Ch03 basic logical_concepts

Lecture Notes © 2008 McGraw Hill Higher Education 1

Critical Thinking

Chapter 3

Basic Logical Concepts


What We Are Concerned With:

In evaluating any argument, one should always ask two key questions: Are the premises true? Do the premises provide good reasons to accept

the conclusion? We will only be concerned with the latter

question in chapter 3. For any given argument, the answer to the

latter question has nothing to do with the answer to the former question.


Example:

Take this argument: 1. Premise 1: If the moon is made of green cheese then you will

score perfectly on the next exam. 2. Premise 2: The moon is made of green cheese. 3. Conclusion: Therefore, you will score perfectly on the next exam.

Even though premises 1 & 2 are false, they still provide good reason to accept the conclusion. Why?

Because, if they were true, the conclusion would have to be true (i.e., If a green cheese moon really did ensure that you ace the

next exam, and it really was green cheese, then you really would ace the next exam.)

So, you will never be able to show that premises don’t “provide good reason” for a conclusion by pointing out that they are false.

Save “truth evaluation” for chapter 8.


Deduction vs. Induction

Deductive Arguments try to prove their conclusions with rigorous, inescapable logic. Example:

1. All humans are mortal.2. Socrates is a human.3. Therefore, Socrates is mortal.

Inductive Arguments try to show that their conclusion are plausible (likely or probable), given their premises:

1. So far, every class, the professor has worn a tie. 2. Therefore, next class, the professor will wear a

tie.


Avoid a Misconception

Misconception: Deductive arguments go from the general to the specific/particular-- Inductive argument go from the specific/particular to the general.

This is wrong!! They can do that:

Deductive: All Males are mortal. (general) I am Male. Therefore I am mortal. (particular)

Inductive: The last two winter days were cold. (particular) Therefore, all winter days are cold. (general)

But they can also do the exact opposite: Deductive: Lincoln was president from 1861-1865 (particular);

therefore everyone born while he was president was born in the 1800’s. (general)

Inductive: I have got A’s in all my classes so far (general). Therefore, I will get an A in this class (particular).


Telling the difference between Deductive and Inductive Arguments

Indicator Words: Deductive: certainly, definitely, this entails that,

conclusively Inductive: probably, likely, one would expect, odds

are, reasonable to assumeLike before, indicators are not perfect. They are not always present, and they can sometimes be misleading. (e.g., The speaker may say “it certainly follows” but be exaggerating, knowing that it only “probably follows.”)



The Strict Necessity Test: If it would make a good (valid) deductive argument, assume it is. If it

wouldn’t, assume it isn’t. (We will learn more about how to tell later.) Exceptions to the Strict Necessity Test

Sometimes, even though the premises don’t deductively support the conclusion, it is clear that the speaker of the argument intended them to.

Maybe they say “it necessarily follows” or “it is guaranteed” Maybe the context shows that they meant it to be deductive.

When that happens, call it deductive. Examples of exceptions (deductive but not valid arguments):

1. Magellan's ships sailed around the world. 2. It necessarily follows that the earth is a sphere.

1. If I am in Wilkes-Barre, then I'm in PA.2. I am not in Wilkes-Barre.3. Therefore, I am not in Pennsylvania



The Common Pattern Test: There are many common patterns that valid

arguments “use.”For example: If P then Q. P. Therefore Q.

This is called Modus Ponens

If an argument follows one of these patterns, it is deductive.

We will learn other patterns shortly.



The Principle of Charity Test. If it is unclear what kind of argument it is then… …if it would make a bad deductive argument assume it is not

a deductive argument. If it’s a choice between “bad deductive” and “good inductive,”

go with the latter—it’s nicer and more likely what the person had in mind.

Example: Andy told me that he ate at Maxine’s yesterday, but it burned

down a month ago. It is certain that he was lying or mistaken. Since it is possible that Maxine’s was rebuilt quickly, this

argument is deductively invalid. Assume that the author intended it to be inductive.


Common Patterns of Deductive Reasoning

Hypothetical SyllogismCategorical SyllogismArgument by EliminationArgument Based on Mathematics. Argument from Definition



Hypothetical Syllogism: three line argument that contains at least one conditional. If A then B. A. Therefore B. (Modus Ponens) If P then Q. if Q then R. Therefore if P then R. (chain

argument) If A then B. Not B. Therefore not A. (Modus Tollens)

Invalid versions (that are still deductive): If A then B. Not A. Therefore not B. (denying the antecedent)

If I am female then I am a person. I am not female. Therefore I am not a person.

