Part II: Examples

In the following section, we will be looking through a couple of examples depicting instances of each of the fallacies listed above:

i. Affirming the Consequent

1. Suppose you know that if a certain dog can smell things up to one mile away, then it will chase after some animal. Now suppose that the dog goes chasing after some animal. From this information, if you were to further conclude that the dog must’ve been able to smell things up to one mile away, you would be committing the fallacy of Affirming the Consequent. After looking at a concrete example of this fallacy in use, it should be easier to see exactly what makes this fallacious; while the dog did in fact chase after an animal, we are not told whether the dog smelled the animal or whether it used one of its other senses to detect the animal. Because of this, we cannot assuredly conclude that the dog can smell things up to one mile away, as we have no evidence in favor of the truth of this proposition. To see this more clearly, we can formalize the inference as:

1. If a certain dog can smell things up to one mile away, then it will chase after some animal.
2. The dog goes chasing after some animal
3. Therefore, the dog must’ve been able to smell things up to one mile away

2.    (1) All men are mortal 

       (2) Socrates is mortal

       (3) Therefore, Socrates is a man

It is important to note that while the above syllogism has a true conclusion, it is still fallacious. A fallacious type of inference can sometimes give true conclusions, but many times it does not. In logic, we care about types of inference that always give true conclusions (if the premises are true). Here’s an example where Affirming the Consequent fails to give a true conclusion:

3.    (1) All dogs are animals

       (2) Aristotle is an animal

       (3) Therefore, Aristotle is a dog. 

We can visualize where we went wrong using Euler Circles:

ii. Denying the Antecedent

1. Once again, as in the last lesson, suppose that you now know that if someone falls asleep with a lit candle, their room will catch fire. One day, upon asking your parents about your distant cousin, they tell you that they know nothing about your cousin besides the fact that they have never fallen asleep with a lit candle. If you were to follow the path of Denying the Antecedent, you would reach the conclusion that your cousin’s room never caught on fire. This assumption could never be assured by the two premises you received, as there are a multitude of other ways in which a room could catch on fire. Lightning could’ve stricken the house, or something could have been left on top of the radiator, etc. Given that possibilities of this variety exist, you cannot assume from the premises provided that your cousin’s room has not caught on fire. To see this more clearly, we can formalize the inference as:

1. If someone falls asleep with a lit candle, their room will catch fire
2. Your cousin has never fallen asleep with a lit candle
3. Your cousin’s room never caught on fire

2. (1) All universities have students

    (2) No high schools are universities

    (3) Therefore, no high schools have students 

We can visualize where we went wrong using Euler Circles: