Propositional Logic
Symbolic logic
AB
If A, then B
Capital letters stand for simple sentences
A simple sentence is an affirmative single statement
“It’s warm in here” : W
“Yesterday it was not raining”: compound sentence
Not R
~R
~ stands for “Not”
Capital letters symbolize simple sentences
Five additional symbols stand for other basic
operators that affect statements.
In English, these five are:
“not”
“and”
“or”
“if…then…”
“if and only if”
Truth-functional operators
don’t just connect statements to one another, but also
determine the truth value of compound statements
Clinton had bypass surgery:
C
Clinton didn’t have bypass surgery: ~C
True
False
For any sentence p, ~p has the opposite truth value
p
T
F
~p
F
T
Lower case p, q, r, s,… are variables
standing for any sentence, simple
or compound
“~”
the tilde represents Negation
“not”
“no”
“never”
“It is false that”
~J
John did not finish his work
It’s false that John finished his work
“ · ” the dot represents Conjunction
Conjoining: asserting two things as both the case:
“and”
“but”
“however”
“although”
“moreover”
“;”
G·S
Garfield and Snoopy are dogs.
Garfield’s a dog and so is Snoopy.
Garfield’s a dog, but Snoopy is too.
Apple and Ford both make computers
A·F
~(A · F)
It’s false that both Apple and Ford make computers
Not both companies make computers
Apple makes computers A
Honda makes computers H
A· H
A conjunction is True only if both
statements are True.
T·F
F
p
T
q
T
p·q
T
T
F
F
F
T
F
F
F
F
Truth-table definition
of the dot.
“v” the wedge represents Disjunction
A disjunction is an “either…or” statement.
“Either…or”
“…or…”
“…unless…”
Either Bill or Jane will get elected: B v J
Jane will win unless Bill does:
JvB
“Or” has a weak (inclusive) and a strong (exclusive) sense.
Weak “or” :
Jones is either getting a raise or a promotion
Strong “or”:
Bill is either 20 or 21.
“v” represents the weak sense of “or”
I’ll either buy a Civic or a PT Cruiser:
CvP
(C v P) · ~(C · P)
I’ll either buy a Civic or a PT Cruiser, but not both
A disjunction is False only if both statements are False
p q
T T
T F
F T
F F
p v q
T
T
T
F
Truth-table definition of “v”
~ ( A v B)
Neither apples nor bananas are vegetables
~(A · D) Adam and Dylan won’t both be late.
It’s false that Adam and Dylan will both be late.
“And” has some other meanings which are not
truth-functional, such as
You come near me again and I’ll call the police
If you come near me again, I’ll call the police.
“  ” the horseshoe represents Conditionals
“if…then…”
“…in case…”
“…given that…”
“…only if…”
C  P If you come near me again I’ll call the police
The car will start only if there’s gas in the tank: S  G
Kaine will win given that Warner campaigns for him.
WK
When is a material conditional statement true?
If you get an A on the final, you’ll get an A in the course
F  C
T
T
F  C
T
F
F  C
F
T
T
F
T
F  C
F
F
T
p
T
T
F
F
q
T
F
T
F
p  q
T
F
T
T
Truth-table definition of
the horseshoe
A conditional statement is false only
when the antecedent is True and
the consequent is False.
A Sufficient Condition is expressed as the antecedent
of a conditional.
A Necessary Condition is expressed as the consequent
of a conditional.
S > N
You have to have a College ID to get in to the show.
G>C
“  ” the triple bar represents “if and only if”
“If and only if” is called the biconditional: two conditionals
p if q
and p only if q
You are President of the USA iff you are Commander in Chief
of the Armed Forces
You are President if you are Commander in Chief
and
You are President only if you are Commander in Chief.
You are President if you are Commander in Chief
and
You are President only if you are Commander in Chief
You are President if you are Commander in Chief
CP
You are President only if you are Commander in Chief
PC
(C  P) • (P  C)
P  C
(P
T
T
F
F

T
F
T
T
C)
T
F
T
F
• (C  P)
T T T T
F F T T
F T F F
T F T F
P
T
T
F
F
 C
T T
F F
F T
T F
The truth table definition of the triple bar is the
equivalent of the truth table for a biconditional
The triple bar means “…is equivalent to…”.
A triple bar statement is True only if its two parts
have the same truth-value.
Truth-table definitions of the logical operators
p
T
T
F
F
p ~p
T F
F T
p
T
T
F
F

T
F
T
T
q
T
F
T
F
•
T
F
F
F
p
T
T
F
F
q
T
F
T
F
p
T
T
F
F

T
F
F
T
q
T
F
T
F
v
T
T
T
F
q
T
F
T
F
The Tilde reverses the value of the statement it affects.
A conjunction is True only if both conjuncts are true.
A disjunction is False only if both disjuncts are false.
A conditional is false only when the antecendent is True and the
consequent is False.
A biconditional is True when the two statements have the same
truth-value.
~[(A v B) • (D  E)]
negation
(A  C) v [(H  G)  (N v M)]
disjunction
[(F  B) v (K v G)] • ~ P
conjunction
[(A v B)  (~B  A)]  [(A • B) • (~B• ~A)]
biconditional
What kinds of statements are these?
What is the main operator?
What is wrong with the punctuation in each of these examples?
(A  C) v [(H ~ G)  (N v • M)]
~[(A v B) • (D  E)
[(A v B)  (~B  A)]  [(A • B) • ~B• ~A)]
[(F  B) v {(K v G)] • ~ P}
1. Cats are not friendly
~C
2. Frank’s a jerk and so is Paul
F.P
3. Either Al will buy dinner or
we’ll go hungry
AvH
4. If they get married,
they’re crazy
M>C
5. They’d have to be crazy to get married
6. If Bob doesn’t take his umbrella,
he’ll be sorry
M>C
~U > S
7. If Eve and Abel can spell, then it’s a sign of inferiority
(E . A) > I
8. Either Humpty Dumpty controls the meaning of all the
words he uses or else some of them are too expensive
and he just has to accept that.
C v (E . A)
9. It’s a most provoking thing if someone can’t tell a
cravate from a belt.
~T > P
10. Either you can’t tell that unbirthday presents are better,
or else you don’t understand basic arithmetic
~T v ~U
11. Either Poland and Holland were both invaded by Hitler or
else Switzerland was.
(P . H) v S
12. Poland was invaded by Hitler and either Holland or
Switzerland was too.
P . (H v S)
13. If you are President then you’re commander-in-chief,
and vice versa
(P > C) . (C > P)
14. They have neither croissants nor Danish today
~(C v D)
15. If they had croissants, I would not even want Danish as long
as they have coffee.
C > (O > ~D)
16. Danish and croissants are high in fat; if you want
something healthy, you should have a blueberry muffin
(D . C) . (H > B)