If A then B. B. Therefore A. (affirming the consequent). If we’re on Neptune then we are in the solar system. We are in

the solar system. Therefore, we’re on Neptune. These are still called deductive because usually people think they are valid when they put them forward.



Categorical Syllogism: a three-lined argument in which each statement begins with the word all, some or no.

Example Forms: All a’s are b’s. All b’s are c’s. Therefore, all a’s are

c’s. Some a’s are b’s. All b’s are c’s. Therefore some

a’s are c’s. Example:

1. All oaks are trees2. All trees are plants. 3. So all oaks are plants.



Argument by Elimination: seeks to logically rule out various possibilities until only a single possibility remains.

Example forms: A or B. Not B. Therefore A. P or Q. if A then ~P. A. Therefore Q.

Example: 1. Either Joe walked to the library or he drove. 2. But Joe didn’t drive to the library. 3. Therefore, Joe walked to the library.



Arguments based on Mathematics: argument in which the conclusion depends largely or entirely on some mathematical calculation or measurement.

Example forms: There are four a’s and two b’s. Therefore there

are six things all together. Example:

1. Eight is greater than four. 2. Four is greater than two. 3. Therefore, eight is greater than two.



Arguments from definition: an argument in which the conclusion is presented as being true in virtue of the definition of some key word or phrase.

Examples:Bob is a bachelor. Therefore Bob is

unmarried.Janelle is a cardiologist. Therefore, Janelle

is a doctor.


Common Patterns of Inductive Reasoning

Inductive generalizationPredictive argumentAugment from authorityCausal ArgumentStatistical ArgumentArgument from Analogy



Inductive generalization: Generalization: statement that attributes characteristics to

all or most members of some group or class. Inductive generalization: drawing a generalization as a

likely conclusion from observations. Common Form:

So far, the b’s I have seen have had the property p. Therefore, all b’s must have the property p.

Example: 1. All dinosaur bones found so far have been over 65 million

years old. 2. Therefore all dinosaur bones found will be over 65 million

years old.



Predictive argument: Prediction: a statement about what will happen in the future. Predictive argument: an argument that has, as a conclusion, a

prediction. Common form:

So far, all the b’s I have seen have had property P. Therefore, the next b I see will have property P.

Example: Most U.S. presidents have been tall. Therefore, the next president will be tall.

Although many inductive arguments are predictive (and vise-versa) not all predictive arguments are inductive. Counter example:

1. If Amy goes, then Ted goes. 2. Amy will go. 3. Therefore, Ted will go.



Augment from authority: asserts a claim and supports that claim by citing some presumed authority or witness.

Common form: P said that A was true, therefore A is true.

Example: The Encyclopedia said that bats eat bugs; therefore it is likely that

bats eat bugs. Since sources are not 100% reliable, conclusions of such

arguments are not guaranteed and thus such arguments are inductive.

However, the better the source, the better the argument. Not all arguments from authority are inductive. Example:

1. If the bible says it, it’s true. 2. The bible says you should love your neighbor.3. Therefore, you should love your neighbor.



Causal Argument: asserts or denies that something is the cause of something else.

Common form: X is true. The likely cause of X being true is Y

being true. Therefore, Y must be true. Example:

I can’t log in. The network must be down. Not all causal arguments are inductive:

Counter example:1. Whenever iron is exposed to oxygen, it eventually rusts.2. This iron pipe has been exposed to oxygen. 3. Therefore, it will eventually rust.



Statistical Argument: rests on statistical evidence (evidence that some percentage of some group has some particular property).

Common form: 90% of b’s have property p and x is a b, therefore x probably has

property p. Example:

1. 83% of Notre Dame students are Catholic.2. Bob is a Notre Dame student. 3. Therefore Bob is probably Catholic.

Because stats aren’t necessarily reflective of the population, such arguments are usually inductive arguments. But not always:

If 65% polled will vote Democratic, then Hillary will win. 65% polled will vote Democratic. Therefore, Hillary will win.



