Dr. Alexis Erich S. Almocera
What is an argument?
It is basically a series of statements that comprises the premises that you would have to assume, and a conclusion which you would have to evaluate as true or false. The argument becomes valid when the conclusion logically follows from the premises.
But what if this argument is invalid? This is what you call a fallacy.
Let’s take a look at two fallacies that assume “if p then q”. The first is the fallacy of the inverse. It works by rejecting the “if” part in order to conclude the “then” part. Normally, you would have a valid argument by affirming p in order to affirm q. But in the fallacy of a converse, you do it the other way around. It is illegal to affirm q to conclude p; just because p implies q does not always mean the reverse.
Let’s compare modus ponens with the fallacy of the inverse. Modus ponens is valid when you have the conditional “if p then q” and you affirm p, then by modus ponens, you affirm q. But, in the fallacy of the inverse, you reject p in order to reject q. This argument is invalid, and the conclusion “not q” is not always logically true. We can compare the fallacy of the converse with the valid statement modus tollens. Modus tollens is valid when you’re given the conditional “if p then q” then, it is legal to reject q in order to reject p. But instead, when you affirm q to affirm p, now you have an invalid argument. This is the fallacy of the converse; just because p implies q doesn’t mean that it’s converse will hold.
Here’s an example of a fallacy of the inverse compared with modus ponens. In modus ponens, you would normally say “All dogs are hairy. My pet cotton is a dog. Therefore, Cotton is hairy.” Now, that is a valid argument. However, it is invalid to say “All dogs are hairy. My pet cotton is not a dog. Therefore, cotton is not hairy.” You can verify this by drawing an Euler diagram. In the fallacy of the inverse, it is invalid because cotton is not a dog doesn’t mean that cotton will not necessarily be hairy.
Here’s an example for the fallacy of the converse. It is valid to say “All dogs are hairy. My pet cotton is not hairy. Therefore, cotton is not a dog”. In the fallacy of the converse, it is invalid to say “All dogs are hairy. My pet cotton is hairy. Therefore, cotton is a dog”. This is invalid because just because she said that cotton is hairy doesn’t mean it’s going to be a dog. It might be a hairy cat.
So, here are the fallacies of the inverse and the fallacy of the converse when displayed in a truth table. Unlike valid arguments where there’s only one truth value, mainly true, these two fallacies are not always true. In the highlighted case where p is false and q is true, both fallacies become false.
So, the fallacies are not tautologies, and that is just the tip of the iceberg. There are many other fallacies that have defects in content. Even though there may not be a problem with the logical structure. For example, the fallacy Argumentum Ad Hominem has its conclusion justified by attacking the arguer, whether to its character or to its personality.
Argumentum Ad Populum appeals to the majority of the audience. It’s like the bandwagon principle. A product is good because 90% of the consumers agree that it is good. This is an example of Ad Populum—an appeal to popularity.
Another fallacy is Appeal to Authority. Historically, there is an argument that you should take lots of vitamin C. That’s why we have daily supplements and that is because someone of a reasonable [or a high] authority rank said that. Just because someone who occupy—who has won a Nobel Prize said that “vitamin C is good for you” doesn’t necessarily mean that—doesn’t necessarily agree with the scientific evidence of the effects of vitamin C.
Another fallacy is False Cause. It’s like saying “X is correlated to Y so, X causes Y”. An example: whenever I wake up, it will rain today. So, my waking up caused the rain. This is a false cause.
And, there’s also the fallacy of Hasty Generalization. The conclusion is justified by very few supporting examples. A famous example links autism with vaccination, despite this not being true. The argument states that due to a small number of cases that seem to link autism with vaccinations, then we would say that autism should be linked to vaccination. That is a hasty generalization. It does not consider reliable and sufficient evidence.
And here’s one more: Argumentum Ad Baculum, when in doubt, use force. This comic is self explanatory.
There are some video resources on fallacies and their examples—two, to name a few.
Now, I would like to talk about deductive and inductive reasoning. Consider the following classical statement: :All men are evil. Socrates is a man. Therefore, Socrates is evil’. This is a classical example of inductive reasoning. You start with something general “all men are evil”, and you come up with a specific case: “Socrates being a man must be evil.” Inductive reasoning goes in the other direction. So, in a deductive argument, the premises yield a logical conclusion.That is, you start with something—you assume a general statement and you come up with a specific case. In an inductive argument, the premises merely support a plausible conclusion.
