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### CS322

Last time

- Course overview
- Propositional logic
- Truth tables
- AND, OR, NOT
- Logical equivalence

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

Tautology

- A tautology is something that is true no matter what
- Examples:
- T
- p ~p
- pp

- 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
- ~(pp)

- 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

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 pq

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, pq 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 pq ?

Why is implication used that way?

- pq 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 pq, 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 pq:
- Its converse is qp
- 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 piffq or pq
- This is called the biconditional and has this truth table:
- What is the biconditional logically equivalent to?

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?
- pq ~r (premise)
- qp r (premise)
- pq (conclusion)
- p (q r)(premise)
- ~r (premise)
- pq (conclusion)

Common argument tools

- Modus ponens is a valid argument of the following form:
- pq
- p
- q

- Modus tollens is a contrapositive reworking of the argument, which is also valid:
- pq
- ~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 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?

Next time…

- Predicate logic
- Universal quantifier
- Existential quantifier

Reminders

- Read Chapter 3

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