>> For this lesson, we're going to build on last class dealing with the truth table operator definitions. And now we're going to be working on building [inaudible] truth tables and using them to evaluate statements, and then eventually evaluating arguments. The function of a truth table is to cover all possible combinations of truth values that could happen in a statement. Now with simple statements, it's kind of easy to know the possibility. So for instance, if I say, "Matthew likes ice cream," there's only two possibilities for that. Either it's a true statement or it's a false statement. But if we add another component to this, so if we say, "Matthew likes ice cream and pizza," there's a few more ways these truth combinations could go. It could be true that I like ice cream, and true that I like pizza. It could be true that I like ice cream and false that I like pizza. It could be false that I like ice cream and true that I like pizza. Or it could be false that I like ice cream and false that I like pizza. So the complexity of the evaluation of all the components, got a little bit more. Then if we added another variable, "and," so now Matthew likes ice cream and pizza and let's add a third component. Let's go Cheetos. We've got to add some more truth variables to that. So, we have -- it could be true that Matthew likes ice cream, true that Matthew likes pizza, true that Matthew likes Cheetos. True that Matthew likes ice cream, true that Matthew likes pizza, false that Matthew likes Cheetos. True that Matthew likes ice cream, false that Matthew likes pizza, true that Matthew likes Cheetos. True that Matthew likes ice cream, false that Matthew likes pizza, and false that Matthew likes Cheetos. Now we've got to go through all these false combinations of ice cream. So it could be false that I like ice cream. It could be true that I like pizza. True that I like Cheetos. False that I like ice cream, true that I like pizza, and false that I like Cheetos. False that I like ice cream again, false that I like pizza, true that I like Cheetos. And finally, false that I like ice cream, false that I like pizza, and false that I like Cheetos. So as you add variables, the combinations get more and more complex. There's actually a formula for setting up the amount of combinations you could have. So what we're going to learn today is the actual formula for figuring out how many different combinations of truths and falses we could have in any given amount of statement combinations. Today we're going to evaluate statements, compound statements, like this one here where you have two operators. Sometimes you can have one operator, two operators, more operators. So we're going to evaluate compound statements using truth tables, and we'll learn something to say about the actual statements. And then the next class, we'll use these same truth tables to figure out how to evaluate arguments. The textbook formula for getting the rows, goes like this. L equals 2 to the Nth power, where L means the amount of rows, and N means the amount of different letters. By rows, be sure we're clear on this. A row goes left to right. I'm going to use row/column language a lot, so be clear on this. And a column goes up and down. So this is a row that's going side -- or horizontal, and a column goes vertical. So that formula, L equals 2 to the Nth power, if you had say three different letters, 2 to the third power, 2 times 2 is 4, times 2 is 8. So the amount of lines is going to be 8. So in this case, you would have that L lines, but in this case specifically we mean rows, you're going to have 8 different rows. We'll talk about what those rows look like. But first, let me give you a more common sense understanding of that. If that formula looks kind of too much to memorize, here's the common sense version. The first letter counts for two. Each different letter doubles the number before it. Just read and place the letters from left to right. So for instance, if you saw a statement that went P horseshoe Q, triple bar R, you're going to read our letters from left to right. So we have a P, a Q, and an R. P, Q, R. The first letter counts for two. So we're going to have two rows here. So here's our first row. Here's our second row. Each different letter, doubles the number before it. So we had 2. P gave us 2. Q is going to double the number 2. So now we're at 4. Three, 4. We have 1, 2, 3, 4. R, we have 4, now we have a new letter. We're going to double the number before that, so now we're up to 8. Five, 6, 7, 8. So here in this case, we have 8 rows. If you had two letters, so if we went P, wedge Q, P would count for 2, Q doubles the number before it, so this would have 4 rows. One, 2, 3, 4. Let's do one more just to be sure we're all on the same page. P horseshoe Q, and R wedge S. So we'd go P, Q, R, S. P would give us two rows. Q doubles that to 4 rows. R doubles that to 8 rows. And S doubles that to 16 rows. So, the first letter counts for 2. Each different letter doubles the number before it. Just read and place the letters from left to right. So in this example, P, R, Q, R, this has three different letters. If you notice that R in green, it happens twice. So that one isn't counting twice -- it's just counting once. So the three different letters you have are a P, an R, and a Q. And we're