Critical Thinking 04 Soundness

A Standard of Critical Thinking

Soundness

Review

Logical conjunctions, often expressed in English by ‘and’, is true when the component claims it joins are true, otherwise it is false. It is symbolized by ‘&’. It’s logical form is P & Q.

Logical negation, often expressed in English by ‘not’, is true when the component claim is false, false when the component claim is true. It is symbolized by ‘~’ and has the logical form ~P.

Review

The Principle of Noncontradiction, states that no thing can, at the same time and in the same manner, both have and not have the same property.

Contradiction, a special form of conjunction in which a claim and its negation are joined—they are always false. The logical form of a contradiction is P & ~P.

Review

The Standard of Consistency—accept only those beliefs which are consistent with each other and any accessible evidence.

Reductio ad ridiculum, appealing to ridicule (making fun of an opposing view) rather than providing reasons against it—it is a fallacy.

Equivocation, to use a term ambiguously or vaguely in an argument—it is a fallacy.

Reviewdouble negation—any even number of negations cancel each other out.to prove a conjunction false prove that one of the component claims is false.to evaluate by

contradiction—isolate the subject and predicate, generate lists of things that fall under each, stopping when you determine that they are not identical.

proof by counterexample—Choose an item that is not on both lists, explain how the definition says it should be, then explain why it is not, indicate the inconsistency, and reject or revise the definition.

Line of Reasoning

An explanation showing that the

definition should be true of a specific example (thing or

event).

Another explanation showing that the

definition is not true of the same example.

Reject the original

definitionOriginal definition.

A Method for Reasoning with Contradictions

Proof by Counterexample

Another Line of Reasoning

Line of Reasoning

2. reasons3. conclusion

4. other reasons5. other conclusion

6. P & ~P

1. Claim

Reductio ad absurdamIndirect Proof, Proof by

Counterexample

Reductio ad absurdam

Another Line of Reasoning

7. Rejection

The Logical Form of a Reductio

Reductio ad absurdam, Indirect Proof, Proof by

Counterexample1.claim2.reasons3.conclusion4.other reasons5.other conclusion6.contradiction

7.rejection

Reviewto avoid equivocating—define key terms by giving them one (to disambiguate) clear (to avoid vagueness) meaning.

to avoid equivocating—use the Principle of Charity to settle on the best interpretation, whether normative or descriptive.

to avoid reductio ad ridiculums—use the Principle of Sufficient Reason and attempt to provide reasons for each claim.

Logical Complexity

The Conditional

A Logically Simple Truth

Sarah attends Stanford.

1 True

2 False

Two logical possibilities

Given that we’ve filled

in the indices,

made the ceteris paribus

explicit, and defined key

terms.

Combining Logically Simple Truths

Sarah attends Stanford.

Sarah goes into debt.

1 True True

2 True False

3 False True

4 False False

Four states of affairs

(states) or possible worlds

Logical Conjunction

Sarah attends Stanford

ANDSarah goes into

debt

1 True True True

2 True False False

3 False False True

4 False False False

Logical Conditional

IFSarah attends

StanfordTHE

NSarah goes into

debt

1 True ? True

2 True ? False

3 False ? True

4 False ? False

Logical Conditional

IFSarah attends

StanfordTHE

NSarah goes into

debt

1 True True True

2 True ? False

3 False ? True

4 False ? False

Logical Conditional

IFSarah attends

StanfordTHE

NSarah goes into

debt

1 True True True

2 True False False

3 False ? True

4 False ? False

Logical Conditional

IFSarah attends

StanfordTHE

NSarah goes into

debt

1 True True True

2 True False False

3 False True True

4 False True False

Logical Conditional

IFSarah attends

StanfordTHE

NSarah goes into

debt

1 True True True

2 True False False

3 False True True

4 False True False

Taking ‘if’ seriously

Logical Conditional

If Sarah goes to Stanford then she will incur debt.

If Sarah will incur debt then she goes to Stanford.

≠

Either Sarah will incur debt or she goes the Stanford.

Either Sarah goes the Stanford or she will incur debt.

=

Both Sarah will incur debt and she goes the Stanford.

Both Sarah goes the Stanford and she will incur debt.

=

Order Matters

Logical Conditional


Antecedent Consequent

The Case of Iffy AdviceSarah Scatterleigh weighed her options. She could transfer to Stanford, which had a stronger program for her major and a better track record of placing graduates into the job market. But Stanford cost quite a bit more than the school she was presently attending, Jefferson University. She sought advice from her friend, Johnny Nogginhead, musing that If I go to Stanford then I’ll go into debt.But, replied Johnny, You don’t go to Stanford.I know, said Sarah, I said If I go to Stanford….But you don’t, retorted Johnny, you go to Jefferson!I never said I didn’t, said an exasperated Sarah, I know I don’t go to Stanford, my point is that going to Stanford might mean going into debt.Why didn’t you just say that, said Johnny.

