Transcript Logic Slides 2

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Logic Slides 2
Kinds of Arguments
PHIL 211
Cosmos to Citizen
Dr. Mike Miller
Mount St. Mary’s College
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There are really only two types of arguments:
Good and Bad
But before you can assess if an argument is good or bad, you
must first understand the two different kinds of good arguments:
Deductive and Non-Deductive
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A Deductive argument is one whose premise,
if true, provides conclusive evidence for the
truth of the conclusion.
This means that if we accept the truth of the premises of a properly formed
deductive argument (sometimes this requires a little imagination), it is impossible
for us to say the conclusion is false without admitting we are acting illogically.
Consider the following deductive argument:
All cats are mammals.
All mammals are animals.
Therefore, all cats are animals.
The argument is set-up in such a way that if we believe the premises (and that
shouldn’t be too hard because they are certainly true), then the conclusion
must also be true. This argument is classified as a ‘good’ deductive argument.
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Aristotle said that a person who refuses to accept the conclusion of
a good deductive argument as true (the type of argument where the
premises really do support the conclusion and the premises are
true), then that person is no better than a vegetable (that is, alive but not thinking)! Aristotle went on to say that it is impossible to
have a rational conversation with this kind of person.
Don’t be fooled, however, into thinking that every deductive argument is ‘good.’
There are plenty of ‘bad’ deductive arguments out there. Consider the following:
All animals are fish
All fish are mammals
Therefore, all animals are mammals.
Although this argument is ‘bad’ (because the premises are false) the premises
actually would support the conclusion if they were true. They are not true, but if
they were true they would provide conclusive evidence for the truth of the
conclusion. Therefore, this argument is deductive even though it is also a ‘bad’
argument.
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We will study 7 different ‘forms’ of deductive arguments later in this
presentation (6 of which are discussed in detail in Weston, Chapter 6):
• Categorical Syllogism (CS)
• Modus Ponens (MP)
• Modus Tollens (MT)
• Hypothetical Syllogism (HS)
• Disjunctive Syllogism (DS)
• Dilemma
• Reduction ad Absurdum
These different kinds of deductive arguments are called ‘forms’ because it is the
structure of the argument (how the premises and conclusion are written) and not
the content of the premises or conclusion that make the argument deductive.
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Now, let’s turn our attention to non-deductive arguments.
A Non-Deductive argument is one whose premises, if
true, provide reliable, although not conclusive,
evidence for the truth of the conclusion.
This means that in every non-deductive argument if we admit that the
premises are true (again this often takes an act of imagination) then the
most we can say about the conclusion is that is likely to be true.
Conclusions of non-deductive arguments are never absolutely certain,
even if they may be almost certain.
Non-deductive arguments usually appeal to some notion of similarity
between what has happened or been observed in the past and what will
happen in the future.
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Consider the following non-deductive argument:
Since the Mount softball team has won their last 19 games
this season (all against highly ranked opponents) and State
U has lost 14 straight, it is likely that the Mount will win
their game against State U tomorrow.
If the premises are true, the conclusion is very likely true. (If I were a betting
man I would certainly place a bet.) However, since it is always possible that a
highly favored team may lose a game they are expected to win the conclusion of
this argument does not follow with absolute certainty (even if the premises are
true). Therefore, the argument is non-deductive.
Please note that non-deductive arguments are not necessarily worse than deductive
arguments because their conclusions are never certain. They are just different.
Both deductive and non-deductive arguments are useful and have their advantages.
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Is the following argument deductive or non-deductive?:
If you don’t eat lunch you will collapse like a car without fuel.
This argument is non-deductive because the conclusion does not necessarily follow
even if the premise was true. Many people skip lunch and they don’t collapse.
The following is also a non-deductive argument:
Since no rational person believes in Santa Claus,
no rational person should believe in Jesus either.
The argument is non-deductive because even if the premise was true the
conclusion does not necessarily follow. That is, the premise does not
exclude the possibility of a rational person believing in Jesus because no
rational adult believes in Santa. (If you are interested, the argument is ‘bad’
because the difference between believing in Jesus and believing in Santa is
greater than any possible similarity.)
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There are two different types of non-deductive arguments:
Inductive:
Arguments about causes:
• Generalizations
•Agency (who did it?)
• Analogies
•Motivation (why it was done?)
• Arguments from Authorities
•The Cause of action or event
The next 17 slides will focus upon these non-deductive arguments (all
of which are discussed in detail in Weston, Chapters 2–5).
We will first discuss 3 different types of inductive arguments
(generalizations, analogies, and arguments from authority), and then
make some general comments concerning arguments about causes.
