In unit two of this course in our study of the rules governing the validity of deductive arguments, we've looked at other ways of representing information, other ways besides sentences. We've looked at representing information using a truth table or using a Venn diagram. Now today in this final lecture of unit two, I'd like to talk about the advantages of these other ways of representing information. What's the point of using truth tables or using Venn diagrams? Why did we learn these techniques? That's the topic of today's lecture. Okay, so let's consider some different ways that we can represent the same information. Consider for example this list of seven sentences. Here's seven sentences. They all represent the very same information, they all mean the very same thing, they just say the same thing in different languages. One is in French, one is Italian, one is in Spanish and so on, but even though they're in different languages and they look different, they all express the same information. They represent the same information. Well similarly, just as we can use different sentences to represent that information, we can also use something that is not a sentence at all to represent that information. For instance, here we can use a truth table to represent the very same information that we were representing by means of those different sentences in the previous slide. So look at this truth table for a moment. Now, first [COUGH] column of the truth table is the proposition Walter likes bourbon, the second column of the truth table is the proposition Walter likes vodka, and the third column is the conjunction of those two propositions. Which could naturally be expressed in English by the sentence, Walter likes bourbon and vodka. And now what we can do, once we have this truth table here that shows how the truth or falsity of the conjunction depends on the truth or falsity of each of the conjuncts, we can circle a particular line of that truth table to indicate that this is the situation that we're actually in. Okay, now when we circle that particular row of the truth table, what we're doing is expressing the very same information that we expressed using one of these seven sentences earlier, right? There's no difference in what information we're expressing, what information we're representing. We're just representing it without using a sentence. We're representing it by circling a row of the truth table, okay? So it's just a different way of representing the same information. Now you might wonder, well, what's the point of representing that same information in different ways? Well, the point of representing the same information using different sentences in different languages is just that you can make yourself understood by different groups of people. You can say it in French to make yourself understood in France. You can say it in Spanish to make yourself understood in Spain or in Latin America. You can say it in Russian to make yourself understood in Russia, and so forth. So, what's the point of using the truth table, and there's no country where they speak truth table, so what's the point of using a truth table to express the very same information? Well, the point is that by using a truth table to represent the very same information, you represent that information in a way that makes very clear exactly what deductive arguments that use that information are valid and which ones are invalid. So for instance, by looking at this truth table, looking at our representation of the information that uses this truth table, we can see very clearly that the argument from the premise, Walter likes bourbon and vodka, to the conclusion, Walter likes bourbon, is going to be a valid argument. There's no possible way for the premise to be true while the conclusion is false, right? If the premise is true, then the conclusion is also going to have to be true. But we can also see that the deductive argument from the premise, Walter likes bourbon, to the conclusion, Walter likes bourbon and vodka, is invalid. There is a possible way for the premise, Walter likes bourbon, to be true while the conclusion, Walter likes bourbon and vodka, is false. So by looking at the truth table, you can see very plainly why some deductive arguments that involve the proposition, Walter likes bourbon and vodka, are valid. And other deductive arguments involving that same proposition are invalid. And that's something that you can't see just as clearly by looking at any of the seven sentences that we can use to represent that information. So these sentences have some advantages as a way of representing the information that they all represent. But they all have a disadvantage relative to the truth table, which shows us plainly why certain arguments that use that information are valid and others are invalid. So that's the advantage of using a truth table to represent information that could be represented more naturally by means of sentences. It's not that we make ourselves understood by more people when we use a truth table. It's rather that when we use a truth table, we can see relations of deductive validity that we can't just see when we use sentences. Okay, now how about Venn diagrams? Well, consider again a piece of information that could be represented using any of these seven sentences, right? One is in French, one is in Spanish, one is in Italian, and so on. All seven of these sentences mean the same thing. They express the very same information. And they represent it, even though they represent the same information, they're useful in different situations. You might use one when you're in Russia, and you want to be understood by Russians. You might use another when you're in India, and you want to be understood by a certain population of people in India. You might want to use another when you're in Brazil or Portugal, and you want to be understood by Portuguese speakers. But notice, none of these sentences represent the information that they represent in a way that makes it completely clear which deductive arguments that use that information are valid and which are invalid. Okay, we can make that point clear by representing the same information using a Venn diagram. First, we construct a circle to represent the category of Walter's drinks. And second, we construct a circle to represent the category of imported things. And since these seven sentences all say that all of Walter's drinks are imported, we can then shade out the part of the circle representing Walter's drinks that's outside the circle of imported things. To show us that if Walter drinks anything, then whatever it is that he drinks must be imported, it must be inside this region here. Okay, and now we have a Venn diagram that represents the very same information that was represented by those seven sentences. But what's the point of representing this information using a Venn diagram? Well the Venn diagram shows us which deductive arguments that use that information are valid and which are invalid. So for instance, consider the argument from all Walter's drinks are imported to all imported drinks are Walter's. Is that argument valid or invalid? Well, if you just look at the sentences, it might not be obvious whether it's valid or invalid, but if you look at the Venn diagram, you can see quite clearly that this argument is invalid. For all of Walter's drinks to be imported is for this part of the Venn diagram to be shaded in. But for all imported drinks to be Walter's would be for this part of the Venn diagram to be shaded in. And the question is, is there some way for the premise to be true while the conclusion is false? And the answer is clearly, yes. This part of the Venn diagram could be shaded in, even if this part of the Venn diagram is not shaded in. And so, this is an example of how we can use the Venn diagram to show very plainly and visually why certain arguments that use the information, all Walter's drinks are imported. Certain of those arguments are valid and certain of them are invalid. Again, so the point of using the Venn diagram to represent information is in that respect very similar to the point of using a truth table to represent information. When you want to understand whether a particular deductive argument is valid or invalid, sometimes it helps to translate the information in that argument into a form where you can plainly see the relations of validity or invalidity, into a form like a truth table or a Venn diagram. Where you can plainly see those relationships because ordinary language doesn't always expose those relationships. Okay, so that's why truth tables and Venn diagrams are useful devices for understanding whether deductive arguments are valid or invalid. And there are many other useful devices like that, but truth tables and Venn diagrams are the two simplest ones, and so the two that we focused on in this course. Okay, well, have fun with the quizzes and have fun with the rest of the course. See you next time.
