LogicThe Science of Good Reasons

http://www.harryhiker.com/logic.htm

Reflects on the nature of thinking itself; The most fundamental branch of

Philosophy; Is the study of HOW we reason; Is prescriptive:

◦ i.e., develops rules for correct reasoning◦ Applying logic:

enables us to make clear and powerful arguments (or to be able to analyze another’s and avoid being

“sold” a bill of goods)

Logic:

What is an ARGUMENT?

◦ http://www.youtube.com/watch?v=kQFKtI6gn9Y

“An argument is a connected series of statements to establish a definite proposition”

The use of one or more reasons to support an idea or action.

Logic

Premise (a

proposition/statement)

Premise (a statement)

Conclusion (another statement)

NOT always in that order

The “form” of an argument

Which statement is supported by the other statements?◦ Conclusion◦ Key words identifying the conclusion:

“So…” “Therefore…” “Ergo…” “Consequently…” “Hence”

◦ “Hint Words” are not always included in the statement, sometimes they are implied

How to identify premises/conclusions?

Since… Because…

“Hints” of premises

The mental process that occurs when we move from premises (reasons) to a conclusions.

Using existing information to develop new information.

Inference

Deductive vs. Inductive reasoning

From general to particular (specific)

Is conclusive, NECESSARY inferences

IF the reasons are true, the conclusion MUST be true

Focuses on rules for determining VALIDITY of an argument

From particular to general

Conclusions are only PROBABLE

IF the reasons are true, the conclusion is PROBABLY true (i.e., it might be false)

Formal: ◦ Rules concerning the “form” i.e. structure of

arguments◦ Dealing with VALID inferences (are the premises

linked in such a way that the conclusion follows from them)

Informal (a.k.a., Critical Thinking)◦ Day-to-day situations◦ Rhetoric◦ Emotional appeal◦ Relevance / ambiguity

Two Major Categories of Logic

Deal with declarative statements,◦ i.e., sentences used to assert something about

something else Declarative statements are the only ways

that we can say something about the world. Declarative sentences can be either true or

false.

In Logic we

IF the premises are true –in a VALID argument - it will be impossible for the conclusion to be false.

A SOUND argument is a VALID argument that uses TRUE premises.

When we reason correctly,

Does NOT guarantee that the conclusion is NECESSARILY TRUE!

Determined by the FORM of the argument:◦Are the premises organized in such a

way that they can indeed lead to the conclusion?

◦Validity is NOT concerned with the truth of the premises,

◦Validity is concerned with possibility or reliability of the INFERENCE.

A Valid syllogism…

An argument with two premises that lead to a conclusion.

A Syllogism can be made with premises (statements) that are:◦Categorical◦Hypothetical / Conditional (If a, then c.)

◦Disjunctive (A or B)

A syllogism

Use Categorical statements:◦All S are P.◦No S are P.◦Some S are P.◦Some S are not P.

2 premises (categorical statements) Leading to a conclusion (also a categorical)

Categorical Syllogisms

A SUBJECT: that about which something is said.

All giraffes are animals. ◦ (giraffes = subject)

A PREDICATE: that which is said about something.

All giraffes are animals. ◦ (animals = predicate)

The COPULA: connects together or separates the S and the P.

All giraffes are animals. ◦ (is/is not)

A Proposition is a simple declarative sentence with:

By QUALITY,◦ Are we AFFIRMING the predicate of the subject?◦ Are we NEGATING (i.e., denying) the predicate of

the subject? (Ex. 2)

By QUANTITY,◦ Are we saying the predicate applies to ALL of the

subject, i.e., is UNIVERSAL?◦ Are we saying the predicate applies to only SOME

of the subject, i.e., is PARTICULAR?◦ ALL & SOME are QUANTIFIERS (Ex. 1)

Quality & Quantity of Categorical Statements

QUALITY

Standard Categorical Statements

Affirmative (+) Negative (-)QUANTITY

Universal

Particular

All S is (are) P. A

No S is P. E

Some S is P. I Some S is not P. O

All women are human. No cats are dogs.

Some men are bald. Some students are not athletes.

These codes come from the Latin words "Affirmo" and "Nego".

Affirmo: I affirm. Note the A and the I◦ A and I sentences AFFIRM a connection

between subject & predicate Nego: I deny. Note the E and the O

◦ E and O sentences NEGATE (deny) link between subect & predicate

Ex. 3 & 4

Standard Propositional Codes.

a.  The two premises.

All A is B (first premise) Some B is C (second premise) Therefore, Some C is A

b.  The Conclusion. In the above syllogism, Therefore, Some C is A

The parts of a categorical syllogism:

The major term: always the P (predicate) of the conclusion

The minor term: always the S (subject) of the conclusion. 

The middle term: never in the conclusion but appears twice in the

premises. (the middle term connects together or keeps apart the S and P in the conclusion).

