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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
slide12
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
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