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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
Valid Argument Invalid argument
all fish have lungs
Whales are fish
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Whales have lungs
All fish have scales
Whales have scales
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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
Valid Argument Invalid argument
all fish have wings
Whales are fish
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Whales have wings
All fish have scales
Whales have scales
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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
If it is snowing the
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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If it is snowing the
mail will be late
It is snowing------------------------
It is not the case the
mail will be late
If it is snowing the
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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If it is snowing the
mail will be late
It is snowing
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It is not the case themail will be late
If it is snowing the
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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All whales are mammals
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The counterexample set:
All whales are mammals
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It is not the case that all whales are
mammals
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Are these sentences consistent?
All whales are mammals
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It is not the case that all whales are
mammals
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Are these sentences consistent?
All whales are mammals
-----------------------------
It is not the case that all whales are
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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Are these sentences consistent?
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