Generalization
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We generalize every day, arguing that what has happened before is likely to
happen again. For example,
The last two pies I bought from David’s Bakery were
fantastic. I bet the next one will be fantastic too.
You are generalizing if you make a conclusion about a group (the
population) from a claim about some part of it (the sample).
As we increase our experience our generalizations typically get better because we
have more examples from which to generalize (that is, we have a bigger sample).
Generalizations can be very good arguments. Surveys are common examples of
generalizations. Some surveys are outstanding. Others are very bad. The
difference often depends upon the similarity between the sample and the population.
If generalizations are made with faulty logic, they commit the fallacy of hasty
generalization.
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Imagine that you wanted to conduct a survey about what people in Emmitsburg
like to eat for dinner. To be a good generalization the survey must abide by the
following rules:
• Rule 8: Give more than one example
Generally, the larger the population, the more samples are needed. If
your survey makes a conclusion about what 2,000 people like to eat for
dinner, you cannot simply ask 6 people and draw a conclusion.
• Rule 9: Use Representative examples
Make sure that your samples actually represent the population. Even if
you ask 200 women (which could certainly be a large enough sample),
they are not representative of all the people in Emmitsburg.
• Rule 10: Background information is crucial
The generalization made must not be misleading or bias. Unfortunately
many surveys are. It takes a lot of work to make a good survey. Be
careful that a generalization doesn’t mislead you.
• Rule 11: Consider Counterexamples
Ask questions to help you determine if the conclusion needs to be
revised, limited, or given up entirely.
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Are These Good Generalizations?
Your sociology professor asks you to conduct a
survey about attitudes of students on campus
about sex before marriage. So, you ask 28 of
your friends if they think sex before marriage is a
great idea or not. 20 say “No” and 8 say “Yes.”
Is this a good survey?
Should the auto plant accept the batch of nuts and bolts
from its supplier? The inspector chooses 10 pairs from
the 20,000 items, inspects them under a micro-scope,
finds that all are acceptable, and passes the lot.
Was that a right thing to do?
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Be careful not to believe something just because you hear it many times from
many different sources. That is, be wary when somebody says, “I know a
person who . . .” Urban legends (sometimes known as the ‘person who’
fallacy) are everywhere, and almost always false. For example, have you
heard the following . . .
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Not sure if what you heard is an urban legend? Check out:
http://urbanlegends.about.com
Reasoning by Analogy
When you Reason by Analogy you argue
from a similarity between two cases.
Generally, an argument by analogy tries to argue that since two things (an
item or an event) are similar in many ways, the second thing discussed
must also have a certain characteristic that the first one has.
A argument involving an analogy typically sound like this:
Since objects A and B have properties a, b, c, . . . n and object A has
property x, we can infer that object B also has property x.
The Mayor did not give the city’s firefighters a raise last
year when they asked for one. Therefore, the Mayor should
not give the city’s sanitation workers a raise this year.
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Note that analogies are not always expressed as formal arguments,
but simply sketches of arguments. Since many things in analogies
are left unstated, you must often:
• Infer the conclusion.
• Come up with the similarities and/or the differences between
the things being compared.
• Determine what general principle is being applied to both
sides of the analogy. This principle is the ‘glue’ which the
person giving the analogy uses to unite both sides.
For an analogy to be ‘good’:
• All the premises must be true.
• Rule 12: The Analogy requires a relevantly similar
example. If there are more significant differences than
similarities between the two sides, the analogy is a bad one.
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Is the following a ‘good’ analogy?
Blaming soldiers for war is like blaming firemen for fires.
What is the unstated conclusion?:
What are the similarities?:
What are the differences?:
What’s the general principle at work here? (That is, what is the ‘glue’
that unites the two examples in the analogy?):
Is it a good analogy?
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You wouldn’t buy a kitten at a pet store to give to your
dog. Why, then, do you consider it acceptable to buy
white rats for your boa constrictor?
What is the unstated conclusion?:
What are the similarities?:
What are the differences?:
What’s the general principle at work here? (That is, what is the ‘glue’
that unites the two examples in the analogy?):
Is it a good analogy?
Arguments from Authority
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No one is an expert on everything. Sometimes we must rely on the expertise of
other people, organizations or referenced works to tell us what we need to know.
Here’s a typical argument from Authority:
Dr. Renpher, a two time winner of the Noble Prize for
biology, says that the corpus luteum secretes progesterone.
Therefore, the corpus luteum does secrete progesterone.
However, as you know, you should not trust everything that everyone says.