Ex. 6

Major, Minor, Middle Terms

A distributed term covers 100% of the things referred to by the term. An undistributed term covers less than 100% of the things referred to by the term (few, many, almost all).

For instance, All men are mortal.

In this statement, "men" is distributed; for it covers 100% of the things referred by the term "men".

In Some men are Italian, "men" is undistributed; for the term covers less than 100% of the things referred to by the term "men".

Distribution of terms(how “much” of the term are we talking about?)

Consider the example from the last slide:◦ All men are mortal.

How much of the predicate (i.e., mortal things) are we talking about in that statement?◦ All mortal things?◦ Only some of those things that are mortal?

Since we can’t be talking about all mortal things in that statement, the predicate is UNDISTRIBUTED.

Distribution of terms...

QUALITY

Standard Categorical Statements

Affirmative (+) Negative (-)QUANTITY

Universal

Particular

All S is (are) P. A

No S is P. E

Some S is P. I Some S is not P. O

All women are human. No cats are dogs.

Some men are bald. Some students are not athletes.

D U D D

U U U D

Universal Affirmative statements (A statements): the subject is distributed, the predicate is undistributed.

Universal Negative statements (E statements): both the subject and the predicate are distributed.

Particular Affirmative statements (I statements): neither subject nor predicate is distributed (both are undistributed).

Particular Negative statements (O statements): the predicate alone is distributed.

Exercise 5

FIRST, CONSIDER THE QUALITY OF THE STATEMENTS:

Are BOTH premises negative? If YES, quit: it’s invalid(No conclusion follows from two negative premises)

If NO, continue,

Steps to take to determine the validity of a Categorical Syllogism

 If YES, quit: it’s invalid.(Two affirmative premises cannot lead to a negative conclusion)

If NO, continue,

Are BOTH premises affirmative AND the conclusion negative?

If NO, quit: it’s invalid.(Conclusion MUST be negative if a premise is negative.)

If YES, continue,

Is one premise negative AND the conclusion NEGATIVE?

Is the MIDDLE TERM distributed in AT LEAST ONE premise?

 If NO, quit: it’s invalid.(The middle term must be distributed AT LEAST ONCE.)

If YES, continue,

NOW FOCUS ON THE DISTRIBUTION OF TERMS:

If NO, quit: it’s invalid.(A term distributed in the conclusion MUST also be distributed in the premises.)

If YES, the form of the argument (the syllogism) is valid.

Ex. 7

Is a term that is distributed in the conclusion also distributed in the premises?

contain hypothetical or conditional statements.  e.g. If it is raining, then the ground is wet.  If you study, then you’ll get a good grade.  If Sue is late, then she must be sick.  If we keep building bombs, then we’ll use

them some day

Hypothetical Arguments

Antecedent: the first simple sentence, usually preceded by if.

  Consequent: the second simple sentence, usually preceded by then.

If (the antecedent) then the (consequent)

The Conditional / Hypothetical Statement

The ONLY valid options

(AA) is a good thing! (DC) a nice place!

If P, then Q. P. (AA) Therefore Q(AC)

AFFIRMING the antecedent in the 2nd premise + AFFIRMING consequent in the conclusion.

If P, then Q. Not Q. (DC)

Therefore not P. (DA)

DENYING the consequent in the 2nd premise + DENYING the antecedent in the conclusion.

The Invalid Options

Would you want to be a Dumb A**?

Would you like to have ACne?

If P, then Q. Not P (DA) Therefore, Not Q.

(DC)

DENYING the antecedent in the 2nd premise + DENYING the consequent in the conclusion

If P, then Q. Q. (AC) Therefore, P. (AA)

AFFIRMING the consequent in the 2nd premise + AFFIRMING the antecedent in the conclusion.

Either X or Y.   X is true, or Y is true

Either I will study or I will watch TV.  Either Buddha was right or Christ was right.  Either it is raining or the sprinklers are on.

Disjunctive Statements

A disjunctive statement asserts that at least one disjunct is true.

(there are STRICT disjuncts, where only one can be true at the same time, but

most of our disjunctive statements are “weak” i.e., BOTH could be true)

A disjunctive syllogism is valid if one premise denies one disjunct and the conclusion affirms the other.

The parts are called “disjuncts”

The Valid Disjunctive Syllogism

DENY 1 DISJUNCT IN PREMISE 2

AFFIRM THE OTHER DISJUNCT IN THE CONCLUSION

Either A or B.Not A.Therefore B.

Either A or B.Not B.Therefore A.

ONLY IF the form of the argument is VALID, AND the premises are BOTH TRUE is the conclusion true!

That is what constitutes a SOUND argument!

A VALID argument DOES NOT guarantee a true conclusion

(at least one of the premises could be false)

REMEMBER!!!