Video Transcript: Negation and Disjunction

Hi, I'm David Feddes. And this talk is about negation, and disjunction. Before we get into those two terms, let's just remember what a truth functional connective is. It's a way of connecting  propositions such that the truth value of the resulting complex proposition can be determined  by the truth value of the propositions that compose it. So when you have a connective, then  you can predict what the truth values are going to be. If you know the truth values of the  individual propositions that were put together by that connective, an atomic proposition is a  statement makes just one claim. And there's no truth functional connective in that atomic  statement such as and or not, or if then, floor has been mopped, the dishes have been  washed, those are atomic propositions. A complex proposition links more than one proposition with truth functional connective words, and or not. If then, examples of a complex proposition  snow is white and snow is cold, the floor has been mopped and the dishes have been washed. So there's four truth functional connectives that we're going to be considering we've already  looked at conjunction. In the future we'll be looking at the conditionals symbolized by the  horseshoe, but in this talk, we're going to focus on negation, symbolized by the tilde, and  disjunction symbolized by the wedge or lowercase v. Let's start out by looking at negation.  negation is the truth functional operator that switches the truth value of proposition from  false to true, or from true to false. So let's take the statement dogs are mammals, to negate  that you say dogs are not mammals, or it is not the case that dogs are mammals. And  negation can be stated in more than one way there are various wordings the negation of  Satan is good is Satan is not good, or Satan isn't good. Or it's not true that Satan is good, or  it's not the case that Satan is good. You might want to say as Satan is bad, but that's a  different kind of proposition. So if you had your atomic proposition as Satan is good, then the  negation of that is expressed in various ways not isn't Not true, not the case. Each proposition is the negation of the atomic proposition. Satan is good. So what is involved with negation  when it comes to its impact on propositions? Unlike other truth functional connectives,  negation does not connect propositions Why would we call it a connective if it doesn't  connect? Well, it's called a truth functional connective, because it changes the truth value of  propositions in a truth functional way. That is, if we know the truth value of proposition that  we negate, then we know the truth value of the negated proposition. If we know that a  statement was true, and we negate it, then we know that negation is false. If we know that a  proposition is false, and we negate it, then we know that the negation is true. The truth table  for negation is pretty simple. In the top row, you have the proposition and then the negation  of the proposition P and then not P. And so you have two possibilities P is either true or false.  If P is true, then not P is false. If P is false, then not P is true, simple enough. Now, in  propositional logic, a constant is a capital letter we use to represent an atomic proposition. So let's say we're looking at two atomic propositions. Or one atomic proposition Satan is good  we'd symbolize that with a capital S. If you want to say Satan is not good, then we write Tildy  S and that symbolizes Satan is not good. If you say that the statement Michael is good will  symbolize that with the constant M. If you want to symbolize Michael is not good, then you'll  say Tildy M and that means Michael is not good. So if you put together a complex proposition,  here's how you do it. S means satan is good. M means Michael is good. So what is tilde S dot  M mean? Well, if S means satan is good, then tilde S means Satan is not good. The raised dot  means and and Michael is good. So you get a result Satan is not good. And Michael is good.  That's what tilde S dot M means. Now if you want to translate the other way. If we want to  take a complex proposition and turn it into symbolic logic. Let's give a few examples of that.  Samson loved Delilah. But Delilah did not love Samson. How do we symbolize that? Well, first  of all, we have to have our atomic propositions. So we'll use the constant S. For Samson loved Delilah. That's our first atomic proposition. The second atomic proposition is Delilah loved  Samson. You say no, she didn't. And the sentence says Delilah did not love Samson, but just  remember, we're symbolizing atomic propositions and an atomic proposition cannot contain a  truth functional connective, it can't contain a truth functional operator and the word not is an  operator. So you can't have an atomic proposition that contains not or a kind of negation. You  can have the negation later. So again, Samson loved Delilah. And the second atomic  proposition is Delilah loved Samson. And then if you want to symbolize that, you say S and  remember the word but is a raised dot. There are several ways of expressing the conjunction, 