Little Johnny, the six-year-old kid down the street, says the Japanese
economy will recover in three months only if their banking system
no longer leverages pre-tax buyouts on foreign debt.
Maybe he is right, but do you think it reasonable for a six-year-old be an expert of
foreign monetary policy? I would trust what a six-year-old has to say about dinosaurs
or Dr. Seuss books, but not technical and abstract material – unless (by some unusual
circumstance) there was some reason to think he actually is an expert.
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A ‘good’ argument from authority must follow these rules:
Rule 13: Sources should be cited
A detailed claim is more likely to be believed if a reference is given. If
you are unsure if the fact is true, you can look it up yourself. Unnamed
sources are generally not to be trusted.
Rule 14: Seek informed sources
Sources must be qualified about the fact being discussed. You should
question the veracity of claims made by someone if you have no
evidence to think that they are experts in that field. Also, beware of
anyone making claims to know what cannot be known (such as what
Princess Diana was thinking just before her car crashed in Paris).
Rule 15: Seek impartial sources
Those that have something to gain or lose in a dispute are generally not
the best sources of information. Impartial sources of information are
usually the best.
Rule 16: Cross check sources
When experts disagree, look for other authorities to back-up the claim.
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Would you accept the following arguments? Why or why not?
Your mother: You can get AIDS by touching someone with AIDS.
Is your Mom an authority about medicine? What have
you learned about AIDS? Is she right?
Friend: My uncle in San Francisco says that President Carter failed second
grade. I wouldn’t accept this claim, unless I knew your uncle was a
presidential historian.
55 year-old salesman: I think this car is the coolest car on the
planet! Everyone in the dorm is going to love it.
The salesman is probably biased because he wants to make a sale.
And is it likely that he knows what a college student would love?
The Lancet Journal (April 4, 2003): Women who use oral contraceptives have
a 60% greater chance for cervical cancer than those who do not.
The Lancet is a respectable journal with peer reviewed articles. I would believe it.
Don’t forget your own authority. Your experience counts too.
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There is a difference between attacking what a person says and the character of
the person speaking. The two are not the same. Since you can’t assume that bad
people always give bad arguments, the following rule applies:
Rule 17: Personal attacks do not disqualify a source
That is, when considering the quality of an argument or a claim, the personal
quality of the person speaking makes no difference! Consider the following:
John’s argument to put a stop light at 5th and Main Street can’t be
right because John is a real jerk! Just yesterday he hit an old lady!
Don’t make this mistake (called an ad hominem fallacy). John’s argument might be
a good one, even if John really is a jerk. You must consider John’s argument and
John’s character separately.
Note: Good people do not necessarily make good arguments either. However, character
does matter in a court of law, as when a witness has been convicted of a felony. Why is
this an exception to the general rule?
Arguments about Causes
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Arguments about cause and effect are very common. In these arguments you are
looking for the correlation between two events or kinds of events. For example . . .
The facts: Spot barked. Sean woke up.
Is it fair to say that Spot’s barking woke Sean up?
Effective arguments about causes:
Rule 18: Explain how cause leads to effect
Good arguments not only show the correlation between events A and B, they
explain why A caused B. Obviously, the cause must precede the effect and take
place close to the effect in space and time.
Rule 19: Propose the most likely cause
The most likely causes are ones that fit with well-established beliefs. The best
arguments are those where it is impossible for the cause to happen and the effect
not happen, given normal conditions.
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Be certain not to forget the following rules when making arguments about causes.
If someone breaks any one of the following rules they are making a mistake in
reasoning called a false cause fallacy:
Rule 20: Correlated events are not necessarily related
I won the lottery because I wore my lucky socks.
Really? Putting on the socks may have happened before you won the lottery, but
how did the socks cause you to win?
Rule 21: Correlated events may have a common cause
Cleo is irritable because she can’t sleep properly.
Well, maybe Cleo is irritable and unable to sleep because she
drinks 6 cups of espresso every day.
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Rule 22: Either of two correlated events may cause the other
Sitting close to the TV will give you bad eyesight.
Whoever made this brief argument has probably reversed the cause and the effect.
I think it more likely that having bad eyesight leads one to sit too close to the TV.
Rule 23: Causes may be complex
If people really want the help those who are starving in
the world today, they should try to eat less themselves.
Then there would be more left over for everyone else.
Well, less consumption of food would help alleviate world hunger, but it
certainly couldn’t end it alone. The problem with global poverty is simply
too complex to be solved by one solution.
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We are now near the end of our discussion about non-deductive arguments
(arguments by example, arguments by analogy, arguments from authority, and
arguments about causes).
