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Critical Reasoning forBeginners: Five

Marianne Talbot

Department for Continuing Education

University of Oxford

Michaelmas 2009

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Recap on last week

evaluating inductive arguments.

.inductive generalisations andcausal generalisations

. arguments from analogy andauthority

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Inductive generalisations:

Is the premise true?How large is the sample?How representative is the sample?Beware informal heuristicsIs there a counterexample?

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Arguments from analogy:

are the two things similar?are they similar in respect of

something relevant?

can we find a disanalogy?

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Arguments from authority:

who exactly is the source ofinformation?

is this source qualified in theappropriate area?

is the source impartial in respect ofthis claim?

do other experts make other claims?

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This week we shall be looking at

the distinction between validity

and truth

at why validity is important

.and at evaluating deductive

arguments

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A good deductive argument is

SOUND if and only if it:

(a)is validAND

(b)has true premises

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Is the argument sound?

True premises False premises

Valid

Invalid

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Is the argument sound?

True premises False premises

Valid Sound Unsound

Invalid Unsound Unsound

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The truth of the premises is not a matter

for logicians or those interested in

critical reasoning.

.there are many ways in which we

determine the truth or falsehood ofpremises

and these ways do not fall into thescope of a class on critical reasoning

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Validity, on the other hand

is very much of interest to logicians

because validitypreserves truth

if an argument is valid, then if itspremises are true

we can be certain its conclusion is true

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Validity, in fact, is of interest to anyone

who is concerned about truth

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because we often dont know the truth

of our premises.

and we often test the truth of our

premises by

constructing valid arguments and..

testing the truth of the conclusion

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If we can show that

the conclusion of a valid argument is

false

what do we thereby discover?

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Hypothesis: Smoking causes cancer

Prediction: if smoking causes cancer

then every smoker will get cancer

Test: each smoker gets cancer

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All women are passive

Mrs. Thatcher is a woman

----------------------------------

Therefore Mrs. Thatcher is

passive

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So what is this relation of validitythat everyone is so concerned with?

Here is the best theory thatphilosophers and mathematicians

can come up with

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An arguments is valid

if and only if

there is no possible situation in

which

all its premises are true

. and its conclusion false

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Beware: it is the possibility of the

combination

of true premises and false

conclusion.

.that is ruled out by an arguments

being valid

(this is why validity preserves truth)

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Note: it is thepossibilityof the

combination

of true premises and false conclusion

that is ruled out by an arguments being

valid

Not just the actualityof the

combination of true premises and falseconclusion

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So, faced with an argument whose validity we

are trying to determine, we must ask

not (just):

ARE the premises true and the conclusion

false together in actuality?

But

COULD the premises be true and the

conclusion false together in some

situation?

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Please say whether or not you think

arguments of the following sort could be

valid:

(i) The premises of the argument are false(ii)The premises of the argument are true

and the conclusion is true

(iii)The premises of the argument are trueand the conclusion false?

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If the premises COULD be true

. TOGETHER WITH the conclusions

being false

then the argument is invalid

otherwise it could be valid

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Could this argument be valid?

2 + 2 = 5

--------------

grass is green

Is there a situation in which the premise

could be true and the conclusion false?

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Could this argument be valid?

grass is green

---------------------

2 + 2 = 4

Is there a situation in which the premise

could be true and the conclusion false?

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Is the argument valid?

True

conclusion

False conclusion

TruePremises

False

Premises

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Is the argument valid?

True

conclusion

False conclusion

TruePremises Possibly valid Invalid

False

Premises Possibly valid Possibly valid

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We shall have a look at this more closely by

using Venn diagrams to determine, of some

arguments, whether or not they are valid

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Premises actually true and

conclusion actually true

Valid Argument Invalid argument

all cats meow

Bo does not meow

--------------------

Bo is not a cat

All cats meow

Dogs are not cats

-------------------

Dogs dont meow

In both cases the premises are actually true and so is the conclusion.But in the first case the truth of the premises guaranteesthe truth of theconclusion. In the second case the conclusion couldbe false despite the

truth of the premises.

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Premises actually false and

conclusion actually true


all fish have lungs

Whales are fish

--------------------

Whales have lungs

All fish have scales

Whales have scales

-------------------

Whales are not fish

In both cases the premises are actually false, and the conclusion isactually true. But in the first case if the premises weretrue the truth ofthe premises would be guaranteed. In the second case even if the

premises were true the conclusion couldstill be false.

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Premises actually false and

conclusion actually false


all fish have wings

Whales are fish

--------------------

Whales have wings

All fish have scales

Whales have scales

-------------------

Whales are fish

In both cases the premises and the conclusion are actually false. But inthe first case if the premises weretrue the truth of the conclusion wouldbe guaranteed. In the second case even if the premises were true the

conclusion couldstill be false.

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We have used Venn diagrams to

determine the validity of the argument we

have so far looked at.

.another way to determine validity is tocreate a counterexample set and

determine consistency

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To determine the counterexample set we

set out the argument logic book style

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If it is snowing the

mail will be late

It is snowing------------------------

The mail will be late


mail will be late

The mail will be late---------------------------

It is snowing

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Then we negate the conclusion by

tacking it is not the case that in front of

it

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mail will be late

It is snowing------------------------

It is not the case the

mail will be late


mail will be late

The mail will be late---------------------------

It is not the case it is

snowing

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We then consider whether the set of

sentences consisting of

the premises and the negation of the

conclusion is consistent

.i.e. whether they could all be true

together

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If the counterexample set is consistent

then the original argument is invalid

if the counterexample set isntconsistent then the original argument is

not valid.

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mail will be late

It is snowing

------------------------

It is not the case themail will be late


mail will be late

The mail will be late

---------------------------

It is not the case it issnowing

Could these sentences be consistent i.e. could they all be true together?

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Is this argument valid?

All whales are mammals

-----------------------------


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The counterexample set:


-----------------------------

It is not the case that all whales are

mammals

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Are these sentences consistent?


-----------------------------


mammals

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-----------------------------


mammals

So the original argument is..???

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Is this argument valid?

If it is Friday Marianne is wearing

jeans

It is Friday

-----------------------------------

Marianne is wearing jeans

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The counterexample set:

If it is Friday Marianne is wearing

jeans

It is Friday

-----------------------------------

It is not the case that Marianne is

wearing jeans

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If it is Friday Marianne is wearingjeans

It is Friday

-----------------------------------

It is not the case that Marianne iswearing jeans

So the original argument is???

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Next week we shall be looking at

common fallacies

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