Here’s a mathematical example comparing deductive argument and inductive argument regarding even numbers. In a deductive argument, you would start by saying that “All multiples of two are even. 12 is a multiple of two. Therefore, 12 is even.” In an inductive argument, the idea of an even number is based on examples. We know that 2 is an even number, 12 is an even number, 22 is also an even number. So, inductively you can say that all numbers ending in 2 are even. Although, we still need to show if this conclusion is indeed true. And it is indeed true; all numbers ending in 2 are multiples of 2. Therefore, they are even.
Here’s our famous example: If p is prime consider the number 2 to the power p minus 1. Do you think it’s always prime? Now, we can verify for a few examples of p. It is true when p is 2, 3, and 5. So by inductive reasoning, we believe that 2 to the power p minus 1 is prime for any prime number p. However, the following counter example, 2 to the power of 11 minus one, states that the conclusion must be false. So, you see, in an inductive reasoning, you only get a plausible conclusion that is supported by a few or enough premises. If even just one counter example can be found then the conclusion is definitely false.
Let me introduce you to Pierre de Fermat, one of the famous mathematicians in the 17th century. He was interested in a problem that basically extends the Pythagorean theorem. Now, I would like to bring your attention to a very famous quote. He wrote in a margin of a mathematics textbook, “It is impossible to separate a cube into two cubes, or fourth power into two fourth powers, or in general any power higher than the second into two like powers. I have discovered a marvelous proof of this, which this margin is too narrow to contain.” This is a historical example of an inductive argument, called Fermat’s Last Theorem.
Now, think of the theorem as a major discovery or the logical conclusion of a deductive argument. Historically, at the time when Fermat’s Last Theorem was first stated, it was not actually a theorem, but it was rather, a conclusion by inductive argument. We can dissect Fermat’s quotes in the following manner and you can see how that translates to an inductive argument. Consider the statement that there are no natural numbers a, b and c such that a to the power of n plus b to the power of n equals c to the power of n. Now, this is true when n equals 3, also true when n equals 4.
Mathematicians have also verified this for other values of p. But at the time, it was not sure when the statement p is true for all n greater than 2. This is what you call Fermat’s Last Theorem. And it gets a little bit interesting. The good news is Fermat’s Last Theorem is now a theorem. Statement q is a logical conclusion from a deductive argument. However, it took 358 years for mathematicians to figure out that Fermat was right all along.
Sir Andrew Wiles provided the first successful verification of Fermat’s Last Theorem.
A little story about Andrew Wiles. He did publish—he did announce the proof of Fermat’s Last Theorem, however, some mathematicians later realized that there was a problem with his arguments—a hole in his proof. So, it took a few more years and the help of one of Wiles’ students in order to finally plug that hole. So, it was really a concerted effort of mathematicians around the world who are interested in number theory to show that Fermat was right all along, and that took 358 years.
I would like to show you how logic can be used to solve mathematical problems. Actually, there is an algorithm or a series of steps popularized by George Polya in his book: “How to Solve It”. You can consider this as a manual for solving all of your mathematical problems. It’s basically a four-step process. The very first thing you need to do is to understand the problem. Of course, you’re faced with a problem, so it is natural to understand it. Once you have an understanding, then you need to come up with the appropriate strategy. And it turns out that inductive and deductive reasoning are two of many strategies. You can also do pattern searching or reducing the problem into smaller bite-sized pieces or finding a similar problem. Once you have a strategy, carry out the plan, see if it works, look back and see how this might be applicable to other related problems.
So, you’ve learned what it’s like for an argument to be invalid and you have encountered fallacies of the converse, fallacies of the inverse, and other real-life fallacies. You’ve also learned how to argue in a deductive reasoning—using deductive reasoning—and inductive reasoning. And, you have also encountered some historical examples of inductive reasoning. In particular are the 350 years old mathematical problem. So, I hope that this helps you think better and think more logically.
I wish you all the best. Thank you very much.