Over the past couple of weeks, we've seen how we can use truth tables and Venn diagrams to predict and explain the validity of arguments. But the arguments that we've looked at are all arguments that are given in ordinary language that we can understand. But of course not all arguments are like that. Some arguments are given in foreign languages that we don't understand. Or they're given in technical languages that we don't understand. Could you use Truth Tables or Venn diagrams to predict and explain the validity of those arguments? Even those arguments that are given in languages that you don't yet understand? Well, today I'm going to show you that you can, you can do that. And I'm going to show it by giving some examples. So let's start with a simple but fanciful example. So, suppose you're an anthropologist, you're a cultural anthropologist. You're studying some foreign culture and you're trying to translate their language. Because their language has been heretofore untranslated, so you go in and you're translating their language. You're figuring out what they mean by various words that they use. And you've translated most of their words, but there's this word they use, SPooG. And you haven't yet figured out what that means. And so you're observing their behavior and finally you come up with a hypothesis. [COUGH] and you're pretty confident of this hypothesis. Your hypothesis is that they use SPooG as a truth functional connective. And it's a truth functional connective that has this truth table. It connects two propositions to make a larger proposition. And that resulting proposition P SPooG Q, for whatever propositions P and Q SPooG is connecting. That resulting proposition is going to be true whenever P is true or whenever both P and Q are false. So the only scenario in which P SPooG Q is going to be false is where P is false and Q is true. So that's your hypothesis about what they mean by SPooG. Okay, now one day you hear one of the members of this foreign culture give an argument. And, using the translation manual that you've developed, you translate their argument as follows. John is riding his bicycle. SPooG Jill is walking to the park. Premise two, Jill is walking to the park therefore, John is riding his bicycle. Now, is this argument valid or invalid? Well, you can use the truth table for SPooG to figure out whether the argument is valid or invalid and also to see why. So, let's consider how the truth table would go. This is the truth table again for SPooG, applied to the case that we're considering. Well, premise one tells us that John is riding his bicycle SPooG, Jill is walking to the park. So if premise one is true, then we're either in this scenario, in this scenario, or in this scenario. Premise two tells us that Jill is walking to the park. So if premise two is true, then we're either in this scenario, or in this scenario. Okay, so what do premises one and two put together, what do they tell us? Well, premises one and two tell us that, first of all, we're not in this scenario. because in this scenario it's not going to be true that Jill is walking to the park. Again, premises one and two put together tell us that we're not in this scenario. because in this scenario, it's not going to be true that John is riding his bicycle SPooG Jill is walking to the park. Again, premises one and two tell us that we're not in this scenario. because in this scenario, it's not going to be true that Jill is walking to the park. So, if premises one and two are both true, that tells us that we've got to be in this scenario. But in this scenario, it's gotta be true that John is riding his bicycle. And so using the truth table for SPooG, you can figure out that this argument is valid. And you can explain why it's valid. See? So you can use the truth table for a truth functional connective that you don't have any understanding of independently of the truth table for it, right? It's not a connective that you use in your language. You can just use the truth table to figure out that a particular argument is valid. Let's try that with a more complicated case. Okay, so now suppose you hear some member of this foreign culture arguing as follows, at least according to your translation. John is riding his bicycle SPooG, Jill is walking to the park or Frank is sick. Frank is not sick, John is not riding his bicycle. Therefore, Jill is walking to the park. Okay, now is that argument valid or not? Okay, so let's look at this more complicated truth table. Okay so, premise three tells us that John is not riding his bicycle. So that tells us that we've got to be in one of these four scenarios. Premise two tells us that Frank is not sick. So that tells us that we've got to be in one of these four scenarios. And premise one tells us that John is riding his bicycle, SPooG, Jill is walking to the park, or Frank is sick. So that tells us that we've gotta be in one of these scenarios right here. Okay, so based on that information, what can we figure out? Well, we can immediately figure out that we're not in one of these top four scenarios. because in all of those top four scenarios, Frank is riding his bicycle, right? So premises one, two, and three together, rule out the top four scenarios. We can also figure out that we're not in scenarios five or seven because in scenarios five or seven, Frank is sick. So we can rule out those scenarios. And we can rule out the sixth scenario because in the sixth scenario it's not true that John is riding his bicycle SPooG, Jill is walking to the