In a certain sense, ‘if’ means the antecedent isn’t true.

Logical Interpretations of ‘if’


Either Sarah doesn’t go to Stanford OR she does AND will incur debt.

It is NOT that Sarah could go to Stanford and not incur debt.

Logical Interpretations of ‘if’

IF Sarah attends Stanford THEN Sarah goes into debt

1 True True True

2 True False False

3 False True True

4 False True False

Either

Sarah doesn’t attend Stanford

OR(she does

ANDgoes into

debt)

1 False True True True True

2 False False True False False

3 True True False False True

4 True True False False False

It’s NOT that

(Sarah attends Stanford

AND doesn't go into debt)

1 True True False False

2 False True True True

3 True False False False

4 True False False True

Truth Values match line for

line (across

all possible worlds)

Logical conditional, often expressed in English by ‘if…then….’, is true when the antecedent is true and the consequent is false, otherwise it is true. It is symbolized by ‘⊃’. It’s logical form is P ⊃ Q.

Logical Form of Conditionals

IF P THEN QP ⊃ QP → Q

P ⊃ Q

1 TrueTrue

True

2 TrueFals

eFalse

3 FalseTrue

True

4 FalseTrue

False

to prove a conditional false

Prove that the antecedent is true while the consequent is false.

to interpret conditionals logically

translate it as ‘Either not p or (p and q)’ or ‘It not that (p and not q)

indicators for conditionals

a. if p then qb. q if pc. p only if qd. not p unless qe. supposing p, q

f. imagine p ... qg. assuming p, qh. all p are qi. whenever p, qj. when p, q

A Standard of Critical Thinking

Validity

Argument, a set of claims in which some claims (premises) are offered to show the truth (or falsehood) of another claim (the conclusion). A line of reasoning.

Arguments

Lines of reasoning

If it is red then it has color,if it has color then it emits or reflects a wavelength of light,thus if it is red then it emits or reflects a wavelength of light.

When a government abuses rights it ought to be removed.The king abuses rights and so he ought to be removed.

If it is a mammal then it gives live birth.It lays eggs.So it’s not a mammal.

If living pigeons didn’t all come from rock pigeons then they must have come from other kinds of pigeons.There are no other kinds of pigeons.This established they all come from rock pigeons.

If anything is a dog then it is a mammal.If anything is a mammal then it is an animal.which proves that if anything is a dog then it is an animal.

When water is heated to 212° it boils.It’s not boiling, which demonstrates it hasn’t been heated to 212°.

All dogs are mammals.All mammals are animals.Hence all dogs are animals.

indicators for premises

a.asb.as shown byc.becaused.deduce frome.derive fromf. finally, the last reasong.first, second, third,…

nexth.follows fromi. forj. inasmuch as

k.indicated byl. is the reason thatm.it is the case thatn.may be deduced fromo.may be derived fromp.may be inferred fromq.one reason being…r. sinces. the fact thatt. the reason

indicators for conclusions

a.as a resultb.consequentlyc.demonstratesd.entailse.establishesf. henceg.I conclude thath.impliesi. in conclusionj. infer

k.it follows thatl. justifies m.meansn.proveso.showsp.soq.thenr. therefores. thus

Sets of Claims

Are Without Any Particular Order

Today is either Tuesday or Thursday.

She has chemistry today. She will recognize her teacher.

She will recognize her classmates.

The class meets in the same room.

She’s not dreaming.

The alarm did not go off.

Today is Monday.

Sarah’s beliefs

She went to the right room.

She is late.

Arguments

Have Order (‘∴’ means therefore)

1.If it is red then it has color,2.if it has color then it emits or reflects a wavelength of

light,3. ∴ If it is red then it emits or reflects a wavelength of

light.

1.When a government abuses rights it ought to be removed.

2.The king abuses rights and 3. ∴ He ought to be removed.

1.If it is a mammal then it gives live birth.

2.It lays eggs.3. ∴ It’s not a mammal.

1.If living pigeons didn’t all come from rock pigeons then they must have come from other kinds of pigeons.

2.There are no other kinds of pigeons.

3. ∴ They all come from rock pigeons.

1. If anything is a dog then it is a mammal.

2. If anything is a mammal then it is an animal.

3. ∴ If anything is a dog then it is an animal.

1.When water is heated to 212° it boils.

2.It’s not boiling, 3. ∴ It hasn’t been heated to 212°.

1. All dogs are mammals.2. All mammals are animals.3. ∴ All dogs are animals.

Validity, if the premises are true then the conclusion is true.

Validity, either the premises are false, or they are true and so it the conclusion.

…it is not possible that the premises are true while the conclusion is false.

