Reasoning Many people would sooner die than think. In fact, they do. --Bertrand Russell

Reasoning

Many people would sooner diethan think. In fact, they do.

--Bertrand Russell

“Blue Diamond” Experiment

I am thinking of one of those colors and one of those shapes.

If a symbol has either the color I am thinking about , or the shape I am thinking about, or both,

then I accept it,

but otherwise I reject it.

I accept the blue diamond.

Does anything follow about my acceptance, or rejection, of the other symbols?

Blue Diamond

• In accepting the blue diamond, we know the experimenter is thinking of:

1) blue and diamond

2) blue and circle, or

3) yellow and diamond,

but we do not know which.

Blue Diamond—Case 1

1. Since he accepts all other symbols that have either the color or the shape he is thinking about (and otherwise rejects the symbol), he accepts all blue shapes and any color diamond. (He rejects only the yellow circle.

• In case 2, he accepts all blue shapes and any color circle. (He rejects only the yellow diamond.



• He accepts any yellow shapes and any color diamonds. (He rejects only the blue circle.)

Blue Diamond—Conclusion

• Since we don’t know which of the of the above three scenarios he is thinking of, we can’t possible know which of the other symbols will be rejected.

• We do, however, know that he rejects one.

• His acceptance of the blue diamond does not provide enough information for us to be certain about any other shape.

Two Kinds of Reasoning

Two Basic Types of Valid Argument

StatementsFundamental Structures of Reasoning

• Statement—a particular kind of sentence that brings a message which can be said to be either true or false.

1. LAMP is located in Montgomery.2. LAMP is a private school.• Statement (1) has a true truth value.

• Statement (2) has a false truth value.

statement

self-supporting statement

supported statement

by authority by experience by deductionself-reportby logicalstructure

by definition

tautology

self-contradiction

Questions, Commands, Nonsense

Is LAMP in Montgomery?• A question has no truth value.

Go to school.• An imperative or command has no truth value.

This statement is false.• If this is true, it is therefore false.• If this is true, it is therefore true and

consequently false. It is nonsense and not useful in logic.

Self-Supporting Statements

• The distinction between self-supporting statements and supporting statements is the means by which the truth value of the statement is determined.

• self-report: a statement by a person concerning his or her own desires, beliefs, or feelings.

• I believe that the earth is flat.

– The statement refers to the beliefbelief of the speaker, not whether the earth is in fact flat.


• Statements which are true or false by logical structureThe shape is a circle, or the shape is not a

circle.

The statement is necessarily true because it covers all of the possibilities.

This is called a tautology.

• Statements which are true or false by logical structure

Dr. Summers is the teacher, and Dr. Summers is not the teacher.– Self-Contradiction—both statements cannot

be true.


Logical Truism


• Statements that are true or false by definition.– Some statements are true or false because of

the definitions of the words in the sentence.– All triangles are three-sided figures.– This triangle is an octagon.

Logical Truism

Supported Statements

• A supported statement does not stand or fall by itself. It requires evidence from outside investigation before it can be declared true or false.– The Greeks formed a treaty with the

Spartans.– It is raining outside.– The flower will wilt.


– Henry A. Loveless invested substantial portions of his personal wealth to support education in Montgomery’s African American community.

• In order to determine if this is true, we must refer to a trustworthy, authoritative source.


– It is raining outside.

• To determine the true value of this statement depends on experience.

• We are trusting not an outside source, but our own senses: sight, touch, sound, smell, taste.

– The live flower will wilt.

• We reason to some conclusion based on other statements: deduction.

• All live flowers will eventually wilt. The flower is live. The flower will wilt.


statement

self-supporting statement

supported statement

by authority by experience by deductionself-reportby logicalstructure

by definition

tautology

self-contradiction

Relationships Between Statements

Consistency Inconsistency

Implication Independence

Equivalence


• Consistency: When two statements can be true at the same time, we say they are consistent.

LAMP is in Montgomery.Montgomery Academy is in

Montgomery.

No conflict exists between the two statements.

LAMP is in Montgomery.LAMP is in Birmingham.• Both statements cannot be true.

Therefore, they are inconsistent.


• Implication: Two statements are related by implication when the truth of the first requires or necessitates the truth of the second. If A implies B, and if A is true, the statement B must also be true.

A: All fish live in water.B: Some fish live in water.

The truth of A requires the truth of B.


• Logical equivalence: If two statements are logically equivalent, then the first must imply the second, and the second must imply the first.

• They must both be true, or they must both be false.• It is not possible for one to be true and the other

false.

A: No circles are squares.B: No squares are circles.

A: Some mammals are egg-layers.B: Some egg-layers are mammals.


• Independence: If the truth or falsity of one statement has nothing at all to do with the truth or falsity or another statement, they are independent.– Neither statement can necessitate the truth of the

other statement (implication).– Neither statement can necessitate the falsity of the

other (inconsistency)

All LAMP students applied to the school.All tricycles have three wheels.



