Hello, and welcome to Presentation 3c: Some Counterintuitive Facts about Validity. You will recall that an argument is valid if and only if there is no logically possible situation where all the premises are true and the conclusion is false at the same time. Equivalently, we can say that an argument is valid if and only if it is logically impossible for the premises to be true while the conclusion is false. The notion of validity is supposed to capture the fairly intuitive notion of one claim following from some others. But it’s important to remember that the term “validity” has a precise stipulated definition, and whether an argument is valid or not depends on whether it satisfies this definition, not on whether it satisfies any intuitive related notions. This stipulated definition has some consequences that might seem counterintuitive. Understanding these consequences will help us better understand and learn to correctly apply the notion of validity. So let’s look at some of these counterintuitive consequences. Counterintuitive consequence #1: Circular arguments are valid. For our purposes here, let’s take a circular argument to be an argument where the conclusion is also one of the premises. Here is an example of a circular argument: The earth is flat. Therefore, the earth is flat. Here’s another example: There are fairies in your backyard. Rocks have souls. Therefore, there are fairies in your backyard. In both cases, the conclusion is also one of the premises. So, these arguments satisfy our definition of a circular argument---they are circular. Of course, these are terrible arguments. They shouldn’t convince anyone. However, counterintuitively, these arguments are valid. And the same is true of any circular argument. But how come? Let’s look at our definition of validity again. An argument is valid if and only if there is no logically possible situation where all the premises are true and the conclusion is false at the same time. One way to think of it is this: In order for an argument to be invalid, then it has to be possible for the premises to be true and the conclusion false. In other words, there has to be an invalidating counterexample. But in a circular argument, the conclusion is just one of the premises. And when an argument’s conclusion is also one of its premises, it’s logically impossible for the premises to be true while the conclusion is false. This is because the conclusion is just one of the premises, so if they are all true, the conclusion must be true as well. Let’s look at an example to make this clearer. Take the argument “The earth is flat. Therefore, the earth is flat.” In order for the argument to be invalid, it must be possible for the premises to be true while the conclusion is false. But suppose all the premises are in fact true. The conclusion is the same as the premise. So, it must also be true! So, it’s not possible for the premises to be true while the conclusion is false. And so, the argument satisfies our definition of validity—it is valid. Here is another way to think about it. In order for an argument to be invalid, there has to be an invalidating counterexample––that is, a possible situation in which the premises are true and the conclusion is false. So what would that look like in this example? The premises would be true. And the conclusion would be false. But that’s impossible because the conclusion just is the same as the single premise. One can’t be true while the other is false. So there is no possible invalidating counterexample. Let’s move on to counterintuitive consequence #2: Arguments with necessarily true conclusions are valid. In some cases, it is logically impossible for an argument’s conclusion to be false. In other words, there is absolutely no way that the conclusion can be false––it’s guaranteed to be true no matter what. Another word for a necessarily true statement is a tautology. A tautology is true no matter what. It cannot be false. Here are some examples: “Either I will eat lunch or I will not eat lunch.” “Everyone who eats lunch eats something.” “If you eat lunch, you will eat.” These statements are necessarily true––they’re true no matter what actually happens, no matter who ends up eating lunch or not. They are tautologies. Feel free to pause this video to take a look at this comic about tautologies from xkcd. How many tautologies can you find? Are some of them merely implied? Can you explain why it’s supposed to be funny? Here are some examples of arguments with necessarily true conclusions––conclusions that cannot be false: Rainbows are made of ice cream. Therefore, either it is raining here now or it is not raining here now. This conclusion can’t be false because it is either raining here now or it is not. If it is, the conclusion is true. If it is not, the conclusion is again true. Either way, the conclusion is true. Here’s another argument with a necessarily true conclusion: Wood chips are nutritious. There are unicorns. Therefore, if the earth is flat, then the earth is flat. The conclusion says that if the earth is flat, then the earth is flat. Whether or not the earth is in fact flat, it’s true that if it is flat then it is flat. So the conclusion is true no matter what––there’s no way for it to be false. These arguments are not very good ones. For one, the conclusion has nothing to do with the premises. However, they are both valid! And the same is true of any argument with a necessarily true conclusion––they’re all valid. But why? Let’s return to our definition of validity. The definition of validity says that an argument is valid if and only if there is no logically possible situation where all the premises are true and the conclusion is false at the same time. In the case of an argument with a necessarily true conclusion, there is no possible way for the conclusion to be false. So, there is no possible way for the premises to be true and the conclusion false. So, it satisfies our definition of validity! Let’s look at one of our examples to make this clearer. In this argument, there is no possible way for the conclusion to be false. Whether or not the earth is flat, it is true that if it is flat then it is flat. So, there is no possible way for the premises to be true while the conclusion is false. So, the argument satisfies the definition of validity. Here is another way to think about it. Recall that in order for an argument to be invalid, there has to be an invalidating counterexample––that is, a possible situation in which the premises are true and the conclusion false, a way for the premises to be true and the conclusion false at the same time. But when the conclusion of an argument is a necessary truth, there is no way for the conclusion to be false. And so there is no possible situation in which the premises are true and the conclusion false––that is, there is no invalidating counterexample. Let’s move on to counterintuitive consequence #3: Arguments with inconsistent premises are valid. Inconsistent premises are premises that cannot all be true at the same time. More generally, a set of statements is inconsistent when all members of the set cannot be true at the same time. For example, this set of statements is inconsistent: If horses are mammals, then they are warm-blooded. Horses are mammals. Horses are not warm-blooded. Let’s look at an example of an argument with inconsistent premises. Rocks have souls. Rocks do not have souls. Therefore, the moon is made of cheese. It isn’t possible for both premises to be true at the same time. At least one must be false. So the premises are inconsistent. Here’s another example: It’s raining here now and it’s not raining here now. Therefore, the earth is flat. Here, it’s not possible for all the premises to be true at the same time because the single premise is a contraction—it just can’t be true, no matter what. Now, these are terrible arguments. But they are valid. Let’s see why. Let’s look at our definition of validity again. An argument is valid if and only if there is no logically possible situation where all the premises are true and the conclusion is false. In other words, an argument is valid if and only if it’s logically impossible for the premises to be true and the conclusion false. In the case of an argument with inconsistent premises, there is no possible way for its premises to all be true. So, there is no possible way for its premises to be true while its conclusion is false. So, the argument satisfies the definition of validity. Put otherwise, in order for an argument to be invalid, there has to be an invalidating counterexample––a possible way in which the premises are true and the conclusion false. In this argument, an invalidating counterexample would be a logically possible scenario in which the two premises are true and the conclusion is false. But in an argument with inconsistent premises, like this one, there is no possible way for the premises to all be true together. So there is no possible way for them to be true together while the conclusion is false. So there are no invalidating counterexamples. So the argument is not invalid––it is valid. So, all arguments with inconsistent premises are valid. Incidentally, this is, roughly, why anything follows from a contradiction, even patently false claims. Any argument with inconsistent premises is valid, regardless of its conclusion. This xkcd comic pokes fun at this fact. Feel free to pause the video to take a look at it. Can you explain why it’s supposed to be funny using the concept of a valid argument? The moral of the story is that the notion of validity has a precise definition. It is not simply the everyday notion of a good argument. When considering whether an argument is valid, make sure to consult that definition. That’s all for now. Thanks for watching!