Another Emergent Property

Wetness emerges as a property of water when hydrogen and oxygen are properly combined—though neither are wet themselves. In a similar manner, validity emerges when claims are properly structured into an argument.

Validity

Validity

Versus Truth

Validity Truth

Applies to whole arguments

Applies to claims, both simple and

complex

Does not apply to claims

Does not apply to arguments

As Technical Terms

Valid Arguments

1If anything is a dog then it is mammal.

2If anything is a mammal then it is an animal.

∴3If anything is a dog then it is an animal.

Premises

Conclusion

✔ 1If anything is a dog then it is mammal.

✔ 2If anything is a mammal then it is an animal.

✔ ∴3If anything is a dog then it is an animal.

animals

mammals

dogs

��

��

��

��

��

��

��

��

��

Here, by premise 1, no dog can be at the bottom of the blue (it is outside of mammals). By premise 2 no mammal can be at the bottom of the green (it is outside of the animals). So there is no place left for a dog to be.

Valid Arguments

✔ 1If anything is a dog then it is mammal.

✔ 2If anything is a mammal then it is an animal.

✔ ∴3If anything is a dog then it is an animal.

So this argument is valid

Valid Arguments

1 If anything is a hectagon then it has more sides than a chiliogon.

2 If anything is a chiliogon then it has more sides than a megagon.

∴3 If anything is a hectagon then it has more sides than a megagon.

Premises

Conclusion

If:hectagon means 1,000,000 sided, and;chiliogon means 1,000 sided, and;megagon means 100 sided;then the conclusion would have to be true.

Valid Arguments

1 If anything is a hectagon then it has more sides than a chiliogon.

2 If anything is a chiliogon then it has more sides than a megagon.


So this argument is valid

Valid Arguments

But:megagon means 1,000,000 sided, andhectagon means 100 sided (chiliogon does mean 1,000 sided)so the premises are in fact false.

✘ 1 If anything is a hectagon then it has more sides than a chiliogon.

✘ 2 If anything is a chiliogon then it has more sides than a megagon.


Valid Arguments




But this argument is valid, because if the premises were true then the conclusion

would be true too.

Valid Arguments




Test it by replacing ‘hectagon’, ‘chiliogon’ and ‘megagon’ with ‘triangle’, rectangle’,

and ‘octogon’.

Valid Arguments

Valid Arguments

Both Arguments have the Form:

1.P ⊃ Q2.Q ⊃ R3. ∴ P ⊃ R

✔ 1 If anything is a camel then it has four legs.

✔ 2 If anything is a pig then it has four legs.

✘ ∴3 If anything is a pig then it is a camel.

This argument is invalid, because even if the premises are true then the conclusion is

not.

Invalid Arguments

This Argument Has the Form:

Invalid Arguments

P ⊃ RQ ⊃ R

∴ P ⊃ Q

It is not a valid form.

to determine validity

check to see if the form of the argument fits one of the valid patterns.

Yes If it is…

If it is not…

…then it is invalid

Is it possible for the premises to be true while the conclusion is false?

Validity: The Test

No …then it is valid

Yes If it does…

If it does not…

…then it is valid

Does the argument have a valid form?

Validity: A Quick Check

No …then it is invalid

A Complex Standard of Critical Thinking

Soundness

Soundness, valid arguments with true premises.

argumentsA Taxonomy

Arguments

valid arguments

sound arguments

1. If anything is a dog then it is has four legs.

2. If anything is a cat then it is has four legs.

3. ∴ If anything is a dog then it is a cat.

1. If anything is a pig then it is a quadruped.

2. Trakr is a quadruped.3. ∴ Trakr is a pig.

1. If anything is a camel then it has four legs.

2. If anything is a pig then it has four legs.

3. ∴ If anything is a pig then it is a camel.

1. If anything is a hectagon then it has more sides than a chiliogon.

2. If anything is a chiliogon then it has more sides than a megagon.

3. ∴ If anything is a hectagon then it has more sides than a megagon.

1. If anything is a dog then it is mammal.2. If anything is a mammal then it is an

animal.3. ∴ If anything is a dog then it is an

animal.

1. If anything is a camel then it is a has humps.

2. Thor has no humps.3. ∴ Thor is not a camel.

1. If anything is wild then it is free.

2. Peter is not wild.3. ∴ Peter is not free.

Yes Are the premises true?

it is sound

Is the argument valid?

Soundness

it is unsound

Yes

No

No

to determine soundness

check to see if the form of the argument fits one of the valid patterns, then check to see if the premises are true.

Some Common Forms

Validity

Chain Arguments

1. If anything is a dog then it is mammal.2. If anything is a mammal then it is an

animal.3. ∴ If anything is a dog then it is an

animal.

