Integrated Math III
Name: _______________________
Logic and Reasoning #2
Translate each argument into symbolic form. Create a logical chain of events to establish that the argument is valid or
invalid.
Example:
Given these known facts, can you correctly reason to the claim?
Known facts:
If all wealthy people are happy, then money can buy happiness.
If money can buy happiness, then love does not exist.
Given:
Love exists.
Claim:
Therefore, not all wealthy people are happy.
First translate into symbols:
Let
p = all wealthy people are happy
q = money can buy happiness
r = love exists
So, in symbols the given information is written as:
(1)
(2)
(3)
(4)
p q
q ~ r
r
 ~p
Can we use these statements to form a logical chain of events and therefore establish “proof” of the claim?
Reorder the statements to create the chain. The conclusion of each statement must become the hypothesis of the following statement.
Remember that the contrapositive of any statement is equivalent and therefore can replace the statement.
~ q ~ p
(from #3)
(contrapositive of #2)
(contrapositive of #1)
~p
(from #4)
r
r ~ q
This is a logical chain and forms a valid argument.
It does prove the claim!
Translate each problem into symbolic form. What is the natural claim? Is this claim based on valid or invalid reasoning?
Example:
Given these known facts, what is the natural claim?
Can you correctly reason to this claim?
Known facts:
If Merlo the Magnificent performs magic on Wednesday, then he will receive his royalty check from the Magician’s
Guild.
Merlo did not perform magic on Wednesday.
Claim?
Most people would claim:
Merlo will not receive his royalty check from the Magician’s Guild.
Is this a valid claim?
First translate into symbols:
Let
p = Merlo performs magic on Wednesday
q = He receives his royalty check from the Magician’s Guild
So, in symbols the given information is written as:
(1)
(2)
p q
~p
(3)
 ~q
Can we use these statements to form a logical chain of events and therefore establish “proof” of the claim?
Reorder the statements to create the chain. The conclusion of each statement must become the hypothesis of the following statement.
Remember that the contrapositive of any statement is equivalent and therefore can replace the statement. The converse and inverse are
NOT equivalent and must NOT replace the statement.
~p
~ p ~ q
(from #2)
(inverse of #2)
 ~q
(from #3)
This chain is based on poor logic (inverse error!) and does NOT form a valid argument.
It does NOT prove the claim!
Problems:
Translate each problem into symbolic form. Given the information, what can you conclude (if anything)?
1.
If the sun rises in the west, then morning shadows point east.
Given: Morning shadows point east.
2.
If Sherlock Holmes gathers all of the clues, then he will be victorious.
Given: Sherlock Holmes gathers all of the clues.
3.
If Linda takes the bus, then she will be late for her job interview.
Given: Linda does not take the bus.
4.
If Hemlock Bones decodes the secret message, then Agent Otto will not be captured.
Given: Agent Otto will be captured.
5.
If Felicia goes out with David, then she will have a good time
Saturday night.
Given: Felicia will not have a good time Saturday night.
Translate each argument into symbolic form. Create a logical chain of events to establish that you can or cannot conclude
the information based upon the given and the logic chain.
1.
If you listen to both sides of a story, then you will make an intelligent decision. If you are open-minded, then you will listen to
both sides of the story.
Given: You did not make an intelligent decision.
Therefore, you are not open-minded.
2.
If Evette is innocent, then Alfa is telling the truth. If Romeo is telling the truth, then Alfa is not. If Romeo is not telling the
truth, then he has something to gain.
Given: Romeo has nothing to gain.
Therefore, Evette is not innocent.
3.
If the bill passes, then the new highway will not be built. The park can be used if the new highway is not built.
Given: The park can’t be used.
Therefore, the bill will not pass.