going to read that from left to right. P gives us two rows. So we have P, R, Q. P gives us two rows, because it's the first letter. R doubles 2, and now we're at 4 rows. Q doubles 4, and now we're at 8 rows. So this statement gives us a table with 8 rows. So we'd have -- the way that would all look now is P horseshoe R, wedge Q dot R. What we'll do is we'll set up a key to the left, the P, R, Q. This is where you're reading the letters from left to the right. The first letter I see is a P, so I put that on the far left of this key. The second letter I see is an R. Put that next in the key. And the third letter I see is a Q. Put that next in the key. And now we already established that there were 8 rows here, so we're going to go 1, 2, 3, 4, 5, 6, 7, 8 rows. So we have 1, 2, 3, 4, 5, 6, 7, 8 rows total. This is Step 1 for setting up your key. So the left side of this, where it's just a PRQ off to the side with no operator's in between, that's the key that we're working on. I'll keep walking you through what we're doing with that key. Next slide. So here is that same thing rewritten. You have PRQ, and that statement was P horseshoe R wedge, Q dot R. That was the statement we just looked at, at the last slide. I just kind of wrote it with the computer, so it's a little more clean. So we're going to solve this statement. What does the key need to look like? So take a second and think through how many rows do we need to have here? There's going to be 8. P counts for 2. R doubles it to 4. Q doubles it to 8. Or if you want to get fancy, you can use the formal formula which is L equals 2 to the Nth power. L meaning the number of rows. Two to the third power, because there's 3 different letters, P, R, and Q. Two to the third power is 8. So there's 8 different rows. Okay, so what is our next move? Formally, you're going to take L equals 2 to the Nth power. You're going to divide L in half. And that gives the number of trues and falses that you're going to put in the far left column. The number slash order, that gives the number slash order of trues and falses. So you're going to divide that number in half, and that gives the number of number slash order of trues and falses in the far left column. So in this case, L to the 2 -- L equals 2 to the Nth power. Here, L is 8. You're going to divide 8 in half, and that's going to give you the number of trues and falses in the far left column. So, under the far left column of P, you'd put true, true, true, true. There's 1, 2, 3, 4. And then false, false, false, false, 5, 6, 7, 8. Every column shift to the right, divides the previous number in half. Every column shift right, divides the previous number in half. So every column shift right, divides the previous number in half. So we were under the P at 4 and 4. Going to R, you divide the previous number in half, and that's going to take you to 2 and 2. True, true. And then you go false, false. True, true. False, false. You shift to the right one more to Q. That divides the previous number in half. Two gets divided in half. Now we're at 1. True, false, true, false, true, false, true, false. And that sets up our key for the truth table. We're not finished. We've got a lot more work to do, but that's kind of the Stage 1. It sets your key. Just like in your map, you have a legend or a key? This is that key. Okay, if you're not a fan of this kind of technical formula, here's another common sense. So let me give you the two technicals back to back, and then the two common senses back to back. I'm going to erase tis before I go into the two of them. Okay, so the two technical back to back would be for establishing your key, L equals 2 to the Nth power, where L is the number of rows. The Nth power is the number of different letters. So in this case, you have the number of rows you're going to create is 2 to the 3rd power because there's 3 different letters. So that's going to be 8 rows. Your next move is to divide the number 8 in half. That's the L. L gets divided in half. That's going to be your far left column, the order of trues and falses. So, 8 divided by 2 is 4. So you have 4, 4. Then, the next column shift to the right, you have -- you're going to divide the previous number you worked with. So it was 4, in half, so now you're at 2, 2, 2, 2. Next column shift to the right, you're going to divide the previous number in half. We were at 2, now we're at 1. So it's 1, 1, 1, 1, 1, 1, 1, 1. That's the formal ways it's done. Now the thing that I find a little more on the common sense side would be, the first letter counts for 2. Every new letter doubles the number before it. So, the letter P counts for 2 rows. R doubles that to 4 rows. Q doubles that to 8 rows. The next step you're going to do is start on the far right column of the key, and make every other one true, false, true, false, until you fill it up, just like this image. Then, every column shift to the left, you're going to double the number before it. So from 1, 1, 1, 1, now you're at 2 trues and 2 falses, 2 trues and 2 falses. You shift left one more time and it's going to go from 2, 2, 2, 2, to 4, 4, until it's filled up. So the second way just works on a pattern of doubling. To get the amount of rows, you double -- you start with 2, double the number before it till you hit your -- fill up your rows. And then you