Argument Analysis

Truth-Table Test for Validity

Example

Consider the argument (P & ~Q) (Q → P).├

Premise: (P & ~Q)Conclusion: (Q → P)

Write Down ALL the PossibilitiesP QT TT FF TF F

Write Down PremisesP Q (P & ~Q)T TT FF TF F

Write Down ConclusionP Q (P & ~Q) (Q → P)T TT FF TF F

Write Down Truth-Table for PremiseP Q (P & ~Q) (Q → P)T T FT F TF T FF F F

Write Down Truth-Table for Conclusion

P Q (P & ~Q) (Q → P)T T F TT F T TF T F FF F F T

Find ALL the Lines Where ANY Premise is F

P Q (P & ~Q) (Q → P)T T F TT F T TF T F FF F F T

Ignore Them!P Q (P & ~Q) (Q → P)T T F TT F T TF T F FF F F T

Now Make Sure There Are Only T’sP Q (P & ~Q) (Q → P)T T F TT F T TF T F FF F F T

Multiple Premises

Sometimes arguments have multiple premises, like: (P v Q), ~P Q├

Premise: (P v Q)Premise: ~PConclusion: Q

Write Down ALL the PossibilitiesP QT TT FF TF F

Write Down the Premises and Conclusion

P Q (P v Q) ~P QT TT FF TF F

Write Down Truth-TablesP Q (P v Q) ~P QT T T F TT F T F FF T T T TF F F T F

Find ALL Lines Where ANY Premise Is F

P Q (P v Q) ~P QT T T F TT F T F FF T T T TF F F T F

Ignore Them!P Q (P v Q) ~P QT T T F TT F T F FF T T T TF F F T F

Make Sure You Only Have T’sP Q (P v Q) ~P QT T T F TT F T F FF T T T TF F F T F

Failing the Test

Not all arguments pass the test. This argument is called “affirming the consequent”: (P → Q), Q P├

Premise: (P → Q)Premise: QConclusion: P

Make ChartP Q (P → Q) Q PT T T T TT F F F TF T T T FF F T F F

Find False PremisesP Q (P → Q) Q PT T T T TT F F F TF T T T FF F T F F

Make Sure You Have All T’sP Q (P → Q) Q PT T T T TT F F F TF T T T FF F T F F

Failing the TestWhenever an argument form fails the truth-table test for validity, then some arguments with that form are invalid.

P = There is no food in Hong Kong.Q = Michael will move out of the country

Failing the TestWhenever an argument form fails the truth-table test for validity, then some arguments with that form are invalid.

Premise: If there is no food in Hong Kong, then Michael will move out of the country.Premise: Michael moved out of the country.Conclusion: There is no food in Hong Kong.

Failing the TestHowever, not every argument of this form is invalid.

P = One person is happy.Q = Two people are happy.

Failing the TestHowever, not every argument of this form is invalid.

Premise: If one person is happy, then two people are happy.Premise: Two people are happy.Conclusion: One person is happy.

TautologiesSome arguments/ argument forms have no premises.

If an argument with no premises is valid, then we call the conclusion a tautology.

Example: (P v ~P)├Premise: Conclusion: (P v ~P)

Tautology

P (P v ~P)T TF T

Contradiction

The opposite of a tautology is a contradiction. Its truth-table is always false.

Contradiction

P (P & ~ P)T T F F TF F F T F

Interesting CaseP Q (P & ~P) QT TT FF TF F

Interesting CaseP Q (P & ~P) QT T F TT F F FF T F TF F F F

Ignore Them All?P Q (P & ~P) QT T F TT F F FF T F TF F F F

Yes!P Q (P & ~P) QT T F TT F F FF T F TF F F F

Interesting Case

Since there are no F’s on the table, the argument is valid.

Every argument with a contradictory premise is valid (though none are sound).

Every argument with a tautology for a conclusion is valid.

The Deduction TheoremWhenever P Q is valid, (P → Q) is valid (and vice versa). ├ ├

If you really want, you can avoid the truth-table test for validity.• If you need to figure out if φ ├ ψ is valid, just write a truth table for

(φ → ψ).• If you need to figure out if φ, χ ├ ψ is valid, just write a truth table for

(φ → (χ → ψ))• If your conditionals are tautologies, you know the argument’s valid.

Earlier We Found:P Q (P v Q) ~P QT T T F TT F T F FF T T T TF F F T F

P Q ((P v Q) → (~ Q → P))T TT FF TF F

P Q ((P v Q) → (~ Q → P))T T T T T TT F T F F TF T F T T FF F F F F F

P Q ((P v Q) → (~ Q → P))T T T T F T TT F T F T F TF T F T F T FF F F F T F F

P Q ((P v Q) → (~ Q → P))T T T T F T T TT F T F T F T TF T F T F T T FF F F F T F F F

P Q ((P v Q) → (~ Q → P))T T T T T F T T TT F T T F T F T TF T F T T F T T FF F F F F T F F F

P Q ((P v Q) → (~ Q → P))T T T T T T F T T TT F T T F T T F T TF T F T T T F T T FF F F F F T T F F F

Argument Analysis

Premise vs. Conclusion

Real-world arguments are not like logic arguments. They almost never tell you which sentence is the conclusion of the argument. You’re left to figure that out yourself.

