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Week 1 - Wednesday. CS322. 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

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Last time
Last time

  • Course overview

  • Propositional logic

    • Truth tables

    • AND, OR, NOT

    • Logical equivalence



Logical warmup
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
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
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
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




What are you implying
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


Cs322
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
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
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
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
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 pq

  • This is called the biconditional and has this truth table:

  • What is the biconditional logically equivalent to?



Arguments1
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
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
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
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
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
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
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
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
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
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
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
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 circuits1
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
Common gates

  • The following gates have the same function as the logical operators with the same names:

  • NOT gate:

  • AND gate:

  • OR gate:


Example
Example

  • Draw the digital logic circuit corresponding to:

    (p ~q)  ~(p  r)

  • What’s the corresponding truth table?



Next time
Next time…

  • Predicate logic

    • Universal quantifier

    • Existential quantifier


Reminders
Reminders

  • Read Chapter 3