Modus Ponens

Modus Tollens

1. When a government abuses rights it ought to be removed.

2. The king abuses rights .3. ∴ He ought to be removed.

1. If it is a mammal then it gives live birth.

2. It lays eggs.3. ∴ It’s not a mammal.

A Common Form

Chain Argument

Chain Arguments

1.If anything is a dog then it is mammal.

2.If anything is a mammal then it is an animal.


The Parts of a Chain Argument

Chain Argument

If anything is a mammal then it is an animal.

If anything is a dog then it is mammal.

∴ If anything is a dog then it is an animal.

1.Conditional Premise2.Conditional Premise3.Conditional Conclusion

The Structure of a Chain Argument

Chain Argument

1.If anything is a dog then it is a mammal.2.If anything is a mammal then it is an animal.


The conclusion has the same antecedent as the first premise…

The conclusion has the same consequent as the last premise…

The antecedents and consequents of the premises link up as in a chain.

Chain Arguments

Also called ‘hypothetical syllogisms’

1.P ⊃ Q2.Q ⊃ R3. ∴ P ⊃ R

Chain Arguments

Can have indefinitely many premises

1. P ⊃ Q2. Q ⊃ R3. R ⊃ S4. S ⊃ T5. ∴ P ⊃ T

to calculate the number of possible worlds

raise two to the power of the number of claims being evaluated, here there are three: P, Q, & R

23 = 2 • 2 • 2 = 8

Proving Chain Arguments Valid

(P ⊃ Q) (Q ⊃ R) (P ⊃ R)

1 True True True True True True

2 True True TrueFals

eTrue

False

3 TrueFals

eFals

eTrue True True

4 TrueFals

eFals

eFals

eTrue

False

5Fals

eTrue True True

False

True

6Fals

eTrue True

False

False

False

7Fals

eFals

eFals

eTrue

False

True

8Fals

eFals

eFals

eFals

eFals

eFals

e

Step One: Assign values to the simplest atomic claims, P, Q, & R

P PQ Q R R

(P ⊃ Q) (Q ⊃ R) (P ⊃ R)

1 True True True True True True True True True

2 True True True TrueFals

eFals

eTrue

False

False

3 TrueFals

eFals

eFals

eTrue True True True True

4 TrueFals

eFals

eFals

eTrue

False

TrueFals

eFals

e

5Fals


False

True True

6Fals

eTrue True True

False

False

False

TrueFals

e

7Fals

eTrue

False

False

True TrueFals

eTrue True

8Fals

eTrue

False

False

TrueFals

eFals

eTrue

False

Step Two: Determine the values of the next simplest or molecular claims.


(P ⊃ Q) (Q ⊃ R) (P ⊃ R)

1 True True True✔True True True✔True True True


eFals

eTrue

False

False

3 TrueFals

eFals

eFals


4 TrueFals

eFals

eFals

eTrue

False

TrueFals

eFals

e

5Fals

eTrue True✔True True True✔

False

True True

6Fals

eTrue True True

False

False

False

TrueFals

e

7Fals

eTrue

False ✔

False

True True✔Fals

eTrue True

8Fals

eTrue

False ✔

False

TrueFals

e ✔Fals

eTrue

False

Step Three: Determine if there is a possible world where the premises are both true while the conclusion is false.


(P ⊃ Q) (Q ⊃ R) (P ⊃ R)



eFals

eTrue

False

False

3 TrueFals

eFals

eFals


4 TrueFals

eFals

eFals

eTrue

False

TrueFals

eFals

e

5Fals


False

True True

6Fals

eTrue True True

False

False

False

TrueFals

e

7Fals

eTrue

False ✔

False

True True✔Fals

eTrue True

8Fals

eTrue

False ✔

False

TrueFals

e ✔Fals

eTrue

False

There is no possible world where the premises are true while the conclusion is false.

So Chain Arguments are valid.


An Unnamed Fallacy

(P ⊃ R) (Q ⊃ R) (P ⊃ Q)


2 TrueFals

eFals

eTrue

False

False

True True True

3 True True True✔Fals

eTrue True ✘ True

False

False

4 TrueFals

eFals

eFals

eTrue

False

TrueFals

eFals

e

5Fals


False

True True

6Fals

eTrue

False

TrueFals

eFals

eFals

eTrue True

7Fals

eTrue True✔

False

True True✔Fals

eTrue

False

8Fals

eTrue

False ✔

False

TrueFals

e ✔Fals

eTrue

False

There is a possible world where the premises are true while the conclusion is false.

So arguments of this form are invalid.

A Common Form

Modus Ponens

modus ponens


2.The king abuses rights .3. ∴ He ought to be removed.

The Parts of a Modus Ponens Argument

Modus Ponens

The king abuses rights .

When a government abuses rights it ought to be removed.

∴ He ought to be removed.

