Critical Thinking

Counter-Examples

Counter-examples function in one of two ways. 

Firstly, they can be used to question or refute a general claim or argument and secondly, they can be considered and then responded to in order to strengthen one’s own reasoning. 

A general claim is one that is intended to hold true in every case. They tend to involve reasoning indicators such as ‘all’, ‘every’ or ‘always’, or when phrased negatively, ‘none’, ‘no’ or ‘never’. If such a claim is correct, it will be impossible to find exceptions. This is especially true of general claims that are true by definition, such as ‘all bachelors are male’. Here, a counter-example (i.e. a non-‘male and unmarried’ bachelor) would be impossible. But the majority of general claims do not display this degree of certainty. Because such claims as:

 

• All swans are white.
• All second-hand car dealers are dishonest.
• All knowledge comes from experience.

 

are meant to hold true in every case, all that would be required to refute them would be one counter-example – i.e. a non-white swan, an honest second-hand car dealer, and an instance of knowledge that did not come from experience. For this reason, counter-examples often force us to limit the force of our conclusion. The above claims would be better phrased as:

 

• Most swans are white.
• Some second-hand car dealers are dishonest.
• The majority of knowledge comes from experience.

 

The harder it is to provide a counter-example for a general claim, the greater the likelihood that it will be true. Conversely, the more counter-examples that can be provided against such a claim, the less plausible the claim will be. Because arguments often depend upon general claims for their effectiveness (although these are often left implicit) looking for counter-examples is an effective method for testing whether or not an inference is sound:

 

•   The Bugatti Veyron is the fastest (253 mph and 0-60 in under 2.5 seconds) and most expensive (£880,000) production car ever built.
•   It has to be the greatest vehicle on the road.

 

Here, the reasoning only works if we accept the (assumed) general principles that:

 

1. An expensive product is always better than an inexpensive one.
2. A faster vehicle will always be better than a slower one.  

 

And more generally:

 

3. That in all cases speed and price are what determine how good a car is.

 

But it is easy to find counter-examples for each of these. 

For the first point, we would just need to bring in an example of an inexpensive product that was superior to an expensive one - for example, buying a pricey new bottle opener that is significantly inferior to the cheaper one already owned.  Price is clearly not always a good indication of quality! 

For the second, we would just need to find an example of a slower vehicle that was superior to a faster one - a Rolls Royce Silver Shadow, widely regarded as one of the greatest cars ever made, has a top speed of 115mph; certainly slower than, but not inferior to the majority of modern, less expensive cars on the market. 

And for the third, we would just need to show that speed and price do not always determine how good a vehicle is. A popular car magazine recently listed fuel consumption, safety, comfort, ease of handling and cost of insurance as being the decisive factors for doing this; none of which the Veyron is likely to score highly on! 

 

Whilst none of these points fully refutes the inference (or conclusion) drawn, they nevertheless cast doubt upon the reasoning used to arrive at it.

Looking for counter-examples, then, is an essential step for determining how strong an argument is.    

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