An ARGUMENT is:

1.       A set of statements.

2.       A connection is claimed to exist between the statements.

3.       The claim is that the truth of one statement (the conclusion) somehow follows from the truth of the others (the premises).

 

There may be many premises, or as few as one. The claim regarding the connection between the premises and the conclusion may be correct or incorrect.

 

A STATEMENT is, in this context, a complete idea, claiming that some state-of-affairs exists or doesn’t exist. Statements are usually expressed by declarative sentences, but are not identical to the sentence that expresses them, since:

1.       Different sentences may make the same statement (e.g., “Ich liebe dich” and “I love you”).

2.       The same sentence may express different statements if uttered in different contexts (e.g., “The current President is taller than I am.”) or even if uttered in different tones of voice (e.g.,”That’s very cute.”).

 

TYPES OF CONNECTION CLAIMED TO EXIST BETWEEN STATEMENTS

Deductive: The truth of the premises would guarantee the truth of the conclusion.

Example:          All people are mortal.

                        Socrates is a person.

                        Therefore, Socrates is mortal.

 

Inductive: The truth of the premises would make the truth of the conclusion likely, but not certain.

Example:          Most Greeks drink wine.

                        Socrates is a Greek.

                        Therefore, probably, Socrates drinks wine

 

DEDUCTION

There are two ways a deductive argument can go wrong.

Factual error:                 One or more premise is false.

Formal error:                  The claim regarding the relationship between the premises and the

conclusion may be false. Such an argument is called INVALID.

VALID

A well-formed deductive argument, in which the desired relationship exists between the premises and the conclusion, is called VALID. The conclusion strictly (without qualification) follows from the premises. To assert the premises and deny the conclusion is self-contradictory

 

VALIDITY can sometimes be graphically represented.

 

 

All people are mortal.

Socrates is a person.

Therefore, Socrates is mortal.

 

The diagram illustrates that if everything in the set of all people has the property of being mortal, and Socrates is in that set, then he must have that property.

 

 

 

SOUND

A deductive argument that is valid (no formal errors) and has all true premises (no factual errors), is called SOUND. The conclusion of a sound argument must be true, since the premises are true, and their truth guarantees the truth of the conclusion.

 

NOTE:  VALID and SOUND are used only in reference to deductive arguments. Inductive arguments must be judged by other criteria. Inductive arguments are never valid, therefore never sound. Can you explain why? (Hint: review the definitions!)

 

TRUTH IS NOT VALIDITY

VALID means that IF the premises are true, that would be sufficient to guarantee the truth of the conclusion. Therefore…

1.       A valid argument may NEVER have all true premises and a false conclusion. A quick-and-dirty test of validity is to ask whether you can imagine ANY circumstances in which the premises could all be true, and the conclusion could still be false. If “yes,” then the argument is invalid.

2.       A valid argument may have any other mix of true/false premises and conclusion. This is because “valid” only describes the relationship between statements, i.e., IF the premises are true, then the conclusion would have to be true as well.

3.       Truth is the property of individual statements. Validity is a potential relationship between statements. Sentences such as “this conclusion is valid” or “this argument is true” would have no clear meaning within the framework of logic.

4.       Since “valid” describes a relationship, an argument with true premises and a true conclusion may still be invalid if the premises and conclusion are not properly related.

 

EXAMPLES -- TRUTH IS NOT VALIDITY

Keeping in mind the four principles explained above, especially the first one, which of these arguments are valid, and which are invalid?

 

1.         All people are mortal.                             2.         All alligators are purple.

            You are a person.                                              Shakira is an alligator.

            Therefore, you are mortal.                                   Therefore, Shakira is purple.

 

3.         All terrorists are birds.                            4.         All birds have wings.

            All penguins are terrorists.                                  All doves have wings.

            Therefore, all penguins are birds.             Therefore, all doves are birds.

 

LOGICAL FORM

The logical form of statement or argument is the “skeleton,” or structure. If you retain only the words that name relationships and quantity, and replace all the other words with placeholders, the logical form is revealed.

 

This argument…                                    …has this logical form

 

All people are mortal.                             All X are Y

Socrates is a person.                             Z is an X__

Therefore, Socrates is mortal.                 \   Z is a Y

 

 

Infinitely many arguments may share the same logical form. Since the logical form is the set of relationships that exist within the argument, it is the logical form that makes an argument valid or invalid. Any two deductive arguments with the same logical form will therefore be the same as regards validity.

