UNIT 2 MODULE 2
Analyzing Premises, Forming Conclusions
First, we define a Trivial Valid Conclusion
Fact: No matter how poorly formulated an argument may be, it is always possible to form a valid conclusion by merely restating one of the premises and calling it the conclusion. Such a conclusion is called a Trivial Valid Conclusion.
Examples of arguments having Trivial Valid Conclusions:
It is raining.
Therefore, it is raining.
My feet hurt and I'm having a bad hair day.
Therefore, my feet hurt.
If I work hard, then I will succeed.
I succeeded.
Therefore, I succeeded.
In the work that follows, we will analyze collections of premises in order to recognize whether valid conclusions are possible. In all cases, we will exclude trivial conclusions.
We begin this module with a summary of important results observed in Unit 2, Module 1.
SUMMARY: SOME COMMON PATTERNS OF VALID REASONING
Direct Reasoning
Contrapositive Reasoning
Disjunctive Syllogisms
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Transitive Reasoning
SUMMARY: SOME COMMON PATTERNS OF INVALID REASONING
Fallacy of the Converse
Fallacy of the Inverse
Disjunctive Fallacies
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False Chains
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These patterns of reasoning are especially useful for exercises like the following example:
EXAMPLE 2.2.1
Select the statement that is a valid conclusion from the following premises,
if a valid conclusion is warranted.
I use my computer or I don't get anything done.
I get something done.
A. I use my computer.
B. I don't use my computer.
C. I use an abacus.
D. None of these is warranted.
EXAMPLE 2.2.1 Solution
This argument problem differs from the earlier examples in
that we aren't given a conclusion for the argument.
This means that if we try to use a truth table to analyze the argument,
we may not be sure what statement to use in the "conclusion" column.
However, it is easily solved by referring to the patterns of reasoning summarized above.
We symbolize the premises:
Let p be the statement "I use my computer." Let q be the statement "
I don't get anything done." Then the symbolic representation for the two premises
has this form:
Now we observe that this is the premise arrangement for one form of Disjunctive Syllogism,
which is a form of valid reasoning. The pattern tells us that
we can form a valid conclusion:
This means that a valid conclusion is warranted, namely "I
use my computer," which is choice A.
EXAMPLE 2.2.2
Select the statement that is a valid conclusion from the following premises,
if a valid conclusion is warranted.
If I win the Lotto, then I'll reform my life. I reformed my life.
A. I didn't win the Lotto.
B. I'll run for President as the Reform Party nominee.
C. I won the Lotto.
D. None of these is warranted.
See solution
EXAMPLE 2.2.3
Select the statement that is a valid conclusion from the following premises,
if a valid conclusion is warranted.
If we strive, then we excel. We didn't strive.
A. We excelled.
B. We didn't excel.
C. We didn't inhale.
D. None of these is warranted.
See solution
EXAMPLE 2.2.4
Select the statement that is a valid conclusion from the following premises,
if a valid conclusion is warranted.
If we win, then we celebrate. We aren't celebrating.
A. We won.
B. We didn't win.
C. We stink.
D. None of these is warranted.
See solution
EXAMPLE 2.2.5
Given:
i. If my car doesn't start, then I'll be late for work; and
ii. I'm not late for work.
select the statement that is a valid conclusion,
if a valid conclusion is warranted.
A. My car started.
B. I rode the bus.
C. I'm late for work.
D. None of these is warranted.
See solution
EXAMPLE 2.2.6
Given:
i. All nurses are kind; and
ii. Florence isn't a nurse.
select the statement that is a valid conclusion,
if a valid conclusion is warranted.
A. Florence is a city in Italy.
B. Florence isn't kind.
C. Florence is kind.
D. None of these is warranted.
See solution
EXAMPLE 2.2.7
Given:
i. No kittens are fierce; and
ii. Fluffy isn't fierce.
select the statement that is a valid conclusion,
if a valid conclusion is warranted.
A. Fluffy is a kitten.
B. Fluffy has fleas.
C. Fluffy isn't a kitten.
D. None of these is warranted.
See solution
EXAMPLE 2.2.8
Select the statement that is a valid conclusion from the following premises,
if a valid conclusion is warranted.
If you want a better grade, then you bring an apple for the teacher.
If you bring an apple for the teacher, then you expose the teacher to dangerous
agricultural chemicals.
A. If you expose the teacher to dangerous agricultural chemicals,
then you want a better grade.
B. If you don't expose the teacher to dangerous agricultural chemicals,
then you don't want a better grade.
C. You want a better grade.
D. None of these is warranted.
See solution
EXAMPLE 2.2.9
Select the statement that is a valid conclusion from the following premises,
if a valid conclusion is warranted.
All people who get many tickets are uninsurable.
All careless drivers get many tickets.
All people who are uninsurable have bad credit ratings.