1.Conditional Premise2.Premise Affirming the Antecedent of the Conditional

3.Concluding the Consequent of the Conditional

The Structure of a Modus Ponens Argument

Modus Ponens


2.The king abuses rights .3. ∴ He ought to be removed.

A conditional premise.

The conclusion is the consequent of the conditional premise.

A premise which affirms the antecedent of the conditional premise.

Modus Ponens

Also called ‘Affirming the Antecedent’ and ‘Conditional

Elimination’

1.P ⊃ Q2.Q3. ∴ P

Modus Ponens

Can be extended by Chain Argument

1. P 2. P ⊃ Q3. Q ⊃ R4. R ⊃ S5. ∴ S


raise two to the power of the number of claims being evaluated, here there are two: P & Q

22 = 2 • 2 = 4

Proving Modus Ponens Valid

(P (P ⊃ Q) Q

1 True True True True


3 False False True True

4 False False False False

Step One: Assign values to the simplest atomic claims, P & Q

P P Q Q


(P (P ⊃ Q) Q

1 True True True True True


3 False False True True True




(P (P ⊃ Q) Q

1 True ✔ True True True ✔ True






(P (P ⊃ Q) Q






So Modus Ponens Arguments are valid.

An Attendant Fallacy: Affirming the Consequent


So arguments which affirm the consequent are invalid—and so such arguments are fallacies

(Q (P ⊃ Q) P


2 False True False False True

3 True ✔ False True True ✘ False


A Common Form

Modus Tollens

modus tollens

1.If it is a mammal then it gives live birth.

2.It lays eggs.3. ∴ It’s not a mammal.

The Parts of a Modus Tollens Argument

Modus Tollens

It lays eggs.

If it is a mammal then it gives live birth.

∴ It’s not a mammal.

1.Conditional Premise2.Premise Denying the Consequent of the Conditional3.Concluding the Denial of the Antecedent of the Conditional

The Structure of a Modus Tollens Argument

Modus Ponens

1.If it is a mammal then it gives live birth.2.It lays eggs.3. ∴ It’s not a mammal.

A conditional premise.

The conclusion is the denial of the antecedent of the conditional premise.

A premise which denies the consequent of the conditional premise.

Modus Tollens

Also called ‘Denying the Consequent’

1.P ⊃ Q2.~Q3. ∴ ~P

Modus Tollens

Can be extended by Chain Argument

1. P ⊃ Q2. Q ⊃ R3. R ⊃ S4. ~S 5. ∴ ~P



22 = 2 • 2 = 4

Proving Modus Tollens Valid

(~Q (P ⊃ Q) ~P

1 False True True False


3 False False True True

4 True False False True

Step One: Assign values to the simplest atomic claims, P & Q, keeping track of negation.

~PP Q~Q


(~Q (P ⊃ Q) ~P

1 False True True True False



4 True False True False True



(~Q (P ⊃ Q) ~P




4 True ✔ False True False ✔ True




So Modus Tollens Arguments are valid.

(~Q (P ⊃ Q) ~P




4 True ✔ False True False ✔ True

An Attendant Fallacy: Denying the Antecedent


So arguments which affirm the consequent are invalid—and so such arguments are fallacies

(~P (P ⊃ Q) ~Q


2 False True False False True

3 True ✔ False True True ✘ False

4 True False True False True

Logical Complexity

Disjunction

A Logically Simple Truth

The coast is foggy.

1 True

2 False

Two logical possibilities

Given that we’ve filled

in the indices,

made the ceteris paribus

explicit, and defined key

terms.

Combining Logically Simple Truths

The coast is foggy. The coast is sunny.

1 True True

2 True False

3 False True

4 False False

Four states of affairs

(states) or possible worlds

Logical Conjunction

The coast is foggy. AND The coast is sunny.

1 True True True

2 True False False

3 False False True

4 False False False

Logical Conditional

IF The coast is foggy.THE

NThe coast is

sunny.

1 True True True

2 True False False

3 False True True

4 False True False

Logical Disjunction

The coast is foggy. OR The coast is sunny.

1 True ? True

2 True ? False

3 False ? True

4 False ? False

Logical Disjunction


1 True ? True

2 True ? False

3 False ? True

4 False False False

Logical Disjunction


1 True ? True

2 True True False

3 False True True

4 False False False

Logical Disjunction


1 TrueTrue/False

True

2 True True False

3 False True True

4 False False False

The Ambiguity of ‘or’

Logical Disjunction

Exclusive ‘or’ Inclusive ‘or’

Either the Giants win the division or the A’s do (but

not both)

Either the Giants make the playoffs or the A’s do (or

both)

Either heads or tails (but not both)

Either by plane or by car (or both)

Latin: aut Latin: vel

Logical Disjunction


1 True True True

2 True True False

3 False True True

4 False False False

Logic settles on an inclusive way

Logical disjunction, often expressed in English by ‘Either…or….’, is false when the both components are false, otherwise it is true. It is symbolized by ‘V’. It’s logical form is P V Q.

