1. Week 2 - Friday

2.  What did we talk about last time?  Predicate logic  Negation  Multiple quantifiers

4.  There are two lengths of rope  Each one takes exactly one hour to burn completely  The ropes are not the same lengths as each other  Neither rope burns at a consistent speed (10% of a rope could take 90% of the burn time, etc.)  How can you burn the ropes to measure out exactly 45 minutes of time?

5.  When doing a negation, negate the predicate and change the universal quantifier to existential or vice versa  Formally:  ~(  x, P(x))   x, ~P(x)  ~(  x, P(x))   x, ~P(x)  Thus, the negation of "Every dragon breathes fire" is "There is one dragon that does not breathe fire"

6.  Argue the following:  "Every unicorn has five legs"  First, let's write the statement formally  Let U(x) be "x is a unicorn"  Let F(x) be "x has five legs"   x, U(x)  F(x)  Its negation is  x, ~(U(x)  F(x))  We can rewrite this as  x, U(x)  ~F(x)  Informally, this is "There is a unicorn which does not have five legs"  Clearly, this is false  If the negation is false, the statement must be true

7.  The previous slide gives an example of a statement which is vacuously true  When we talk about "all things" and there's nothing there, we can say anything we want

8.  Recall:  Statement: p  q  Contrapositive:~q  ~p  Converse:q  p  Inverse:~p  ~q  These can be extended to universal statements:  Statement:  x, P(x)  Q(x)  Contrapositive:  x, ~Q(x)  ~P(x)  Converse:  x, Q(x)  P(x)  Inverse:  x, ~P(x)  ~Q(x)  Similar properties relating a statement equating a statement to its contrapositive (but not to its converse and inverse) apply

9.  The ideas of necessary and sufficient are meaningful for universally quantified statements as well:   x, P(x) is a sufficient condition for Q(x) means  x, P(x)  Q(x)   x, P(x) is a necessary condition for Q(x) means  x, Q(x)  P(x)

11.  So far, we have not had too much trouble converting informal statements of predicate logic into formal statements and vice versa  Many statements with multiple quantifiers in formal statements can be ambiguous in English  Example:  “There is a person supervising every detail of the production process.”

12.  “There is a person supervising every detail of the production process.”  What are the two ways that this could be written formally?  Let D be the set of all details of the production process  Let P be the set of all people  Let S(x,y) mean “x supervises y”   y  D,  x  P such that S(x,y)   x  P,  y  D such that S(x,y)

13.  Intuitively, we imagine that corresponding “actions” happen in the same order as the quantifiers  The action for  x  A is something like, “pick any x from A you want”  Since a “for all” must work on everything, it doesn’t matter which you pick  The action for  y  B is something like, “find some y from B”  Since a “there exists” only needs one to work, you should try to find the one that matches

14.  Is the following statement true?  “For all blue items x, there is a green item y with the same shape.”  Write the statement formally.  Reverse the order of the quantifiers. Does its truth value change? a a c c g g b b d d f f i i k k e e h h j j

15.  Given the formal statements with multiple quantifiers for each of the following:  There is someone for everyone.  All roads lead to some city.  Someone in this class is smarter than everyone else.  There is no largest prime number.

16.  The rules don’t change  Simply switch every  to  and every  to   Then negate the predicate  Write the following formally:  “Every rose has a thorn”  Now, negate the formal version  Convert the formal version back to informal

17.  As show before, changing the order of quantifiers can change the truth of the whole statement  However, it does not necessarily  Furthermore, quantifiers of the same type are commutative:  You can reorder a sequence of  quantifiers however you want  The same goes for   Once they start overlapping, however, you can’t be sure anymore

19.  Quantification adds new features to an argument  The most fundamental is universal instantiation  If a property is true for everything in a domain (universal quantifier), it is true for any specific thing in the domain  Example:  All the party people in the place to be are throwing their hands in the air!  Julio is a party person in the place to be   Julio is throwing his hands in the air

20.  Formally,   x, P(x)  Q(x)  P(a) for some particular a Q(a)Q(a)  Example:  If any person disses Dr. Dre, he or she disses him or herself  Tammy disses Dr. Dre  Therefore, Tammy disses herself

21.  Much the same as universal modus ponens  Formally,   x, P(x)  Q(x)  ~Q(a) for some particular a   ~P(a)  Example:  Every true DJ can skratch  Ted Long can’t skratch  Therefore, Ted Long is not a true DJ

22.  Unsurprisingly, the inverse and the converse of universal conditional statements do not have the same truth value as the original  Thus, the following are not valid arguments:  If a person is a superhero, he or she can fly.  Astronaut John Blaha can fly.  Therefore, John Blaha is a superhero. FALLACY  A good man is hard to find.  Osama Bin Laden is not a good man.  Therefore, Osama Bin Laden is not hard to find. FALLACY

23.  We can test arguments using Venn diagrams  To do so, we draw diagrams for each premise and then try to combine the diagrams Touchable things This

24.  All integers are rational numbers  is not rational  Therefore, is not an integer rational numbers integers rational numbers integers

25.  All tigers are cats  Panthro is a cat  Therefore, Panthro is a tiger cats tigers cats Panthro cats tigers cats tigers Panthro

26.  Diagrams can be useful tools  However, they don’t offer the guarantees that pure logic does  Note that the previous slide makes the converse error unless you are very careful with your diagrams

28.  Proofs and counterexamples  Basic number theory

29.  Assignment 1 is due tonight at midnight  Read Chapter 4  Start on Assignment 2