Consistency Inconsistency

Implication Independence

Equivalence

Why do we care about the relationships between statements?

When statements taken together lead to a conclusion, they form


Reasoning

Identical twins often have different IQ test scores. Yet such twins inherit the same genes. So environment must play some part in determining IQ.

Argument1. Identical twins often have different IQ test

scores.2. Identical twins inherit the same genes.3. So environment must play some part in

determining IQ.

Since it is wrong to kill a human being, it follows that abortion is wrong, because abortion takes the life of (kills) a human being.

1. It is wrong to kill a human being.

2. Abortion takes the life of (kills) a human being.

3. Abortion is wrong.

Since it is wrong to kill a human being, it follows that capital punishment is wrong, because capital punishment takes the life of (kills) a human being.

1. It is wrong to kill a human being.

2. Capital punishment takes the life of (kills) a human being.

3. Capital punishment is wrong.

Informal Logic

Since it is wrong to kill a human being, it follows that capital punishment is wrong, because capital punishment takes the life of (kills) a human being.

Since it is wrong to kill a human being, it follows that abortion is wrong, because abortion takes the life of (kills) a human being.

Formal Logic:Syllogism

1. Killing a human being is wrong.

2. Abortion takes the life of (kills) a human being.

3. Abortion is wrong.

1. Killing a human being is wrong.

2. Capital punishment takes the life of (kills) a human being.

3. Capital punishment is wrong.

Reasoning

• Reasoning—or any argument into which reasoning is cast—is either

–cogent (sound)

–fallacious (unsound)

Cogent Reasoning

Cogent reasoning must satisfy three criteria:1. We must start with justified or warranted

premises.2. We must include all available relevant

information.*3. Our reasoning must be correct or valid.

*This is an extremely stringent requirement that is beyond the reach of most of us most of the time. The point is to come as close as possible to satisfying it given the constraints of time and resources.

Remember this word. We will use it later.

Two Basic Kinds of Valid Argument

• Deductive– If the premises are true, then its conclusion must be

true also.– If the premises of a deductively valid argument are

true, the its conclusion cannot be false.• Inductive

– The premises of an inductively valid argument provide good but not conclusive grounds for accepting its conclusion.

– The truth of the premises does not guarantee that the conclusion is true, but it does make the conclusion probable.

– Inductive arguments are sometimes called “probability arguments”

Inductively Valid Argument

1. So far, none of the presidents of U.S. have been women.

2. Probably none of the future presidents of U.S. will be women.

The premise of the argument is true. The reasoning for the argument’s conclusion is

valid. However, the conclusion may turn out to be false,

even if the premise is true, since voters may choose a woman in a primary and then the general election.

Inductive Logic

• Reasons from particular instances to general theories.

• If you observe enough apples falling from trees, you will conclude that apples always fall down instead of sideways or up.

• You might then form a more general hypothesis that includes other falling bodies, like pears.

• Thus, science progresses—the scientific method.

Inductive Logic

The literary character most renowned for his “powers of deduction,” Sherlock Holmes, actually operates generally by using inductive logic rather than by deductive logic.

Perhaps Holmes arrived at his conclusion this way:

1. I went to sleep in a tent, but now I can see the stars.2. My intuitive working hypothesis, based on analogies to

similar experiences I have had in the past, is that someone has stolen our tent.

3. In testing that hypothesis, let’s rule out alternative hypotheses:

a. Perhaps the tent is still here, but someone is projecting a picture of stars on the roof of the tent. This is unlikely, based on my past experience of human behavior and the equipment that experience tells me would have to be present in the tent and obviously isn’t.

b. Perhaps the tent blew away. This is unlikely, as my past experiences lead me to conclude that that amount of wind would have awakened me, though perhaps not Watson.

c. Etc. etc.

4. No, I think my original hypothesis is probably correct. Someone has stolen our tent.

Valid Deductive Argument

1. Every U.S. President is a U.S. citizen.

2. George Bush is a U. S. President.

3. George Bush is a citizen.

We know that (1) and (2) are true. It is inconceivable that (3) is not true.

It would be inconsistent (contradictory) to believe both premises and not the conclusion.

Difference between a deductively valid argument and an inductively valid argument

The conclusion of a deductively valid argument is as valid as its premises.

Thus, this is the fundamental difference between a valid deductive argument and a valid inductive argument.

Inductive arguments have an element of doubt, however slight, because they make claims not already made by their premises.

Deductive conclusions are already contained in their premises, although often implicitly.

Deductive Validity

“to reason from generally accepted opinions about any problem set before us and shall ourselves, when sustaining an argument, avoid saying anything self-contradictory”

--Aristotle

Logical TruismsPrinciples of Consistency and Noncontradiction

• In ordinary conversation, we expect that others will follow certain principles.

• If you tell me today that you love English, but tomorrow you say that you hate English, since I know you are an honest person, I assume that something has changed. If nothing has changed, you are holding contradictory positions.