Conclusion Discourse Markers

• Thus• So• Therefore• Hence• Consequently

Premise Discourse Markers

• Since• Because• For• [colon] :

Identifying the Conclusion

Most companies would agree that as the risk of physical injury occurring on the job increases, the wages paid to employees should also increase. Hence it makes financial sense for employers to make the workplace safer: they could thus reduce their payroll expenses and save money

Identifying the Conclusion

Most companies would agree that as the risk of physical injury occurring on the job increases, the wages paid to employees should also increase. Hence it makes financial sense for employers to make the workplace safer: they could thus reduce their payroll expenses and save money

Conflict

Thus

The English word ‘thus’ has two meanings: it can either mean the same thing as ‘therefore’– and in this sense it indicates a conclusion.

But it can also mean ‘in this way’ or ‘by doing this.’ That’s what it means in this passage.

Identifying the Conclusion

Premise: Companies can reduce their payroll expenses and save money by making the workplace safer.Conclusion: It makes financial sense for employers to make the workplace safer.

Argument Analysis: Example 2

This past winter, 200 students from Lingnan University traveled to the legislative council building to protest against proposed cuts in funding for various university programs. The other 4,000 Lingnan students evidently weren’t so concerned about their education: they either stayed on campus or left for winter break. Since the group who did not protest is far more numerous, it is more representative of university students than are the protesters. Therefore the legislative council need not heed the appeals of the protesting students.

Locating Discourse Markers

This past winter, 200 students from Lingnan University traveled to the legislative council building to protest against proposed cuts in funding for various university programs. The other 4,000 Lingnan students evidently weren’t so concerned about their education: they either stayed on campus or left for winter break. Since the group who did not protest is far more numerous, it is more representative of university students than are the protesters. Therefore the legislative council need not heed the appeals of the protesting students.

SoThe English word ‘so’ has two meanings. Sometimes it means the same thing as ‘therefore.’

Other times ‘so’ means something like ‘as’ or ‘in the same way’ or ‘to the same degree.’

The other 4,000 students weren’t so concerned about their education = they weren’t as concerned as the protesters about their education.

Argument 1

Premise: 4,000 Lingnan students either stayed on campus or left for winter break instead of protesting. Conclusion: These students did not share the protester’s worries.

Argument 2

Premise: The group who did not protest is far more numerous. Conclusion: That group’s opinions are more representative of university students than are the protesters.

Argument 3

Premise: The 4,000 Lingnan students who either stayed on campus or left for winter break instead of protesting do not share the protester’s worries.Premise: That group’s opinions are more representative of university students than are the protesters. Conclusion: The legislative council need not heed the appeals of the protesting students.

4000 did not attend protests.

4000 disagree with protesters.

Non-protesters are more numerous.

Non-protesters are more representative

LegCo can ignore protesters.

Hidden Assumptions

Hidden Assumptions

Most arguments are not deductively valid as stated. Sometimes they are inductively valid, but other times they omit premises that the writer or speaker expects you to fill in.

Argument 1

Premise: 4,000 Lingnan students either stayed on campus or left for winter break instead of protesting. Conclusion: These students did not share the protester’s worries.

Argument 1

Premise: 4,000 Lingnan students either stayed on campus or left for winter break instead of protesting. Hidden Premise: If students don’t attend a protest, then they do not share the worries or values of the protesters.Conclusion: These students did not share the protester’s worries.

Argument 2

Premise: The group who did not protest is far more numerous. Conclusion: That group’s opinions are more representative of university students than are the protesters.

Argument 2

Premise: The group who did not protest is far more numerous. Hidden Premise: A larger group of university students is more representative of student opinions than a smaller group.Conclusion: That group’s opinions are more representative of university students than are the protesters.

Finding Hidden Premises: Charity

Charity is the virtue of giving freely to those who are in need.

‘Charity’ in argument analysis means to interpret the person making the argument in a way that makes the argument most likely to be true.

(We must, however, be faithful to the argument made, and not interpret in a way that the author clearly didn’t mean.)

CharityIf someone makes an argument like this:

Premise: AConclusion: C

They are probably assuming something like: if A then C; if A then probably C; if A, then usually C; C is the best explanation for A…

CharityIf someone makes an argument like this:

Premise: X is FConclusion: X is G

Then they are probably assuming something like: Everything that’s F is also G; Most things that are F are also G; Things that are F are probably also G…

Example

Premise: Group X of students is more numerous than group YConclusion: Group X is more representative than group Y

Probable assumption: Usually when you have two groups A and B of some larger group C, if A is more numerous than B, A is more representative of C’s views.

Charity and Goals

Consider the following argument:

Premise: Giving students a fail grade will damage their self-confidence. Conclusion: We should not fail students.

Charity and Goals

Consider the following argument:

Premise: Giving students a fail grade will damage their self-confidence.Hidden Premise: We should not damage students’ self-confidence.Conclusion: We should not fail students.

Charity and Goals

Consider the following argument:

Premise: Giving students a fail grade will damage their self-confidence.Hidden Premise: We should not damage students’ self-confidence.Hidden Premise: If we should not do P and Q causes P, then we shouldn’t do Q.Conclusion: We should not fail students.

Charity and GoalsIn general, if someone makes an argument:

Premise: P causes QConclusion: We should [not] do P

They are probably assuming:1. Q is good [bad].2. We should [not] do things that have good [bad] outcomes.