A. Invalid Reasoning

In the Warm Up activity, you saw examples of invalid reasoning. In the first argument it was fairly clear that the reasoning was flawed, but determining why may have been more of a challenge. The word "nothing" was used twice, but in each instance it had a different meaning. The first time "nothing" was used ("Nothing is better"), its use suggested that of all things, there is not one that is better. The second time "nothing" was used ("better than nothing"), its use suggested that having something is better than having nothing at all. The two different uses of the word led to the inappropriate conclusion.

Incorrect reasoning will lead to unsupported conclusions. Even if the conclusion is correct, it still has no justification.

Example 1

Incorrect reasoning was used in each of the following statements. Explain why the reasoning for each is incorrect.

  1. I took a pill and my headache went away, so the pill worked.

    There are many reasons why a headache might go away. It may be a coincidence that the pill was taken and the headache then disappeared.



  2. This example uses the rule , which is not true when a and b are negative.

  3. Mayor Henderson is successful because he is the best candidate.

    This statement uses circular reasoning. The conclusion is used as supporting evidence for the conclusion.

  4. 80% of people believe the Sasquatch doesn't exist, so it can't be real.

    A majority of people agreeing with an idea does not guarantee the idea is correct. Even though 80% of people don't believe in the Sasquatch, it is still possible for the Sasquatch to exist.

  5. You are with us or against us.

    This statement ignores any alternatives, most notably a neutral stance.

  6. This sentence is false.

    This is a self-referencing statement that leads to a paradox, a statement that leads to a contradiction. If the statement is true, then it must also be false. If it is false, then it must also be true. This inconsistency means the statement is neither true nor false and can be thought of as nonsense.

Sometimes statements leave you confused and unsure what to think.

Consider the following scenario. Pinocchio was a puppet whose nose grew each time he lied. Suppose Pinocchio said "My nose will be growing."

What do you expect will happen?

This is another example of a paradox. In this example, if Pinocchio is telling the truth, his nose shouldn't grow, but then his statement is a lie. But if he is lying, his nose should grow meaning he was actually telling the truth.

Paradoxes are typically produced by incorrect reasoning or by self-referencing. Sometimes paradoxes can be dismissed as nonsense, while other times they can lead you to find errors in reasoning.

For further examples of invalid reasoning see pp. 36–41 of Principles of Mathematics 11.