Philosophy 103Linguistics 103

Yet, still, even further more, expanded,

Introductory Logic: Critical Thinking

Dr. Robert Barnard

Last Time:• Definitions

– Lexical– Theoretical– Precising– Pursuasive

• Logical Form• Form and Validity

Plan for Today

• Deductive Argument Forms• Formal Fallacies• Counter-Example Construction

Validity and Form• Deductive Validity – IF the premises are true

THEN the conclusion MUST be true.• Deductive Soundness – the deductive

argument is valid AND premises are all true• Form - The structure of an argument. Validity

is a Property of Form.

Common Deductive Logical Forms

• Modus Ponens• Modus Tollens• Disjunctive Syllogism• Hypothetical Syllogism• Reductio Ad Absurdum

Common Logical Forms

• Modus Ponens

If P then Q, P --- Therefore Q

• Modus Tollens

If P then Q, Q is false --- Therefore P is false

Modus Ponens Example

If P then Q, P --- Therefore Q

If Peter is from Ohio then Peter is an AmericanPeter is from Ohio --- Therefore Peter is an American.

Modus Tollens ExampleIf P then QQ is false

Therefore P is false

If Paul is a potter then Paul has worked with clay Paul has not worked with clay. Therefore Paul is not a potter.

Common Logical Forms• Disjunctive Syllogism P or Q, P is false --- Therefore Q

• Hypothetical Syllogism

If P then Q , If Q then R --- Therefore If P then R

Disjunctive Syllogism Example

P or QP is false

Therefore Q

Pizza is yummy or Quiche is manly.Pizza is not yummy.Therefore Quiche is manly.

Inclusive OR vs Exclusive OR

Assume: Tom is a Lawyer or Tom is a Doctor

If Tom is a Lawyer does that require that he is not a Doctor?

Inclusive-OR: No - (Lawyer and/or Doctor)

Exclusive- OR: Yes - ( Either doctor or lawyer, not both)

Hypothetical Syllogism Example

If P then QIf Q then R

Therefore If P then R

If Pigs fly then Cows kiss.If Cows kiss then Otters sing.Therefore If Pigs fly then Otters sing

Common Forms• Reductio Ad Absurdum

(Reduces to Absurdity)

a) Assume that Pb) On the basis of the assumption if you can prove ANY contradiction, then you may infer that P is false

Case of : Thales and Anaximander

Thales and Anaximander

• Arché - Table of Elements

- Thales: Water - Anaximander: Aperion

The Presocratic Reductio1. Everything is Water (Thales’ Assumption)2. If everything is water then the universe contains an

infinite amount of water and nothing else. (From 1)3. If there is more water than fire in a place, then the

water extinguishes the fire. (observed truth)4. We observe fire. (observed truth)5. Where we observe fire there must be more fire than

water. (from 3 & 4)6. Therefore, everything is water and something is not

water (Contradiction from 5 and 1)7. Thus, (1) is false.

Common Formal Fallacies

• Affirming the Consequent• Denying the Antecedent• Illicit Hypothetical Syllogism• Illicit Disjunctive Syllogism

Common Formal Fallacies

• Affirming The Consequent

If P then Q, Q --- Therefore P

• Denying the Antecedent

If P then Q, P is false --- Therefore Q is false

Affirming the Consequent

If P then QQ is true

Therefore P

1. If it rained last night then the grass is wet

2. The grass is wet.3. Therefore, it rained last night.

Denying the Antecedent

If P then QP is false

Therefore Q is false1. If Tom is not hungry then Tom ate lunch2. Tom is Hungry3. Therefore Tom did not eat lunch.

Common Formal Fallacies

• Illicit Disjunctive Syllogism -P or Q, P is true -- Therefore not-Q

-P or Q, Q is true -- Therefore not-P• Illicit Hypothetical Syllogism(*)

If P then not-Q , If Q then not-R --- Therefore If P then not-R

* - there is more than one form of IHS

Illicit Disjunctive Syllogism

P or QP is true

Therefore not-Q

John is Tim’s father or Sally is Tim’s mother

John is Tim’s FatherTherefore Sally is not Tim’s mother

Illicit Hypothetical Syllogism

If P then not-QIf Q then not-R

Therefore If P then not-R

1. If I like fish then I won’t eat beef2. If I eat beef then I won’t eat cheese3. Therefore, If I like fish then I won’t eat

cheese.

Testing for Validity

The central question we ask in deductive logic is this: IS THIS ARGUMENT VALID?

To answer this question we can try several strategies (including):

a)Counter-example (proof of invalidity)b)Formal Analysis

Counter-Example Test for Validity

1) Start with a given argument2) Determine its form

(Important to do correctly – best to isolate conclusion first)

3) Formulate another argument:a) With the same formb) with true premisesc) with a false conclusion.

An example counter-example…

1. If Lincoln was shot, then Lincoln is dead.

2. Lincoln is dead.3. Therefore, Lincoln was

shot.

The FORM IS:1. If Lincoln was shot,

then Lincoln is dead.2. Lincoln is dead.3. Therefore, Lincoln was

shot.

1. IF --P-- , THEN --Q--.

2. --Q--

3.Therefore -- P--

NEXT: We go from FORM back to ARGUMENT…

1. IF --P-- , THEN --Q--.

2. --Q--3. Therefore -- P--

1. IF Ed passes Phil 101, then Ed has perfect attendance.

2. Ed has perfect attendance.

3. Therefore, Ed Passes Phil 101

NO WAY!

Ed’s Perfect Attendance does NOT make it necessary that Ed pass PHIL 101.

SO: Even if it is true that 1. IF Ed passes Phil 101, then Ed has perfect

attendance.

2. ..AND that..Ed has perfect attendance.

IT DOES NOT FOLLOW THAT ED MUST PASS PHIL 101!

It is possible to have perfect attendance and not pass

•It is also possible to pass and have imperfect attendance

This shows that the original LINCOLN argument is INVALID.

This is ED…

Another Example?1. All fruit have seeds2. All plants have seeds3. Therefore, all fruit are plants

Form:All F are SAll P are STherefore All F are P

Another example….cont.Form:All F are SAll P are STherefore All F are P

1.All Balls (F) are round (S).

2.All Planets (P) are round (S).

3. Therefore, All Balls (F)are (P)lanets.

Formal Evaluation?

The counter-example test for validity has limits.• Counter-Examples should be obvious.• Our ability to construct an Counter-Example is

limited by our concepts and imagination.• Every invalid argument has a possible counter-

example, but no human may be able to find it.