Logical Form of Disjunctions

Either P OR QP V Q

P V Q

1 TrueTrue

True

2 TrueTrue

False

3 FalseTrue

True

4 FalseFals

eFalse

to prove a disjunction true

Prove that one of the component claims is true.

to interpret a disjunction

specify if you are using it inclusively or exclusively.

Tautology

Eleven is a prime number oreleven is not a prime number.

Either Jacqui thinks black is more alluring than pink or she doesn’t.

The music is loud or the music is quiet.*

Jupiter is bigger than Mars or it is not bigger than Mars.

The Constitution of the United States was adopted on either September 17, 1787 or July 4, 1776.*

Romeo and Juliette is a tragedy or it is not a tragedy.*

New York either is or isn’t the largest city in the US.*

Hockey is better than basketball but it is not better than basketball.*

Putting Negation and Disjunction Together

Same-sex schools are optimal unless same-sex schools are less than optimal.

Drinking milk is healthy or unhealthy.*

The jellyfish has tentacles—or not!The child looks at the jellyfish or looks away from it*.

Which claim is not a disjunction?

Tautologies

The square is white

VThe square is not

white

1 True ? False

2 False ? True

The Logical Form of a Tautology: P V ~P

Given that disjunctions are false when all component claims are false, what is the truth value of this

disjunction?

Tautologies

The square is white

VThe square is not

white

1 TrueTrue

False

2 FalseTrue

True

Tautologies are true in all possible worlds.


Tautology

P V ~P

1 TrueTrue

False

2 FalseTrue

True


Tautologies are true in all possible worlds.

Tautology, a special form of disjunction in which a claim and its negation are joined—they are always true. The logical form of a tautology is P V ~P.

Tautology

T V ~T

1 True True False

2 False True True

The Logical Form of the Principle of Sufficient Reason

The Principle of Sufficient Reason covers all possible worlds.

For every claim, give a reason why it is true or not true.

Is drinking milk healthy for humans?

Controlling the Question

Has the Constitutional right to bear arms outlived its

usefulness?Are single-sex schools better for

education?What is the best method of

education?

What Constitutional rights should we keep?

What are the healthiest drinks for humans?

Controlling the QuestionOpen

QuestionsYes-or-no Questions

Is drinking milk healthy for humans?

Has the Constitutional right to bear arms

outlived its usefulness?

Are single-sex schools better for

education?

What is the best method of education?

What Constitutional rights should we

keep?


Open Questions Yes-or-no Questions

What is the best method of education?

What Constitutional rights should we

keep?


Are Topic or Theme Questions

A Topic Question: an open question. Such questions require disjunctive reasoning to treat the alternates.

Reasons supporting this alternate or

refuting the others.





the question

1st alternate:

Which alternate has the better

reasons?

Disjunctive Reasoning

2nd alternate:

Final alternate:

.

.

....

Reasons supporting electrolyte solutions or


Reasons supporting milk or refuting the

others.

Reasons supporting water or refuting

the others.Water:

Which alternate has the better

reasons?

Disjunctive Reasoning

Milk:

Electrolyte Solutions

.

.

....


Some Common Forms Involving Disjunctions

Validity

Disjunctive Argument

1. Either Kierkegaard can be a Christian or a philosopher.

2. He cannot be a philosopher.3. ∴ So he must be a Christian.

Simple Dilemma

Dilemma

1. If Johnny’s friendship is for pleasure then he is not a true friend.

2. If Johnny’s friendship is for utility then he is not a true friend.

3. Either Johnny’s friendship is for pleasure or utility.

4. ∴ Johnny’s friendship is not a true friendship.

1. If existence precedes essence then humanity is free.

2. If there is no God then we we alone can justify ourselves, without excuse.

3. Either existence precedes essence or there is not God.

4. ∴ Either humanity is free or is without any justifications or excuses but those they provide.

A Common Form



1.Either Kierkegaard can be a Christian or a philosopher.

2.He cannot be a philosopher.3. ∴ So he must be a Christian.

The Parts of a Disjunctive Argument


He cannot be a philosopher.

Either Kierkegaard can be a Christian or a philosopher.

∴ So he must be a Christian.

1.Disjunctive Premise.2.Premise Denying one of the disjuncts of the Disjunction.

3.Concluding the remaining Disjunct.

The Structure of a Disjunctive Argument

1.Either Kierkegaard can be a Christian or a philosopher.

2.He cannot be a philosopher.3. ∴ So he must be a Christian.

A disjunctive premise.

The conclusion is the remaining component claim of the disjunctive premise.

A premise which denies one of the component claims of the disjunctive premise.



