Counter Examples
A useful way to check the validity of an argument is to create counter examples. You can see this in the example below.
Example
P1 - All birds have wings.
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P2 - All parrots have wings.
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C - Thus, all parrots are birds.
Counter
Example
P1 - All birds have wings.
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P2 - All bats have wings.
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C - Thus, all bats are birds.
Both arguments are invalid but it is far more obvious in the example about bats. If you are asked to examine the validity of an argument it can be useful to create a counter example.
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Remember that validity relies to the structure of arguments and that arguments with the same structure are either all valid or all invalid. If we are concerned that an argument might be invalid, then we should try to see if it would be possible for an argument with the same structure to have true premises but a false conclusion. That is what is means to come up with a counter example.
To create a counter argument for the above example we must create an argument with the same structure. Below is the structure of the argument explained logically uses letter in place of the premises.
Tips
"A" stands for Antecedent and is the first part of a conditional statement. The "C" stands for Consequent which is the part that comes second in a conditional statement.
Part of the reason that some arguments are difficult to asses its that we get distracted by the content of the argument and our prior knowledge of the world. In the two following examples given we can be easily distracted by our prior knowledge of birds and our prior knowledge of rain into an incorrect evaluation of the arguments.
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Had the arguments not had that particular content we would not have been so easily fooled. This suggests an informal strategy for testing the validty of arguments: find another argument with the same form as the one under consideration but with content that makes the validity or invalidity more obvious.
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For example...
Example
All birds have feathers.
An eagle has feathers.
Therefore, an eagle is a bird.
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This argument has the same form as this argument...
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Example
It is raining then we're in Scotland.
We aren't in Scotland.
Therefore, it isn't raining.
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This argument has the same form as this argument...
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Counter
Example
All dogs have eyes.
A human being has eyes.
Therefore, a human being is a dog.
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However, this argument has 2 true premises followed by false conclusion which we know is impossible for a valid argument.
This argument is therefore invalid.
Counter
Example
If you are a dog then you are a mammal.
A lizard is not a mammal.
Therefore, a lizard is not a dog.
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This argument makes the validity more obvious. If all dogs are mammals then stands to reason that anything which isn't a mammal can't be a dog. By substituting the content o the argument with obvious true premises we have been able to make the evaluation of the argument easier.
Counter examples will only work if you ensure that the example you find has obviously true premises.
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However, this technique does require a bit of imagination and you will be at loss if you cannot think of a suitable counterexample. That is why most logicians resort to more methodical approaches such as Venn diagrams and truth tables.
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In Higher Philosophy we only need to know about counter examples.
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We should create an argument that follows the same structure, but try to make it have true premises and a case conclusion. If we can do this we can prove it is invalid.
Example
P1 - If A then C.
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P2 - C.
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C - Therefore, A
Counter
Example
P1 - If the Queen is a man then she is mortal.
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P2 - The Queen is mortal.
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C - So the Queen is a man.
We can clearly see that the argument on the right is invalid because it has true premises but a false conclusion. The Queen is not, as far as we know, a man! Arguments with true premises and a false conclusion have bad structure and so are invalid.
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Let's go over some more examples.
Example
P1 - All cats are feline.
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P2 - Whiskers is a cat.
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C - So, Whiskers is feline.
Counter
Example
P1 - All Queens are female.
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P2 - Elizabeth is a queen.
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C - So, Elizabeth is Female.
Same structure and true premises = true conclusion.
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this argument must be valid.
Example
P1 - All cats are feline.
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P2 - Whiskers is not a feline.
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C - So, Whiskers is not a cat.
Counter
Example
P1 - All Queens are female.
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P2 - Elizabeth is not a queen.
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C - So, Elizabeth is not Female.
Same structure and true premises = false conclusion.
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This argument must be invalid.
Problems
with
Counter
Examples
A lot of the time they are entirely theoretical counter examples (made up).
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Some extreme and ridiculous counter examples can show that a moral theory isn't universal, but it doesn't show that it is better than the alternatives of that it is a bad theory to use.
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it can be questioned whether or not just because something can be imagined it follows that it can ever become a reality. e.g. a zombie that looks exactly like a normal living human but has no consciousness.
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Despite this it is generally accepted that theoretical counterexamples are good enough (sufficient).
Tasks
Use what you have learnt about to complete the following tasks.
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There are copies of the tasks on your GoogleClassroom.