Reason & Argument Lecture 2. Lecture Synopsis 1. Making Arguments Explicit 2. Validity & Soundness 3. Counter-Examples (& a word about Induction)

Reason & ArgumentLecture 2

Lecture Synopsis

1.Making Arguments Explicit

2.Validity & Soundness

3.Counter-Examples (& a word about Induction)

Recap

Last time:

Key Concepts

‘Argument’: a statement, along with some supposed reasons for accepting the statement.

‘Conclusion’: the statement for which reasons have supposedly been presented.

‘Premises’: the claims which are supposed to be reasons for accepting the argument’s conclusion.

Recap

Spotting premises & conclusions:

They don’t always come in the same order.

‘Indicators’

Premises: for, since, because, due to the fact that, etc.

Conclusions: so, hence, thus, therefore, it follows that

But be careful! These can indicate other connections.

(1) Making Arguments Explicit

Often arguments don’t come neatly packaged as premises and conclusions:

Suppressed or Un-stated Premises

Implicit Conclusions

Missing sub-conclusions

Extraneous Information

Procedure

How to reconstruct an argument

Identify premises and a conclusion.

Eliminate extraneous material.

Fill in suppressed premises.

Fill in suppressed (sub-)conclusions.

Do all this even if your ultimate goal is to dispute the argument. (The ‘Principle of Charity’)


Examples

(S1) Wealthy ancient Greeks always had siblings.

(S2) So Plato had a sibling.


Examples

Let’s try that again:

• (S1) Wealthy ancient Greeks always had siblings.

• (S3*) Plato was a wealthy ancient Greek.

• (S2) So Plato had a sibling.


Examples

(S1) Until the 1960s, men with long hair weren’t allowed to enter Disneyland.

(S2) So if Bob visited Disneyland in 1955, he didn’t get in; although

(S3) Bob was actually living in Selby at the time.


Examples


• (S1) Until the 1960s, men with long hair weren’t allowed to enter Disneyland.

• (S3*) Bob had long hair in 1955.

• (S2) So if Bob visited Disneyland in 1955, he didn’t get in...


Examples

(S1) The killer had a 1-inch scar above his left eye

(S2) There are 5 people who had a motive for committing the murder

(S3) Smith had a motive to commit the murder

(S4) Smith has a 1-inch scar above his left eye

(S5) So Smith is a suspect


Examples


(*S6) Anyone who had a motive to commit the murder is a suspect

(S3) Smith had a motive to commit the murder

(S5) So Smith is a suspect

(S1) The killer had a 1-inch scar above his left eye

(S4) Smith has a 1-inch scar above his left eye

(*S7) Therefore Smith is the killer


(2) Validity & Soundness

Could we convict Smith on the basis of this argument? If not, why not? How do we answer such questions?

Features of a good argument (n.b. technical terms!):

Validity: an argument is valid if and only if when the premises are true, the conclusion is true.

Soundness: an argument is sound if and only if it is valid & its premises are true.

Criticising an Argument

Thus, to show an argument is faulty, you can either show that:

1.It is invalid: the conclusion does not follow from the premises.

2.It is unsound: one or more of its premises is false.

(Notice that if an argument is invalid, then it is automatically unsound.)


Valid & Sound?

How should you decide whether an argument is valid and/or sound?

Validity:

Consider the form or pattern of the argument

Always valid - ‘syllogisms’,

Always invalid - ‘fallacies’

Think of a counter-example (Section 3)

Soundness:

Try to think of a counter-example (Section 3)


Valid ‘Patterns’ of Inference

Check whether an argument is valid by looking at its form:

(S1) If John paid the electricity bill today, then we do not have enough money to pay the gas bill.

(S2) John paid the electricity bill today.

(S3) So we do not have enough money to pay the gas bill.


Modus Ponens

The form of that argument...

If X, then Y

X

So Y.

Arguments taking the form ‘modus ponens’ are syllogisms - they are always valid. (n.b. this does not mean they are always sound!)


Example

(S1) If John paid the electricity bill today, then he would have looked miserable when you saw him.

(S2) John did not look miserable when you saw him.

(S3) So John did not pay the electricity bill today.


Modus Tollens


If X, then Y

not-Y

So not-X


Example

(S1) If John paid the electricity bill today, then we do not have enough money to pay the gas bill.