but the raised dot is how you express that S raised dot and then tilde D, because we know  Delilah didn't love Samson. But if D means Delilah loved Samson and tilde D means she  didn't. So if you see S dot tilde D means Samson love Delilah, but Delilah did not love  Samson. Another example, Joshua and Caleb did not fear the Giants. Well, you'll begin by  asking what are the two atomic propositions? Joshua feared the Giants. Remember, again, you can't have Joshua did not fear the giants as an atomic proposition, because not is an operator. So it always has to be stated without the not in it when you're looking for the atomic  propositions. So Joshua feared the Giants is one atomic proposition and Caleb feared the  Giants is the other will symbolize those with J and C. And then we express the proposition  Joshua and Caleb did not fear the Giants by having tilde J raised dot tilde C negation of  Joshua, fear the Giants and negation of Caleb feared the Giants. So you see how that's done.  You take the overall complex proposition, you break it down into the two or more in some  cases, but in this case, two atomic propositions Joshua feared the Giants. Caleb feared the  Giants, then you'd say what are the operators? Well, there's two negations. Joshua didn't fear  them. And Caleb didn't fear them. So you put the negation in front of both of those. And then  you can join them put them together by using the dot for the conjunction. Well, that's enough  about negation. For now we'll get more practice on it in exercises and as we move into other  aspects of logic, but let's talk now about disjunction. And disjunction can be expressed by the  English word or, or either or. So if you have a complex proposition, that's a disjunction. You  could say Charlie or Julia tracked mud into the house. And you'd symbolize that by C the  constant for Charlie tracked in the mud, or J the constant for Julia tracked in the mud. And the  symbol for disjunction is the wedge or as we type it, the lowercase v. And that's the  disjunction of two atomic propositions Charlie tracked mud into the house, Julia tracked mud  in the house, give each of those propositions their constant, and put a V or a wedge in  between them, and you've got your disjunction. Now, when we're in logic is symbolic logic,  there is what's called an inclusive or an exclusive or a disjunction. That's true when either  disjunct is true, and even when both disjunct are true, is what we mean by an inclusive or so  if you say either Charlie or Julia tracked mud. Now see wedge J, is true if Charlie tracked in the mud, it's true of Julia tracked in the mud. It's also true is both tracked in the mud. You know,  when you read the English sentence, you might say, Well, it means either or, and it couldn't  be both. But if you're mean and inclusive, or then you do mean that it's true, the statement  either or is still true, even if both of them did it. The only case in which it's not true, is if  neither of them did it. So that's what's meant by an inclusive or there's also an exclusive OR  that we're really not going to use much in logic, at least not as a as a symbol with that wedge  V. A disjunction. That's true only if one or the other. But not both of its disjunct it's true. That's what we mean by an exclusive OR we mean that the statement would be false. If if both of  the disjunct were true, so you say B plays first or second in the race? Well, Bob plays first in  the race, or Bob plays second in the race clear now, but Bob can't finish both first and second  at the same time, that's impossible. So an exclusive OR means you say Bob did this or he did  that. But you know, he didn't do both. As I said a moment ago, in our symbolic logic, the  wedge means inclusive or not exclusive. In formal logic, the wedge is always inclusive. So C  wedge j is true if C is true, or if j is true, and it's true, if both C and J are true. And there's a  reason why the logic masters do that, you can make an exclusive R and build it out of other  connectives. But if you needed to symbolize an inclusive or you can't build that out of the  other connectives that we're working with, so if you use the various connectives we're talking  about, you can build and symbolize in logic, the exclusive kind of order, which means either  this one can be true, or that one can be true, but not both. That can be symbolized using our  connectives. And so we have to avoid using our understanding of an inclusive or with the  wedge it always, or I mean an exclusive OR with the wedge, the wedge is always inclusive,  which means that if both are true, in the disjunction, then the statement is true, C or J is true,  if both are true, as well as when either are true. So, a truth table of that just a reminder, truth  tables represent how the truth value of a complex proposition depends on the truth values of  the propositions that compose that complex proposition. And here's a truth table for  disjunction. And the top line you have the two propositions P and Q, and then the complex  proposition, P or Q. And there's four different scenarios. You could have in the first line, P is 

true and Q is true, second line P is true, Q is false. Third line the other way around P is false Q  is true. And the fourth line P and Q are both false. What are the truth values? Well, if P and Q  are true in a disjunction, then P or Q is true. If P is true, and Q is false, then P or Q is true. If P  

is false and Q is true, then P wedge Q is true. So in all three scenarios if they're both true,  then the disjunction is true. If one is true, either one, then the disjunction is true. The only  time a disjunction is false, is if both parts of the disjunction both atomic propositions in the  disjunction are false, then the disjunction is indeed false. So let's work through a few  examples. Here's a disjunction where S symbolizes a Satan is good. M symbolizes Michael is  good. What does S wedge M mean? It means satan is good, or Michael is good. Now, let's  work in the other direction. What does it mean? If we have those same two atomic  propositions? What does it mean to say Tildy S wedge M. What's that mean? It means satan is not good because Tildy is negation. So if the S means Satan is good, then tilde S means Satan is not good. The wedge means or an M simply means Michael is good. Here's another  example. What does this one mean? S wedge tilde M. Well S means Satan is good's that'd be  the first part. You say Satan is good wedge means or so Satan is good or, and then you use  the tilde for negation. That means Michael is not good. So you'd say Satan is good, or Michael  is not good. If you take that complex proposition, that will be false, because it's false to say  Satan is good. And it's false to say Michael the archangel is not good. So in either case, S or  not M. The result is false. But remember in the truth table, in disjunction, both have to be  false in order for the complex proposition to be false, in this case, both are false. It's false to  say Satan is good. It's false to say Michael is not good. So even if you put them together with  an or, the result is still a false statement. Here's another example. Either God wrote the Bible  or people wrote the Bible. So we have to first identify what are our atomic propositions? Well,  let's use G to symbolize God wrote the Bible. Let's use The constant P to symbolize people  wrote the Bible. And then how do we express that? Well, simply G wedge P. God wrote the  Bible or people wrote the Bible, to do it the other direction, and take this proposition God, not  people wrote the Bible. Well, our atomic propositions are going to be, God wrote the Bible, G  and people wrote the Bible, P, and if we're going to symbolize that, if you want to say God  wrote the Bible, then we're going to say, G, but we also want to say that people did not write  the Bible. So we would say G and remember now, throwing you for a loop, no, now we're  going to conjunction. Remember, this isn't a disjunction. It's saying that God wrote the Bible  and people didn't. So G symbolizes God wrote the Bible, the raised dots symbolizes and and  the tilde P means not people writing the Bible. So those are your two functional connectives  conjunction uses the raised dot P and Q. And negation is the tilde means not P disjunction is  the wedge means P or Q and remember that that's an inclusive disjunction where it's true if  either P or Q is true, and it's also true if P and Q are true. So it's an inclusive disjunction.  When you see the wedge later on, we'll get into that horseshoe business. The meanwhile let's focus and do some exercises on negation and disjunction in your assignments.