park, and Frank is sick. So based on the information that we get from premises one, two and three of the argument, we can deduce that we're in this last scenario. We've gotta be in this last scenario. And in this last scenario it's false that Jill is walking to the park, in other words, Jill is not walking to the park. But that's not what the conclusion of our argument says. So this argument, this more complex argument is not valid. And we can prove that as we just did using the truth table for SPooG and disjunction. Okay, now let me show how we can do the same thing with quantifiers. So imagine as you're translating this foreign language, you hear a word. It's always said in a high-pitched tone, an excited tone, Jid! And you wonder what does Jid mean? Well, after watching the way they use that word, you come up with a hypothesis. Your hypothesis is that when they use the word jid, they're using a quantifier. And the quantifier works like this. When you say Jid F or G, you're saying that there aren't any F that are outside the G category. But there is an F that's inside the G category. So this diagram represents, according to your hypothesis,this diagram represents what the members of this foreign culture mean by Jid!. Okay, so now suppose you hear one of the members of this foreign culture give the following argument. Jid! Giraffes are herbivores, you translate it. Then they say Jid! Herbivores are mammals. And then they draw the conclusion there are some giraffes, and all of them are mammals. Now, is that argument valid? Well, let's use the Venn diagram for Jid! to figure out whether or not it's valid. So the first premise says, Jid! Giraffes are herbivores. So, how do we represent that? Well, remember, we have to shade out the portion of the Giraffes' circle that's outside the Herbivores' circle. And we have to draw an X that's in the Giraffes' circle and in the Herbivores' circle. We don't know quite where to draw that X, so let's hedge our bets right now and draw it over here, okay? The second premise though tells us that Jid herbivores are mammals. Now, how do we represent that? Well, remember by the Venn diagram, we have to shade out that part of the Herbivore circle that's outside the Mammal circle. And draw an X inside the Herbivore circle that's also inside the Mammal circle. Right, now we know this X we drew on the edge has to be inside the Mammal circle, not outside the Mammal circle, okay? So we're going to put this X right here on the edge of the Mammal circle and the Giraffe circle. because we don't know specifically which one it's supposed to go into. But wait, we do, and here's why we do. Remember, that when we drew the X on the border of the Mammals' circle and the Herbivores' circle right here, inside the Giraffes' circle. We weren't sure if that X was supposed to go in here or in here. Now we know that this X is supposed to go in here. So, maybe they're also Herbivore Mammals that are not Giraffes. But what we can definitely conclude from premises one and two, is that there are Herbivore Mammals that are Giraffes. So we know that there's an X that's supposed to go in this region right here. There's an X in that region right here and then this whole region is shaded out, there's nothing in this whole region. That's what we know from premises one and two using the Venn diagram for Jid, excuse me, for Jid! Okay, now, let's apply that to our argument. Well, what does our argument say? It says, there are some giraffes and all of them are mammals. But that's exactly what we can read off from the Venn diagram that we just constructed, right? If you look at the Venn diagram that we just constructed, there's an X that's inside the Giraffe circle and inside the Mammal circle. And every part of the Giraffe circle that's outside the Mammal circle is shaded out. So there aren't any giraffes that are not mammals. Okay, so we just figured out that this argument is valid. And we figured it out even though our only understanding of Jid, was from the Venn diagram that we constructed. Now, I have another question. Given this Venn diagram How would you translate Jid into English? Well, often people use the word all to mean the same thing that Jid means according to this Venn diagram. People speaking ordinary English use the word all to mean precisely this quantifier, right? When they use the word all they often mean. That there are things in the category that they're modifying by all. Like if I say, all ravens are birds, often what I'm understood to mean is there actually are some ravens. And all the ravens there are, are birds. So there are some things in the F circle, but they're all inside the G circle. There are some things in the circle of Ravens, let's say, but they're all also inside the circle of birds. There are no Ravens outside the Birds' circle, but there are Ravens inside the Birds' circle. That's often what people would understand me to mean if I said all ravens are birds. Now, we've been using the quantifier all. We've been understanding the quantifier all in a different way. So that it doesn't imply that there actually are members of the category that's being modified by all. So the way we've been using the quantifier all, if you say all ravens are birds, all you mean is there aren't any ravens that are not birds. But that's not the same as saying there are ravens, and all of them are birds. So it looks like we can use the Venn diagram for Jid! To translate that quantifier, in a foreign language into a familiar quantifier from ordinary English.