A. All careless drivers have bad credit ratings.
B. If your car is repossessed because you have bed credit, then you are a car-less driver.
C. All people are uninsurable get many tickets.
D. None of these is warranted.
See solution
The previous two examples illustrate the following procedures that can be employed in order to use Transitive Reasoning to form a conclusion.
TO FORM A VALID CONCLUSION:
1. We may replace any premises, or the conclusion, with equivalent statements.
In particular, conditional statements may be replaced with their contrapositives
(but not with converses or inverses).
2. We may rearrange the order in which the premises are listed.
In particular, in order to use Transitive Reasoning, we will rearrange them premises
so that the antecedent of the first premise is a variable that appears only one time
in the entire premise scheme. We will then continue rearranging the order of the
premises, and perhaps replacing premises with equivalent statements, so that
the antecedent of each premise is exactly the same as the consequent of
the preceding premise.
If at any point it is impossible to continue this linkage of premises, then the argument involves a false chain,
and so it is not possible to form a valid conclusion that uses every premise
(although it may be possible to form valid conclusions that use only a subset of the original set of premises).
EXAMPLE 2.2.10
Select the statement that is a valid conclusion from the following premises,
if a valid conclusion is warranted.
If I win the Lotto, then I won't need a job.
If I have lots of bills, then I will need a job.
A. If I have lots of bills, then I didn't win the Lotto.
B. If I didn't win then Lotto, then I have lots of bills.
C. If I don't need a job, then I won the Lotto.
D. A valid conclusion is not warranted.
See solution
EXAMPLE 2.2.11
Select the statement that is a
valid conclusion from the following premises,
if a valid conclusion is warranted.
If I invest wisely, then I won't lose my money.
If I don't invest wisely, then I buy junk bonds.
If I read Investor's Weekly, then I won't buy junk bonds.
A. If I invest wisely, then I read Investor's Weekly.
B. If I buy junk bonds, then I don't invest wisely.
C. If I lose my money, then I don't read Investor's Weekly.
D. If I eat junk food, then I invest weakly.
E. None of these is warranted.
See solution
EXAMPLE 2.2.12
Select the statement that is a valid conclusion from the following premises,
if a valid conclusion is warranted.
All poodles love noodles. All pooches love noodles.
A. All pooches are poodles.
B. All poodles are pooches.
C. No poodles are pooches.
D. None of these is warranted.
EXAMPLE 2.2.13
Select the statement that is a valid conclusion from the following premises,
if a valid conclusion is warranted.
All compact cars are uncomfortable.
All compact cars get good gas mileage.
A. No compact cars get good gas mileage.
B. All compact cars are cheap.
C. All cars that get good gas mileage are uncomfortable.
D. None of these is warranted.
See solution
EXAMPLE 2.2.14
Select the statement that is a valid conclusion from the following premises,
if a valid conclusion is warranted.
People who dislike cats are degenerates.
All pirates own parrots.
People who like cats never own parrots.
A. If you are a pirate, then you are a degenerate.
B. All degenerates own parrots.
C. All cats lick parrots.
D. None of these is warranted.
See solution
EXAMPLE 2.2.15
Select the statement that is a valid conclusion from the following premises,
if a valid conclusion is warranted.
No body builders are weak.
All professional wrestlers are body builders.
All 300-pound men with bleached hair and sequined tights are professional wrestlers.
Plato is weak.
A. Plato is a 300-pound man with bleached hair and sequined tights.
B. Plato is not a 300-pound man with bleached hair and sequined tights.
C. Plato is the Masked Warrior and hits people with chairs.
D. None of these is a valid conclusion.
See solution
EXAMPLE 2.2.16
Select the statement that is a valid conclusion from the following premises,
if a valid conclusion is warranted.
Sylvestor isn't a parakeet.
Elephants never squawk.
All parakeets squawk.
No elephants are tiny.
A. Sylvester is an elephant.
B. Sylvestor isn't tiny.
C. All parakeets are tiny.
D. None of these is warranted.
See solution
EXAMPLE 2.2.17
Select the statement that is a valid conclusion from the following premises,
if a valid conclusion is warrented.
If you aren't a good stirrer, then you aren't handy with a swizzle stick.
If you are a graduate of Billy Bob's Big Bold School of Mixology, then you are a bartender.
No good stirrers have weak wrist muscles.
If you don't have weak wrist muscles, then you have a firm handshake.
All bartenders are handy with a swizzle stick.
A. If you are a graduate of Billy Bob's Big Bold School of Mixology, then you don't have a firm handshake.
B. If you don't have a firm handshake, then you aren't a graduate of Billy Bob's Big Bold School of Mixology.
C. If you have a firm handshake, then you are a graduate of Billy Bob's Big Bold School of Mixology.
D. None of these is warranted.
See solution
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