In each of the argument types the premises, if true, provided reliable, although not
conclusive, evidence for the truth of the conclusion.
No one can be absolutely sure that non-deductive arguments provide a certain
truth because the content of the arguments does not allow such confidence. (The
content of an argument is what the argument is about, whether it be a neighbor’s
barking dog, the outlook for the Japanese economy, or the safety of 20,000 bolts.)
Think about it, how can anyone be certain that things that have not yet happened
will happen, or that things that are related in some ways are actually related in
additional ways? And, how can we be sure that even an unbiased expert might
not make a mistake? Or for that matter, can we be confident that what appears to
cause something to happen actually did so – if we didn’t clearly see it with our
own eyes?
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The content of non-deductive arguments simply can’t give us absolutely
certain truth. But this does not mean that non-deductive arguments are not any
good. In fact, we use non-deductive arguments all the time (we would be hard
pressed to make it through the day without doing so). And even though the
conclusions of non-deductive arguments are never certain, some nondeductive arguments provide conclusions that are very, very, very likely to be
true. For instance, would anyone doubt this inductive argument?:
The sun has risen for over 15 billions years.
Therefore, the sun will rise tomorrow.
I trust you remember that deductive arguments are different. For in their case
the premises, if true, provide conclusive evidence for the truth of the
conclusion. This means that if the content of non-deductive arguments makes
them what they are, then deductive arguments are made by their ‘form.’
Deductive Arguments
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The form of a deductive argument is its structure, without regard to its content.
As a matter of fact, you can recognize deductive arguments by ‘stripping’ them
of all their content. To do this, you ignore what the argument is about, and look
how it is structured.
For example, take a look at the following three arguments.
All Dogs are Mammals
All Pick-ups are trucks
All Zobots are Quizars
All Mammals are Animals
All trucks are motor vehicles
All Quizars are Venmores
Therefore, all Dogs are Animals
Therefore, all pick-ups are motor vehicles
Therefore, all Zobots are Venmores
All three arguments have the same form, even though they are about
different things. Do you recognize this form?
All A are B
All B are C
Therefore, all A are C
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The type of deductive argument we just looked at is called a Categorical
Syllogism (CS). The CS form has two premises and one conclusion, each
of which include the qualifiers ‘some,’ ‘no’ and/or ‘all.’ For instance, the
following are all examples of Categorical Syllogisms:
Some dogs are mean.
No mean things are loved.
Therefore, all dogs are not loved.
All cars are blue.
Some things in the Smithsonian Museum are blue.
Therefore, some cars are in the Smithsonian Museum.
No Gigowitz is Reggotin. All Reggotin things are Peffin. Therefore, some Gigowitz are Peffin.
Some US presidents have been born in Nevada.
Some US presidents have the middle name of ‘Herbert’
Therefore, some Presidents born in Nevada have the middle name of Herbert.
As it turns out, all of the arguments above are ‘bad’ (either the premises do not
actually support the conclusion, or at least one premise is false), but even then
all of the arguments above are categorical syllogisms. Do you understand why?
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Since the form and not the content of all deductive arguments makes the arguments
deductive, logicians often use symbols to make the form of deductive arguments
more clear. That is, using symbols help focus on the form and not the content.
In deductive arguments letters often take the place of words (any letter will do, as
long as a unique letter takes the place of one term every time it is used in the
argument). For example:
All Apples are Fruit
All Alaskans are named Fred
All A are F
All Fruits are Delicious
All Fred’s are Dead
All F are D
Therefore, all Apples are Delicious
So, all Alaskans are Dead
Thus, all A are D
When you are trying to recognize the type of deductive argument pay attention to
the form, and not the content. So, each of the three arguments above has the CS
form.
In other deductive arguments letters can also take the place of sentences. For
example, ‘If you hit a grand slam, then we will win the game’ becomes ‘if H, then
W’ (or ‘if S, then G’ or ‘if P, then Q’ – the letters don’t matter if you are consistent).
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It is important to understand that valid does not mean true. That is, an argument
may be valid but its conclusion false. For example, the following categorical
syllogism is valid – that is, if the premises were true then the conclusion must be
true as well – but the argument’s premises are obviously false. Thus, we should
not accept the conclusion is true because of the argument:
All cats are brown. All brown things are amphibian.
Therefore, all cats are amphibian.
We will next discuss 6 different forms of deductive arguments in the following slides.
Please note that the following 6 forms discussed in this set of slides (unlike the
categorical syllogism just discussed) are always valid – meaning that if the premises
are true the conclusion must be true too!