“ARGUMENTS, LIKE MEN, ARE SOMETIMES PRETENDERS” (Plato)

We may sometimes be misled by invalid arguments superficially resembling valid arguments. Compare the four argument forms illustrated below. Each of these arguments begins with the conditional first premise, “If P, then Q.” In such a premise, “P” is called the ANTECENDENT; “Q” is called the CONSEQUENT. In each argument, the second premise either asserts or denies either the antecedent or the consequent. In each case, the name of the argument form is derived from what happens in the second premise.

 

1.

2.

3.

4.

If P, then Q

P         

\ Q     

 

If P, then Q

Q         

\ P     

 

If P, then Q

Not P   

\ Not Q           

 

If P, then Q

Not Q   

\ Not P           

 

Affirming the antecedent

Affirming the consequent

Denying the

Antecedent

Denying the

consequent

This form is valid.

 

The first premise states that whenever P is true, Q is also true. The second premise states that P is presently true. If both those premises were true, there is no way the conclusion could be false. Since you have P, you have Q.

This form is invalid.

 

The first premise states that whenever P is true, Q is also true. The second premise states that Q is presently true. But no premise states that Q is never true unless P is true. Therefore, even if both premises are true, the conclusion could be false. For all you know, based on the premises, you can have Q without P.

This form is invalid.

 

The first premise states that whenever P is true, Q is also true. The second premise states that P is presently false. But no premise states that Q is never true unless P is true. Therefore, even if both premises are true, the conclusion could be false. For all you know, based on the premises, you can have Q without P.

This form is valid.

 

The first premise states that whenever P is true, Q is also true. The second premise states that Q is presently false. If both those premises were true, there is no way the conclusion could be false. If you have P, you’d have Q.

 

Further Explanation: Sufficient vs. Necessary

“If P then Q” means that whenever you have P, you will also have Q. It does not mean that you cannot have Q without P. Both of the invalid forms illustrated above make the same mistake – they confuse “sufficient” with “necessary.”

·         “If P, then Q” means that P is sufficient to guarantee Q. It does not mean that P is necessary for Q.

·         If we had said “Only if P, then Q,” we would be saying P is necessary for Q, but not that P is sufficient for Q.

·         If we had said “If and Only If P, then Q,” we would be saying P is both necessary and sufficient for Q.

 

INDUCTION

In an inductive argument, the relationship claimed to exist between the truth of the premises and the truth of the conclusion is probability, not certainty.

 

Induction is based on an assumption called the UNIFORMITY OF NATURE. This principle stipulates that well-established patterns observed in the past will persist in the present and future. Therefore, the past can be used to predict what will happen in the near and remote future. Without this assumption, it would be impossible to learn from experience, and therefore neither science nor common sense would be possible.

 

The conclusion of an inductive argument may commonly be modified by such words as “probably,” or “it is likely.” But sometimes we treat highly probable conclusions as if they were certain, and omit any such restrictions. For example, your conclusion that a speeding bus would kill you if you stepped in front of it, is based on patterns observed in the past, and therefore inductive. But would you normally feel compelled to treat it merely as a probable conclusion?

 

Relationship Between Premises/Conclusion

The logical form of an inductive argument may help illustrate what the argument is trying to accomplish, but it will not show whether the premises and conclusion are properly connected.

 

If the truth of the premises would succeed in making the truth of the conclusion probable in the manner claimed, the relationship is called STRONG (not “valid”). Remember, induction is an attempt to apply the idea of the Uniformity of Nature. Therefore, the common sense rule of induction is that we want as close a match as possible between the evidence we present in the premises, and what we predict in the conclusion. All other criteria for judging inductive arguments are an elaboration of this rule.

 

Here is a summary of the differences between deductive validity and inductive strength.

DEDUCTIVE VALIDITY

INDUCTIVE STRENGTH

Relationship of certainty

Relationship of probability

All or nothing

Matter of degree

Logical form is crucial

Logical form is not decisive

Content is irrelevant

Content is crucial

 

There are many flavors of inductive arguments, but we will examine only two very basic types.

 

INDUCTIVE GENERALIZATION moves from an observation of some members of a set (i.e., a sample) to a prediction about the entire set (i.e., population).

 

EXAMPLES:

All the swans I observed were white.                                Most of the swans I observed are white.