(S2) If we do not have enough money to pay the gas bill, Lisa will be angry.

(S3) So if John paid the electricity bill today, then Lisa will be angry.


Hypothetical Syllogism


If X, then Y

If Y, then Z

So if X, then Z


Conditionals

Modus Ponens

Modus Tollens


These are all valid patterns of inference for reasoning with conditional or hypothetical sentences (i.e. sentences of the form ‘If X, then Y’)


Other Valid Patterns

Disjunctive Syllogism

X or Y (a ‘disjunction’, X & Y are ‘disjuncts’)

not-X

So Y

Example:

(S1) Either we must pay our electricity bill or we must go bankrupt

(S2) We must not go bankrupt

(S3) So we must pay our electricity bill


Invalid Patterns

‘Affirming the Consequent’

Conditional sentence: ‘If X, then Y’

If X = the ‘antecedent’

then Y = the ‘consequent’

If X, then Y

Y

So X


Affirming the Consequent

(S1) If he is a thief, then he would look uncomfortable.

(S2) He looks uncomfortable.

(S3) So he is a thief.

Problem: There are other antecedents (‘reasons why’) for the consequent ‘he would look uncomfortable’.


Denying the Antecedent

If X, then Y

Not-X

So not-Y

(S1) If he says he posted the letter, then he posted it.

(S2) But he hasn’t said he posted the letter.

(S3) So he hasn’t posted the letter.

Problem: The argument only tells you what will happen if he says he posted the letter. He didn’t say he posted it, so the argument cannot tell us anything.


Valid & Invalid Patterns

Valid:

Modus Ponens

Modus Tollens


Disjunctive Syllogism (and many others...)

Invalid:

Affirming the Consequent

Denying the Antecedent


(3) Counter-examples

A counter-example = a case where a premise appears false (argument unsound) or the conclusion does not follow (argument invalid).

(P1) All Muslims are extremists.

(P2) All extremists are terrorists.

(C) Therefore all Muslims are terrorists.

Counter-examples for Soundness

A Muslim who isn’t an extremist

An Extremist who isn’t a terrorist

C-Es for Soundness

If you eat chocolate every week, then you are obese.

Beth Ditto eats chocolate every week.

So Beth Ditto is obese.

Counter-example for soundness = I eat chocolate once a week, I am not obese.


C-Es for Validity

Try to come up with an argument of the same form, but which is obviously mistaken.

Example:

(S1) If the ambassador were unhappy with her position in Portugal, she would be willing to take the position in Ireland.

(S2) The ambassador was willing to take the position in Ireland.

(S3) So the ambassador must have been unhappy with her position in Portugal.


C-Es for Validity


If X, then Y

Y

So X

Recognise this pattern?

A counter-example:

(S1) If Julia Roberts is a lawyer, then she has been to university.

(S2) Julia Roberts has been to university.

(S3) So Julia Roberts is a lawyer.

Mnemonic: ‘If that argument were valid, it could be used to show that... [insert absurd or false claim]’


Counter-examples

For soundness:

Think of a case that shows a premise is false.

For validity:

Show how the same form or pattern of reasoning, when applied to different claims, leads to an obviously false conclusion.


A word about ‘Induction’

Not every good or strong argument is valid.

Valid arguments always involve deductive reasoning - we deduce a conclusion from a set of premises.

But there is also inductive reasoning, where a conclusion is supported by (for example) a large number of observations.

Induction

An example of induction:

Every zebra we have ever observed has black and white stripes.

So all zebras have black and white stripes.

Induction is ampliative, unlike deduction

Goes beyond or adds to the information in the premises/observations

Strictly invalid, thus it carries no guarantee of the truth of its conclusion

Arguments can still be inductively strong


What You Have Learned Today

1. How to Make Arguments Explicit

Identify premises & conclusions, eliminate extraneous material, add implicit premises/conclusions/sub-conclusions

2. Validity and Soundness

Validity: when the premises are true, the conclusion must be true. Soundness: it is valid, and the premises are true.

3. Counter-examples

For soundness: a case where a premise is false. For Validity: an argument of the same form with a false conclusion.

Documents

Reason & Argument Lecture 2. Lecture Synopsis 1. Making Arguments Explicit 2. Validity & Soundness 3. Counter-Examples (& a word about Induction)