week 1 - wednesday. course overview propositional logic truth tables and, or, not logical...
TRANSCRIPT
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
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 Dual
Commutative p q q p p q q p
Associative (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 p
Negation p ~p t p ~p c
Double Negative ~(~p) p
Idempotent p p p p p p
Universal Bound p t t p c c
De Morgan’s ~(p q) ~p ~q ~(p q) ~p ~q
Absorption p (p q) p p (p q) p
Negations of t and c
~t c ~c t
Back to Implications
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 qp 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 qT T T
T F F
F T F
F F 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