CS322

CS322Week 1 - Wednesday

Last time

Course overview Propositional logic

Truth tables AND, OR, NOT Logical equivalence

Questions?

Logical warmup

You come to a fork in the road Two men stand beneath a sign that

reads: Ask for the way, but waste not your

breath One road is freedom, the other is death Just one of the pair will lead you aright For one is a Knave, the other a Knight

What single yes or no question can you ask to determine which fork to take?

De Morgan’s Laws

What’s an expression that logically equivalent to ~(p q) ?

What about logically equivalent to ~(p q) ?

De Morgan’s Laws state: ~(p q) ~p ~q ~(p q) ~p ~q Essentially, the negation flips an AND to

an OR and vice versa

What are you implying? You can construct all possible outputs using

combinations of AND, OR, and NOT But, sometimes it’s useful to introduce

notation for common operations This truth table is for p q

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

If… We use to represent an if-then statement Let p be “The moon is made of green

cheese” Let q be “The earth is made of rye bread” Thus, p q is how a logician would write:

If the moon is made of green cheese, then the earth is made of rye bread

Here, p is called the hypothesis and q is called the conclusion

What other combination of p and q is logically equivalent to p q ?

Why is implication used that way? p q is true when:

p is true and q is true p is false

Why? For the whole implication to be true, the

conclusion must always be true when the hypothesis is true

If the hypothesis is false, it doesn’t matter what the conclusion is

“If I punch the tooth fairy in the face, I will be Emperor of the World”

What’s the negation of an implication?

Contrapositive

Given a conditional statement p q, its contrapositive is ~q ~p

Conditional:“If a murderer cuts off my head, then I will

be dead.” Contrapositive:

“If I am not dead, then a murderer did not cut off my head.”

What’s the relationship between a conditional and its contrapositive?

Converse and inverse

Given a conditional statement p q: Its converse is q p Its inverse is ~p ~q Consider the statement:

“If angry ham sandwiches explode, George Clooney will become immortal.”

What is its converse? What is its inverse? How are they related?

Biconditional Sometimes people say “if and only if”, as in:

“A number is prime if and only if it is divisible only by itself and 1.”

This can be written p iff q or p q This is called the biconditional and has this truth

table:

What is the biconditional logically equivalent to?

p q p q

T T TT F FF T FF F T

Tautologies and Contradictions

Tautology

A tautology is something that is true no matter what

Examples: T p ~p p p

The final column in a truth table for a tautology is all true values

The book sometimes writes a statement which is a tautology as a t

Contradiction

A contradiction is something that is false no matter what

Examples: F p ~p ~(p p)

The final column in a truth table for a contradiction is all false values

The book sometimes writes a statement which is a contradiction as a c

Laws of Boolean algebraName Law DualCommutative p q q p p q q pAssociative (p q) r p (q r) (p q) r p (q r)Distributive p (q r) (p q) (p r) p (q r) (p q) (p r)Identity p t p p c pNegation p ~p t p ~p cDouble Negative ~(~p) p Idempotent p p p p p pUniversal Bound p t t p c cDe Morgan’s ~(p q) ~p ~q ~(p q) ~p ~q

Absorption p (p q) p p (p q) pNegations of t and c

~t c ~c t

Arguments

Arguments

An argument is a list of statements (called premises) followed by a single statement (called a conclusion)

Whenever all of the premises are true, the conclusion must also be true, in order to make the argument valid

Examples

Are the following arguments valid? p q ~r (premise) q p r (premise) p q (conclusion)

p (q r) (premise) ~r (premise) p q (conclusion)

Common argument tools Modus ponens is a valid argument of the following

form: p q p q

Modus tollens is a contrapositive reworking of the argument, which is also valid: p q ~q ~p

Give verbal examples of each We call these short valid arguments rules of

inference

Generalization

The following are also valid rules of inference: p p q

q p q

English example: “If pigs can fly, then pigs can fly or swans can breakdance.”

Specialization

The following are also valid rules of inference: p q p

p q q

English example: “If the beat is out of control and the bassline just won’t stop, then the beat is out of control.”

Conjunction

The following is also a valid rule of inference: p q p q

English example: “If the beat is out of control and the bassline just won’t stop, then the beat is out of control and the bassline just won’t stop.”

Elimination The following are also valid rules of inference:

p q ~q p

p q ~p q

English example: “If you’re playing it cool or I’m maxing and relaxing, and you’re not playing it cool, then I’m maxing and relaxing.”

Transitivity

The following is also a valid rule of inference: p q q r p r

English example: “If you call my mom ugly I will call my brother, and if I call my brother he will beat you up, then if you call my mom ugly my brother will beat you up.”

Division into cases The following is also a valid rule of

inference: p q p r q r r

English example: “If am fat or sassy, and being fat implies that I will give you trouble, and being sassy implies that I will give you trouble, then I will give you trouble.”

Contradiction Rule

The following is also a valid rule of inference: ~p c p

English example: “If my water is at absolute zero then the universe does not exist, thus my water must not be at absolute zero.”

Fallacies

A fallacy is an argument that is not valid

It could mean that the conclusion is not true in only a single case in the truth table

But, if the conclusion is ever false whenever all the premises are true, the argument is a fallacy

Most arguments presented by politicians are fallacies for one reason or another

Common fallacies

Converse error If Joe sings a sad song, then Joe will

make it better. Joes makes it better. Conclusion: Joe sings a sad song.

FALLACY Inverse error

If you eat too much, you will get sick. You are not eating too much. Conclusion: You will not get sick.

FALLACY

Digital Logic Circuits

Digital logic circuits

Digital logic circuits are the foundation of all computer hardware

Circuits are built out of components called gates

A gate has one or more inputs and an output

Gates model Boolean operations Usually, in digital logic, we use a 1

for true and a 0 for false

Common gates The following gates have the same function as

the logical operators with the same names:

NOT gate:

AND gate:

OR gate:

Example

Draw the digital logic circuit corresponding to:

(p ~q) ~(p r)

What’s the corresponding truth table?

Upcoming

Next time…

Predicate logic Universal quantifier Existential quantifier

Reminders

Read Chapter 3