I have heard of no exceptions to this                                I have heard of few exceptions to this

pattern.                                                                         pattern.

Therefore, I conclude that probably all                              Therefore, I conclude that probably most

swans are white.                                                            swans are white.

 

Logical Form =              All (or most) observed X have property Y.

                                    No (or few) exceptions are known______

                                    \ probably All (or most) X have property Y.

 

INDUCTIVE ANALOGY moves from an observation that an individual resembles some members of a set in regard to particular properties, to a prediction that the individual will resemble those same members as regards other properties.

 

EXAMPLE

Emily, Adrienne, Iris, Alicia, Tori are all swans, all bug-eaters, all female, all white, all migratory.

Jennifer is a swan, eats bugs, is female, is white.

Therefore, I predict that Jennifer is probably migratory.

 

Logical Form =              A, B, C, D, E, all have properties V, W, X, Y, Z.

                                    N has properties V, W, X, Y

                                    \ probably N also has property Z.

 

NOTE: The easiest way to distinguish these two forms is to look at the conclusion.

·         In an inductive generalization, the conclusion will be about an entire population.

·         In an inductive analogy, the conclusion will be about an individual.

 

EVALUATING INDUCTIVE STRENGTH

1.       Are the premises true?

2.       How broad is the sample? The variety in the sample should be a good match for the variety in the population. How narrow is the sample? A sample may be broad in some ways, but narrow in others.

3.       Sample size may help, if it helps make a better match for variety.

4.       How sweeping and confident is the conclusion? The more sweeping or confident, the better the evidence needs to be.

5.       Have we made a conscientious effort to examine all the relevant evidence?

6.       For an analogy, have looked at the whole pattern, weighing significant similarities against significant differences?

 

INFORMAL LOGIC

Informal fallacies:

·         “Fallacy” = A consistently misleading type of argument.

·         “Informal” = logical form alone will not reveal the problem.

A few of the more common types are illustrated below.

 

Circular Reasoning

The conclusion is somehow concealed in the premises, through ambiguity or extraneous premises, and the conclusion is merely derived from itself.

 

Examples:         The Book of Gertrude says it is the word of God.

                        Now, everything the Book of Gertrude says, must be true, since after all, it is the

divinely inspired word of God. Therefore, since the Book of Gertrude says it is the

Word of God, it must be so!

 

                        We have free will if we can freely choose among alternatives. But every day, we

DO freely choose among alternatives. Therefore, we do have free will (hint…

define “free will”)

 

Equivocation

Key words or phrases are used in different senses (i.e., with different meanings) within the same argument, in order to make irrelevant premises seem consistent.

 

Example:          Might makes right, since after all, morality dictates that better men should prevail.

                        If the losers had been better men, they would not have lost. When the mighty

                        conquer the weak, it is clearly a case of good men triumphing over men who are

                        not as good. What could be more morally desirable?

 

Appeals to Emotion

Attempts to make bad reasoning seem persuasive through emotional manipulation. Although sentiment may have a proper role in some decisions, it is useless for evaluating factual or logical claims. Some common varieties of this fallacy include:

            Appeal to fear/force:       “If you question this belief, you’ll go to hell for eternity.”

            Appeal to pity:               “How can you question my word, after all I’ve been through.”

            Ad Hominem:                “How can you support the economic policies of a drunk?”

            Guilt by Association:      “He’s a Muslim? You mean, like those terrorists?”

            Flattery:                        “I know you’ll agree, because you are decent, intelligent people.”

            Guilt-tripping:                 “You are calling me a liar, after all I’ve done for you?”

 

Other Topics in Informal Logic

 

Semantic Dispute

Caution! Don’t confuse this one with equivocation. This is a situation in which there is no real disagreement, but people are misled into thinking there is a problem. They are misled because important terms are unclear. Clarification would make the dispute evaporate.

 

Yank:    “Football should be an Olympic event.”

Brit:      “What are you talking about? It already is!”

Yank:    “No way! If it were, we’d be winning. It’s practically our national sport.”

Brit:      “Go on! You were eliminated early from the World Cup competition.”

Yank:    “Oh, wait a minute…I don’t think we’re talking about the same sport. What you call

football, we call soccer. And what we call football is rarely played in Europe.”

