Video Transcript: Not Both and Neither Nor

Hi, I'm David Feddes and this logic talk is about not both, and neither nor those two phrases  are sometimes challenging to translate into the language of symbols and of logic. And we're  going to talk about how to do that today. Let me just before getting into that, say again that  I'm basing these talks on the writing of Professor Dr. Mathew Van Cleave. And if you're really  good at reading, and you take things in by reading and looking at it on paper, you really don't  

even need to watch these videos. But for many people, hearing and seeing helps to learn  better than just reading. So I hope that these talks are helpful to some of you who can't just  read Dr. Vancleave on the page and get it right away. So let's think about not both, and  neither nor. Not both, and neither nor means different things. Here are a couple of sentences,  Carla will not have both cake and ice cream. The other is Carla will have neither cake nor ice  cream. Now, those are two different statements. And a way to understand the difference  between them is to ask ourselves, what scenarios does each of those sentences rule out?  Let's start with the first one, Carla will not have both cake and ice cream. And when we look  at that sentence, we can see four different situations imagine those if we think about having  cake and having ice cream. If she has cake, and she has ice cream, then it's false, that she  won't have both cake and ice cream, you can see that in the first line of this little table. If she  has cake and does not have ice cream, then it's true that Carla did not have both cake and ice cream. If she doesn't have cake, but does have ice cream, then it's also true that Karla didn't  have both cake and ice cream. And if she doesn't have cake, and she doesn't have ice cream,  maybe she's just on a diet and not going to eat either one, then it's also true that she will not  have both. So there's only one scenario in which it's false. And that is if she has cake and she  has ice cream, then it's false that she didn't have both. Now, how do we symbolize that? Well,  let's break it down into its atomic propositions first of all, and those are Carla has cake. And  Carla has ice cream. And if she doesn't have both, then you say that the conjunction is false.  The conjunction would be Carla has cake and Carla has ice cream. And if you say she doesn't  have both, then you put the tilde and then parentheses around C conjunction I. So you see  she doesn't have both cake and ice cream is symbolized in that way. Now not both P and Q is  translated as the negation of a conjunction as we've just seen. So you take the quantity P and Q in parentheses, and then you negated where the tilde the negation is the main operator  and then it applies to everything in the parentheses. So you're saying not both P and Q not  taking ice cream, and you put a parenthesis around that and then you're negating the total  conjunction. Now, remember that parentheses do matter here, not P and Q in parentheses  has a different meaning from not P and Q. If you have not P and Q in parentheses, it means  that P and Q are both are not both true one of them could be true, but both at the same time  aren't true. Not P and then the raised dot and Q means that P is not true, and Q is true. You  see the difference. In the one case, with the tilde in front of the stuff in parentheses, it means  that P and Q are both true at the same time. And the other way of stating it says that P is  true, and Q is so parentheses matter. atomic propositions Carla has cake. Carla has ice cream. Not C and I in parentheses means Carla will not have both cake and ice cream. Not C and I  means Carla will not have cake but she will have ice cream. So the way you put your  parentheses matters very much. Now, we talked earlier about an inclusive or an exclusive or  in a different talk and an inclusive or was what we meant by the wedge and are inclusive or  meant that if either of those is true, then the statement P or Q is true, but an exclusive OR  means that only one of the two is true. And so we said that whenever you see a wedge that  symbolizes an inclusive or but how do you go about symbolizing an exclusive OR eXClusive or  is something like Bob placed either first or second in the race. And you know that he didn't do  both, but he did do one of the two. So you take your atomic propositions Bob placed first,  make F the constant for that Bob placed second, make S, the constant for that and how now  do we symbolize an exclusive or where one or the other is true, but not both of them at the  same time? Well, F or S, F wedge S is inclusive. And that means that Bob could get first he  could get second, or he could get both first and second. So that's not an adequate way of  symbolizing that Bob placed either first or second in the race with an exclusive or the wedge  is an inclusive and so we need to find a different way to state the exclusive or, Bob plays first  or second in the race. We can symbolize that as F or S. But if we want to make it an exclusive  OR then we also have to make another statement, Bob did not place both first and second. 

See what I mean. In the first one, we're saying that he could have placed first or second. But  we also want to say he didn't do both at the same time. And in a race, it's kind of hard to be  both first and second. And the way of showing that is to say the conjunction would be false.  

First and Second as a conjunction in parentheses, that's false. So he can be first, he can be  second. That's what we say by F wedge S but it can't be both. That's what we say by Tilde  with F dot S in parentheses. So, the way to symbolize an exclusive OR is to put those two  statements together, Bob place first or second, F wedge S, Bob did not place both first and  second tilde F dot S in parentheses, and then you just put those two together and you've got  your exclusive OR symbolized so you can join, put in parentheses F wedge S as you see there, then the raised dot is your main operator and then you have a tilde F dot S which means not  both F and S at the same time you didn't get both first and second at the same time. So again in English you've got your atomic propositions Bob place first and Bob placed second and  you're saying one of the other and not both at the same time that your exclusive or now how  do we express neither nor in symbolic language? We say Carla will eat neither cake nor ice  cream maybe she's on a diet whatever the reason, but she's gonna eat neither cake nor ice  cream. First thing you do of course is you take your atomic propositions, you can't have a not  in those. So you have to just state positively Carla will eat cake with a C for your constant  Carla will eat ice cream with an I for your constant when you say she will eat neither cake nor  ice cream. Using those atomic propositions you make your little table and if she eats cake and she eats ice cream if C and I are both true, then it's false. That Carla would neither eat cake  and ice cream. Now if she eats cake and doesn't eat ice cream, then that overall statement is  still false because we were saying she'll either cake nor ice cream. So if she ate cake, the  statements false if she didn't eat cake tilde C but did eat ice cream we're talking about the  third row now that it's false to say that she will eat neither eat cake and ice cream. The only  time it's true to say that Carla will eat neither cake nor ice cream is if C and I are both false or  if they're both negated tilde C tilde I. If both of those atomic propositions are negated then  the statement is true that Carla will neither eat cake or ice cream. So to translate a neither  nor statement, you do it using a conjunction of two negations. Tilde C she won't eat cake. Dot  is The and of the conjunction and then tilde I negation of she'll eat ice cream. And when you  take those two conjuncts not C and not I and join them together in a conjunction then you've  expressed She will eat neither cake nor ice cream. Got it? Let's move on. Neither Nor  statements are expressed as a conjunction of two negation. So this is just kind of a general  form of that tilde P dot tilde Q and you're saying not P and not Q. Now, an interesting thing is  this can be proved to be equivalent to be the negation of a disjunction. The two statements  you see in yellow are saying exactly the same thing. You can work it out If you knew how to  do it, and you can show that not P and not Q means exactly the same as not the quantity P or  Q. Because you're saying neither P nor Q is true. So in Symbolic Logic, tilde P dot tilde Q, is  saying exactly the same thing as tilde parenthesis P wedge Q parentheses isn't a joyful thing  to look at these symbols. But the logic behind it and the facts of the matter are pretty clear.  When you say this isn't true, and this isn't true, then you're also saying, neither of those is  true, the statement this or this would be false. And that's simply what you mean by the  negation of P wedge Q.