Propositional Logic

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Propositional Logic
Symbolic logic
If A, then B
A ? B
Capital letters stand for simple sentences
A simple sentence is an affirmative single
statement
Its warm in here W
Yesterday it was not raining compound sentence
Not R
stands for Not
R
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Capital letters symbolize simple sentences
Five additional symbols stand for other basic
operators that affect statements.
In English, these five are not and or
ifthen if and only if
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Truth-functional operators
dont just connect statements to one another, but
also determine the truth value of compound
statements
True
Clinton had bypass surgery C
Clinton didnt have bypass surgery C
False
For any sentence p, p has the opposite truth
value
p p T F F T
Lower case p, q, r, s, are variables standing
for any sentence, simple or compound
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the tilde represents Negation
not no never It is false that
J John did not finish his work Its
false that John finished his work
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the dot represents Conjunction
Conjoining asserting two things as both the
case and but however although moreover

G S Garfield and Snoopy are
dogs. Garfields a dog and so is
Snoopy. Garfields a dog, but Snoopy is too.
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Apple and Ford both make computers A F
(A F)
Its false that both Apple and Ford make computers
Not both companies make computers
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Apple makes computers A Honda makes computers
H
A H T F
A conjunction is True only if both statements are
True.
p q p q T T T
F
Truth-table definition of the dot.
T F F
F T F
F F F
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v the wedge represents Disjunction
A disjunction is an eitheror statement.
Eitheror or unless
Either Bill or Jane will get elected B v J
Jane will win unless Bill does J v B
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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.
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v represents the weak sense of or
Ill either buy a Civic or a PT Cruiser C v P
(C v P) (C P)
Ill either buy a Civic or a PT Cruiser, but not
both
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A disjunction is False only if both statements
are False
p q p v q T T T
Truth-table definition of v
T F T
F T T
F F F
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( A v B)
Neither apples nor bananas are vegetables
(A D)
Adam and Dylan wont both be late. Its 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 Ill call the police
If you come near me again, Ill call the police.
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? the horseshoe represents Conditionals
ifthen in case given that only if
C ? P If you come near me again Ill call the
police
The car will start only if theres gas in the
tank S ? G
Kaine will win given that Warner campaigns for
him. W ? K
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When is a material conditional statement true?
If you get an A on the final, youll get an A in
the course F ? C F ? C
F ? C F ? C

T T T F F
T F F
T
F
T
T
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p q p ? q T T T T F
F F T T F F T
Truth-table definition of the horseshoe
A conditional statement is false only when the
antecedent is True and the consequent is False.
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A Sufficient Condition is expressed as the
antecedent of a conditional.
A Necessary Condition is expressed as the
consequent of a conditional.
S gt N
You have to have a College ID to get in to the
show.
G gt C
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? 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.
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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
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(P ? C) (C ? P)
P ? C
T T T T T T T
T T T
T F F F F T T
T F F
F T T F T F F
F F T
F T F T F T F
F 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.
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Truth-table definitions of the logical operators
p v q T T T T T F F T T F
F F
p q T T T T F F F F T F F F
p p T F F T
p ? q T T T T F F F T T F T F
p ? q T T T T F F F F T F T F
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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.
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negation
(A v B) (D ? E)
(A ? C) v (H ? G) ? (N v M)
disjunction
conjunction
(F ? B) v (K v G) P
(A v B) ? (B ? A) ? (A B) (B A)
biconditional
What kinds of statements are these? What is the
main operator?
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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
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C
1. Cats are not friendly
F . P
2. Franks a jerk and so is Paul
3. Either Al will buy dinner or well go
hungry
A v H
4. If they get married, theyre crazy
M gt C
5. Theyd have to be crazy to get married
M gt C
6. If Bob doesnt take his umbrella, hell be
sorry
U gt S
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7. If Eve and Abel can spell, then its a sign of
inferiority
(E . A) gt 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. Its a most provoking thing if someone cant
tell a cravate from a belt.
T gt P
10. Either you cant tell that unbirthday
presents are better, or else you dont
understand basic arithmetic
T v U
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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 youre
commander-in-chief, and vice versa
(P gt C) . (C gt P)
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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 gt (O gt D)
16. Danish and croissants are high in fat if you
want something healthy, you should have a
blueberry muffin
(D . C) . (H gt B)
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