 

Sorites Paradox

This is a problem that arises when we are trying to make a decision about where to draw the line between two opposites. There may be no clear and natural place to draw the line, but circumstances require we do so anyway. The paradox cannot be resolved by observing physical facts more closely, because it is a conceptual problem. A prudent way to overcome the problem is to ask why we need to draw the line. Usually, we have some specific, secondary purpose we are trying to serve. Our decisions should therefore serve those secondary purposes as well as possible, since there is no other reason for drawing the line in the first place.

 

Examples:         Consider when and why we have to draw the line between such opposites as guilty/innocent, normal/abnormal, sane/insane, alive/dead, liable/not liable, person/non-person. How do such distinctions affect our attitudes toward important subjects, such as abortion, euthanasia, “do not resuscitate” orders, psychiatric policy, criminal law?

 

Reductio Ad Absurdum

Literally, “reduce to absurdity.” This is one of the main argumentative strategies used by philosophers. You attempt to show that the logical consequences of some position are contradictory, or otherwise unacceptable.

 

Example:          The Cretan said, “Anything a Cretan says, is a lie.”

                        Objection: If that were so, his own statement, “Anything a Cretan says, is a lie,”

would be a lie. So, if what he said is true, then what he said is false. But that’s

self-contradictory, and so it’s obviously not true.

 

 

 

Deduction -- Review Exercises:

 

Answer TRUE or FALSE, and give a FULL EXPLANATION of your answer. Define important terms as part of your explanation.

 

1.       A valid deductive argument may have a false conclusion.

2.       A valid deductive argument may have one or more false premises.

3.       If an argument is invalid, it must have a false conclusion.

4.       If an argument is sound, the conclusion must be true.

5.       If an argument is valid, any other argument with the same logical form will be valid.

6.       If an argument is sound, any other argument with the same logical form will be sound.

7.       If a deductive argument has true premises and a true conclusion, it must be sound.

8.       I a deductive argument has true premises and a false conclusion, it may still be valid.

9.       I a deductive argument has false premises and a true conclusion, it may still be valid.

10.   If the premises are true, and the conclusion strictly follows from the premises, then the argument is sound.

 

For each of the four arguments below, do the following four things.

a.       Show the logical form, as illustrated on page 3.

b.       Give the name of the argument (affirming or denying the antecedent or consequent).

c.       Tell whether it is a valid or invalid form.

d.       Explain WHY it is valid or invalid.

 

11.   If I studied hard, I will get a good grade. I did study hard. Therefore, I will get a good grade.

12.   If I studied hard, I will get a good grade. I did not study hard. Therefore, I will not get a good grade.

13.   If I studied hard, I will get a good grade. I did get a good grade. Therefore, I studied hard.

14.   If I studied hard, I will get a good grade. I did not get a good grade. Therefore, I did not study hard.

 

 

Review of Informal Logic

 

Define each of the following terms. I know you can copy – let’s see if you can explain in your own words. Give an example of each. Real-life examples are preferable (hey, this is an election year! You should have NO trouble finding examples!)  Explain why the reasoning is a fallacy.

 

1.       Circular reasoning.

2.       Sorites paradox.

3.       Reductio Ad Absurdum.

4.       Equivocation.

5.       Semantic Dispute.

 

 

Induction -- Review Exercises:

 

Answer TRUE or FALSE, and give a FULL EXPLANATION of your answer. Define important terms as part of your explanation.

 

1.       If an inductive argument is strong, it can still have a false conclusion.

2.       A good inductive argument must have true premises, and be valid.

3.       If an inductive argument has true premises and a true conclusion, then it is strong.

 

 

Choose the one best answer, and defend your choice.

 

4.“Most of the Hindus I have known have been vegetarians. Gupta is a Hindu. I’d bet he’s a

    vegetarian.”

 

4.1        This argument is an example of:

a.       Inductive generalization.

b.       Inductive analogy.

 

4.2               If we conducted our observations only in vegetarian restaurants, that would make our conclusion:

a.       Weaker, because the sample is more narrow.

b.       Weaker, because the sample is more broad.

c.       Stronger, because the sample is more narrow.

d.       Stronger, because the sample is more broad.

 

4.3               If all the Hindus we observed to be vegetarians also came from the same area as Gupta and worshipped at the same temple as Gupta, that would make the argument:

a.       Weaker, because the sample has more in common with Gupta.

b.       Weaker, because the sample is more narrow.

c.       Stronger, because the sample has more in common with Gupta.

d.       Stronger, because the sample is more broad.