Arguing from Analogy: taking two things that are alike in some way(s) and using that to support claims that they are similar in (an)other way(s).

Common form: 1. These things are similar is such-and-such ways. 2. Therefore, they are similar in some further way.

Examples: 1. Hershey Park is a great amusement park and it has a great roller coaster. 2. Dorney Park is a great amusement park.3. Dorney Park probably has a great roller coaster.

Since, being similar in one way does not guarantee being similar in another, most analogies are inductive. However, some are deductive:

1. Cars and cigarettes both cause thousands of deaths. 2. Thus, if cigarettes are regulated, so too should cars. 3. But cars should not be regulated. 4. Therefore cigarettes should not be regulated either.


Deductive Validity

A valid deductive argument is an argument in which it is impossible for all the premises to be true and the conclusion false. In other words: If the premises are true, the conclusion must be true. The conclusion follows necessarily from the premises. The premises provide logically conclusive grounds for the

truth of the premises. The truth of the premises would guarantee the truth of the

conclusion. It is logically inconsistent to assert all the premises but deny

the conclusion. It is impossible for the premises to be true and the conclusion

to be false.


Deductive Validity

Valid arguments: As we said before, the validity of an argument has nothing to do with the truth of its premises. If the premises would guarantee the conclusion if the premises were true, then the argument is valid.

Valid argument: 1. All squares are circles.2. All circles are triangles. 3. Therefore, all squares are triangles.

Get it out of your head that “valid argument” means “good argument.” An argument can be a good argument and be invalid (for example, it could be a good inductive argument) and could be a bad argument but be valid (because it has false premises like the one above).


Deductive Invalidity

Invalid deductive arguments: deductive arguments whose premises do not guarantee their conclusion. (i.e., they have bad deductive form.)

Notice, in the following argument, even though the premises and conclusion are true, it is invalid:

1. All dogs are animals.2. Lassie is an animal.3. Therefore, Lassie is a Dog.

What if “Lassie” here referred to my pet iguana. The premises would still be true, and yet the conclusion would be false.

If you get rid of your assumption (which is not in the argument) that we are talking about the TV dog, the invalidity becomes clear. Let’s use a different name.

1. All dogs are animals.2. Jub-Jub is an animal.3. Therefore, Jub-Jub is a Dog.


Soundness

An argument is sound if and only if it is valid and has all true premises. If that is the case, you are forced to accept the

conclusion. If an argument is valid, but has a false

premise, we say that it is a deductively unsound argument.

Since all inductive arguments are invalid (their premises don’t guarantee their conclusion), all inductive arguments are unsound.


Inductive Strength

Strong inductive argument: an inductive argument the premises of which, if true, make the conclusion likely or probable. i.e.,:

If the premises are true, the conclusion is probably true. The premises provide probable, but not logically conclusive

grounds for the truth of the conclusion. Common form:

1. Most b’s are p. 2. X is a b.3. Therefore, (probably) X is p.

Example:1. All recent presidents have been college grads. 2. Therefore the next president will be a college grad.


Inductive Weakness

Weak inductive argument: inductive argument, the premises of which, do not provide good evidence for its conclusion.

Example:1. All previous popes have been men.

2. Therefore, probably the next pope will be female.


Induction and Premise Truth

The inductive strength or weakness of an argument has nothing to do with the truth or falsity of its premises.

Example: Inductive strength with false premises:1. All previous presidents have been female.2. Therefore, the next president will be female.

Example: Inductive weakness with true premises: 1. Most presidents have been married.2. Therefore the next president will probably be male.

But you will never have a strong inductive argument with true premises but a probably false conclusion.


Inductive Strength Comes in Degrees

This argument… 1. The last 5 presidents were male2. Therefore the next president will be male.

…is weaker than this one: 1. All the presidents have been male2. Therefore the next president will be male.

Recall, validity doesn’t come in degrees. A deductive argument is either valid or invalid. No two valid arguments differ in “how valid” they are.

But some inductively strong arguments are inductively stronger than others.


Cogency

Even if an argument is inductively strong, it can still have a false premise and be a “bad argument.”

Cogent argument: inductively strong argument with true premises.

Uncogent argument: inductive argument that is either weak, has false premises, or both.

Ch03 basic logical_concepts

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