More commonly called ‘Disjunctive Syllogism’ and also called ‘modus tollendo

ponens'

1. P V Q2. ~Q3. ∴ P

1. P V Q2. ~P3. ∴ Q

Can be run either way

Disjunctive Syllogism

Can be extended indefinitely

1. P V Q V R V S2. ~Q3. ~R4. ~S5. ∴ P



22 = 2 • 2 = 4

Proving Disjunctive Argument Valid

~P (P V Q) Q

1 False True True True

2 False True False False

3 True False True True

4 True False False False

Step One: Assign values to the simplest atomic claims, P & Q, minding the negations.

~P P Q Q


~P (P V Q) Q


2 False True True False False

3 True False True True True

4 True False False False False



~P (P V Q) Q



3 True ✔ False True True ✔ True




So Disjunctive Arguments are valid.


~P (P V Q) Q



3 True ✔ False True True ✔ True


A Common Form

Simple Dilemma

Simple Dilemma

1.If Johnny’s friendship is for pleasure then he is not a true friend.

2.If Johnny’s friendship is for utility then he is not a true friend.

3.Either Johnny’s friendship is for pleasure or utility.

4. ∴ Johnny is not a true friend.

The Parts of a Simple Dilemma

Simple Dilemma

If Johnny’s friendship is for pleasure then he is not a true friend.

1. Disjunctive Premise.2. A Conditional Premise whose antecedent is one of the disjuncts of the

Disjunctive Premise and whose consequent is the same as the other Conditional Premise.

3. Another Conditional Premise whose antecedent is the other disjunct of the Disjunctive Premise and whose consequent is the same as the other Conditional Premise.

4. Concluding the Consequent of the Conditional Premises.

∴ Johnny is not a true friend.

If Johnny’s friendship is for utility then he is not a true friend.

Either Johnny’s friendship is for pleasure or utility.

The Structure of a Simple Dilemma

1.If Johnny’s friendship is for pleasure then he is not a true friend.

2.If Johnny’s friendship is for utility then he is not a true friend.3.Either Johnny’s friendship is for pleasure or utility.4. ∴ Johnny is not a true friend.


The conclusion is the consequent of the conditional premises.

Both Conditional Premises share a consequent.

Simple Dilemma

One antecedent is a component of the disjunction.

The other antecedent is the other component of the disjunction.

Simple Dilemma

Also called ‘Disjunctive Elimination’

1. P V Q2. P ⊃ R3. Q ⊃ R4. ∴ R

Simple Dilemma


1. P V Q V R2. P ⊃ S3. Q ⊃ S4. R ⊃ S5. ∴ S

Destructive Dilemma

Simple Dilemma combined with Modus Tollens

1. ~R2. P ⊃ R3. Q ⊃ R4. ∴ ~P V Q



23 = 2 • 2 • 2 = 8

Proving Simple Dilemma Valid

(P V Q) (P ⊃ R) (Q ⊃ R) R

1 True True True True True True True

2 True True TrueFals

eTrue

False

False

3 TrueFals

eTrue True

False

True True

4 TrueFals

eTrue

False

False

False

False

5 False

TrueFals

eTrue True True True

6 False

TrueFals

eFals

eTrue

False

False

7 False

False

False

TrueFals

eTrue True

8 False

False

False

False

False

False

False

Step One: Assign values to the simplest atomic claims, P, Q, & R

P PQ QR RR


(P V Q) (P ⊃ R) (Q ⊃ R) R

1 True True True True True True True True True True


eFals

eTrue

False

False

False

3 True TrueFals

eTrue True True

False

True True True

4 True TrueFals

eTrue

False

False

False

TrueFals

eFals

e

5 False

True TrueFals

eTrue True True True True True

6 False

True TrueFals

eTrue

False

TrueFals

eFals

eFals

e

7 False

False

False

False

True TrueFals

eTrue True True

8 False

False

False

False

TrueFals

eFals

eTrue

False

False



(P V Q) (P ⊃ R) (Q ⊃ R) R

1 True True True✔True True True✔True True True✔True


eFals

eTrue

False

False

False

3 True TrueFals

e ✔True True True✔

False

True True✔True

4 True TrueFals

eTrue

False

False

False

TrueFals

eFals

e

5 False

True True✔Fals

eTrue True✔True True True✔True

6 False

True TrueFals

eTrue

False

TrueFals

eFals

eFals

e

7 False

False

False

False

True TrueFals

eTrue True True

8 False

False

False

False

TrueFals

eFals

eTrue

False

False



So Simple Dilemmas are valid.


