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Counter examples
How do you give counter examples?

Counter examples

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Posted 10/22/09 - 06:32 AM: Subject: Counter examples	#1
How do you give a counter example in an examination. It says "give a counter example e.g. describe possible situation where premise is true and conclusion false." For example an argument like Every cat is a mammal No pets are mammals So no cats are pers * Pets are mamals e.g. someone may have a cat as a pet so the statement is invalid. Is that a counter example? A cat as a pet? Or All runners train hard Some people who train hard do push ups when training So some runners do push ups when training *Not all runners train hard e.g. a runner who is lazy and doesn't train. Is that a counter example a runner who doesn't train hard?

Kwalish Kid
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1 of 1 people found this post helpful Posted 10/22/09 - 06:39 AM:	#2
I think it is best to describe specific individuals that meet the conditions of the premises but not those of the conclusion. So you can talk about Fluffy, who is a cat and a mammal, but it owned as a pet my Mrs. Smith. Your last example doesn't fit. Based on what you said at the start of your post, you should be giving an example of a runner that does train hard. For the last example, you might have to give at least two people. So you have to describe a runner that trains hard but doesn't do push-ups and someone else who does train hard and do push-ups. But this only works if you only have to provide a single counter example. Otherwise, you have to describe a circumstance where all runners don't do push-ups, even though they train hard.
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Posted 10/22/09 - 06:50 AM:	#3
Kwalish Kid wrote: I think it is best to describe specific individuals that meet the conditions of the premises but not those of the conclusion. So you can talk about Fluffy, who is a cat and a mammal, but it owned as a pet my Mrs. Smith. Your last example doesn't fit. Based on what you said at the start of your post, you should be giving an example of a runner that does train hard. For the last example, you might have to give at least two people. So you have to describe a runner that trains hard but doesn't do push-ups and someone else who does train hard and do push-ups. But this only works if you only have to provide a single counter example. Otherwise, you have to describe a circumstance where all runners don't do push-ups, even though they train hard. Oh right thanks I get the first one. You know the second one. It says "some runners" so even if I found a runner who didn't do push ups it wouldn't disprove the argument so should I attack the first bit that says all runners train hard?

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Posted 10/22/09 - 06:54 AM:	#4
In the first example, the argument is valid, but the second premise is false. A counter example to the premise might be 'Fido is a pet mammal'. The second argument is invalid - the conclusion does not follow from the premises. This is where a good counter example clarifies the invalidity, by changing one of the terms while keeping the same structure. All runners train hard. Some people who train hard have no legs. So some runners have no legs. ??? Do you see the difference between attacking the premises and attacking the argument?
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Posted 10/22/09 - 11:00 AM:	#5
Given the definition of counterexample you are working under, you cannot give a counterexample to a valid argument. You can only falsify a premise, which is not what the question is asking for. If an exam question asks for a counterexample to a valid argument, it is a trick. The answer is "no counterexample can be given as this is a valid argument." If the argument is not valid, you must give an example where the premises are true and the conclusion is false (as per the definition of counterexample provided). For the second case, you need only show that a world where no runners do push-ups when training is possible under the premises. And it is possible in a world where non-runners also train hard. QED.
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