CS322 Week 2 - Friday
Dec 31, 2015
CS322Week 2 - Friday
Last time
What did we talk about last time? Predicate logic
Negation Multiple quantifiers
Questions?
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
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
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
c
g
b
d
f
i
k
e
h
j
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
Arguments with Quantified Statements
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
All integers are rational numbers
is not rational
Therefore, is not an integer
rational numbers
integers
2
rational numbers
2
2
rational numbers
integers
2
Diagrams showing invalidity
All tigers are cats
Panthro is a cat
Therefore, Panthro is a tiger
cats
tigers
cats
Panthro
cats
tigers
cats
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
Upcoming
Next time…
Proofs and counterexamples Basic number theory
Reminders
Assignment 1 is due tonight at midnight
Read Chapter 4 Start on Assignment 2