Now, once again, the following 6 deductive forms are always valid. That does not
mean, however, that you should always consider their conclusions true. It just means
that if the premises are true, then the conclusion also has to be true.
This is an important point. For if you give a valid deductive argument and can prove
your premises to be true, then the person to whom you are giving your argument must
accept your conclusion as true! Now that is power!
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A deductive argument that is easy to recognize is called Modus Ponens (Latin
for ‘method of putting’). It has the following form:
If P then Q. P. Therefore, Q (where P and Q stand for any proposition).
Here are two modus ponens (MP):
The mayor said that if it rains the town picnic will be cancelled.
It is raining. Therefore, the picnic is cancelled.
If Ohio State football is on television, then Grandpa
is watching the game at home. Ohio State is on
television, so we can find Grandpa at home.
Can you write each in its symbolic form?
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Modus Tollens (meaning ‘the method of taking’) is somewhat similar to MP.
However, MT has the following form:
If P, then Q. Not-Q. Then not-P
Don’t be confused by ‘not-Q’ or ‘not-P.’ It just means the logical opposite of
whatever Q and P stand for. For example:
Q: I can ride a bike
not-Q: I cannot ride a bike
I like Brad Pitt
I hate Brad Pitt
I don’t enjoy rollercoasters.
I love rollercoasters (Do you understand
why not-Q has a ‘positive sound’ to it here?)
Here are two examples of MT:
If Biff wins the election, then he will throw a party at his house tonight. There is
no party at Biff’s house tonight. He must not have won the election.
If it is raining, then the street will be wet. The street is dry. So,
it must not be raining.
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A Hypothetical Syllogism (HS) has the following form:
If P, then Q
If Q, then R
Therefore, if P, then R
For instance,
If Gabby wins the lottery, she will want to pay back her
debts. If she wants to pay back her debts, she will give me
the $1,000 she owes me. So, if Gabby wins the lottery, I’ll
get my $1,000 back.
Hypothetical syllogisms can have any number of premises, as long as each has
the form ‘if P, then Q.’ For instance: If P, then Q. If Q, then R. If R, then S. If
S, then T. If T, then U. Therefore, if P, then U.
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The Disjunctive Syllogism (DS) has the following form:
P or Q
Not-P
Therefore, Q
Here are a couple of examples:
Either he is alive or dead.
He’s not alive.
Therefore, he’s dead.
Either Jesus is a madman, or he is
God.
Either the British must come by land or
by sea.
Jesus is not a madman.
They did not come by sea.
Therefore, Jesus is God
Therefore, they must come by land.
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Dilemma are also deductive arguments. Traditionally, dilemmas involve
choices between two bad consequences. In logic, the consequences can both
be bad, good or indifferent. Dilemma have the following form:
P or Q
If P, then R.
If Q, then S.
Therefore, R or S.
Either I stay up and read philosophy, or I go to sleep.
If I read philosophy, I’ll learn something useful.
If I go to sleep, I’ll feel better in the morning.
Therefore, either I’ll learn something useful or I’ll
feel better in the morning.
This argument doesn’t tell you what to do (other arguments are needed
to help you there), but it does make clear the consequences of your
choices.
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The last deductive argument we will examine is called Reductio ad Absurdum
(or, a reduction to the absurd). Those that use this argument form well prove
their point by showing that the exact opposite of what they want to prove leads
to a contradiction (or an absurd conclusion).
Imagine my brother thinks that I’m harming my 4 year-old daughter by not letting
her watch movies like Star Wars and Harry Potter. He thinks kids should be
aware that life is often dangerous and hard, and if you keep them too protected
they will grow up unprepared for the reality of life. I want to prove him wrong.
I would start my argument by assuming that I should let my daughter watch Star
Wars because it will ‘toughen her up.’ If true, I probably should let her watch
Rocky and Jaws as well. Maybe I should sit her down to watch The Godfather
while I’m at it. And if I really wanted to prepare my daughter for life, I should not
feed her for a day, or make her walk home from preschool once in a while. But
these things would be absurd. So, I won’t let her watch movies like Star Wars.
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Be assured, deductive and non-deductive arguments are often
more complex than the ones we have looked at. However, once
you recognize how the arguments work, you should be able to
recognize them no matter how complex they might be.
Keep your eyes and ears open and you will find examples of the
arguments we just examined all over the place. Keep a special
lookout for examples of these arguments in our class readings.
In this set of slides we discussed several mistakes in reasoning
(called fallacies). In the 3rd set of slides we will discuss a number
of other common fallacies.
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Please contact me with any questions about the
information in these slides or the related assigned reading:
• Weston, Chapters II – VI
• Logic Handout, p. 4-5