Propositional Logic Symbolic logic

If A, then B A 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 ~ stands for “Not”

~R

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 True

Clinton didn’t have bypass surgery: ~C False

For any sentence p, ~p has the opposite truth value

p ~pT FF 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 workIt’s false that John finished his work

“ · ” the dot represents Conjunction

G · S Garfield and Snoopy are dogs.Garfield’s a dog and so is Snoopy.Garfield’s a dog, but Snoopy is too.

Conjoining: asserting two things as both the case: “and”“but”“however”“although”“moreover”“ ; ”

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 AHonda makes computers H

A · H

T · F

F

A conjunction is True only if bothstatements are True.

p q p · qT T T

T F F

F T FF 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: J v B

“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:

C v P

(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 p v q T T TT F T

F T TF F 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 F C F C F C

T T T F F T F F

T F T T

p q p qT T TT F FF T TF F T

Truth-table definition ofthe 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 andYou 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 C) • (C P) P CT 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.

Truth-table definitions of the logical operators

p • q T T TT F FF F TF F F

p v q T T T T T F F T T F F F

p ~pT FF 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

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)]

(A C) v [(H G) (N v M)]

[(F B) v (K v G)] • ~ P

negation

disjunction

conjunction

[(A v B) (~B A)] [(A • B) • (~B• ~A)]

biconditional

What kinds of statements are these? What is the main operator?

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

A v H

4. If they get married, they’re crazy

M > C

5. They’d have to be crazy to get married M > C

6. If Bob doesn’t take his umbrella, he’ll be sorry

~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 > P10. 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 orelse 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 wantsomething healthy, you should have a blueberry muffin

(D . C) . (H > B)