Back To CourseMath 102: College Mathematics
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Amy has a master's degree in secondary education and has taught math at a public charter high school.
A logical fallacy is reasoning that is not based on pure facts. In other words, it is bad logic because the arguments are not sound. It is different from a logical conclusion in that a logical conclusion comes from proven facts, and you can trust the original statement. Sometimes, reasoning based on things besides pure facts produces a valid or true result, but in many cases, it produces an invalid or false result.
For example, in a presidential campaign, each group tries to change people's opinions about their candidate by trying to increase their candidate's popularity. They are using the logical fallacy of popularity to try to convince people that their candidate is the right one. Usually, the popularity of a presidential candidate is a good indicator that he or she will be elected. In this case, this type of reasoning is valid because the most popular candidate is the one that is elected.
But, how about the case where a group of people are trying to convince one person that a bowl of corn starch is sugar? The group of people may be eating from the same bowl and making sounds like they really are enjoying the 'sugar.' But, does that mean that it really is sugar? No, it doesn't. In this case, the reasoning and logic are false.
How does this relate to math? Well, mathematical proofs and theorems are based on logic. The reasoning behind them has to be solid for a theorem or proof to be true and usable. So, learning about logical fallacies, or reasoning that is not solid, will help you prevent making mistakes when proving your mathematical work. We're going to look at the logical fallacies of appeals to ignorance, emotion, and popularity in this video lesson. So, let's see what they're all about and how it ties in with math.
An appeal to ignorance is saying something is true or false just because evidence to the contrary is not known. For example, back in the day, people thought the world was flat. We know today that the world is round. But, back in the day, people were ignorant of that fact, so they believed that the world was flat. Nobody had sailed around the world and lived to tell about it, so people thought that once someone sailed off into the horizon, they fell off the earth.
You can have a valid appeal to ignorance, though. If, for example, you go to someone's house and you don't see any signs that they have children, it is probably true that they do not have children. If you don't see children's toys, children's clothes, or hear any children, you appeal to ignorance because you don't have evidence that the contrary may be true.
You can identify an appeal to ignorance when someone tells you something is true or false just because they don't have all the facts. If someone came up to you and told you that it is a fact that giant squids don't exist because she has never seen one in real life, you would know that she is making an appeal to ignorance in her logic. She believes something to be true just because she hasn't seen evidence to the contrary.
Likewise in math, you can't prove that something is true just by saying you can't think of anything else that fits. For example, if a young child came up to you and said that 2 + 4 = 24 because in his mind the plus means putting the numbers next to each other, does that make it true and is that sound reasoning? You would say no, that is not true. But for the little child, it is true because he doesn't know the real meaning of the plus sign. The little child has committed the logical fallacy of appealing to ignorance. This is just a small example, but mathematicians have to be careful not to fall prey to this logical fallacy as well. Just because they are mathematicians does not mean that they know everything.
An appeal to emotion is saying something is true or false based on emotions. We encounter this type of logical fallacy all the time when watching commercials. Think of a commercial for an expensive sports shoe. The premise behind the shoe is that it makes you feel good about yourself and you'll be popular and you'll be able to do all kinds of jumps and sports kicks with them.
What the commercial is doing is trying to get you to purchase the shoe and believe in its powers because of your emotions. If you feel strongly enough about the shoe and its supposed power to make you a sportier, more popular person, then you'll purchase the shoe. While this premise might be true for some, it may not be true for others. Just because you purchase the shoe does not guarantee that you will become popular and sportier. If you are already popular and sporty, then it might be true that it will enhance your performance.
You can identify appeals to emotion if the evidence behind the statement is based on emotion. If you start feeling emotional because of what they are saying, then they are appealing to your emotions in order to persuade you to do or believe in something.
Mathematicians have to be careful here to not appeal to emotion when working on a mathematical proof. The same goes when working through the problems. An answer to a problem might feel like the right answer, but unless you can prove it with sound reasoning, you cannot trust that answer. It's like telling your teacher that your answer is right because you felt a strong emotion when you came across it. Unless you have work to back it up, you really can't prove that your answer is the right one without committing this logical fallacy.
Last, but not least, we have an appeal to popularity. This type of logical fallacy is saying that something is true or false based on popular opinion. It may be true that if the majority of people find the donuts in one particular doughnut shop to be especially good, then that doughnut shop really does make awesome doughnuts. But, what if everybody around you started telling you that 2 + 2 = 5? Does it make it true just because everyone around you says so? No, that is not sound reasoning, and it is logically false. Whether the end result turns out to be true or false, keep in mind that unless the reasoning is sound and based on pure facts, it is a logical fallacy.
You can identify an appeal to popularity by seeing if the evidence behind the statement is based on popularity. This is a logical fallacy because this type of argument is not always sound. Sometimes popular opinion is correct, and other times it may not be true. For instance, several centuries ago, it was a popular idea that the planet was supported by a giant turtle, who in turn was supported by elephants. Basing an argument on popularity is not sound because it does not always produce a correct conclusion. We know today that just because a whole bunch of people believe something to be true, does not make that something true. When working with math problems, you can't prove your work by saying that something is true because all your friends also got the same answer.
In this video lesson, we looked at three different types of logical fallacies, or unsound arguments not based on pure facts. We looked at appeals to ignorance, emotion, and popularity. An appeal to ignorance is saying something is true or false because evidence to the contrary is not known. An appeal to emotion is saying something is true or false based on emotions. And an appeal to popularity is saying something is true or false based on popular opinion. All three of these logical fallacies can have some statements that are valid. But, they may not be sound arguments because history has shown us that statements based on these types of logic may very well be false.
You can identify each by looking at the premise behind the logic of any statement. If it is because of a lack of evidence, then you know the logic is appealing to ignorance. If you feel your emotions getting worked on, then you know the logic is appealing to your emotions. And if you see that it is because of popular opinion, then the logic is appealing to popularity.
All of this relates to math because math proofs, theorems, and your answers to math problems are all based on proper logic. Without sound reasoning, we couldn't trust that the proofs and theorems we rely on actually work each and every time. It is important to be able to identify logical fallacies so we can avoid them when working with math problems.
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Back To CourseMath 102: College Mathematics
14 chapters | 108 lessons