start on the far right column, go every other one, true, false, true, false. And then every shift to the left, you double the number before it. So it's every other 1 on the Q section. The R is 1 shift left, so you go every other 2. The P section is 1 shift left. You double the Rs, and now we're at every other 4. So you go, 1, 1, 1, 1, 2, 2, 2, 2, 4, 4. Okay, so that's the key. Whatever way works for you, so long as you can establish that key, you're good to go. The reason we do this key this way, is because it covers every possible combination of truth values that could exist. So for instance, if let's say, we want P to be true, R to be false, and Q to be true. So we want a true, false, true. If you look through this table, there it is. The third one down is true, false, true. When I'm doing this class with students face to face and we can get real time engagement, I'll ask students to just throw out you know, a couple combinations so you know that I'm not messing with you. I don't have this pre-staged. But in this case, I can't call on any of you. So, I'm just going to throw out two more. So another one would be like, let's say P is false, R is true, Q is true. So let's find that one. There it is. P is false, R is true, and Q is true. And one last one. Let's make them all false. There that is at the very bottom, false, false, false. So every possible combination of truth values has been established in this truth table. And that's what that fancy formula, L equals 2 to the Nth power, and then you know, divide L in half, and then divide the previous number in half to get your things. That's what that fancy formula does, is make it so that every possible combination of truth values is covered. That's the key. Once you have the key set up, you have to start solving the statement. So, first thing you're going to do is plug in the key. What we did, is we took the letter P, and just plugged it in every time we saw a P. So you'll notice the P and the statement we're working through with the operators in it, has 4 trues and 4 falses, just like the P in our key. Then, we take our R and plug that in. In the key, we set it up so that there are 2 and 2. And now the R in our statement is 2 and 2, 2 and 2. And then there's a second R, and we plug in the key to that second one. And it's 2 and 2 as well. And then the final one we have is the Q. That one is every other one. And sure enough, our statement matches, true, false, true, false, true, false, true, false, every other one. So, you just take your key and you plug it in. Next step, you solve the most inside parenthesis first, and work your way out. In this case, we have two sets of parentheses. The wedge is the most exposed. That's going to be our last thing we solve. That's the main operator. We're going to start with the horseshoe, and we're going to start with the dot, because those are both enclosed. If I had -- if I was ambidextrous, I'd do them both at the same time, but I'm not, so we're just going to pick one and do it first. In this case, we're going to start with the left one. So, what you'll do is go through each one at a time, and solve them through. So you'd start with the horseshoe. You'd go true, horseshoe, true. We have our truth table operator definitions we already learned. A true, horseshoe, true is true. A true, horseshoe, true is true again. A true, horseshoe, false is false because the horseshoe is only false when the antecedent is true and the consequent is false. True, horseshoe, false, is false again. False, horseshow, true, is true. False, horseshoe, true, is true. False, horseshoe, false, is true. And false, horseshoe, false is true. So there's that one. Let me erase what I just wrote, because the computer's going to plug in what I just did. So, it's the exact same thing. It's just going to be in typeset now. True, true, false, false, and then all the rest were true. Okay, now we're going to move to the next one. We're going to do the right side. So true, dot, now we're working with the dot operator, the and. That's only true when both sides are true. So the first row is true, true, true. The next row is a false, true, is false. The next row is true, false, false. The next row is false, false. That's false. The next row is true, true. That's true. The next row is false, true. That's false. Next row is true, false. That's false. And the last row is false, false. That's false. Again, I'm going to erase this. Okay, so there's your filling in for the dot operator. Okay, now we're going to work through the most exposed operator. The main operator. That's the last thing we do. Before we do that, before we solve that wedge, let's go ahead and talk about what we're doing here. The first step we took was used the P, and the R, to solve that horseshoe. Then we used the Q and the R, to solve the dot. We solved those two things separately. They were most enclosed in parentheses. Our final step is take the horseshoe that we solved, and take the dot that we solved, and use those two things to evaluate the wedge. So the left side of the wedge is going to be the horseshoe operator. And the right side of the wedge is going to be the dot operator. Okay. True, wedge, true is true. True, wedge, false is true because it's only false when both sides are false. False, wedge, false is false. False, wedge, false