CS322
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Week 2 - Friday CS322
Last time • What did we talk about last time? • Predicate logic • Negation • Multiple quantifiers
Logical warmup • 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?
Negating quantified statements • 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"
Negation example • 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
Vacuously true • 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
Conditionals • 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
Necessary and sufficient • 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)
Multiple quantifiers • 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.”
Example • “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)
Mechanics • 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
Tarski’s World Example a b • 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? c d e f g h i j k
Practice • 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.
Negating multiply quantified statements • 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
Changing quantifier order • 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
Quantification in arguments • 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
Universal modus ponens • Formally, • x, P(x) Q(x) • P(a) for some particular 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
Universal modus tollens • 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
Inverse and converse errors strike again • 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
Venn diagrams • 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
Diagrams showing validity rational numbers rational numbers • All integers are rational numbers • is not rational • Therefore, is not an integer integers integers rational numbers
Diagrams showing invalidity cats • All tigers are cats • Panthro is a cat • Therefore, Panthro is a tiger tigers cats Panthro cats cats tigers tigers Panthro Panthro
Be careful • 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
Next time… • Proofs and counterexamples • Basic number theory
Reminders • Assignment 1 is due tonight at midnight • Read Chapter 4 • Start on Assignment 2