(P V Q) (P ⊃ R) (Q ⊃ R) R

1 True True True✔True True True✔True True True✔True


eFals

eTrue

False

False

False

3 True TrueFals

e ✔True True True✔

False

True True✔True

4 True TrueFals

eTrue

False

False

False

TrueFals

eFals

e

5 False

True True✔Fals

eTrue True✔True True True✔True

6 False

True TrueFals

eTrue

False

TrueFals

eFals

eFals

e

7 False

False

False

False

True TrueFals

eTrue True True

8 False

False

False

False

TrueFals

eFals

eTrue

False

False

A Common Form

Dilemma

Dilemma

1.If existence precedes essence then humanity is free.

2.If there is no God then we we alone can justify ourselves, without excuse.

3.Either existence precedes essence or there is no God.

4. ∴ Either humanity is free or is without any justifications or excuses but those they provide.

If there is no God then we we alone can justify ourselves, without excuse.

The Parts of a Dilemma

Dilemma

If existence precedes essence then humanity is free.

1. Disjunctive Premise.2. A Conditional Premise whose antecedent is one of the disjuncts of the

Disjunctive Premise.3. Another Conditional Premise whose antecedent is the other disjunct of the

Disjunctive Premise.4. Concluding a Disjunction of the Consequents of the Conditional Premises.

∴ Either humanity is free or is without any justifications or excuses but those they provide.

Either existence precedes essence or there is no God.

The Structure of a Dilemma

1. If existence precedes essence then humanity is free.2. If there is no God then we we alone can justify ourselves, without

excuse.3. Either existence precedes essence or there is no God.4. ∴ Either humanity is free or is without any justifications or

excuses but those they provide.


The conclusion is a disjunction of the consequents of the conditional premises.

Dilemma

One antecedent is a component of the disjunction.

The other antecedent is the other component of the disjunction.

Dilemma

Also called ‘Constructive Dilemma’

1. P V Q2. P ⊃ R3. Q ⊃ S4. ∴ R V S

Simple Dilemma


1. P V Q V R2. P ⊃ S3. Q ⊃ T4. R ⊃ U5. ∴ S V T V U

Destructive Dilemma

Dilemma combined with Modus Tollens

1. ~R V ~S2. P ⊃ R3. Q ⊃ S4. ∴ ~P V ~Q

Destructive Dilemma

Dilemma combined with Modus Tollens and a Tautology

1. ~R V ~R2. P ⊃ R3. Q ⊃ R4. ∴ ~P V ~Q



24 = 2 • 2 • 2 • 2 = 16

Proving Dilemma Valid

(P V Q) (P ⊃ R) (Q ⊃ S) R V S

1 True True True True True True True True

2 True True True True True False True False

3 True True True False True True False True

4 True True True False True False False False

5 True False True True False True True True

6 True False True True False False True False

7 True False True False False True False True

8 True False True False False False False False

9 False True False True True True True True

10 False True False True True False True False

11 False True False False True True False True

12 False True False False True False False False

13 False False False True False True True True

14 False False False True False False True False

15 False False False False False True False True

16 False False False False False False False False

Step One: Assign values to the simplest atomic claims, P, Q, R, & S

RP Q SP Q R S


(P V Q) (P ⊃ R) (Q ⊃ S) R V S

1 True True True True True True True True True True True True

2 True True True True True True True False False True True False

3 True True True True False False True True True False True True

4 True True True True False False True False False False False False

5 True True False True True True False True True True True True

6 True True False True True True False True False True True False

7 True True False True False False False True True False True True

8 True True False True False False False True False False False False

9 False True True False True True True True True True True True

10 False True True False True True True False False True True False

11 False True True False True False True True True False True True

12 False True True False True False True False False False False False

13 False False False False True True False True True True True True

14 False False False False True True False True False True True False

15 False False False False True False False True True False True True

16 False False False False True False False True False False False False



(P V Q) (P ⊃ R) (Q ⊃ S) R V S

1 True True True ✔ True True True ✔ True True True ✔ True True True




5 True True False ✔ True True True ✔ False True True ✔ True True True

6 True True False ✔ True True True ✔ False True False ✔ True True False



9 False True True ✔ False True True ✔ True True True ✔ True True True


11 False True True ✔ False True False ✔ True True True ✔ False True True








(P V Q) (P ⊃ R) (Q ⊃ S) R V S

1 True True True ✔ True True True ✔ True True True ✔ True True True




5 True True False ✔ True True True ✔ False True True ✔ True True True

6 True True False ✔ True True True ✔ False True False ✔ True True False



9 False True True ✔ False True True ✔ True True True ✔ True True True


11 False True True ✔ False True False ✔ True True True ✔ False True True







So Dilemmas are valid.

A Relevant Fallacy

Fallacies

False Dilemma, to provide a non exhaustive disjunction as a premise—it is a fallacy.

to avoid false dilemma

Provide an exhaustive list of the possible answers to the topic question, listing explicitly those you may not wish to treat.

Ethics

Assignment

How do you come to an ethical decision?

What is hypocrisy?

What does it mean to be ethical?

Education

Critical Thinking 04 Soundness