is false. True, wedge, true is true. True, wedge, false is true because it's only false when both sides are false. True wedge false is true, and true wedge false is true. Once again, I'm going to erase this because it's going to change it over to the typeset. So there you have the true, true, false, false, true, true, true, true. And that's going to be our final answer. So, looking at this wedge operator, we've got to say something about it. This is the truth value of the whole statement. The main operator is the truth value of the whole statement. And there's some things you can say. Looking at all of the rows under the main operator, so the full column under the main operator, if they're all true, the whole statement is logically true or it's a tautology. So if everything under there is true, all 8 of those, if they're all true, the statement is logically true or it's a tautology. If all of those were false, the statement would be logically false, or self-contradictory. If there is at least one true and one false, the whole statement is contingent. In this case, the statement is contingent because you have one true, one true, that's your false. You have at least one true. In this case, you have 1, 2, 3, 4, 5, 6 -- in this case, you have six true, but 6 is at least one true. And here you have at least one false. In this case, you really have two false. But if there's at least one true and at least one false, then the statement is contingent. Alright, let's try another one. Okay, so you have open bracket, parenthesis, P horseshoe Q, closed parenthesis, dot P, closed bracket, horseshoe Q. How many rows will you need? So this has two different letters. So it's going to get 4 rows, because the first letter's going to count for 2. The second letter's going to double the number before it. Remember, there got to be different letters. So here, P happens twice and Q happens twice. So you only have two different letters. Alright, let's set up those rows. Then you go ahead and plug in your key, go to the far right, every other one is true, false, true, false. And then every column shift to the left, you're going to double that to true, true, false, false. Unless you want to do it the formula way, and then you would go backwards with it. Whatever works for you, just stick with it. Find a way of memorizing it and stick with it. Okay, so we're going to plug in the key. I'm going to do it on the far right, and then go right to left. So Q, is going to go true, false, true, false. And then P is going to go true, true, false, false. Now, you plug in your appropriate truth values. So the P is going here. And here. Your Q is going to go, for that Q, the first Q you see, and the second Q you see. So the P's going to be used twice and the Q's going to be used twice. Next step, you're going to solve what's most inside of separators. So here, you have this second, so P horseshoe Q and P. The P horseshoe Q is inside parentheses and inside the square bracket. That second P after the and, is only inside of the square bracket. So that's more exposed. The Q on the far right, after the horseshoe, after the bracket, is the completely exposed Q. And then the P horseshoe Q is inside parentheses and inside the bracket, so that's the most enclosed one. It's the most inside of separators. That's the one we're going to do first. So, true, horseshoe, true, is true. True, horseshoe, false, is false. False, horseshoe, true, is true. False, horseshoe, false, is true. Once again, for the horseshoe, it's only false when the left side is true and the right side is false. Okay, now I'm going to erase that and plug it in with the computer. So you have true, false, true, true. Then you're going to solve the next most inside separators. So, we use the P, horseshoe, Q in the very beginning. We use the P and the Q to solve the horseshoe. Now, we're going to use the horseshoe and the P to solve the dot. So the horseshoe of the P horseshoe Q, and then the P that's on the right side of the dot, is going to solve that dot. So, we'll go horseshoe P. The horseshoe goes true, false, true, true. The P goes true, true, false, false. So, with the dot, we have true, true, is true. With the dot we have false, true is false. We have true, false is false. And we have true, false is false. Alright, there's the written version of it. Then the final step is the most exposed operator. So we're going to use the dot, and the Q, to solve the horseshoe. So the dot is true, false, false, false. And the Q is true, false, true, false. So, true, horseshoe, true is true. False, horseshoe, false is true. False, horseshoe, true is true. False, horseshoe, false is true. We'll erase and let it go to the computer typeset. Alright. So, what do we call this statement? Remember the three options we had? We had logically true or tautology, logically false or self-contradictory, or contingent. This one is a tautology or in other words, logically true. It has all true, truth values, under the main operator. That whole column is filled with trues. So that's Step 1, just evaluating statements. What we're going to do next time is use this to evaluate entire arguments. But for now, just keep practicing working on statement evaluation, and then we'll work on argument evaluation next time around. Take care. See you next time. Bye.