Logic

LOGIC

Statements

• Logic is the tool for reasoning about the truth or falsity of statements.– Propositional logic is the study of

Boolean functions– Predicate logic is the study of

quantified Boolean functions (first order predicate logic)

Arithmetic vs. Logic

Arithmetic Logic

0 false

1 true

Boolean variable statement variable

form of function statement form

value of function truth value of statement

equality of function equivalence of statements

Notation

Word Symboland vor wimplies 6equivalent ]not ~not 5parentheses ( ) used for grouping

terms

Notation Examples

English Symbolic

A and B A v B

A or B A w B

A implies B A 6 B

A is equivalent to B A ] Bnot A ~A

not A 5A

Statement Forms

• (p v q) and (q v p) are different as statement forms. They look different.

• (p v q) and (q v p) are logically equivalent. They have the same truth table.

• A statement form that represents the constant 1 function is called a tautology. It is true for all truth values of the statement variables.

• A statement form that represents the constant 0 function is called a contradiction. It is false for all truth values of the statement variables.

Truth Tables - NOT

P 5P

T F

F T

Truth Tables - AND

P Q PvQ

T T T

T F F

F T F

F F F

Truth Tables - OR

P Q PwQ

T T T

T F T

F T T

F F F

Truth Tables - EQUIVALENT

P Q P]QT T T

T F F

F T F

F F T

Truth Tables - IMPLICATION

P Q P6Q

T T T

T F F

F T T

F F T

Truth Tables - Example

P true means rain

P false means no rain

Q true means clouds

Q false means no clouds

Truth Tables - Example

P Q P6Q P6Q

rain clouds rainclouds T

rain no clouds rainno clouds F

no rain clouds no rainclouds T

no rain no clouds no rainno clouds T

Algebraic rules for statement forms• Associative rules:

(p v q) v r ] p v (q v r) (p w q) w r ] p w (q w r)

• Distributive rules:p v (q w r) ] (p v q) w (p v r) p w (q v r) ] (p w q) v (p w r)

• Idempotent rules:p v p ] p p w p ] p

Rules (continued)

• Double Negation:55p ] p

• DeMorgan’s Rules:5(p v q) ] 5p w 5q5(p w q) ] 5p v 5q

• Commutative Rules:p v q ] q v pp w q ] q w p

Rules (continued)• Absorption Rules:

p w (p v q) ] p p v (p w q) ] p • Bound Rules:

p v 0 ] 0p v 1 ] pp w 0 ] pp w 1 ] 1

• Negation Rules:p v 5p ] 0p w 5p ] 1

A Simple Tautology

P Q is the same as 5Q 5PThis is the same as asking: PQ ] 5Q 5PHow can we prove this true?By creating a truth table!

P QT TT FF TF F

A Simple Tautology (continued)

Add appropriate columns

P Q 5P 5Q

T T F F

T F F T

F T T F

F F T T


Remember your implication table!

P Q 5P 5Q PQ

T T F F T

T F F T F

F T T F T

F F T T T


Remember your implication table!

P Q 5P 5Q PQ 5Q5P

T T F F T T

T F F T F F

F T T F T T

F F T T T T


Remember your implication table!P Q 5P 5Q PQ 5Q5P PQ ] 5Q5P

T T F F T T T

T F F T F F T

F T T F T T T

F F T T T T T

Since the last column is all true, then the original statement:

PQ ] 5Q5P is a tautology

Note: 5Q5P is the contrapositive of PQ

Translation of English

If P then Q: PQ

P only if Q: 5Q5P or

PQ

P if and only if Q: P ] Qalso written as P iff Q

Translation of English

P is sufficient for Q: PQ

P is necessary for Q: 5P5Q or QP

P is necessary and sufficient for Q: P ] Q

P unless Q: 5QP or 5PQ

Predicate Logic

• Consider the statement: x2 > 1• Is it true or false?• Depends upon the value of x!• What values can x take on (what is the

domain of x)?• Express this as a function: S(x) = x2 > 1• Suppose the domain is the set of reals.• The codomain (range) of S is a set of

statements that are either true or false.

Example

• S(0.9) = 0.92 > 1 is a false statement!• S(3.2) = 3.22 > 1 is a true statement!

• The function S is an example of a predicate.

• A predicate is any function whose codomain is a set of statements that are either true or false.

Note

• The codomain is a set of statements• The codomain is not the truth value of the

statements• The domain of predicate is arbitrary• Different predicates can have different domains• The truth set of a predicate S with domain D is

the set of the x ε D for which S(x) is true:{x ε D | S(x) is true}

• Or simply: {x | S(x)}

Quantifiers

• The phrase “for all” is called a universal quantifier and is symbolically written as œ

• The phrase “for some” is called an existential quantifier and is written as ›

Notations for set of numbers:

Reals Integers

Rationals Primes

Naturals (nonnegative integers)

Goldbach’s conjecture

• Every even number greater than or equal to 4 can be written as the sum of two primes

• Express it as a quantified predicate• It is unknown whether or not Goldbach’s

conjecture is true. You are only asked to make the assertion

• Another example: Every sufficiently large odd number is the sum of three primes.

Negating Quantifiers

• Let D be a set and let P(x) be a predicate that is defined for x ε D, then

5(œ(x ε D), P(x)) ] (›(x ε D), 5P(x))

and

5(›(x ε D), P(x)) ] (œ(x ε D), 5P(x))

Technology

Logic