• If you claim that you always drive yourself to school, and I see you arrive at school in a car your mother is driving, your behavior is contradicting your claim and you are being inconsistent.

Logical Truisms

• law of the excluded middle– requires that a thing must either possess a given

attribute or must not possess it.– A thing must be one way or the other; there is no

middle.– The middle ground is excluded.

• A shape is either a circle or not a circle.• A figure is either a square or not a square.• Two lines on a plane either intersect or they do not intersect.

• Caution: If an argument is formed as an “either-or” proposition when in fact a middle ground exists, then it is a logical fallacy.

Logical Truisms

• law of noncontradiction– closely related to the law of the excluded middle.– A thing cannot both be and not be at the same time.– A shape cannot be both a circle and not a circle.– A figure cannot be both a square and not a square.– Two lines in a plane cannot both intersect and not

intersect.

Logical Truism

• Aristotle distinguished between the law of the excluded middle and the law of noncontradiction.

• The Stoics of the third century BC acknowledged them as a single rule:

“Either the first or not the first”—meaning always one or the other but never both.

Deductive Reasoning

• The basic steps in any deductive proof, either mathematical or metaphysical, are the same.

• We begin with true (or agreed upon) statements called premises.

• We concede at each step that the next statement or construction follows legitimately from the previous statement.

• Arriving at the final statement, or conclusion, we know it must necessarily be true due to our logical chain of reasoning.

Syllogism

• In a syllogism, each proposition is one of Aristotle’s four proposition types.

• The propositions in the first two lines are the premises.

• The proposition in the third line is the conclusion.

• If the argument is valid and you accept the premises as true, then you must accept the conclusion as true.

Deductive arguments may have the same form, or structure.(1) 1. If you obtain a valid drivers license in

Alabama, then you must pass a drivers test.

2. You obtained a valid Alabama drivers license.

3. You passed a drivers test.

(2) 1. If the U.S does not stay in Iraq, the terrorists will win.

2. The U.S. will not stay in Iraq.

3. The terrorist will win.

The form they share: Modus Ponens (mood that affirms)

1. If [some sentence], then [some sentence]

2. First sentence

3. Second sentence

Premise

Premise

Conclusion

Modus Ponens

1. If A, then B.

2. A

3. B

The earliest attempts to create artificial intelligence used this form. In 1956, artificial intelligence pioneers implemented modus ponens in their program The Logic Theorist, a program designed to make logical conclusions.

Premise

Premise

Conclusion

Modus Ponens A Parental Favorite

• If you clean your room and take out the garbage, you can go to the movie and buy popcorn.

1. If A and B, then C and D.

2. A and B.

3. C and D.

Premise

Premise

Conclusion

Modus PonensAn Internal Revenue Favorite

If you itemize your deductions, then you enter the amount from Schedule A on line 36.

If you do not itemize your deductions, then you enter your standard deduction on line 36.

Therefore, either you will enter the amount from Schedule A on line 36 or you will enter your standard deduction on line 36.

If A, then B.If not-A, then C.Therefore, B or C.

Premise

Conclusion

Premise

Modus Tollens(mood that denies)

Commonly used and intuitively valid form

1.If A, then B.2.Not A.3.Not B.

Example:1. If women are genetically less intelligent than men,

then men should have higher test score averages than women.2. Men do not have higher test scores than women.3. Women are not genetically less intelligent than men.

Modus Ponens and Modus Tollens

Both are so universal they appear as two figures of argument in Buddhist logic, called the Method of Agreement and the Method of Difference.

Disjunctive Syllogism

Form:

1. A or B.

2. Not A.

3. B

1. Either LAMP is a traditional school or LAMP is a magnet school.

2. LAMP is not a traditional school.

3. Lamp is a magnet school.

Disjunctive Syllogism


2. LAMP is not a traditional school.

3. Lamp is a magnet school.


2. LAMP is not a magnet school.

3. Lamp is a traditional school.

Hypothetical Syllogism

Form:1. If A, then B.2. If B, then C.3. If A, then C.

1. If enough companies make solar energy panels, then solar energy will be readily available.

2. If solar energy is readily available, then we will reduce carbon emissions.

3. If enough companies make solar energy panels, then we will reduce carbon emissions.

No names, just forms

1. All A’s are B’s

2. This is an A.

3. This is a B.

1. All politicians are liars.

2. Jane Smith is a politician.

3. Jane Smith is a liar.


1. No A’s are B’s.

2. It is false that some A’s are B’s.

1. No policemen accept bribes.

2. It is false that some policemen accept bribes.


1. All A’s are B’s.

2. All B’s are C’s.

3. All A’s are C’s.

1. All geeks are nerds.

2. All nerds are dorks.

3. All geeks are dorks.

Contradiction

“No amount of experimentation can ever prove me right; a single experiment can prove me wrong.”

Albert Einstein

Documents

Reasoning Many people would sooner die than think. In fact, they do. --Bertrand Russell