . B)
[G v (B . O)] > [(D . V) . R]
~[(D . V) . R]
~[G v (B . O)]

8. Disjunctive Syllogism DS
p v q ~p
----- q
(K v L) v O
~(K v L)
------ O
(A v (B . D)) ~A
-----
B . D
[(T v Y) > (L . D)] v {[(F . T) v O] > W}
~ [(T v Y) > (L . D)
------
[(F . T) v O] > W

9. Hypothetical Syllogism HS
p > q
q > r
p > r
(N . B) > G
G > ( L v ~E)
(N . B) > (L v ~E)
M > (B > G)
(B > G) > (F . T)
~V > [(O . ~C) > P]
(K v L) > ~V
M > (F . T)
(K v L) > [(O . ~C) > P]

10. Simplification SM
p . q
------ p
(B v N) . (C > L)
-----
B v N
A . (B >C)
------ A

11. Conjunction CN p q
---
p . q
(M . N) v W
D > K
---
((M . N) v W) . (D > K)

12. p > q
p / q MP
p > q
~q / ~ p MT
p v q
~ p / q DS
p > q
q > r / p > r HS
A > B
~A > (C v D) ~B
~C / D
5. ~A
1,3 MT
6. C v D
2, 5 MP
7. D
4,6 DS

13. MP MT DS HS
p > q p > q p v q p > q
p / q ~q / ~p ~ p / q q > r / p > r
E > (K > L)
F > (L > M)
G v E ~G
F / K >M
6. E
3, 4 DS
7. K > L
1, 6 MP
8. L > M
2, 5 MP
7,8 HS
9. K > M

14. MP MT DS HS
p > q p > q p v q p > q
p / q ~q / ~p ~ p / q q > r / p > r
J > (K > L)
L v J
~L / ~K J
K > L ~K
2,3 DS
1, 4 MP
5, 3 MT

15. MP MT DS HS
p > q p > q p v q p > q
p / q ~q / ~p ~ p / q q > r / p > r
1. ~ (S  T) > (~P > Q)
2. (S  T) > P
3. ~P / Q
4. ~ (S  T)
5. ~P > Q
6. Q
2,3 MT
1, 4 MP
3,5 MP

16. p > q p > q p v q p > q
MP MT DS HS
p > q p > q p v q p > q
p / q ~q / ~p ~ p / q q > r / p > r
H > [~E > (C > ~D)]
~D > E
E v H
~E / ~C H
6. ~E > (C > ~D)
7. C > ~D
8. C > E
9. ~C
3,4 DS
xxx
5,1 MP
4, 6 MP
7, 2 HS
4, 8 MT

17. Addition AD p
---
p v q
Constructive Dilemma
(p > q) . (r > s)
p v r
-----
q v s
T v F T
A conjunction of conditionals, plus
the disjunction of their antecedents
yields the disjunction of their consequents.

18. Rules of inference (8)
MP p > q / p // q
MT p > q / ~q // ~p
HS p > q / q > r // p > r
DS p v q / ~p // q
SM p . q // p
CN p / q // p. q
AD p // p v q
CD (p > q) . (r > s) / p v r // q v s

19. Rules of inference (mt, mp, ds, etc.) are “one way” rules.
Rules of equivalence are “two way” rules, allowing
substitution of a statement form for an equivalent
statement form.
Rules of equivalence are written using “::” to indicate two expressions are equivalent to one another.

20. Double Negation
p :: ~ ~p
Two tildes can be added or
deleted from any statement with
no effect on the truth-value.
1.A > ~ (B . C)
2. B . C
~ ~ ( H v K)
~ H
3. ~ ~ (B . C) 2 DN
4. ~A ,3 MT
3. H v K 1 DN
4. K ,3 DS

21. ~ (p v q) ::
~ p . ~ q
“neither”
“not this and not that”
~ (p . q) ::
~ p v ~ q
“not both”
“either not this or else not that”
DM DeMorgan’s Theorem

22. The order of statements around a dot or wedge is of no consequence
CM Commutation
(p . q) : : (q . p)
(p v q) : : (q v p)
The order of statements around a
dot or wedge is of no consequence
to the truth-value of the statement
1. (J > N) v (C v D)
2. ~C
3. ~D / J >N
4. ~ C . ~ D CONJ 2,3,
5. ~ (C v D) DM 4
6. J > N COMM, DS 1,5
6. (C v D) v (J >N) COMM, 1
7. J > N DS 5, 6

23. statements around dots and wedges is of no
Association AS
[ p . (q . r)] : : [ (p . q) . r]
[ p v (q v r)] : : [ (p v q) v r]
The grouping of simple
statements around dots
and wedges is of no
consequence for truth-values
1. A . (B . C) / C
2. (A . B) . C AS 1
3. C . (A . B) CM 2
4. C SM 3