Week 1 - Wednesday

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

 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?

 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

 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 pq p  qp  q TTT TFF FTT FFT

 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 ?

 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?

 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?

 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?

 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? pq p  qp  q TTT TFF FTF FFT

 A tautology is something that is true no matter what  Examples: TT  p  ~p 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

 A contradiction is something that is false no matter what  Examples: F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

NameLawDual Commutative p  q  q  pp  q  q  pp  q  q  pp  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  pp  c  p Negation p  ~p  tp  ~p  c Double Negative ~(~p)  p Idempotent p  p  pp  p  pp  p  pp  p  p Universal Bound p  t  tp  t  tp  c  c De Morgan’s ~(p  q)  ~p  ~q~(p  q)  ~p  ~q Absorption p  (p  q)  pp  (p  q)  p Negations of t and c ~t  c~t  c~c  t~c  t

 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

 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)

 Modus ponens is a valid argument of the following form: p  qp  q pp  q q  Modus tollens is a contrapositive reworking of the argument, which is also valid: p  qp  q ~q~q  ~p ~p  Give verbal examples of each  We call these short valid arguments rules of inference

 The following are also valid rules of inference: pp  p q p q qq  p q p q  English example: “If pigs can fly, then pigs can fly or swans can breakdance.”

 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.”

 The following is also a valid rule of inference: pp q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.”

 The following are also valid rules of inference:  p  q ~q~q  p p ~p~p  q 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.”

 The following is also a valid rule of inference:  p  q q  r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.”

 The following is also a valid rule of inference:  p  q  p  r q  rq  r  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.”

 The following is also a valid rule of inference:  ~p  c  p 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.”

 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

 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 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

 The following gates have the same function as the logical operators with the same names:  NOT gate:  AND gate:  OR gate:

 Draw the digital logic circuit corresponding to: (p  ~q)  ~(p  r)  What’s the corresponding truth table?

 Predicate logic  Universal quantifier  Existential quantifier

 Read Chapter 3